Sysex vs. Pitch Bend                                                9/21/97

Gary Morrison wrote...

>>You might be well-served to see how your "product plans" compare with
>>McCoskey's "FasTrack" JI MIDI sequencer, or possibly even Justonic's
>>Pitch Palette software.  I personally don't know much about the features
>>of either.
>Can someone please give more information on these?

I've done some work with the Pitch Palette, what would you like to know?
It's a windows program that sends sysex re-tuning messages to compatible
synths, including ones by emu, ensoniq, and roland.  It has a ratio-based
scale editor, and a calculator utility for converting between fractions,
cents, and hertz.  It has a primitive sequencer that allows you to add scale
and key changes to midi files.

It even has a live performance mode that can make key changes automatically
if you stick to block chords in root position.  The Justonic people brag
about this feature, but it's really very limited.  Much better is to send key
changes from the lowest octave of your keyboard or from midi foot pedals,
which the Pitch Palette also supports.  I haven't done any work with the live
performance mode yet; so far I've used it only for midi sequencing.

My problem was, as seems to be the bugaboo of the tuning-table method, TIME.
The tuning dumps take too long.  Unless you're very very careful, and even if
you are, your music will stutter and bark at every key change.  At least,
this is the problem I had with the Proteus/2.

To avoid this, I used the roland Virtual Sound Canvas, a software synth that
comes bundled with the Pitch Palette.  Because it is "virtual", it can do
sysex retuning much more quickly.  It doesn't sound like the proteus, but I
never liked proteus sounds anyway.  To sample my work, and for more
information on the Justonic software, you can visit my music page at...


My teacher, Denny Genovese, has written an excellent DOS program called MMT.
It uses the pitch-bend method, cycling thru 8 midi channels for polyphony.  I
believe that with 95% of synths, the pitch-bend method is much faster and
more flexible than the sysex method, if at the cost of maximum polyphony.
95% isn't really fair in this context, however, since a large number of
synths don't even support sysex microtuning.

>>Assuming that that's what you're asking about, there has been a bit of
>>discussion about how to use pitch bend for microtonality.  The tricky part
>>clearly is how to synchronize it with note-ons under very legato playing.
>>Since the either the pitch-bend message must precede the note-on or the
>>reverse, you're going to have a "blurp" in pitch either at the attack of
>>the current note or at the release of the previous note.
>MIDI transmits 3125 instructions per second.  A pitch bend is three
>instructions.  To send 16 of these takes 3*16/3125*1000=15.36 ms, or twice
>as much if you include the note-ons as well.  Is this really a problem?

Not in my experience.  I have used MMT with two Yamaha TX81Z's, the Proteus,
and a Kawai K5m, and I have yet to hear a "blurp" from any style of playing
whatever.  MMT can't play midi files, but it does accept live midi data, and,
if one is bent on playing midi files, a second computer can be used.  Denny
is looking to develop MMT further, and would appreciate any input in that

Before I close this message, I would like to make it clear that I am not an
advocate of the pitch-bend method.  Or of the tuning-table method.  Any
pitch-bend favoritism detected in this letter may be considered fair defense
against those who seem to pick on it without considering cons of sysex or the
work of artists like Denny and Jules Siegel, who use it to great effect.

What I do believe, quite frankly, is that both of these methods suck.  I
don't think that any of us really believe otherwise --  it's all too easy to
let which method is better and why obscure the fact that we're being handing
our hats when it comes to a microtuning solution for midi.  Now, I'm told
that the Symbolic Systems Kyma setup is just about as good as can be, but
I've never seen one.  Has anyone seen one?  Justonic is working on a synth
designed for microtuning called the "Tone Palette".  They claim it is several
orders of magnitude quicker and higher-res than anything else.  They've got a
prototype, but are having problems getting to production.  Find out more in
the bowels of their site at www.justonic.com.

Theory vs. Practice                                                  9/23/97

The recent "pop in microtones" issue is one I had planned to avoid, since it
seemed that any contribution I could make would be destructive.  But, maybe
I've found a way...

There is a difference between compositional models and methods of
performance.  Musicians will always play things differently between one
another, and between performances.  My high school choir sang in crappy
5-limit JI.  The only things that play in 12 equal are a few pianos, some
guitars, and many synthesizers.  No band, choir, or orchestra performs in
12-tone.  Why do you think piano concertos sound so bad?

Actually, all music is microtonal (unless it's played on a Hammond organ or a
very special additive synthesizer) because of the harmonics!  Yeah!

The point of JI is that it offers an expanded compositional model.  Music
composed in such a model will no doubt be played in all sorts of ways by the
musicians of the future.  An understanding of tuning is not a prerequisite
for playing this music, as DFinnamore@aol.com suggests; our future musicians
will be making more powerful music because it will be based on a more
powerful model.

Can we agree that there is a difference between re-tuning music made for
100-cent equal, as powerful a model as it is, and playing music made for 7+
limit Just Intonation, however it is actually tuned in performance?

Tuners                                                             10/19/97

>Before inventing something that is already here, you may want to look at the
>Sanderson Accu-tuner or the Reyburn Cyber Tuner.  These are tuning specific
>machines, though the RCT uses a powerbook.  Unlimited temperament access,
>over 200 can be carried in the machine itself.  Allows all manner of partial
>selection, unlimited scaling curves, accuracy to .1 cent, etc...........

I found the Reyburn Cyber Tuner on the web, but I could not find the
Sanderson Accu-tuner.  Does anybody know where to get information on this?

>There are a number of products on the market today that provide microtonal
>tuning support.  I have looked at the Accutone model 250, with preprogrammed
>scales, and a three light strobe for $400; Korg model MT-1200 wihch can be
>programmed to temperments of your choice for $300; Peterson 5000, with 112
>customer definable temperment files; Precision Strobe Tuner PST2, with the
>rotating LED's that I found useful before and programmable temperments, for
>$325.  All of these assume that you are tuning 12 tones to the octave, but
>let you alter any of those 12 to any note.  With a little fancy temperment
>switching, you can tune any number of tones to the octave with little

Am familiar with the Korg and Peterson, but have never seen the Accutone or
Precision Strobe Tuner models.

As far as a wish list goes, the smaller, simpler, and lighter, the better.
Perhaps ideal would be a device without any scale template at all, just entry
for cents and some rotating LED's.  It is always nice to have reference tones
to compare the visual feedback to something you can hear.  In fact, it would
be really cool just to have a hand-held, battery powered device that only
gave reference pitches (that is, had no listening capability) at a given
frequency in hertz.

The Violab Pitchman is such a tuner, and it can be purchased from Zuckermann
harpsichords at http://www.zuckermann.com/earlykeyboardnotes/pitchman.html/,
but it is stuck to one of several preset temperaments.  I have this tuner and
it is really cool.  I have corresponded with the guy who makes this tuner
about the idea of hertz entry, and he told me to buy a piece of lab equipment
from http://www.telulex.com/.  I looked into this, but eventually decided
that the Peterson 520 was better suited to my needs.  I have this tuner and
it is okay.  But, when it comes down to it, strobes have got to suck.  And,
big heavy boxes do not make me happy.

The Tempered... Flute?                                             11/18/97

>Unfortunately, current instruments do not aid (as well as they might) the
>advancement of the microtonal imagination: we work with conventional designs
>because they are (in acoustic instrumental terms) the best we have, and
>because we play them.  But, in a sense, the design of current instruments
>hinders the development of that conciousness, since they demand a degree of
>technique which will be unnecessary with new and specifically designed
>instruments for a given system (or systems). 

() I feel that the field of instrument design is pretty stagnant.  Not that
it's necessarily been better in the past -- who cares? -- but compared to
what it could be.

() However!  The myth seems to be that certain woodwind and brass instruments
are designed to play in 12 equal.  If this is the case, the designers weren't
very successful.  It is harder to play free-pitch instruments in equal
temperaments than it is in Just Intonation, as a general rule.

() It's just plain hard to control the intonation of woodwinds and brass at
all.  "Good Intonation" is the holy grail of conservatory training.

IN 12.  If there was a keyboard that played in JI, certain people would be
amazed at the overgrowth of ensembles playing in JI.

() Wait!  According to the above, we ought to already have a number of
ensembles playing in JI, as long as they were made of free-pitched
instruments and had good players at the helms!

() We do.  But it's crappy JI.  A good fixed-pitch instrument tuned to JI
could get 'em playing stuff they would never play otherwise.  So build me a
good microtonal keyboard!

Xenharmonic?                                                       11/18/97

>Question: What would be most helpful in music today to the composer, to the
>performer, and to the theorizer...

I think Paul E.'s paper on 22-tet is really cool, but in parts it seems, to
me at least, to advocate 22 as a good idea for a "next paradigm" or "next big
thing" in music.

I have a negative reaction to this, and it has nothing to do with 22.  It has
to do with the idea that there ought to be a new paradigm, a new "main thing"
in music at all.  To me, this idea represents bad old industrialist over-
standardization, a mindset that I blame for the current Xenharmonic crisis.

What crisis?  For me, it's felt as a distinct lacking.  Something's missing
from my toolbox as a musician.  Let's see... I've got dynamics, tone color,
what I wear on stage... cheap instruments that are all... tuned the SAME?

In this I also disagree with Partch's plugging of the Monophonic Fabric,
although he emphasized experimentation and non-conformity enough to
compensate.  I've got my own 'ideal' tunings, and reasons why they're really
good.  It would be great if other musicians found them useful, but it would
be a shame, I think, if they were universally burned into instrument design.

I have read things that evoke my reflex against bad old over-standardization
far more strongly than the abovequoted paper on 22 -> Jules Siegel's
insistence on his "System", for example.  Paul's paper was just the most
recent example, and so I put in my two cents on what "Xenharmonic" means to

Pair-of-dimes Lost                                                 11/19/97

[Gary Morrison wrote...]
>As for new paradigms though, I think the fact that each tuning presents new
>paradigms of composition to be very interesting.  But perhaps that depends
>on what you mean by paradigm.  I take the term to mean a set of bounds that
>you put yourself into.

Individuals can apply various sets of bounds to particular efforts with the
intention of exploring the space created within the bounds.  As you say, this
yields interesting results.  For example, there are many ways to play the
piano.  Bach spent his life looking at the keyboard as a place where a
simulation of independent solo voices could be run.  Only within the last
century has an approach that worked with the entire possible "space" of the
keyboard been invented.  Both approaches give interesting results, but one
might expect the Bach approach to be a subset of the modern approach.  This
is not the case -- it would seem that every set of limits, no matter how
confining, yields unique results!  However...

In my recent post, "Xenharmonic?", I was using the word "paradigm" to mean a
set of bounds placed over an entire field of human interest for a long period
of time that's hard to change.  This, it seems to me, is an embarrassment to

Leafing back thru the pages of history, I often notice that 12-tone music
isn't the only example of this kind of "paradigm" (that is, the broad-
reaching, entrenched kind).  Take chess for example.

There's a 400-year-old effort to figure out all the ways to start the game
(opening books).  We've finally worked ourselves up to the point where a 
motivated modern student of chess could blow the heals off the masters of the 
past; Garry Kasparov is the not just the best human player alive today, he is 
the best human to ever play the game.

But all this applies only to regular FIDE chess.  There are an infinity of
possible "chess variants" (see the link on my web page under Chess for an
excellent reference on chess variants) that produce significant changes in
the way humans look at the game -- even the slightest change to the starting
position obsoletes opening study -- and dozens of them are at least as
worthwhile as regular chess.  There's absolutely no rational for picking FIDE
chess and sticking to it so completely for the last 400 years.

Maybe the reason we take so long with things is because we're just not that
smart.  The fields of FIDE chess are still ripe for the plucking after all
these years and I believe, contrary to what is implied by many alternate-
tuners, that 12-tone equal temperament also has plenty left to offer.

I've never heard a cogent reason why 12-tone should be exhausted.  What I
have heard seems to blame 12-tone for bad composition (and there seems to be
an extra helping of that in the serialist camp).  For the critic, I believe
the vitality of 12-tone is well demonstrated by the CD and midi files
available through my web site.  Inversely, Xenharmonics do not guarantee
interesting music, although they probably make it much easier.

In any case, I do not attempt to debate the wisdom of the collective efforts
of 12 generations of our ancestors.  What I do say, and hope, is that human
evolution is reaching the point where we no longer need to stretch our
paradigms over the entire civilization and lock in the freshness with gobs of
industrial standards to have them be effective.  Can you see a future in
which each individual can exercise intellectual freedom without having to
independently re-discover the possibility of intellectual freedom, become a
carpenter, or live like a hermit?

Carl's opinion on FPI tuning                                       11/20/97

That's Free Pitched Instruments, and Haverstick says it best:

>I don't think our fixed pitch instruments are stuck in 12; our minds

Bingo!  If we had the conservatories filled with Xenharmonic keyboards and
theory classes, then we have ensembles using Xenharmonics in a very effective
way (as opposed to doing it by accident, as they do now).

The only plausible excuse for... enjoying your own music?          11/23/97

>I've heard several musicians, in the sense of both performers and composers,
>say that they rarely listen to their own music.  The context and tone of
>voice in those statements suggests that doing so would make them feel

These people are straight out of Ayn Rand.  It may amuse you to check out the
Fountainhead or Atlas Shrugged if you haven't already.  These are good books,
but they do have some problems.

Not the least of which is the flyer that comes with every book, advertising
what amounts to a religious cult for "Objectivists".  So if you ever have
these books recommended to you by someone with a crazy glint in their eye and
a thin layer of sweat on their upper lip, you know what's up.

Service it to say that vanity is a major cause of avoiding sounding vain.
And the idea that vanity is bad is a way people protect themselves from those
who make them feel inadequate.

>I certainly don't frequently listen to my own music, but when I do from time
>to time, I often do find it enjoyable. 

Why wouldn't you find your music enjoyable?  You wouldn't go to all that
trouble to make un-enjoyable music, would you?

>But not - unless I'm deluding myself somehow - from the perspective of
>vanity.  Instead its from the perspective of reveling in the joys of a
>completed effort.  Usually these are efforts that I enjoyed doing, so I
>relive, in essence, "the good times".  Perhaps it's kind of like looking at
>vacation photos.

I love to listen to your music.  I think I prefer it to looking at your
vacation photos.

>Anybody others of you have that sort of feeling from listening to your own
>music, or perhaps any other thoughts on that topic?

I love to listen to my music.  I make it how I like it.  I think it's also
beneficial to me as a growing musician to listen to what I've done.

As for bringing back the "good old times", I guess I haven't been composing
long enough to get the effect.  I do have some 2 year old stuff on
microcassette, but bringing back the "good old times" with microcassette is
kinda like reminiscing about when you had radiation sickness.

Again, Haverstick hits it on the head...

>The only plausible excuse for any one tuning replacing 12TET, would be
>that exploring it requires the entire efforts of every human being in the
>World for however long it dominates.

...and who would believe that any tuning could require that?

Would you like fries with your information?                        12/12/97

Lots of stuff has been written on the nature of music.  In late 1994, I
explained music in terms of my ideas about cognition.  Since then, I have
found only two cases where something similar was mentioned by someone other
than me -- an old Minsky essay, and the following...

Beethoven: Music.  What is it?  I don't understand it!  What does it do?

Schindler: It exalts the soul?

Beethoven: B.S. It is the power of music to carry one directly into the
mental state of the composer.  What was I thinking when I wrote this?

Schindler: duh? [listening to something - Kreutzer sonata?]

Beethoven: A man is trying to reach his lover, but his carriage is stuck in
the mud.  She will only wait so long.  This is the sound of his agitation!

...from the movie _Immortal Beloved_.  I suppose things like "the language of
music" or "music tames the savage beast" fit along these lines too, but these
are expressions more than things people think about.

So the idea is that music is a sort of 'fake information'.  The brain hears
it, and thinks it's learning.  But it doesn't know _what_ it's learning.  I
believe that all information has a certain character that makes learning
possible (called "The Music of the Spheres", coincidentally).  Music can
imitate this, offering a kind of generic data that the brain accepts,
generating appropriate reinforcements (emotions) just like you were learning
to drive or whatever.

So, while music is a form of communication, in that it can create a specific
mental state in the listener that was first in the mind of the composer (or
maybe out of nowhere if certain conceptualist techniques are used -- alg.
comp, dice rolling, etc.), specific data cannot as yet be transferred (things
like carriage or mud) without words.

How is this information stored?  I believe it's all rhythm.  Pitch is just a
way to fit more rhythm in.  I believe all musical devices (including melody,
fugue, harmony, polyrhythms, etc.) have a useful definition in this model,
and I have classified many of them.  There is the matter of minor being "sad"
and major being "happy", but I'm hoping to god this is cultural.  A single
sustained dyad might cause a reaction, but so does looking at a bright red
wall.  The study of which harmonies effect your liver or which colors heal
your spleen really beg the question.

Comma shift and the speed of the galaxy                            12/26/97

>Is there a way for a multiplicity of tunings to co-exist in a way that
>performers from different camps can play together or is there a need for a
>"standard" tuning for practical reasons such as instrument manufacturing
>and performance?

I don't know of any reason why there can't be a variety of tunings in use.
My good friend, who is in his 3rd year at Juilliard, has 4 trumpets, all of
them approximate 12 to a different degree.  So why can't he have instead 4
trumpets that approximate 4 different tunings?  I would think different
tunings would be a boon for manufacturing- they're always looking for
something to sell.

Keep in mind that only fixed pitch instruments are fixed in pitch.  My
abovementioned friend has a brass quintet.  They played a concert this past
Monday night at a local catholic church.  Any chord longer than .5 seconds
was played in pure 5-limit Just Intonation, despite the fact that none of
them know what that means any better than they know what equal temperament
means.  They do know the old rule: "Raise the 5th and lower the 3rd!".

They played some songs along with the organ, and it wasn't hard to hear the
intonation fight on these tracks.  Afterwards, they all confirmed the
difficulty of keeping in tune with the organ.  Anybody familiar with groups
like the Canadian or Empire brass can confirm that JI is nothing new to
ensembles of this kind.

>Will the fragmentation of opinion among tuning pioneers inhibit a popular
>movement beyond 12TET, so that alternate tunings remain the province of
>academics and isolated pockets of experimentation?

Who wants a popular movement?  A bottom-up enema is needed.  It starts with a
generalized midi keyboard and GOOD MICROTUNING SYNTH in every ear training
room in the conservatory.  Assignments for analyzing the various ET's.
Singing just intervals.

We've got it in Barbershop.  I went to a week-long camp called "Harmony
College" this past August.  I took a class called "Tune it or Die".  It
discussed the problem of where to put the comma so you don't go flat.  The
prof had a PC running Justonic's Pitch Palette (which the Barbershop Society
has been involved with from the beginning), and some software that would
measure the harmonics of a sound (good Barbershop vocal technique aims to
cull every harmonic possible from the human voice).

He also had an old Xen-modified keyboard called the "Just Intonation
Trainer".  It automatically tunes 4-5-6-7 chords in every key, in any
inversion, but only if all 4 notes, and no others, are used (when one plays
a 4 and 6 on the 12-tone physical keyboard, a simple gait opens to a pair of
virtual keyboards, one for the 5, the other for the 7).  It has two
patches and a three-octave range.  Really a very limiting instrument, but
great for singing and hearing -- except when the logic was disabled!

Now, as to where to put the comma...

>Certainly, singers temper intervals.  However, they do not require an
>explicit system to do so.

If they did, they wouldn't.  But I don't think they do, at least not without
a keyboard.  I'm NOT saying they always sing right on in any system.  But
"tempering" usually means adjusting intervals in a specific way, for the
purpose of modulating in a specific way.  Singers just don't do this.

>As odd as it may seem, I must agree with Gregg Gibson that most traditional
>Western music requires temperament, not necessarily always ET but
>necessarily a meantone temperament, to adequately perform it.

I don't think i'd say "most".  Much of it really should have meantone, true
enough.  But at least as much of it can be tuned in JI with good results.

>If the pitches were not adjusted in the midst of playing them, the key of a
>typical piece of classical music would drift, usually downwards, by several
>commas by the time the piece was completed.

The first thing to consider is if this pitch drift is important.  In pieces
that resolve to the tonic, I believe it most definitely is.  Even listeners
who do not know that the song has gone flat will without a doubt have some
feeling of being let down.  In pieces with plenty of modulation, I'm not sure 
it's as important.

The second question is where to put the comma.  In the abovementioned
Barbershop class I took, the following example was given...

Chord:    CMaj      E7       A7       A7       D7       G7      CMaj
          (I)     (III7)    (VI7)    (VI7)    (II7)    (V7)     (I)
Tenor      5........8........12       7        10       7        5
Lead       4        7        10.......5        7        5        4
Bari       3        5        7        4........6        4........3
Bass       2        3        4        3        4........3        2
         "Here     is       why      some     guys     sing    flat"

...I wish I could include the score.  Although there were no ties, the dotted 
lines indicate that the same 12-tone note is repeated in that part.  Now, for 

Chord:    CMaj      E7       A7       A7       D7       G7      CMaj
          (I)     (III7)    (VI7)    (VI7)    (II7)    (V7)     (I)
Tenor    327.03~~~327.03~~~~327.03   381.52   363.36   339.13   322.98
Lead     261.63   286.15    272.52~~~272.52   254.35   242.24   258.39
Bari     196.22   204.39    190.76   218.01~~~218.01   193.79~~~193.79
Bass     130.81   122.64    109.01   163.51   145.34~~~145.34   129.19
            ^                                                      ^

Tildes replace dots in this chart, so as to look better against the decimals.
The question marks point out that our quartet has gone exactly 1 comma flat,

5/2 x 10/12 x 4/5 x 4/6 x 4/3 x 2/3 = 81/80

...if you follow the dotted lines across the first chart.  Where to put it?
Well, this is an art and a skill.  I'd like to give Jim Richards full credit
for the above chart, and also for his suggestion that the comma should be put
between the first and second chords (as that is the biggest modulation).
Usually, the bass is really in charge of the extra comma.  The effect of the
Tenor note changing?  I love it.  I love to hear the change.

The class went one to practice the art and skill of adding the commas in
several well-known Barbershop tunes.

>JI is but one tuning system (actually, several) and has its defects just
>like any other.

Absolutely True!  Everything has pros and cons.  I do suggest that many of
the "defects" don't matter as much as previously believed.  Re-tuning
sustained notes, for example, doesn't bother me a bit!  It only bothers those 
hung up on chords coming from scales...

>You can't deny that most Western music derives much of its beauty and
>contrapuntal versatility from having both vertical and horizontal
>sonorities taken from the same pitch set. These pitch sets have
>characteristic dissonant intervals which allow a "tonic" to be defined and
>when the pitch set changes, so does the tonic. In this way non-diatonic
>notes can anticipate a change of tonality before that tonality also arrives.

And you can't deny that there's a motherload of beauty to be found in music
where the vertical sonorities come from a different pitch set than the
horizontal ones.  The horizontal pitch set holds everything together, while
it is turned 81 shades of purple by the changing harmonies.

Barbershop                                                          6/18/98

>I'm not very familiar with the barbershop style.  Do they generally use the
>dominant 7th as V7 of the key, as opposed to jazz harmony which uses it as
>a sound color that can occur almost anywhere, functionally?

The Barbershop style is jazz.  It uses both otonal and utonal (tho mostly
otonal) 4-5-6-7 chords functionally.  These chords, along with chords like
the 12-14-18-21, form the basic consonances of this music.  They are built on
every beat of the music, just like major and minor triads are built on every
beat of a Bach chorale.

The V7 chord is usually tuned with the harmonic seventh, as opposed to the
16:9 or 9:5 sevenths.  But the music modulates widely (it is composed in
12TET, with all the effects of that tuning assumed) and lots of things happen
throughout the 5 and 7 limits.

The best example I have found of this music is Nitelife's Basin Street Blues
album, available at...


RE: TUNING digest 1457                                              6/28/98

>Apparantly, Bosanquet studied 22-tET extensively, and Ogolevets proclaimed
>it the future of music (along with 17-tET). The section of my paper entitled
>"History of 22" now seems really underinformed. But my chances of ever being
>able to get a hold of those writings seems very small -- anyone know if
>McLaren actually had the primary sources on hand?

I wouldn't say that Bosanquet studied 22 "extensively".  I am certain he was
unaware of the decatonic scales.  What he did know was that 22 was one of a
group of positive 2nd order systems that had good 3rds.  He knew such systems
wouldn't work on his generalized keyboard, which was designed for 1st order
systems, so he developed a generalized mapping for use with 2nd order
systems.  This mapping was based on the diatonic scale, and no keyboard has
ever been built to play it to the best of anyone's knowledge.  His work with
22 was towards the end of his music theory investigations; the last years of
his career were devoted to the study of magnetism.

For those un-acquainted with Bosanquet's ideas and terminology, I offer this

1. The size of an interval can be measured in...
      a) "departure", the difference from the 12tET approximation
      b) "error", the difference from just

2. All tunings can be described by specifying the size of the 3:2, or nearest

3. A tuning is "positive" if its best 3:2 has a positive departure.  A tuning
is "negative" if its best 3:2 has a negative departure.  [Although note
recent post from John Chalmers regarding McLaren's XH17 article and the
possibility of revising this terminology to use error instead of departure].

4. Pitches in a tuning are described by their location on the chain of these

5. Significant things can be told about a tuning by how various commas are
represented.  Special terms are assigned to the two most important commas...
      a) "order", the number of steps of the tuning representing the
          Pythagorean comma
      b) "class", the number of steps of the tuning representing the Syntonic
For example, 12tET has order zero and class zero.  22tET has order two and
class one.

RE: TUNING digest 1462                                              6/30/98

[Paul Erlich]
>Actually, Bosanquet's extensive studies of 22 were toward the beginning of
>his music theory investigtions; I don't have McLaren's article handy, but
>I'm pretty sure about that much. I'll post the references (if no one beats
>me to it) when I get a chance, as it sounds like they'll be new to you.

Rudolph Rasch's definitive account claims...

"Bosanquet spent little ink on tuning and temperament after the publication
of Treatise.  Most important in this respect is the paper on the 22-tone
"Hindoo" scale and higher order systems read before the Royal Society on Feb
8, 1877 and subsequently published in Volume 26 of the Proceedings of the
Royal Society."

"Subsequently".  Here's the history of the paper...

1. "On the Hindoo division of the octave, with some additions to the theory
of higher orders" [abstract] Sent to H.J.S. Smith (Jan 5, 1877)

2. "On the Hindoo division of the octave, with some additions to the theory
of higher orders" [presentation] Read at a meeting of the Royal Society (Feb
8, 1877)

3. "On the Hindoo division of the octave, with some additions to the theory
of higher orders" [full paper] Proceedings of the Royal Society #26 (1878)

4. "On the Hindu division of the octave, with some additions to the theory
of higher orders" [reprint of full paper] Hindu music from various authors
(1882) -> edited by S.M. Tagore!

Bosanquet published his first work on music theory in 1874, his only book in
1876, and #4 above was his last work on music theory.  The stuff before his
book could contain information about 22 tone, but I doubt it.  But send the
references anyhow.

Diatonicity in a nutshell                                          11/21/98

[Paul Erlich]
>a given consonant interval is always approximated by the same number of
>scale steps. 

>I think agree...

Okay.  I have thought it over.  I still agree that this may be responsible
for an important effect, but it is not the rule I'd give if I had to give
only one rule telling where the goodness of diatonicity comes from.

That rule is almost the inverse: that a given number of scale steps be able
to represent different consonances.

For example, 12tET, used like a diatonic scale, gives a complete tetrad on
every scale degree.  It is strictly proper, maximally even, and a MOS between
the two most fundamental intervals known.  And it is a very resourceful
scale!  But it lacks the ability to produce different harmonies using the
same number of scale degrees.  And this is the thing that gives (7 tone)
diatonic music much of its punch.  Changes to the relative minor and so

22tET vs. 12tET                                                    11/21/98

>Please elaborate. If you mean the properties I describe in my paper, note
>that virtually all these properties presuppose representations of ratios of
>small whole numbers. If you don't mean those, then which ones do you mean? 

I think he means that 22tET commits some serious errors against the
consonance of 7-limit intervals.  7:5's are off 17 cents.  That's quite a

>12tET evolved in, and largely superceded the authentic tunings of, a musical
>style where 5-limit ratios were of primary importance (it could do so
>because the quality of its approximations was sufficient).

This is sort of false.  12tET really only happens on pianos and guitars.  And
it's only been happening there for 100 years.  And during this time, solo
music for these instruments has become increasing less using of the 5-limit

>I do examine non-ET 7-limit tunings which contain the scales in my paper,
>but find that they are optimally tuned very close to 22tET. The impetus for
>choosing an ET rather than an open, meantone-type system is certainly a
>"group" property, that of infinite transposability with a finite number of
>notes. I would be more than happy to advocate a non-ET, open 7-limit tuning
>if it had any significant acoustical advantages over 22tET. It doesn't.

22tET seems to be just about the best "meantone" for the decatonic scale,
yes.  But would you accept JI?  Or 41?

Ambiguous or Contradictory?                                        11/21/98

>>4. Can somebody show a method for finding which tunings will support a
>>given rank-order matrix?  Preferably one that works on both just and equal
>>step scales?
>What's a rank-order matrix?  Can it apply to meantone-type or Keenan-type
>scales as well?

A graphic method of determining propriety, no matter how many unique interior
intervals a scale has (Chalmers' formula only works for two) is to construct
its "interval matrix".  This is just its Lambdoma or Tonality Diamond.

To make a "rank-order matrix", take this diamond and replace each interval's
logarithmic magnitude with a number ranking it's size relative to all the
other intervals in the diamond.  Thus, any subset of an equal temperament
that uses all the intervals in the temperament will have identical interval
and rank-order matrices.  Example: diatonic scale in 12tET.

This idea is that several scales, each having a different interval matrix,
may all be perceived as re-tunings of eachother if they share the same rank-
order matrix.  This works for meantone scales, so far as I can figure.
What's a Keenan-type scale?

>>2. Can a proper scale with one and only one ambiguous interval in each
>>mode exist?
>I don't know what modes have to do with it, as all modes of a scale have the
>same intervals, but isn't the usual diatonic scale in 12-tone equal
>temperament an example?

Isn't it amazing how many definitions the word "interval" has?  Here, I meant
a (number of scale steps, acoustic magnitude) pair.  So no, the usual
diatonic is not an example; the tritone is not a type of fifth in the natural

>>3. What about a proper scale with one and only one ambiguous interval in
>>each interval (steps) class?
>An ambiguity occurs between two intervals, therefore it is hard to know what
>you mean by "one ambiguous interval." How about C D E F# G# in 12-tET?

Let's take a look at its "interval matrix"...
      (C)  (D)  (E) (F#) (G#)
2nds   2    2    2    2   *4       
3rds  *4   *4   *4   *6   *6 
4ths  *6   *6   *8   *8   *8
5ths  *8   10   10   10   10

And rank-order matrix...

      (C)  (D)  (E) (F#) (G#)
2nds   1    1    1    1   *2       
3rds  *2   *2   *2   *3   *3 
4ths  *3   *3   *4   *4   *4
5ths  *4    5    5    5    5

Modes are the columns, interval classes the rows.  Ambiguous intervals are
marked by stars.  My questions are then: 1. Can a scale exist that has one
and only one star in each column?  2. Can a scale exist that has one and only
one star in each row?

Aside from being fun, I think these are useful questions.  For me, ambiguous
intervals are the common tones of a type of "melodic modulation".  You don't
want too many or too few.  In fact, I think I care more about ambiguous
intervals than I do contradictory ones.

Wolf: Melodic vs. Harmonic mistuning                               11/22/98

>any sensitivity to mistuning in melody (making comma adjustments) must be at
>least an order of magnitude rougher than the acoustic pleasure tolerance...

>I am more than a bit thrown back. If you have any short term pitch memory,
>you can train yourself to listen for melodic commas and they become glaring
>features in a performance.

Yes, you can hear melodic commas in music.  But they are heard as alterations
of the source scale.  Hearing the 81:80 in 5-limit diatonic music does not
give me the impression that the music is in some strange 8-tone scale.  In
fact, most listeners do not notice these commas [as is well-shown by the
rampant ignorance in music theory regarding how choirs prefer to tune their

On the other hand, shifting the tuning of a music's harmonies by 81:80's will
cause an immediate and glaring change in the timbre of that music.  What Ivor
called "moods" were in fact characteristic changes in the _timbre_ of
harmonic music when mistuned in certain ways.

Partch Bio                                                         11/22/98

>Gilmore goes a bit off the deep end with his occasional psycho-analysis of
>some of Harry's situations/characteristics (I find it hard to believe that
>Harry's homosexuallity was caused by him being circumsized (sp?) at the age
>of eight!)

Not to mention that the book is twice as long as it ought to be, overflowing
with naive reviews of Partch's music and framed in a language that seems to
try to legitimize itself by being 6 and 7 times more "scholarly" than it
would ever have reason to be (you'd think Gilmore was an aspiring

I'm sorry.  But anyone who can criticize Partch's dissatisfaction with the
first staging of The Bewitched has missed the boat in my book.

Erlich's Contest                                                   11/23/98

>>Those of you who have read my paper or followed my posts know that I
>>suggest replacing the 7-out-of-12 scale, which has defined most Western
>>music for centuries if not millenia, with a 10-out-of-22 scale.

I think it's rather inaccurate to say that the 7-out-of-12 scale has
"defined" Western music for centuries.  What's "Western" music?  The 7-tone
MOS is one of the most commonly used scales in the world.  Always has been.

If we do recognize a "Western Music", then we'll notice that it's been using
to awesome effect the 12 tone MOS for over 100 years, and I do not mean
serialism.  It's also made ample use 5, 6, and 8 tone scales.

While the argument for the 10-of-22 scale is thorough, well-presented, and
very compelling, it remains to be proven or disproved only through a body of
music, since that is what the theory is *for*.  Unfortunately, the only
reasonable instrument for decatonic music that exists at the moment is the
guitar, which is simply not my can of worms...

>>I haven't PROVEN that something significantly different from 10 of 22 can't
>>work, but I doubt it.

Can't work for what?  The specific set of rules you chose to generalize
diatonicity?  There must be other sets of rules that capture the essence of
G.D. just as well:

I think that as long as we keep propriety in mind, and make so that the set
of intervals (scale steps) can rotate through the set of acoustic magnitudes
in some systematic way with the scale's circular permutations, and keep
everything to a digestible yet challenging size (see discussion of cognitive
limits below), we have "got it".

>>I know of no 9-limit or 11-limit generalized-diatonic scales, but they
>>might exist (I don't know how important that would be, since the 9-limit
>>and especially 11-limit analogues of the minor chord sound pretty dissonant
>>to me, despite Partch's excellent use of them).

The 9-limit utonalities sound good to me.  The 11's work with the right
(especially electronic) timbres, and/or tasteful amplitude balance and
voicing.  And I don't think that we have to rotate through major and minor to
achieve diatonicity.  We could rotate through higher and lower identities, or
all sorts of things.

If we do insist on complete chords, and I don't think we have to, higher-
limit generalized diatonic scales run into another problem: many of the
desirable effects of diatonicity drop off as the number of tones in the scale
increases.  Some will drop off because of the Miller limit (which has to do
with tracking events over time), and some will drop off due to the Subitizing
limit (which has to do with tracking multiple, simultaneous events).

1. Miller limit

(a) I don't believe in just one point-of-no-return Miller limit, at least not
in the application of how listeners experience melodic symmetries.  Rather, I
think that there may be several types of memory effects that fall in and out
as the number of tones in a melody changes.  Exact numbers would depend to
some extent on how much practice the subject has had at this stuff, but
here's a rough idea of what I'm thinking (we assume octave equivalence)...

TONES   EFFECT                      PROPRIETY            MUSIC
  2-4   too easy                    little importance    chant
        less interesting                                 ritual song
        more "join in" potential

 5-12   tracking starts to slip     most important       polyphony
        mind has fun trying to                           parallel harmony
        keep its place                                   melody over chords

11-22   tracking the entire scale   some importance      parallel harmony
        impossible: mind "chunks"                        melody over chords
        scale into proper subsets                        melody over drone
        and tracks within/between

23-34   inability to focus or       no importance        conceptualism
and up  mind begins to fuse 
        individual stimuli and
        re-interprets as if
        hearing 5-9 tone scale

(b) I'd say that the Miller limit has claimed all it will from generalized
diatonicity by 12 notes.  This would seemingly nix limits higher than 9, at
least for scales based on saturated chords.  But I suspect that most
listeners will need quite a bit of practice (and maybe a few Millers...)
before getting the most out of even a 9 tone G.D. scale.

(c) Miller complained that he couldn't explain the performance of those
subjects with absolute pitch.  There are some very good reasons to believe
that almost everyone is capable of very accurate absolute pitch.  But I do
not believe that the ability to remember the tones (as measured in Miller's
experiments) using absolute pitch means that we are not experiencing a loss
of some type of experience.  For example, someone with a well-developed sense
of absolute pitch may not have a problem correctly tracking 34 tones/oct.
However, I believe she would suffer the same loss of ability at tracking
melodic symmetries at this number of tones as someone without absolute pitch.
With this I admit to some difficulty defining and measuring "melodic

(d) I list "mind begins to fuse individual tones and re-interprets as if
hearing a 5-9 tone scale" as one of the effects of a melody with over 23-34
tones.  I list "conceptualism" as the kind of music you'd make with it.
Here, I am insulting "conceptualist" music (the idea behind a work of music
is extremely important to me as a listener and composer, and "conceptualism"
belittles this).  But there is a way to profit from the brain's tendency to
fuse tones when they are this close in size and this many in number -- the
performance of generalized diatonic music in just intonation!  Choirs have
been doing it for centuries.

2. Subitizing limit

(a) It's been shown that average dudes from all over can count how many
stones you toss on the ground almost instantly- so long as you don't toss
more than six stones at a time.  Since a good deal of the interest of G.D.
scales comes from the interaction between parts in polyphonic composition, it
seems that we'll lose something if we go above the 11-limit.

(b) While this ability should be more easily improved with training than the
Miller limits discussed above (remember Rainman and the toothpicks?), its
carry-over to the tracking of simultaneous parts in a polyphonic music is not
entire.  This is due to the fact that our psychoacoustic bandwidth (keeping
notes with their respective parts) is not as great as our visual bandwidth
(as used for counting stones) -- especially when listening to music produced
on speakers, which lacks the spatial cues of acoustic performance.  I think
six parts is a good practical upper limit for polyphony.  Parallel harmony
shouldn't have a limit so long as we stick to otonalites.

Many Tones                                                         11/26/98

[Paul Erlich]
>On the many-tones issue, I think the tetrachord structure helps to reduce
>the number of independent elements that need to be perceived...

Could you elucidate?

[Paul Erlich]
>compositional technique can make at least as much difference as an order of
>magnitude difference in the number of tones.

Yes compositional context makes a huge difference.  But you can't have your
cake and eat it too.  If you use less, in whatever way, you're using less.
I tried to make the chart reflect this.

[Stephen Soderberg]
>This doesn't take into account (unless I'm missing something -- a distinct
>possibility) the mind's characteristic ability to recognize (and organize
>into) patterns.  It may not be as "instantaneous" as Miller's tests
>suggest (and I admit I don't know the study -- I'm going on the present
>description and similar ones I've heard), but seven stones are nearly
>immediately organized into four-plus-three and so on.

Miller's paper addresses (if in a limited way) the issue of chunking, and is
available at...


My interpretation into the realm of music seems to be that while chunking
works (and I think it works a lot better than Miller does!), some type of
experience is still dropping off.  I tried to represent this in the chart and
the notes that went with it.

[Stephen Soderberg]
>Second, a strictly "melodic" test doesn't take into account the full power
>of many musics to organize material into recognizable, memorable chunks.

This is quite true.  But it is hoped that a strictly melodic test can say
something about the contribution of the melody to these types of musics...

It's difficult to write about chunking clearly, because you can chunk chunks,
and you've got to be careful about what level you're talking about.  How all
this works in the cognitive process is one of the greatest unanswered
questions in psychology, and its application in music is even more sketchy.
I certainly don't have many answers.  I whipped up the chart pretty quickly,
and I'm surprised it is holding up to my second thoughts on the matter as
well as it is.  As it is, I need more time to think about it, especially as
regards the excellent feedback from Stephen Soderberg.

G.D.                                                               11/27/98

>>And I don't think that we have to rotate through major and minor to achieve
>>diatonicity.  We could rotate through higher and lower identities, or all
>>sorts of things.
>I wholeheartedly agree, and would love to find some scales that do anything
>like that.

Harmonic series segments do it.  Although they may fail to do some other
things, I do believe that harmonic series segments are a viable melodic
resource.  David Canright's guitar suite, Morrison's 7HS music, and the work
of Denny Genovese and Jules Siegel all come readily to mind.

CPS's rotate between both major/minor and higher/lower.  But again, they may
fail to do other things, depending on how we've built them.  In particular,
the consonances tend to wander around the scale degrees.

I can imagine the following minimum criteria for generalized-diatonic

{1} The scale is periodic at an equivalence interval, a strong consonance.

{2} There are between 5 and 12 notes in each period of the scale.

{3} There is at least one scale degree that, a. is consonant in a majority
of the scale's modes, b. contains only one consonance, which forms a
consonant triad with [1], and c. has all the occurrences of this consonance
in the scale.

{4} There is at least one scale degree that, a. is consonant in a majority
of the scale's modes, b. contains more than one consonance, each of which
form consonant triads with [1] and [3], and c. has all the occurrences of
these consonances in the scale.

{5} i) The scale is covered by [3]+[4] chords, or ii) has relatively high
stability and efficiency.

{6} Your suggestion here.

The search for strange diatonic-like scales is on!

Tuning and timbre of acoustic keyboard instruments                  12/3/98

>Okay, but the other things were far from equal. On early keyboards, the
>guage used for the wire was narrower, so they would have behaved more like
>ordinary strings than does modern piano wire.

I seriously doubt that the wire back then behaved more like ideal strings
than modern wire.  It was full of imperfections, leading to false beats, and
all the rest.

>Further, I recall seeing sonograms from both IRCAM and Robert Cogan showing
>that the amplitude of the partials above the fundamental in harpsichords,
>clavichords and fortepianos was, overall, higher than in modern pianos, so I
>would suspect that sensitivity to tuning vis-a-vis the overtone stucture
>would have actually been higher.

Harpsichords may have the best harmonics of the bunch, from any era (of
course, today's harpsichords are woefully similar to those made hundreds of
years ago), but clavichord timbre has the least harmonic, weakest, and
funniest amplitude envelope of just about any keyboard timbre imaginable.  It
is unusable for ratios above the 5-limit, and the difference between meantone
and equal temperament (if you can set it at all!) is hardly noticeable.

To me the overwhelming difference between fortepianos and modern pianos is
the fact that the former are single or double strung, whereas the latter are
triple strung.  Take that away (as Michael Harrison did), and you'll find
that the tension, diameter, and everything else about the modern piano
outshines the fortepiano as far as harmonicity and sensitivity to tuning.

>I may also add, as a bit of music-cultural speculation, that the creation of
>the the tuning profession probably had a net effect of densensitizing
>players to the quality of keyboard intonation.

I'll agree with that!

Rothenberg questions                                               12/29/98

Regarding Rothenberg propriety, I asked three questions...

>>1. Can somebody show a method for finding which tunings will support a
>>given rank-order matrix?  Preferably one that works on both just and equal
>>step scales?

Paul E. asked...

>What's a rank-order matrix?

I answered...

>A graphic method of determining propriety, no matter how many unique
>interior intervals a scale has (Chalmers' formula only works for two), is
>to construct its "interval matrix".  This is just its tonality diamond.
>To make a "rank-order matrix", take this diamond and replace each interval's
>logarithmic magnitude with an integer ranking its size relative to all the
>other intervals in the diamond.
>The idea is that several scales, each having a different interval matrix,
>may all be perceived as re-tunings of eachother if they share the same rank-
>order matrix.

This first question remains unanswered.  I asked two more...

>>2. Can a proper scale with one and only one ambiguous interval in each
>>mode exist?
>>3. What about a proper scale with one and only one ambiguous interval
>>in each interval (steps) class?

Paul E. responded...

>An ambiguity occurs between two intervals, therefore it is hard to know what
>you mean by "one ambiguous interval."

I did phrase these last two questions in a confusing way.   Perhaps I should
have asked: Can a proper (but not strictly proper) scale have its ambiguous
interval(s) distributed so that they occur only once in each mode (question
2) or interval class (question 3)?  Despite my confusion of terminology, I
think my subsequent post did make it clear what I was after...

>Let's take a look at its "interval matrix"...
>      (C)  (D)  (E) (F#) (G#)
>2nds   2    2    2    2   *4       
>3rds  *4   *4   *4   *6   *6 
>4ths  *6   *6   *8   *8   *8
>5ths  *8   10   10   10   10
>And rank-order matrix...
>      (C)  (D)  (E) (F#) (G#)
>2nds   1    1    1    1   *2       
>3rds  *2   *2   *2   *3   *3 
>4ths  *3   *3   *4   *4   *4
>5ths  *4    5    5    5    5
>Modes are the columns, interval classes the rows.  Ambiguous intervals are
>marked by stars.  My questions are then:  2. Can a scale exist that has one
>and only one star in each column?  3. Can a scale exist that has one and
>only one star in each row?

Since I asked them, I have figured answers to these last two questions.  The
answer to question 3 is "no".  If you insist on having at least one ambiguous
interval appearance in each interval class, then the least amount you can get
to appear is 1 in the smallest class, 1 in the largest class, and 2 in each
of the other classes (no matter how many there are).

The answer to question 2 is "yes".  Scales can exist that have one and only
one ambiguous interval appearance in each mode.  Take this generic 3 note
scale (rank order matrix)...

       X    Y    Z
2nds   1    1   *2
3rds  *2   *2    3
4ths   4    4    4

To see how this might look, let's tune it in 12tET.  For simplicity let's
assume that the 4th is the interval of equivalence, and that it is 12 steps
of 12tET, and that the other intervals are scaled proportionally to their
rank.  The interval matrix then looks like this...

       X    Y    Z
2nds   3    3   *6
3rds  *6   *6    9
4ths  12   12   12

In standard notation, we can notate this scale like this...

 C   Eb   F#   C
   ^    ^    ^
   3    3    6

In which case mode "X" starts on C, mode "Y" on Eb, and mode "Z" on F#.

Diamonds are Forever                                               12/31/98

The ancient Greeks called it the Lambdoma, Partch called it the Tonality
Diamond, and Novarro must have called it something.*  The structure is old,
and has a funny habit of being re-discovered many times across a wide variety
of disciplines.

[* There are minor differences here.  "Lambdoma" is usually reserved for the
infinite diamond of all consecutive counting numbers, while Partch's diamond
exists at some "limit" and involves adjustments and omissions related to
octave equivalence.]

But what is it, this matrix of order/limit/cap X?

1. A fashionable arrangement of all the ratios in a Farey series of order X?
2. The solutions to an old puzzle involving colored space-filling polytopes?
3. A list of all the modes of a scale relative to one master key?
4. A Cartesian cross between a set of rationals and their reciprocals?
5. A slightly-mutilated Rothenberg interval matrix (aka difference matrix)?
6. The structure, up to cardinality C, with the greatest chord/note ratio?
7. The solution to a sphere-packing problem in X dimensions?

[#1.]  The number of items in a Farey series of order X, and the number of
items in a Lambdoma of cap-X, is X^2.  A Farey series of order X is a list of
all the fractions that you can make using whole numbers up to X.  So X^2
makes sense, because you're taking X things two at a time; there are two
places for a number in a fraction.  X^2 makes sense for the Lambdoma when you
realize the Lambdoma is a square; as a child you learn to count objects in a
grid by counting along the edges and multiplying.

[#2.]  Some time ago, John Chalmers posted to the list about his fondness for
Martin Gardner's recreational math.  Recently, when looking thru my bookshelf
for something to read on my frequent bus trips to and from New York city, I
picked out "Fractal Music Hypercards and More", which proved to be a lot of
fun.  One of the articles in it, "The Thirty Color Cubes" (it doesn't seem to
say what issue of Scientific American it was reprinted from) describes two
classic "domino tiling" puzzles involving a set of cubes.  With 6 colors and
one color to a face, there are 30 ways to paint a cube so it uses each color
only once.

One puzzle is to take one of these cubes and then build a two-by-two model of
it using 8 of the remaining cubes, so that all touching interior faces of the
model match in color.  A second puzzle is to find a set of 5 cubes such that
they can be placed same-color down on a table and have each of the 5
remaining colors showing faces up.

John H. Conway found a graphic method for solving both of these puzzles -- an
arrangement of the 30 cubes in a grid.  The grid has no blocks along this
mysterious diagonal... when I saw this, my diamond warning lights went off.
With 6 factors the diamond has 36 members, 6 of which will contain two of the
same color (the 1/1's), leaving 30 (of course, we are used to the 11-limit
diamond having 29 pitches, as we only eliminate 5 of the 1/1's, giving 31,
and then eliminate the 9/3 and 3/9).

All of the solutions to the second puzzle are given by the rows and columns
in Conway's matrix.  This puzzle is equivalent to selecting o- or utonalities
(minus the unity) from the diamond.  The color touching the table is the
numerary nexus.

The first puzzle is solved by finding the inversion of the prototype cube and
taking the 8 cubes for the model from the row and column that contain the
inversion cube, minus the inversion cube itself and the 1/1's (11 minus 1
minus 2).  I do not know if this first puzzle has a musical analog.

A final note is that Conway's matrix itself meets the "domino condition".
That is, all the touching faces match in color.  Again I do not know the
musical significance (if any).

It seems to me that the diamond should provide solutions for similar puzzles
involving any number of things and a corresponding set of thing-sided
space-filling polytopes.

[#3-4.]  This was my breakthru with the diamond.  I had come up with a bunch
of 7-limit scales (one of which was identical to Kraig Grady's "centaur"
scale), and I wanted to compare them by how "low numbered" the ratios in each
of the modes were.  That is, if I modulated to one of the other notes in the
scale, what kind of relationships would I have?  In essence, I wanted to make
a given note in the scale the "1/1".  This can be done by multiplying the
entire scale by the reciprocal of the given note.  Doing this for each note
of the scale in turn, I wound up with the cross-set of the original scale and
its subharmonic inversion --- a diamond!

[#5.]  This article was inspired by John Chalmers -- whose substantial work
and ready help has played no small part in my understanding of music -- when
he suggested I explain on the list my recent statement that a scale's
Rothenberg interval matrix was "just its Lambdoma or tonality diamond".  I
believe that anyone who read that recent Rothenberg post, and up to here in
this article, should have no problems turning a diamond into an interval
matrix and back (hint: the interval classes in the interval matrix are the
diagonals on the diamond).  Of course, Partch used the diamond to generate a
pitch set, while Rothenberg used it to generate an interval set (it would
make sense to take the interval matrix of a diamond, but not the diamond of
an interval matrix), but the structure is the same.

[#6.]  I am not aware of a proof, but there can be little doubt that the
diamond provides a greater number of saturated chords per vertex than any
X-limit structure with up to as many notes.

[#7.]  A famous problem asks for the most compact arrangement of generic
spheres of unit diameter in N dimensions.  In 2-D, the generic sphere is the
circle, and the closest packing has been proven to be "hexagonal".  If we
tile a plane with circles in this way, and mark the centers of the circles on
the plane, we get the 2-D triangular lattice.  If we then take any given
circle, and all the circles whose centers lie one unit away, and erase their
centers' marks, we will have a hexagonal hole in our lattice.  Interestingly,
the 3-factor diamond is this hexagon when mapped to the 2-D (3-factor)
triangular lattice.

A similar situation occurs in 3-D.  It was conjectured long ago, but only
recently proven, that the closest packing of 3-D spheres in 3-space is the
so-called "face-centered cubic" packing.  In this arrangement, the centers of
a given sphere and all those 1 unit away form the vertices of a
cuboctahedron.  Again, when the 4-factor diamond is mapped to the tetrahedral
(3-D triangular) lattice, the diamond represents this "unit packing" of
spheres (the cuboctahedron).

So it is tempting to say that the closest packing of N-dimensional spheres is
achieved by centering them on the vertices of the N-dimensional triangular
lattice, and that the (N+1) factor diamond represents the unit packing on
this lattice.  It has been shown, however, that the generalized triangular
lattice is only the closest lattice packing in up to 5 dimensions.  Also, it
is unknown, when working with more than 3 dimensions, if non-lattice packings
exist which are more compact than the optimal periodic (lattice) packings.

It is interesting to speculate on the relationship between this and item #6.
How does the diamond map to these higher-dimensional optimal lattices?  Do
they still represent the unit packings?  How is sphere-packing related to the
chords/note ratio?

Forum CD review                                                      1/6/99

Got my forum CD in the mail yesterday.  It is my honest and carefully
considered opinion that the album is quite good!  It really works.  I don't
have a problem ranking it with either of the JI Network compilations.

Here are my favorites...

1. New Awakening (Gary Morrison):  Pure kickass.  Mr. Morrison proves his
enormous potential as a composer.  88CET is cool, but I wish he wouldn't
focus on it exclusively.  The old Ensoniq is sounding rather, well, bad.

2. Limp off to School (John Starrett):  Rock on!  John is one of the most
impressive musicians I've ever had the pleasure of meeting.  This song has
got it all.

3. Citified notions (John Starrett):  Another winner from Starrett.  I find
it strange that nobody seems to be taking credit for these vocals...

4. Just on Time (Bill Alves):  Profoundly beautiful, well performed.  Classic

Here are ones that are very good, but somehow fall short of my favorite

1. Mood of Neptune/Ashes Before the Sky, Stephen James Taylor:  I have to
disagree with the critic at Westword; this tune is cool, and works fine as an
opener (for both CD and ear).  Potential problems with being too dark or
conceptual are totally avoided because it *succeeds* in taking you there.

2. Kaleidophon (Carter Scholz):  Mind expanding!  In fact, regarding recent
talk about the tonality diamond, I believe this is a rhythmic diamond -- all
circular permutations of a rhythm played simultaneously.  [Carter, please
straighten me out if this is not the piece.]  I like all of Carter's stuff.
In fact, his stuff is probably the best algo-comp I've heard.  For me,
however, algo-comp can never quite break out of a sort of realm.  Not that
it's bad, mechanical, intellectual, or any of that.  Just that it lacks a
sort of shine.  Hard to explain.

3. Duet for Morphine and Cymbal (Bill Sethares):  Sethares' work is extremely
important because it explores the extent to which harmony is meta timbre.
Listening to his stuff is like sending your ear to graduate school.  I do
have serious reservations about the strong version of the consonance=beatless
theory, regarding the ear's innate preference for harmonic partials, as
demonstrated by the virtual pitch phenomenon and the persistence of my ear,
despite much listening, to hear many of the adaptive timbres as "coming
apart", causing a loss of "bandwidth", a reduction of the number of
independent parts I can follow compared to a beatless tuning with harmonic
timbres.  Fortunately, Bill doesn't really subscribe to the strong version
himself, and his book "Tuning, Timbre, Spectrum and Scale" is very objective,
first-rate stuff, presenting all sides and never insisting on anything
without good reason.

4. Glass Lake (Sethares):  I like everything on "Xentonality", but I think
"Three Ears" is really something special, and would have had it before Glass
Lake or Morphine.

Worth a listen...

1. Matrix (Denny Genovese):  This is really a great tune.  Unfortunately, the
CD seems to have "revealed the limitations of the analog source."  Also, I am
not particularly fond of this performance.  I have a version done on TX-81Z
that I like much better.  Theremin part clouds the thing.

2. Resuscitation (John Loffink):  Synthesis is good.  Music is disconnected.

3. Snake Dance (Neil Haverstick):  Interesting but unconvincing.

4. Vilano (Ernie Crews):  Nice idea but too long and self-absorbed.  Well-
played, but riffs do not compensate for sparseness of material.

5. Pient Molles (Rick Sanford):  You can hear some of the melodic potential
of Sanford's decatonic scale.  You can also hear yourself slapping three of
these out between lunch and dinner.

6. Evening in Landcox Park (Warren Burt):  I wouldn't spend an evening in
Landcox Park without it.

Barbershop (and recent concept question)                             1/9/99

I'd like to say that Barbershop is some of my favorite music.  I am quite
serious about this; it does everything that I expect good music to do.  I'm
writing because it so often doesn't get taken seriously.

Barbershop combines the calming and beneficial qualities of beatless tuning
with the intellectual rush of western polyphony.  It combines the enhanced
musical bandwidth of the 7-limit with all the desirable properties of the
diatonic scale, and to some extent with the 12-tone Pythagorean MOS.  Being
jazz, it has plenty of good rhythms and exotic harmonic movement.  Being
choral music, it often has lyrics.  Being four-part, it is flexible to write
for and a joy to listen to (as a composer, I find 4-part writing perhaps my
favorite).  As a singer, I find the barbershop technique a fulfillment of the
vocal medium.  And as Erv Wilson and I once agreed, there is hardly a sound
more desirable than the sound of the human voice.

It is the concept of Barbershop that I think often throws people off.  And it
is concept-rich.  In fact, it achieves for me greater power of concept than
any "conceptualist" music I know of.  And a better concept I could not ask
for.  I tend to look at barbershop tunes as 20th century madrigals.  They
offer a unique view at the roots of Jazz.  They are often surprisingly
sexually explicit, but in a completely wholesome way.  It's so refreshing to
find music so benevolent.

Participating in Barbershop has been a whole lot of fun.  Nothing has been
better for my ear, or my voice.  And you get to meet these sweet guys,
married for 50 years and with big white curly mustaches.  These guys are

Barbershop tuning                                                    1/9/99

In my last message, I assumed some things about how Barbershop music is
tuned.  I'd like to clarify.

If you take a bunch of short samples of a typical Barbershop song, you'll
find a justly-tuned 7-limit chord most of the time.

This may not sound very strong, so let me be clear: Barbershop is the most
consistently and deliberately higher-limit music I know of.  It is not some
accidental, sometimes JI, sometimes 7-limit music.  It is an entire style
firmly committed to the 7-limit.  Their theory is fully aware of the tuning
issue, their mission statement contains reference to JI, and even amateur
groups sing nicely in tune.

Barbershop Spectrogram                                              1/10/99

In the spirit of Gary Morrison's suggestion, this post makes use of remote
resources posted at...


The other day, Daniel Wolf recommended a spectrogram program and I downloaded
it.  It works really well.  It seems we have reason to be happy that we live
in a time when people can trade such valuable stuff without having to bother
with value abstraction... but I digress.

To test the program, I thought I'd try some Barbershop.  For the test, I
picked a brief excerpt of song by the Happiness Emporium.  This group did
most of their recordings (including this one, from their album "That's
Entertainment!") back in the 70's, when Barbershop technique wasn't a shadow
of what it is today.  However, they have a nice sound, and I thought I'd give
them a try.

Using Cool Edit, I extracted the CD audio to 16-bit 44.1 WAV format, and
mixed it down from stereo to mono.  I then ran it thru Spectrogram, fiddling
with the settings until I got a clear picture.  The values used to produce
the image full.bmp are:

Freq Scale=Log
FFT Size (points)=8192
Freq Resolution (hertz)=5.4
Band (hertz)=10-22050
Time Scale (ms)=12
Spectrum Average (ms)=1
Toggle Grid=off

This example contains a nice C7 chord, and I cropped it with Cool Edit and
spectrogrized it with the same options as above, except with a time scale of
4 milliseconds.

Note: I downsampled full.wav to 22K with Cool Edit to make it smaller for you
all.  Up.wav remains at 44K.

Note: When I save bmps with the grid on, the grid moves relative to the rest
of the graph.  This is why the bmp files on my web site don't have grids.
Does this happen to you?

Then, I opened Notepad with the idea that I'd record the frequencies of the
first 9 peaks (starting at the bottom of the graph) in this chord that I
could get a signal strength of greater than -40 dbs out of.  I picked _9_
arbitrarily.  I started at the bottom because I knew that's where most of the
parts would be.

I didn't take the readings from the same time on the graph.  Rather, I took
each reading at the strongest point in each peak (it happened that all of
these fell very near each other, in about the third quarter of the total
time).  The crosshairs have a resolution of 1cps, and I did some careful work
with the mouse.  If there was a range of frequencies that shared the same
strength, I took the one closest to the middle of the range, taking the
higher frequency if the range was odd.

After I had done all that and closed Spectrogram, I turned my text file into
the chart up.txt.  I am at a loss to explain the results.  With the transform
limited to a frequency resolution worse than 5 hertz, I should not have
gotten, nor did I expect to get, anywhere near the accuracy I did.

It may be noticed that I do not have the lead part labeled in the up.txt
chart.  At first I thought that maybe the baritone was at peak3 and the lead
at peak4, with peak2 being 5-3.  However, this would be an unusual voicing,
and listening reveals that the Baritone is singing peak2, and the lead is
singing a 5/2 above the bass.  This peak is in the spectrogram (up.bmp) but
not the chart because I couldn't get more than -40 db's out of it.

Rothenberg's Efficiency                                             2/13/99

Rick Sanford wrote...

>I agree, but the term 'tonal' means specifically supporting a tonic-dominant
>relationship in the piece, no?  If not, we could call many drone-type things
>'tonal', which they are not.

Perhaps this is the distinction Doty suggests:

1) What I've always called "tonality"
   - the psychoacoustic phenomenon of
     fitting a group of pitches to a harmonic series

2) What I've always called "key"
   - the cognitive phenomenon of locating a
     stimulus within P (a proper map of pitch space)

The first one, as I understand it, basically depends on the virtual pitch
mechanism, provided roughness is low enough for it to work.  To the extent
that this does not imply a harmonic series, by combination tones or perhaps
some hard-wired preference, tonality can be established by fitting a group to
an inharmonic series.

The second one is best explained by Rothenberg.  See his series of 5 papers
published to Mathematical Systems Theory in the late 1970's (Myhill was on
the Editorial board at the time, I see).  I have read the three of them that
are listed in the Microtonal bibliography...


They are the single most important body of theory on melody known to me,
laying down a framework capable of supplying a useful explanation for just
about every melodic phenomenon there is.  The strength of the explanations
may be easily over-estimated, and maybe Rothenberg suffers from this a little
bit, but on the whole he is a first-rate theorist who applies his model with
impressive accuracy.

R. calls a stimulus that occupies a unique location in P a "sufficient set".
Sufficient sets which have no subsets that are both sufficient and proper are
called "minimal sets".  "Efficiency" is basically the percent of the scale
you need to hear, on average, before a sufficient set occurs.  This amounts
to asking when the listener will be able to measure intervals by scale
degrees.  In 12tET's 5-limit diatonic, the key-minimal set is the diminished
triad.  [It is interesting to speculate on the fact that this contains the
disjunct fifth (and the only ambiguous interval in the scale), and that
scales without modes (like the "wholetone" scale) are necessarily without
ambiguous intervals.]

So, I would say that "drone-type things" tend to meet both of these criteria
quite well.  By the first, one might say that the tonality is static, but
certainly not weak.

By the second, OTOH, the tonality is probably shifting more than in most
western music.  Even tho the scales used in such are often improper, the
drone provides the reference needed to measure intervals in scale degrees. 
But since the scale is improper, the key will constantly be shifting as
various proper subsets are sectioned off.  It could be that Mr. Sanford does
not hear this as being tonal because he is used to scales with a relatively
high efficiency.  Improper scales may work best with low efficiency.

Two tonalities                                                      2/16/99

>>To the extent that this does not imply a harmonic series, by combination
>>tones or perhaps some hard-wired preference, tonality can be established
>>by fitting a group to an inharmonic series.
>I don't understand that.

I was just allowing for adaptive tuning.  The only work that I know of in the
area of explicit adaptive tuning is Sethares', and I have said many times how
cool I think it is.  But, as I have also said, I feel that the inharmonic
timbres and chords lack a quality that I find in harmonic timbres and chords
(and although I haven't done enough listening to say for sure, my first
reaction has been that justly-tuned subharmonic chords also lack this
quality).  I don't know why this is.  I was speculating, and I may have
gotten these from Sethares' book or posts to the list, that it could be one
or both of...

*)That our mind-ear, having evolved to recognize spoken language, might have
many routines that assume harmonic timbres, since the human voice fairly
harmonic, and in fact vowels are to a great extent the selection of certain
harmonics from it.

*)That roughness curves, when adjusted for combination tones, may show a
preference for harmonics, or at for least some sort of additive series.

>Anyhow, do you think the minor third in the minor key is not part of the
>phenomenon of harmonic tonality?

Its function in the minor key is strictly a matter of melodic tonality, as
far as I'm concerned.  I have reservations about the term "melodic tonality"
because the stimulus P need not be melodic.  In fact I think of "key" as
being a spec for relating harmonic and scalar stimuli.  So in a way "harmonic
tonality" is part of "key".

I'm not sure how to measure the "key effects" in a sample of music.  I said
that Rothenberg comes as close as I know.  Also close are Wilson, Boomsliter
and Creel, and you (Paul Erlich).  For "harmonic tonality effects", I will
accept a list of, at each beat in the music, the top three most likely
fundamentals and their relative probabilities.  This is lifted right from

RE: Rothenberg's Efficiency                                         2/16/99

>>"Efficiency" is basically the percent of the scale you need to hear, on
>>average, before a sufficient set occurs.  This amounts to asking when the
>>listener will be able to measure intervals by scale degrees.
>Why is that?  A random scale has 100% efficiency, unless you meant
>"minimal" instead of "sufficient" in the second-to-last sentence above.

I mean sufficient.  There's nothing that says high efficiency is somehow

This involves a distinction I didn't make earlier.  All of R.'s measures
apply to codes, not stimuli.  That is, a random scale may have 100%
information, but it can not have 100% efficiency.  R.'s idea is that people
need a code (I take this to be very hardware-abstracted stuff) to get key-
effects from pitch data.  Thus, the random scale might sound like a mistuned
diatonic scale to an average western listener.  The idea is that after enough
exposure to the scale, one learns a code just for it.  So scales are
evaluated only by the type of code they imply, and the point of the whole
thing is to predict what kind of music best takes advantage of the code.
This is difficult, but I believe R. has made useful predictions with it.

>In any other regular diatonic tuning, the key-minimal set is the tritone.
>Do you think that spells a qualitative difference between the diatonic
>scale in 12tET and that in other regular tunings (I sure don't!)?

The nature of the minimal set shouldn't matter that much.  What should matter
more is the overall efficiency, which Rothenberg claims is stable for the
diatonic scale across all tunings with reasonable fifths.

>What is the point of saying "5-limit" above?

In this case, it probably didn't matter.  But you never know... it's not
quite the same scale at the 5-limit as it is at the 3-limit, is it?

>Do you think it consequential that the Pythagorean diatonic scale is not

I think things ought to be run thru some sort of fit function, or whatever,
before being subjected to R.'s measures.  Something to prevent 1 cent
differences from impropering them...

I've been screaming in favor of the just diatonic scale on this list for two
years, and in general I believe that almost any scale can be successfully
approximated during skillful tours of (pick your limit) JI.

I've also praised the melodic usefulness of harmonic series segments, which
are often improper.  Impropriety isn't bad, but I believe it does influence
how these scales are used.  Melody almost always involves making proper
subsets of the master scale, wether or not the master scale is proper itself.
With harmonics 8-15, I noticed myself treating 13 and 15 as passing tones
before I knew anything about propriety.  They are good passing tones, tho,
and they are fantastic harmonically...

>>[It is interesting to speculate on the fact that this contains the disjuct
>>fifth (and the only ambiguous interval in the scale), and that scales
>>without modes (like the "wholetone" scale) are necessarily without
>>ambiguous intervals.]
>The diatonic scale in meantone tuning is proper (not that I think that
>matters) and has no ambiguous intervals.

It matters for something.  What it is might not be as simple or as strong as
you seem to want it to be.

Here's a few items regarding a possible relation between ambiguous intervals,
minimal sets, and the "disjunct" position in MOS chains:

While the augmented fourth and diminished fifths aren't ambiguous in
meantone, they do represent a blurring between the two classes.

The key-minimal set in meantone is the disjoint 5th, according to you.  In
12tET it is the disjoint 5th plus a third note, according to R.  For MOS's
that are proper but not strictly so, the disjoint fifth was ambiguous in
every case considered by John Chalmers (tho he just looked at it briefly).

>>By the second, OTOH, the tonality is probably shifting more than in most
>>western music.  Even tho the scales used in such are often improper, the
>>drone provides the reference needed to measure intervals in scale degrees.
>I don't understand.  Did you mean the reference needed to locate your
>position within the scale?

Yes.  I am a little weak on the locating part, and need to think about it
some more.  Obviously, and the point is, composition plays a big role in all
this.  If the composer purposely avoids all sufficient sets, then efficiency
is sort of out the window -> except that it determines what must be avoided
and at what cost...

BTW, I just noticed, both the pentachordal and symmetrical decatonic scales
are strictly proper in 22tET :~)

Just guitar                                                         2/18/99

>I believe it may be quite practical to have frets going all the way across
>in a JI guitar.  I have been considering the following arrangement which
>when combined together creates a kind of matrix guitar.  Please let me know
>what you think, as I have not yet fully thought it through.

I built an electric slide version of this instrument.  It has two sides.  One
has strings tuned to harmonics 8-16, and the other has two sets of strings,
one tuned to a harmonic hexad, and the other to a subharmonic hexad.  Both
sides of the body are marked to indicate harmonics 8-16 and subharmonics 6-
12.  Darren, if you have played Denny's Cosmolyra you are familiar with this

The instrument is extremely resourceful, even as a solo instrument, for
anyone who has paid their fingerpicking and slide technique dues.  John
Starrett played the @#$% out of it, if only briefly and unplugged, about a
year ago now.  Rod Poole has also played it, briefly and unplugged, to good

To play it in a band situation is much easier, and almost anybody could pick
it up in an hour or so.

But the design does have drawbacks.  I can't imagine, even with a fretted
version, getting voice leading from inverting chords the way you do on a
regular guitar.  But I'm not a guitarist, so I don't really know...

225:224                                                             2/18/99

>Terry Riley's "The Harp of New Albion" is beautiful (Riley manages to
>increase the effective number of pitches per octave in this 5-limit tuning
>by using the interval 225/128 as a convincing approximation 7/4.)

Does anybody know the best 5-limit shape for taking advantage of the 225:224
at the 7-limit?  When looking for the best 7-limit shape to take advantage of
the comma in the 5-limit, I came up with...

                 /|\       /|'       / \ 
                / | \     / | `     /   \
               / 7/6-------7/4 '   /
              / // \\ \ / //  `.` /
            /|\/     \/| /
           / |/\     /\|/
         / // \\ \ / //
      16/15/---\-8/5 /
        | /     \ | /
        |/       \|/

...can anyone find a better one that's still reasonable as a scale?

Evolution                                                           2/26/99

>>Within a tonal context, we do read 400c as 5/4, but 400c still has a sound
>>of its own. Weather that sound fits within any discernable musical system
>>as itself is another question. I doubt if it could to any human ears (at
>>least at this stage of our aural evolution. Who knows what the next few
>>thousand years may bring!)
>Evolution happens because of survival of the fittest. What kind of eugenics
>program do you forsee that will wipe out those with poor aural resolution?

Evolution works because of mutation and survival.  Fitness is very hard to
define, unless you say it's what survives, which is redundant.  Even survival
is not the whole story, as many systems that tend towards extinction are more
successful that traditionally "fit" systems (they wipe out the "fit" system,
and then themselves).  Then there is the lemming reflex, where suicide is
selected to preserve the population.  Observing this behavior in human cells
(esp. immune system cells) is very interesting.  The thing opens up a whole
new reason to get picky about one's definition of "time".

Anyway, I think it's safe to say that humans have reached the point where
what we think is a critical part of our "genome".  And so we can evolve just
by thinking.  Language is perhaps the earliest and most glaring example.
Music is also important.  I think music can be a powerful learning tool; I
believe it "works" by imitating symmetries found in nature, ones we use for
problem-solving and memory.  We have emotional responses to music because we
think we're learning -- the primary role of emotion is the reinforcement of

To answer your question Paul, looking closely at life on earth reveals to me
a surprising lack of 1:1 relationship between selected traits and fitness.
There's all kinds of piggybacking (porkbarrel legislation ) and the like.
But, assuming you disagree with the above, what kind of eugenics program can
you see to select for those who enjoy music at all?  That is, if it has come
this far, why not to the point where 400 cents can have an independent
harmonic meaning?

And putting the evolution thing aside for a sec, what about just over the
course of a song?  While your probabilistic approach to harmony makes the
most sense of any I know of, I think that compositional practice may narrow
down the options real fast -- what about a song that makes a point of
emphasizing the difference between 400 cents and 5:4?

Good rock now                                                        3/2/99

>Teeny bopper, ice cream, pre-fabs.... yes, they were always here you say...
>Monkees, Menudo, NKOTB..... But now they are all that there is.
>Let's feed the Piano some hay and see if it eats......

I know of three rock groups producing top-of-the-line music right now...

1. Bela Fleck and the Flecktones
2. Ozric Tentacles
3. Phish

...and more producing very good music.

Why not improve the piano rather than sending it to the farm?  Carbon fiber
soundboard instead of wood, tuning machine instead of pins/pinblock, 5 or 6
octaves instead of 8, single and double stringing instead of triple,
generalized keyboard instead of halberstadt, 19 and 31 tones per octave
instead of 12...

RE: piano                                                            3/2/99

>>Why not improve the piano rather than sending it to the farm?
>Because a good synth is a lot cheaper than a cheap piano.

For one, you're comparing apples and oranges.  For two, there are very few
synths I would call good, and hardly a keyboard controller I would call
acceptable.  I'll be spending over $6000 for my midi keyboard and synth in
the next 2 years.

Pianos need not be expensive as they are.  Stupidity is a large part of the
cost.  Aging wooden soundboards for 2 years when $5 worth of fiberglass gives
better sound and lasts twice as long?  Making crappy pianos (anybody own a
Baldwin Acrosonic?) with 8 octaves when for the same price you could have a
passable instrument with 5?

RE: piano                                                            3/4/99

>The main reason for not choosing a synthesizer over a piano is that
>listening to a soundboard and listening to a loudspeaker are very different

Ideally it would be the only reason.  But I have yet to hear a convincing
struck-string timbre from a synthesizer.  Then there is the resonance between
strings.  Generalmusic tried to model this on the Pro2, but failed.  Then
there is control.  A properly-regulated piano gives unmatched control over
volume and *attack characteristics*.  Most of this stuff should be possible
on a synth, but it just hasn't been done.

>While I am also extremely doubtful that carbon fibre or fibreglass is ever
>going to be the basis of a good soundboard

The lighter and more rigid, the better.  The more resistant to humidity
changes, the better.  Even aluminum makes a fantastic soundboard, if you've
ever heard a Challis instrument.

>I have yet to encounter a set of plastic keys which are satisfactory in
>texture or weight when compared to the San Francisco-built Broadwood upright
>with which I grew up.

Plastic keys?  I know of no piano ever produced with plastic keys.  The boys
at DS Keyboards experimented, but found wood to be the best material.  If you
mean key tops, i.e. plastic v. ivory, I don't think there's really a
significant difference.

>Not to mention that the development costs for a carbon fibre soundboard
>would be astronomical.   

How do you figure?  Carbon fiber is inexpensive, and soundboard design is

>Has anyone made a serious effort to market an improved piano? And if not,
>why not? It might even be possible to make some money selling them.

Pianos don't make very much money, even when you mass produce them.  But I'm
sure one could do alright.  Marketing would have to be tactful.

RE: piano (Breed)                                                    3/8/99

>>Maybe for runs.  But ask yourself why nobody plays Rachmaninov on a synth.
>Yes.  Rachmaminov wrote for a particular instrument, so you wouldn't expect
>the music to transfer to another one.  I keep asking myself why people still
>play Bach on a piano, and "conservatism" seems to be the answer.

I'm no Rach fan, BTW.  I know people who want my head for this, but with few
exceptions (perhaps the Toccatas and Goldbergs), I prefer Bach on the piano.
It has a lot more phrasing power than the harpsichord, if you know how to
play it (not that I do).

>>There's more to speed than runs.  Repetition is the most important place to
>>have speed, and the piano has got everything else beat by a mile.
>Rubbish.  Sequencers are the thing.  Record once, loop it, double the

Now there's a recipe for musical results!

>Sure.  Regardless of the keyboards, though, there are some places where
>synths have the upper hand.  One is that it's much easier to retune a
>synth.  This is the killer for me.

For me too.

>Another is that you can get a higher variety of sounds from a synth.

That all depends.  In theory?  Yes.  Have I heard it?  No.

>Only playing one voice at a time as if it were an acoustic instrument isn't
>showing it in its best light.

Multitracking is cool for anything.  Let's talk instruments, that is, what
one person can do in real time.

>For that matter, using the preset voices doesn't give me what I want.

I've been synthesizing since I was in the 5th grade (ESQ-1).  I've never been
able to come up with anything good for much more than shock value.  I had a
chance to play a bit with a GDS synergy (used by Wendy Carlos on Digital
Moonscapes), and it was one exception.  DM is my pick for best synthesis
ever, BTW.

>I'm not much interested in a synth that sounds like any particular acoustic

Nor am I.  But I do want one at least as powerful.

>If you don't like the sound, try sending it through guitar effects.

Maybe the best keyboard sound I've ever got was with Pagano's Baldwin organ
and a MuTron phasor...

Theory vs. Practice (Haverstick)                                     3/9/99

Ideally, John Starrett is right.  But there has been a tendency for theorists
on this list, and I am not immune, to say stuff like "higher values are
better", or "I like this more".  Better for what?  Why do you like it more?
I think theorists have a responsibility to say.  Science ought to make its
point without a lot of sales pitch.  That's one reason I like Wilson's work
so much.

Inversionally-identical chords (Monzo)                              3/12/99

>This relates to the "inversionally similar" chords I posted yesterday
>(actual they should be referred to as "inversionally identical").  Perhaps
>utonalities that are most favored are the ones which are inversionally
>identical when described otonally. Has anyone ever investigated this
>property in chords before?

I'm not sure I know what you're getting at... but if it's what I think it is,
there's no such thing as a inversionally-identical utonality.  What we're
looking at here is the order of the interior intervals of a chord...

7-limit otonality is 5:4 -> 6:5 -> 7:6
10-12-15-18 is 6:5 -> 5:4 -> 6:5

The reason the 10-12-15-18 is inversionally identical is because the list of
its interior intervals has reflection symmetry.  This is never the case for
Partchian O and U tonalities because the same interval never appears twice. 
The Utonality is what you get when you invert the Otonality anyway.

This can be likened to palindromes.  Mom=Mom, Dad=Dad, but Mad =/= Dam.

RE: 4:5:6                                                           3/16/99

>29-equal is another tuning with good triads, IIRC.

It does have good triads, but I don't know about a 4:5:6 chord.  I listened
to it, and was put off by the fact that 9 steps barely approximates 5:4
better than 10.  After listening more, I'm willing to include it (9 is best).

I was really looking for chords that were improving as I went higher up.  But
I did say "usable" and I did include 15 and 22, so to be fair I should also
add 27, which beats 29, and 25, which is hardly worse than 12.  26 doesn't
cut it.

25 has a killer 4:5:7 chord, but it isn't 7-limit consistent, and adding the
6 makes me want to raise the 7 a step...

Of course 43, 46, 49, and 50 should be in there, and probably others are as
good as 15...  but I do have a day job.

[Kees van Prooijen]
>I miss 34 with just this chord.

Wow!  I can't believe I forgot 34, which has always been very interesting on
paper.  I plugged it in, and was thrilled to find it is about as good as 53.
Much better than 41 or 31.  Add to that its near-perfect minor triad, and I
think that it is as good as 53 for 5-limit work.

The only bummer with this ET is that it is only consistent thru the 5-limit,
and I checked for acceptable 4:5:6:7's, and there weren't any.

So here's the updated list.  Gaps indicate a big change.



I should mention that I am using a highly idealized environment here.  In a
band situation, maybe 26 would do, I don't know...

It would be neat to have others confirm or deny this ranking.  I have used
Midi Relay and the GM Reed Organ patch, recording into Cool Edit for the
comparo.  All "4's" were fixed to middle "C".

RE: Wendy Carlos scales                                             3/24/99

--Joe Monzo--
> Also, any further technical info on the Carlos scales would be appreciated 

Did you catch my two posts on non-octave scales in early Feb?

Alpha, 78.0 cents/step

15tET (x1) = 80.0
31tET (x2) = 77.1
46tET (x3) = 78.3
77tET (x5) = 77.9

Beta, 63.8 cents/step

19tET (x1) = 63.2
56tET (x3) = 64.3
75tET (x4) = 64.0

Gamma, 35.1 cents/step

34tET (x1) = 35.3
103tET (x3) = 35.0

88CET, 88.0 cents/step

14tET (x1) = 85.7
27tET (x2) = 88.9
41tET (x3) = 87.8

Equalized Bohlen-Pierce, 146.3 cents/step

8tET (x1) = 150.0
25tET (x3) = 144.0
33tET (x4) = 145.6

Just as the 7th root of 3:2 is related to 12tET, so are Alpha and Beta
related to 15 and 19tET.  Gamma can be considered two interlaced 10th root of
3:2 scales, much as 34tET can be considered two interlaced 17tET scales.

The idea is that when tempering an MOS, the generator usually takes the
punishment.  But why not give the interval of equivalence a taste of the
medicine?  The difference between the versions will be proportional to
D/(G+I) where "D" is the size difference between the two chains, "G" is the
number of members in the chain of generators, and "I" is the number of
members in the chain of intervals of equivalence.

Of course any tuning can be explained as a retempering of any other tuning,
and so much the better if it helps us think about them.

88CET is then the "other" version of the 11-tone 7:4 vs. 2:1 MOS.  27 and 41
are both higher MOS's of this interval pair.  I don't know how to classify
BP, which I find to be a scale of limited (I should say specific) usefulness.

For sufficiently low values of the above formula I think various temperings
of an MOS ought to be quite similar conceptually, differing mainly in "mood".
My experience with 12tET vs. the 7th root of 3:2 backs this up.

Lattice wrap-up                                                     3/24/99

After much thought, I have reached some conclusions regarding the lattice
metric stuff.  I believe strongly in these, and hope that they can contribute
to a consensus on this thread...

1. Triangular is better than rectangular.

2. "Euclidean" distance isn't good for much.

3. Weighting by (n*d) is trivial.  If you add the results of each rung on the
way to the target, you get nonsense.  If you multiply, you get (n*d) of the
original fraction.

4. Because they are useful for measuring dyads, (n+d), whole-number-limit,
and odd-limit all seem like viable ways to weight the rungs.  But I don't see
how anything is gained by weighting on an octave-equivalent lattice.  Simply
counting the rungs in the shortest path to target at the declared odd-limit
(ala Paul Hahn's consistency and diameter) is the pinnacle of city block
metrics.  Somebody please get on and say any reason this would not be so.

5. A 2-axis is probably not desirable.  If it is, some form of weighting
would likely be needed; a weight of log(axis) is suggested.

RE: harmonic entropy (Graham Breed)                                 3/24/99

>As large intervals do appear to be less consonant, the denominator rule

Aha!  The issue of large intervals comes again.  I will follow Keenan and
call this "span".

It is true that with a large enough span, you can play almost anything and it
won't be dissonant.  So in a way, span decreases dissonance.  But, these
large dyads are also more difficult to difficult to tune accurately, which
suggests an increase.

Maybe the answer is that consonance and dissonance aren't mutually exclusive
opposites -- In the case of large dyads, maybe we should say that the
dissonance decreases, and so does the consonance!

Generations Lost                                                    3/28/99

I wrote...

>Sony deserves the boot in the ass for purposely engineering loss into an
>otherwise digital medium.

Just to clarify this, to make sure everybody realizes what flies for a
solution for managing our 'intellectual property', I quote from a tiny blurb
buried in the back of my MZR50 owner's manual...

"Note on Digital Recording

This recorder uses the Serial Copy Management System, which allows only
first-generation copies to be made from premastered software.  You can only
make copies from a home-recorded MD by using the analog (line out)

As information storage changes, the economy must change.  In the 'dark ages'
only a few could read, and they controlled the masses in the most brutal
dictatorship ever- a 1000 years of embarrassment.  The printing press brought
about the great revolution of our time, gave rise to 'intellectual property'.
Now 'intellectual property' is no good, just like the church of the middle
ages, must go.  I don't have any easy answers, but I think we should all be
able to agree that the above is not acceptable.

Augmented                                                           4/21/99

>But the most consonant tuning for the augmented triad is something near
>equal temperament.

Your definition of augmented triad may come from the linear series, and so
you may not consider the following to be "augmented triads"...


...but I think they are all more consonant than the augmented triad of 12tET.

Just one interval... I mean...                                      4/27/99

>>No, but it may be close enough to help the triad by the effect I mention...
>>I find RMS to be good enough conceptually and empirically for most
>Carl, you seem to be contradicting yourself.

The 720 cent fifth has an RMS error of 18, the 15tET triad of ~14.  Where's
the contradiction?

>Compare the 15-tET major triad with one whose pitches are 0 395 720. Which
>one has lower RMS error? Which one has a close-enough-just-interval?

The one-just-interval observation is supposed to make small differences when
the RMS is nearly the same.  It is not supposed to significantly change the
RMS ranking.  In this case, a brief listening test found the 15tET triad
slightly better despite the slightly lower RMS of your chord.  Do you

>Why is it that 6:5 can lock down the periodicity but 7:5 can't?

The 7:5 should be able to, but the power to lock should go down as the limit
goes up.  I'm also suggesting that the :4 intervals are more important.  I
did say all 5-limit intervals could be weighted equally, but perhaps the 5:4
is slightly stronger than the 6:5, and perhaps 3:2 is stronger than either.
But I doubt these distinctions will make much difference until the 7-limit.

Wilson's got class                                                   5/7/99

A while back there was a discussion about 12-note scales with high numbers of
7-limit dyads.  Paul Hahn found the winner, with 32 dyads...

       /|\         /|\
      / | \       / | \
     /  |  \     /  |  \
    /  7/6---------7/4  \
   /.-'/ \'-.\ /.-'/ \'-.\
  |\ /     \ /|\ /     \ /|
  | /       \ | /       \ |
  |/ \     / \|/ \     / \|
    '-.\ /.-'   '-.\ /.-'

...four tetrads (3 otonal), three 8:10:12:15 "ASS" chords, three 10:12:15:18
ASS's, and 2 hexanies.  Unfortunately it is so melodically un-even that its
chords float around scale degrees like small boats.  Recently, I noticed that
I had come across this structure (in this same position) last July, when I
was looking for 12-tone scales that contained bits of the 7-limit diamond.

I had also discovered this scale...

                    ,'/ \`.  ,' /
                 /|\/     \/| /
                / |/\     /\|/
               / 7/4------21/16
              /,'/ \`.\ /,'/
            1/1-/---\-3/2 /
            /|\/     \/| /
           / |/\     /\|/
          / 7/5------21/20
         /,'   `.\ /.' 

...Paul Hahn was the first to correctly count the number of consonant dyads
in this structure; 31.  He was also able to count all possible versions of
the two-hexany 12-plex, and found that only three lacked an interval smaller
than 25:24 (the smallest interval in my version is 525:512, or 43 cents)...

a)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 7/4, 15/8
b)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 105/64, 7/4, 15/8
c)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 42/25, 7/4, 15/8

...of these, he noticed that (a) and (b) are not covered by their tetrads,
but (c) is, although he seemed to prefer (a) because of its lower diameter.
I played a bit with these scales, and found I like (c) the best.  Today,
whilst going thru the Scala archive, I found this...

! wilson_class.scl
 Class Scale, Erv Wilson, 9 july 1967

...which is identical to (c) when transposed by its 5th degree...

                 / | \
               /,'/ \`.\
             /|\/     \/|
            / |/\     /\|
           / 7/4------21/16
          /,'/ \`.\ /,'/
        1/1-/---\-3/2 /
         |\/     \/| /
         |/\     /\|/
          \`.\ /,'/
           \ 6/5 /
            \ | /

...so there you have it.  Nothing new under the sun (at least since 1967).

Academia again                                                      5/11/99

I wrote...

>I've thought about it.  If academia is to be pigeonholed, then it is best
>considered an industry.

Not a bad thing, or good, just interesting.  They are a personnel service.
One that happens to employ a large number of people in its own right.

Seen this way, one can understand why it might seem shady to work for them.
Like being a lawyer, you're on the inside of a narrow but essential
operation; a vertical (rather than horizontal) industry.

Like other industries whose principle source of income is not directly
subject to supply and demand (like donation and tax), bureaucracy flowers.

Departments may be evaluated by the ratio of people they employ to the ratio
of people they place outside.  Hard sciences tend to have low values.  Other
departments have, in the last 50 years, become almost completely self-
absorbed (high values).  Look at what's happened to lit crit.

Unfortunately, composition/theory is definitely in this group.  Like
deconstruction, serialism may be cool in its own right, but nobody to my
knowledge has ever come forward with any reason why serializing things should
be interesting.  Then we have various "process" methods, the point of which
are that results don't count (aside from the initial folk-ethic shock value;
hanging a snow shovel in a museum to say that art can be a valuable part of
everyday human life).

It boils down to this: music of the type made by Bach and Beethoven is not
coming out of academia.  It came out of the south, then from England -- the
folk ethic does produce results!  There are a handful of exceptions, almost
all of them pre-1940, almost all of them with less-than-complete ties to

RE: magic chord                                                     6/22/99

Paul Erlich wrote...

>In 72-tone equal temperament, take the chord C E- F#- Bb-- (where - means
>flattening by 1/72 octave relative to 12-tone equal temperament). The six
>intervals in this chord are:
>One cannot tune the chord in JI and have all the intervals listed in the
>rightmost column. The best one can do is to use 28:25 instead of 9:8; then
>the other five intervals can be just as shown. 28:25 is a 25-limit interval
>but is clearly a case where the limit is of far less importance than the
>proximity to simpler ratios. It is 196 cents, and has a fairly clear
>interpretation as an 8-cent flat 9:8, but it might also be heard as a
>14-cent sharp 10:9. Tuning it closer to 9:8 would increase its consonance
>and the 72-tET version shown here does so with minimal damage to the other
>So the "magic chord" is, allowing for intervals to be tempered by up to 4
>cents, a saturated 9-limit chord. Similarly, the augmented triad in 12-tET
>is, allowing intervals to be tempered by up to 14 cents, a saturated 5-limit
>chord. Neither of these chords can be expressed adequately in just

I agree that there are consonant chords which do not have a natural
expression in JI.  But have you actually compared the two versions of this
chord?  I did, in root position, and they sounded almost exactly alike.
Certainly they shared the same VF.  In no way did the 72tET version sound
more consonant, in fact the just version sounded a little smoother.  I doubt
anyone would resent the substitution of one of these chords for the other in
a reasonably-paced piece of music.

RE: magic chord                                                     6/24/99

Paul Erlich wrote...

>What's the VF here? Anyway, I think that 9:8 is too weak a "consonance" to
>have an audible effect here, but with the augmented triad, I definitely
>prefer the tempered over the just version.

Of course the VF is given as a distribution, and a fairly even one in this
case, which is why the chord sounds as it does.  What is it?  I listened to
all inversions of the just version, rather in haste...

1/1    5/4    7/5    7/4   (1)  1/1  8/5 ...
c      e      f#    a#          70   20  10

1/1   28/25   7/5    8/5   (2)  8/5  7/5 ...
c      c#     f#    g#          70   20  10

1/1    5/4   10/7   25/14  (3)  10/7 1/1 8/7
c      e      g      b          50   40  10

1/1    8/7   10/7    8/5   (4)  8/7 10/7 ...
c      d      g      g#         70   20  10

...of course the 1/1 frequency must be fixed, or you'll get tricked.  Here's
the scala file to do it (1/1 is mapped to "c" above)...

! magic.scl
 magic chord test

I compared only root-position just and tempered versions, but I doubt the VF
for the tempered version differs in any significant way from the above.

As for the augmented triad, to which just version do you "prefer" 3tET...



RE: An augmented baker's dozen                                      6/26/99

>Wilson's CPS scales tend to favor otonal and utonal chords equally;

Only for CPS's where k:n is 1:2 are otonal and utonal resources equal.  Ah-
actually, for all k of any n (power set of n), they are equal, that's true...

>of course, one can use these scales and treat the utonal chords as
>dissonances; but then why not use more otonally-based scales, which are
>bound to contain plenty of resources for dissonance anyway?

For bells, sure.  But for many harmonic timbres, I find the utonal chords
beautifully dark consonances thru the 9-limit.  And unsaturated utonal
chords, which come naturally from the CPS's, can be consonances to the 17-
limit, I think.

>(I think Kraig is about to describe such scales for us, as well as their
>utonal versions.)

Please do, Kraig!

>With 12et I always preferred the minor chord so it is no surprise my
>preference for subharmonics.

Actually, Kraig, I am beginning to believe that the 12tET minor triad
approximates 16:19:24 more often than 10:12:15.  You and Wilson both told me
about the 19-limit approximations in 12-tone... I think you guys were right.

RE: magic chord                                                     6/26/99

>>As for the augmented triad, to which just version do you "prefer" 3tET...
>All -- each one has at least one interval more dissonant than the tempered
>major third.

Don't you mean at least one interval more dissonant than the tempered 8:5?

Chords are more than the sums of their intervals.  I hear the first three of
the above as clearly more consonant than the 12tET augmented triad.

Even so, I find it strange to prefer any one chord to any three.

CPS's and magic chord                                               6/28/99

>>Only for CPS's where k:n is 1:2 are otonal and utonal resources equal.
>>Ah- actually, for all k of any n (power set of n), they are equal, that's
>I meant generally. Clearly some CPSs favor otonality, while others favor
>utonality. What does your last sentence mean?

If n is the set of factors and k fixes the size of the subsets, you'll get
the power set of n if k is allowed to be anything.  The power set will
contain an equal number of otonal and utonal chords, and not only for the
original n!  As Wilson points out in D'Allesandro, the power set also
contains every CPS above it in Pascal's triangle --- in fact a different
triangle top for each of the combinations of (n-1) out of n, recursively
until n hits 1!

>>Actually, Kraig, I am beginning to believe that the 12tET minor triad
>>approximates 16:19:24 more often than 10:12:15.  You and Wilson both told
>>me about the 19-limit approximations in 12-tone... I think you guys were
>I guess my comment to that effect came too late.

Too late for what?  It's good to know your view.  Incidentally, I don't think
it's only JI enthusiasts that hear the chord this way (I certainly don't have
a very good ear at the 19-limit!).  While the 19-limit version has a higher
harmonic series representation than the 5-limit utonal one, as you once said,
both sets of numbers are too high to rule out the effects of their
constituent intervals.  The intervals of the utonal triad do not all evoke
the same VF, whereas the intervals of the 19-limit otonal chord do.  Might
this give a boost to the 19-limit triad's probability of being heard in the
case of 0 300 700?  Was our earlier observation of 'something funny with
fractions with a base 2 under' a manifestation of some sort of VF rally

>>Don't you mean at least one interval more dissonant than the tempered 8:5?
>I was thinking octave equivalence, which seemed appropriate since you did
>not list any inversions.

For the record, I twice disclum my use of only root position just versions.

>>I hear the first three of the above as clearly more consonant than the
>>12tET augmented triad.
>Well, you may be listening in particular for the effects of otonal-in-JI
>chords, which one might call consonance.

I think I must be -- lower roughness and higher tonalness.

>there are other ways of perceiving consonance, and these ways are important
>in music.

Certainly in 12tET music, none of the just versions I list could replace the
augmented triad (although I feel strongly that the same need not be true of
music written for JI).  Your point is well taken, and in recent conversation
with a friend of mine (a classically-trained musician), I was unable to
convince that the 7:5 is 'roughly as consonant' as the 8:5, wether the 5's
are fixed or not -- he claims to hear 8:5's as incomplete triads, and thus as
more consonant than 7:5's.

MOS                                                                  7/1/99

>Beyond that, I don't think he's ever dealt with an MOS where the generating
>interval isn't taken to be a fourth.  We may be on virgin territory!

?  Wilson has played with more generators than I'd care to shake a stick at.
He is especially fond of irrational ones, if you've heard _From on High_ or
talked to Marcus about the tunings.  He has these iterative formulas which
converge on generators that give certain properties to the scale --- often
difference tone properties, ala meta-meantone.

Wilson told me, with a grin a mile wide, that one of his favorite things in
the world is varying the size of the generator and observing the points where
the harmonic approximations rearrange on the linear series.  He said that
sampling this continuum at only the roots of two is often too crude to see
the real action.

>Graham, In Wilson's term "Moment Of Symmetry", what symmetry is it referring

I take MOS to refer to those special sizes of chain, for a given link size
and interval of equivalence (IE), at which every link subtends the same
number of IE-equivalent joints.

I view this basic definition as the cause of the two-interval pattern, and
its affect on the structures of the modes of the scale.

I think these properties are desirable because of the way they relate scale
degrees to acoustic intervals.

>The scale tree is meant to be used both acoustically and/or logrithmically.

It is very interesting that you say that!  I was just thinking the other day,
that the Stern-Brocot tree (aka Farey tree, scale tree...) has got to be the
most important structure in all of music theory.  When applied to
frequencies, it gives the best map of dyadic consonance there is (might as
well define consonance for harmonic timbres).  When applied to logarithms of
frequencies, it gives the best lexicon of melodic resources there is (yet).

RE: propriety                                                        7/2/99

>I think there's more to it than propriety, as my favourite "Blues Scale" as
>outlined on my website was designed to work melodically, but is improper.
>In that case, it sounds like either a 5-note scale with "quartertone"
>inflections, or an uneven 7-note scale, or an even more uneven 8-note scale
>if you have good pitch discrimination.

>Although I do think propriety is a more reasonable property than maximal
>evenness, I think it's too restrictive. My favorite example over the years
>has been the Pythagorean diatonic scale, which is not proper.

Propriety is the most basic measurement of acousti-scalar effects, the first
thing you ask.  It isn't meant to decide the value of a scale, at least not
in so crude a way as these posts seem to assume.

Propriety also isn't a property of scales, it is a property of the mental map
that a listener uses to extract a certain type of information from an audio
stimulus.  Only if you assume that the listener is doing the most efficient
extraction possible, and that the stimulus is strictly tuned in the given
scale, can you use propriety as a property of the scale.

Even then it will apply only to sensations that arise from the tracking of
scale degrees.  Sensations related to absolute pitch, and any myriad of other
things, are an entirely different affair:

Graham, you say your scales work melodically despite being improper.  Work
melodically they might, but this need not include melodic modulation per se,
and I hear very little of it in the examples on your web site.  What melodic
modulation I do hear, I hear as subsets of a diatonic scale, which is the
only map I have learned as a listener.  So my experience jives perfectly with
Rothenberg's theory. 

RE: propriety                                                        7/2/99

>I have advocated a weaker version of propriety which only enforces the
>non-contradiction between generic size and specific size when it involves
>consonant intervals.

I like this a lot.  Much as the drone works in Indian music, harmony can work
in western music --- when a harmony sounds, dissonant notes tend to be heard
as passing tones, measured against the VF (and probably harmonics of the VF)
of the chord wether or not they participate in a proper mapping.

So I look at this version as a way of applying propriety to scales that may
be used harmonically.  As long as there's enough acoustic material to
establish a VF consistently fixed to scale degrees, the other notes don't

RE: propriety                                                        7/6/99

>>Graham, you say your scales work melodically despite being improper.  Work
>>melodically they might, but this need not include melodic modulation per
>>se, and I hear very little of it in the examples on your web site.  What
>>melodic modulation I do hear, I hear as subsets of a diatonic scale, which
>>is  the only map I have learned as a listener.  So my experience jives
>>perfectly with Rothenberg's theory. 
>The scale works *because* it's improper!  The contrast of large and small
>intervals has an emotional affect.

Sure!  This is an example of something that isn't melodic modulation.  I've
been hounding for 2 years on this list how commatic shifts in melody can be
cool, even correct.

>For the uninitiated, the scale is
>             A---------E               
>              \       / \              
>               \     /   \             
>                \   /     \            
>                 \ /       \           
>                  C---------G---------D
>                 / \       / \       / 
>             D/-/---\--A/-/---\--E/ /  
>               /     \   /     \   /   

A nifty scale.  If you remember the first mail I sent you, back in

>>Your website is also very cool.  I really like some of the blues scales.
>When heard as a 5-note scale, it is proper, so melodic modulation will work
>for this mental map.  The scale C-D-E/-G-A/-C happens to be very close to

Right!  And I tend to hear 5-note scales as subsets of 7-note scales.

>I think it's useful to talk of a "propriety index".  This is the ratio of
>the largest to the smallest atomic interval in the scale.  Atomic intervals
>being scale steps, or those intervals that can't be divided.  In this case,
>the largest atomic intervals are A/-C and E/-G.  In just intonation, these
>are 7/6 or 267 cents.  The smallest intervals are E-E/ and A-A/, which at
>36/35 are 49 cents wide.  That gives the 7-note scale a propriety index of
>5.5.  The scale C-D-E/-G-A/-C has an index of 1.3.

Say, that's a very interesting idea, which I'd never thought of.  I'll have
to take a look at it tonight.

I'd always imagined two other measures, that could work together, 1. What
percent of pitch-class space is filled?  2. What percent of pitch class space
is more than singly-filled?  It may be best to formulate this measure
graphically, rather than worry about which numbers to show.  If you take a
tonality diamond, and look down the diagonals, you have the pitch classes. 
One could imagine a graph where each of the pitch classes would be color-
coded strips, with overlapping areas florescent orange, or so.  A little
practice looking at these and the user would probably be able to tell more
about the melodic properties of a scale, faster, than anybody else.

>But this is a long way from saying that impropriety itself is a bad thing.
>Although nobody is saying that any more.

Who ever said that?

RE: propriety graph                                                  7/8/99

>>I'd always imagined two other measures, that could work together, 1. What
>>percent of pitch-class space is filled?  2. What percent of pitch class
>>space is more than singly-filled?  It may be best to formulate this measure
>>graphically, rather than worry about which numbers to show.  If you take a
>>tonality diamond, and look down the diagonals, you have the pitch classes.
>>One could imagine a graph where each of the pitch classes would be
>>color-coded strips, with overlapping areas florescent orange, or so.  A
>>little practice looking at these and the user would probably be able to
>>tell more about the melodic properties of a scale, faster, than anybody
>I don't get it.

I wasn't too clear.  I'm after a graphic representation of the Rothenberg
interval matrix (tonality diamond).  Imagine a line segment representing an
octave, pitch ascending from left to right, on a log scale.  For each scale
degree, plot a point for each occurrence in each mode and color a segment
connecting all the points.  The measures are then 1. What percent of the line
is colored?  2. What percent of the line is more than singly covered?  It
might be better to ask the second question in terms of the first, 2b. What
percent of the colored space contains overlaps?  But as I said, just looking
at the thing will probably tell the most.

RE: propriety graph                                                 7/11/99

>>>One could even keep the modes distinct with a line graph version.  Pitch
>>>is on the y axis, and on the x axis, at equally-spaced intervals, are the
>>>scale degrees.  Then the modes can be shown as lines on the graph by
>>>connecting the right dots at each scale degree.  If a pair of line cross,
>>>there's impropriety.
>>The converse is not necessarily true though, is it?
>That's a good question.  It seems not, but it may be.  I'll get back to you.

Indeed lines may cross on the graph of a proper scale.  I believe that all
improper scales will have crossing lines, but I'm not sure.  In any case, it
isn't crossing that indicates propriety.

What does indicate propriety is the slope of the lines.  They've got to go up
to be strictly proper.  If two intervals are connected by a line parallel to
the x-axis, then they are "ambiguous".  If any go down (from left to right),
there is impropriety.

Here's an Excel 97 spreadsheet of the pentachordal major decatonic scale in


There are two charts.  The larger one is as I described above.  The smaller
chart does show impropriety by line crossing.  I don't think either of them
are as useful as my original, 1D idea.  Examples of this type are on the

RE: Chain-of-minor-thirds tuning                                    7/20/99

Sent to Dave Keenan on July 20...

I checked out the new version of your 11-tone MOS article.  Forgive me if I
haven't been responsive to some of the great work you've been doing lately. 
I can't respond to everything on the list, and it didn't seem that there was
much for me to say... until now!  :)

I'm wondering why are you using diatonic (that is, based on the 7 fifths
semitone) nomenclature for this scale and the chords in your diagram?  Would
it be better to use scale degree numbers up to 11, with 4, 8, or 11 nominal
symbols?  With 4 nominals there may be more accidentals than is convenient,
but with 8 or 11 there are weird effects too, since there will be step sizes
between nominals smaller than the size of the accidentals.

Speaking of the 8-tone subsets of this scale... they are proper, and due to
Rothenberg and Miller's results, might they be heard even when the 11-tone
superset is desired?  What are the harmonic characteristics of the 8-tone
subset?  Scala file is attached.

In the legend of your lattice diagram, there seems to be an extra blue line
between 4 and 9?

How can the error go from 18 cents to 21.5 cents when the generator goes from
6:5 to 315.789?  In the 7:4 you say?  The difference is only 3 * 0.1485.

>That is, it has only two step sizes and has reflective symmetry (about D in
>the notation below).  This is also called Myhill's property.

There is a conflict between this and my understanding of Myhill's property.
I thought Myhill's property was the scale has no more than two sizes of
interval in each interval class.  It seems the above quote talks about
interior intervals of the scale.  Also, I was not aware that Myhill's
required reflective symmetry.  Having only two interior intervals with
reflective symmetry may be enough to guarantee what I thought was Myhill's
property --- I'll take a look at it later (must go fishing now) --- but I
doubt the converse is true?

Thanks for the soapbox!  Do you think we should take this on-list?

RE: Chain-of-minor-thirds tuning                                    7/22/99

>You haven't exactly made a case for why we should use 4, 8 or 11 nominals.
>Perhaps you can spell out what you have in mind. I used diatonic notation
>because it's familiar. Since there are many diatonic chords, why shouldn't
>they have diatonic notation? Although, in effect, it _is_ a 7 nominal + 4
>accidentals notation. Ab, Cb, E#, G# may be considered nominals since there
>is no A, C, E, or G (or their other flats or sharps).

No I didn't make a case, I asked.  However.  I find this 'familiar' stuff to
be way over-rated.  Composing in this scale will be the challenge.  Finding
structures --- took 500 years to get where we are with the diatonic scale. 
That's a huge challenge.  To get the chords under your fingers on your
instrument.  That's a huge challenge.  To learn a new notation is absolutely
trivial.  The skills of sight reading are not revoked by changing the number
of nominal symbols in the notation.  It's a 2-hour matter of flash cards to
learn the notes names on the staff.  Observe what happened when Easley
Blackwood made retention of diatonic notation a priority.  He wound up with
the most obtuse and hard-to-read notation I can imagine.  His decatonic scale
is a piece of cake, for example, with ten nominals and one octave to a
conventional 5-line staff.  The chords pop out, the key signatures are a
cinch, and melodic motion is intuitive.

The 7-fifths semitone exists in your ear, but not in this scale.  By using 7
nominals, you reinforce it in the compositional framework of the scale.  I
suggested 4, 8, and 11 because they represent places where the chain gets
close to closing against the octave, just as 5 and 7 fifths almost close
around the octave.

>In general I don't like using degree numbers because they change when you
>change the number of tones in the chain.

You're using them, tho?

>>Speaking of the 8-tone subsets of this scale... they are proper, and due
>>to Rothenberg and Miller's results, might they be heard even when the 11-
>>tone superset is desired?
>I'm not familiar with these results.  Can you give me a URL?

Miller wrote a paper on short-term memory which has become a classic of
cognitive psychology, and has been discussed before on this list...


As far as Rothenberg goes, I just meant that the 11-tone chain is not proper,
while the 8-tone chain is.

>Consideration of meantone chains leads me to think that having 2 step sizes
>is more important than propriety, since 4, 5, 6, 8 and 12 of meantone are
>all proper, but of these, only 5, 7 and 12 have only 2 step sizes (+
>Myhill's + MOS). Although they are also the only ones that are strictly

Rothenberg was able to explain the selection of 5 and 7 using only two
measures stability and efficiency.  Stability is essentially the percentage
of the scale's intervals which are "ambiguous", that is, which appear in more
than one interval class.  For a definition of efficiency, see TD #52 and 55. 
What I like about Rothenberg's stuff is that he came up with it first,
actually for a speech recognition model, and then applied it to music and got
results that jived.

>Well, for every tone you drop, you lose a utonal and otonal tetrad, so
>going from 11 down to 8 tones you go from 10 down to 4 tetrads. If you're
>only interested in 5-limit it's the same; from 10 down to 4 triads. But
>it's better than 7 tones with only two (one major and one minor).


>>In the legend of your lattice diagram, there seems to be an extra blue
>>line between 4 and 9?
>Yes, there are no 49's in an 11 tone chain of m3's. One would have to go
>to 13 or more tones. But you will see it used in the 9-limit lattice for
>diatonic, at the end of the article.
>In this case there are no 49's because there is no chain of two fifths,
>but in general for temperaments we may have consecutive fifths with
>acceptable errors but they may add up to an unacceptable error in the 49.

Oh wait- there's not an extra line, you've just bent the 49 so it's distinct
from 46 and the 69?

>>How can the error go from 18 cents to 21.5 cents when the generator goes
>>from 6:5 to 315.789?  In the 7:4 you say?  The difference is only 3 *
>I'll have to make this clearer. Thanks. It goes from 18c to 21.5c when the
>generator goes from 316.99 (1/6-kleisma, the one with the lowest highest
>absolute error) to 315.79 (19-tET).


>>>That is, it has only two step sizes and has reflective symmetry (about D
>>>in the notation below).  This is also called Myhill's property.
>>There is a conflict between this and my understanding of Myhill's property.
>>I thought Myhill's property was the scale has no more than two sizes of
>>interval in each interval class.  It seems the above quote talks about
>>interior intervals of the scale.
>Don't you mean exterior? Better clue me into the definition of interior and
>exterior, I'm not too good on this melodic stuff. Isn't an inner interval
>just what I'm calling a step.

A  B  C  D

According to me, AB, BC, CD, and DA is a complete list of this scale's
interior intervals.  Basically, all the 2nds.  I understood Myhill's property
to be no more than two kinds of 2nd, no more than two kinds of 3rd, and so

>I should have said "is equivalent to Myhill's property". I got this from
>the Scala Help for the "MOS" command. The Scala Help agrees elsewhere (FIT
>/MODE) with your understanding of Myhill's. So Manuel, should this be MOS
>=> Myhill's, not MOS <=> Myhill's? And BTW what is pseudo-Myhill's property?

I also noticed "pseudo-Myhill's property" and wondered what it was.

RE: chain of minor thirds scale                                     7/26/99

>You've made a good case.

Well, I don't know about that.

>>You're using them, tho?
>I don't follow. How, or in what sense, am I using degree numbers?

By calling chords "subminor 6th", "diminished 7th", etc.

>Thanks for the Miller URL. Of course this talks about 7 +- 2 notes _total_
>in short term memory at any given time, not 7 +- 2 notes _per_octave_ in a

Indeed.  This was discussed on the list around Thanksgiving (can't remember
if we were on onelist yet).  I don't think any conclusion was reached.

>But I guess it _is_ desirable for a whole octave to fit in STM.

I think there may be a feeling associated with keeping all the octave-
equivalent tones of a scale in STM, but I'm not sure if it's always
desirable, and I think the limit is probably more on the -2 side of 7 for
most people.

>>Rothenberg was able to explain the selection of 5 and 7 using only two
>>measures stability and efficiency.  Stability is essentially the
>>percentage of the scale's intervals which are "ambiguous", that is, which
>>appear in more than one interval class.  For a definition of efficiency,
>>see TD #52 and 55.
>Thanks for this explanation and pointer too. But I'm not sure I understand
>them well enough to apply them to 4, 7, 8 and 11 of m3rds. Could you,

Stability is easy.  Just count the intervals that are not involved in any
ambiguity and divide by the total number of intervals in the scale.  So the
4-tone scale, like all strictly proper scales, has stability of 1.  The 8-
tone scale has stability of 0.35714, which is quite low.  The 7 and 11 tone
scales are improper, so stability is not defined.

Along with stability, I suggest two measures I call instability and
impropriety.  The former is the portion of (octave-equivalent) log-frequency
space which is covered by intervals of the scale (using the smallest and
largest members of each scale degree as endpoints of a segment on the line
for that degree).  The latter is the portion of the covered area which is
more than singly covered.  When I've figured out how to get Excel to draw
these, I'll let you know.

The upshot here is that scales with low stability may tend to be used like
improper scales in music.  So the Rothenberg portion of my argument favoring
the 8-tone scale over the 11-tone one is weak, pending further listening and
line graphs for each... the 11-tone version does have the considerable
advantage that the minor thirds are all confined to one interval class...

Efficiency is the average number of tones you need to hear before you know
which key the scale is being played in, divided by the number of tones in the
scale.  If n is the number of scale tones and S_i the length of a key-minimal
string, then efficiency is given by...

(Sigma) S_i / n(n!)

...which is a bit of a pain.  Rothenberg mentions in a footnote that methods
exist for calculating efficiency without having to find key-minimal strings
at all.  Naturally, he refers to a paper that was never published.

>>What I like about Rothenberg's stuff is that he came up with it first,
>>actually for a speech recognition model, and then applied it to music and
>>got results that jived.
>Sounds great. I really want to learn this stuff, but I don't know when I'll
>make time to go and find the papers.

I should point out that to a large extent I am taking on faith his jiving
results.  For convenience, he only considered scales as subsets of ETs.  The
original papers give results for subsets of 12tET only, and say write for
further info.  I met with Rothenberg in May.  He said he had once made a
printout for every equivalence class in 31tET (!), and had made comparisons
with scales used around the world.  He hasn't touched this stuff in 30 years,
however, and had trouble finding these supplementary materials (and even the
papers which I had brought).

>See what you think of it now.

Super!  The 4-naturals notation looks to be my fav, except I would have
probably just used A-D.  It was clever how you got the 8-naturals notation to
be alphabetical.  Of course, you still have to live with accidentals hopping
scale degrees.

Paraphrase of some Rothenberg                                       7/27/99

The model concerns a code that can be used to extract information from huge
stimulus spaces despite the limitations of the human memory system...

	We have hypothesized that when a listener is presented with a series
 	of unfamiliar tonal stimuli, he must mentally construct a reference
	frame, P, to which all such stimuli are referred.  Many proper P may
	satisfy this requirement.  If the stimuli are sufficiently unfamiliar
	(as when one listens to music of an alien culture) many repeated
	hearings may be necessary during which a listener replaces a familiar
	P with one more appropriate for classifying the stimuli heard.  The
	cardinality of the constructed P will depend upon the numbers of
	distinctions required by the particular musical language or, if the
	stimuli are not musical, upon the fineness of discrimination required
	by the recognition task to be performed.

Once an appropriate P has been found, the next step is to locate a stimulus
within it...

	Given any "interval" (pair) in P, the listener is able to recognize
	the possible positions its elements might occupy in P.  In effect,
	given any pair of elements he can mentally supply (interpolate) a
	possible set of remaining elements of P which satisfy the equivalence
	class and tuning he has learned.  Such an interpolation becomes
	unique after a sufficient number of elements of P are heard.  This is
	equivalent to the identification of x (key) in a give P(x).

	For example, the major triad [C E G] is not a sufficient subset of
 	the C-major scale since there are two other major scales (namely G
	and F) in which it occurs.  But C-major is the only major scale which
	contains the four notes [G B D F], so the "dominant seventh chord" is
	a sufficient subset of C-major.  A minimal set is a sufficient set
	with no sufficient subsets.  Thus, [G B D F] is not a minimal set,
	but [G B F] is, as there is no major scale except C which contains
	[G B F], while each of its proper subsets [G B] [G F] [B F] are
	included in some other major scale (G, F, and F# respectively).

	It is straightforward to verify that sufficient (and therefore
	minimal) sets are invariants of equivalence.  They depend only on the
	equivalence class, not on the particular tuning.

A measure is developed for this locating...

	Consider a language whose alphabet consists of n letters (or
	phonemes).  How many distinct words can be formed using this
	alphabet?  Certain restrictions may exist which can limit the
	sequences of letters that can occur.  The more distinct words that
	can be formed whose length is less than or equal to some maximal
	value, the more efficient the alphabet is said to be.

	A similar situation applies when "words" are formed from sequences of
	intervals.  Since interval sequences are formed from tone sequences,
 	we consider sequences of the elements of some P.  Also, since no new
	intervals are formed when an element is repeated, only non-repeating
	sequences will be considered.  Since we are here concerned only with
	properties deriving from the structure of P, we will use the
	following criterion for the termination of a "word" (other criteria
	apply when "motifs" are considered) When all remaining elements of P
	are determined by a sequence of some of its elements, the addition of
	elements will impart no further information of this type, and the
	"word" will be considered terminated.  That is, any sequence will be
	considered complete as soon as a sufficient set occurs in it.

	We now ask, given a particular equivalence class, how many distinct
	words can be formed using k elements where k varies from 1 to n.
	Consider all non-repeating sequences of n points (there are n! such
	sequences).  Let S_i be the number of elements in each such sequence
	which must appear before a sufficient set is encountered.  The F(P)
	is defined as the average,

	Sigma(S_i) / n!

	F(P) may be interpreted as the average number of elements in a non-
	repeating sequence of n elements of P(x) required to determine the
	key, x.  Efficiency, E, is defined as F(P)/n and Redundancy, R, as
	1 - F(P)/n.  Both numbers lie between 0 and 1.

	It should be noted that this kind of efficiency and redundancy
	differs from the meanings these terms assume in information theory.
	The distinction is important and applies to alphabets in spoken
	natural languages as well as to musical scales.  The redundancy of
	information theory refers to a redundancy in the message, not in the
	code.  In our discussion here, that property of the code which
	determines whether efficient messages _can_ be constructed (if such
	are desired) is considered.  This property is inherent in the code
	itself, and does not apply to the message.

Rothenberg outlines six scenarios...

	  Scale type  Stability    Efficiency
	(a)  proper       high         high
	(b)  proper       high         low
	(c)  proper       low          high
	(d)  proper       low          low
	(e)  improper     ----         high
	(f)  improper     ----         low

And discusses them...

	Notice that in Figure 1, all scales in 12tET with which we are most
	familiar (the major, minor, and Chinese pentatonic) conform to
	situation (a).  In fact, the major scale (of which the "natural"
	minor is a mode) is far higher in both stability and efficiency than
	any other 7-tone scale.  Next among 7-tone scales is the "melodic"
	minor (2,2,2,2,1,2,1).  The Chinese pentatonic stands out among
	scales of 5 and 6 tones.

	However, situation (b) applies to many scales with which we are
	familiar, such as the "whole-tone" and "12-tone" scales.  Note that
	while these are strictly proper scales, from the hearing of a
	sufficient set (any element) alone, it is not possible to code the
	elements of P into scale degrees.  That is, although PxP is coded by
	the proper mapping, there is no way to index elements of P except by
	arbitrary choice.  __Thus, since in these cases intervals are coded
	but tones are not, composition with these scales must involve
	relations which make use of motivic similarities rather than
	relations between scale degrees.__  Hence the tone row basis of
	12-tone music (which is essentially motivic in concept) is not
	surprising.  An examination of Debussy's whole-tone piano prelude
	"Violes" show similar motivic dependency.

	Now consider improper scales.  PxP is not coded except by the
	employment of proper subsets or a fixed tonic.  __Hence information
	is primarily communicated by the scale degrees.__  Thus it is
	important that P be coded as quickly as possible, which is indicated
	by a high redundancy (low efficiency) as in case (f).  It would be
	expected that scales characterized by case (e) would be extremely
	difficult to use, except when the tonic is fixed by a drone or
	similar device and, in fact, we have not discovered such scales in
	any musical culture examined thus far.  In general, the use of
	motivic sequences on different scale degrees of improper scales would
	not be expected (except within proper subsets of such scales), and
	this is strongly supported by examination of Indian and other music
	using improper scales.

	We would also expect that proper scales characterized by low
	stability would tend to be used as improper scales, so that case (c)
	would resemble case (e), and (d) resemble (f), and similar remarks

Makes cross-cultural observations...

	In Java there exist two scale systems, "Slendro" and "Pelog", each
	containing a variety of scales.  It has been observed that all scales
	in the "Slendro" class are strictly proper and that all in the
	"Pelog" class are improper.  In a study conducted with the assistance
	of Mr. Surya Brata of the Ministry of Education and Culture, Jakarta,
	the uses of these scale systems were observed to be in accord with
	the predictions of this model.

References to the Javanese study...

	Kunst, J. (1949), _Music in Java_, The Hague; Martinus Nijhoff.

	Hood, M. (1954), _The Nuclear Theme as a Determinant of Patet in
	Javanese Music_, Groningen, Djakarta J.B. Wolters.

	Hood, M. (1966), "Slendro and Pelog Redefined", Selected Reports,
	Institute of Ethnomusicology, University of California at Los

Does anybody know anything about these references?

The world according to Rothenberg (re Kraig Grady)                   8/5/99

>I appreciate you going through all this as I have never seen Rothenberg
>work. I seem to be missing somerthing as I don't see where any thing like
>MOS is spelt out. It seems to be barely implied.

Rothenberg's theory does not address MOS as such.  However, I do believe that
MOS can be explained with only Rothenberg's model plus extended reference. 
That is, if we take Rothenberg and extended reference to be our requirements
for human melodic enjoyment, then MOS shouldn't miss many good scales.  I
doubt tetrachordality produces significantly different results here, provided
we don't take the "tetra" literally, and that we don't insist on 3:2's or
fifths (that is, we consider tetrachordality of the generator).

>What are musical application of anything having to do with equivalence

This is arguably the most useful part of the model.  It predicts that
different scales can sometimes sound like re-tunings of eachother.  Actually,
it predicts when this will be the case.  How well does it work?  This is one
of the most important parts of the model to test, and Rothenberg has designed
ingenious experiments to test it.  In the 60's he commissioned Moog to built
him a microtonal synth for the experiments, but Moog's design was a dud, the
air-force decided to pull funding, and the experiments are unperformed to
this day.

However, Rothenberg claims to have achieved good results with ethno-
musicology.  Apparently, many of the instruments in the Indonesian gamelan
change their timbre as they age, and the gamelan is periodically re-tuned to
accommodate this.  Eventually there is a point at which, suddenly, the basic
scale seems to have changed.  Rothenberg claims this is the point where the
equivalence class has changed.

>or what is gain by the labeling scales proper and improper.

The difference is in how they tend to be used.  Improper scales are usually
sectioned into principle and ornamental tones, the tonic is often kept by
some device like a drone, or is static, and composition is much less likely
to use modal transposition of themes.  Instead, rhythms trace out the
boundaries of the principle tones, and the principle/ornamental sectioning
can be changed on the fly.

If a scale is proper, a melody can be played in any mode of the scale and it
will still be recognized as the same melody.  Scale degree fun may or may not
be used, depending on the efficiency (see below).

>I really don't know what to do with
>>Sigma(S_i) / n!
>>F(P) may be interpreted as the average number of elements in a non-
>>repeating sequence of n elements of P(x) required to determine the
>>key, x.  Efficiency, E, is defined as F(P)/n and Redundancy, R, as
>>1 - F(P)/n.  Both numbers lie between 0 and 1.

If my language here is accurate, Sigma represents a function which sums the
values of one variable as another is varied over some range, which is given
above and below the Sigma.

In this case, we are considering all non-repeating strings of tones possible
(what is called a peal in change ringing, I believe) in a scale with n tones
per octave (there are n! of them).  Imagine I play one for you, starting on
some random frequency.  Your job is to shout "stop!" as soon as you can name
the tonic frequency.  When you yell stop, I stop, and the number of tones you
heard that far is called S_i (the "i" subscript tells which string the number
S_i came from -- since there are n! possible strings, we vary the "i" from 1
to n!).  Now suppose you want to find the average number of tones you had to
hear for all strings.  You'd sum them all up (Sigma) and divide by the total
number of strings (n!).

That's the equation.  Efficiency is actually the answer of that equation
divided by n.  So if you have a scale with 22 tones per octave, and the
average number of tones you need to hear to identify the key is 3, you've got
19 tones that aren't doing anything to help you identify the key, which means
low efficiency.

The important thing about efficiency is that it measures the difficulty of
finding the tonic in a segment of music.  For improper scales, where scale
degrees (and therefore the whereabouts of the tonic) is all-important,
efficiency must be low.  For proper scales, efficiency can be high or low,
but it may be better high (Paul, do you think your preference for the
pentachordal decatonics can be explained by the lower efficiency of the
symmetric ones?).

Does that help?  Comments, corrections (as always, it should go without
saying) wanted.

RE: Various MOS                                                     8/12/99

>"The process of producing a scale of melodic integrity by the superposition
>of a single interval (generator)."
>"Those points where there are only 2 different size intervals are called
>moments of symmetry."

Joe, may I suggest: "A pythagorean-type scale (the generator need not be the
3:2, and the interval of equivalence need not be the 2/1) is MOS iff the
generating interval occurs at only one scale degree in all modes of the scale
(i.e. 3:2 is always a 5th in the diatonic scale)."?

>"This cycle has the property that any occurrence of an interval will always
>be subtended by the same number of steps."

That's not true.  I think I said it in a post once, and posted a correction
shortly after.  Sorry if I have contributed to a misunderstanding.  This
condition is equivalent to strict propriety.

>So, equal temperaments would all qualify?

What qualifies or not is a set of three things- generator, interval of
equivalence (IE), and number of tones.  It is true that all possible
generators in an x root of n tuning have an x-tone MOS when the interval of
equivalence is n, and some generators will also have MOS's with < x tones, if
x is not prime.

>Plus a succession of fourths would qualify?  How about a succession of
>variable fourths?

That depends on what you mean by "fourths"?  In "variable fourths," Wilson
means intervals that subtend four scale degrees, but may be of different
acoustic sizes.  To be MOS or not, the acoustic size of the generator must be
known; the scalar size of the generator is determined later, by the number of
places the chain is carried out to.

Remember that not all chains of an interval are MOS- only those chains where
the chained interval (generator) always covers the same number tones in the
IE-reduced scale are MOS.

RE: interval boarders                                               8/24/99

>I regret not following the dyadic complexity formula thread (of a couple
>months back) more attentively, and I'm wondering if there was any sort of a
>rough outcome, or general consensus - anyone who was on top of it care to
>post a (synoptic) recap?

The conclusion was that ratios of small whole numbers are more consonant that
ratios of large whole numbers, and anything that measures the size of the
numbers in a ratio will reflect this.  Size of the denominator, arithmetic
mean of numerator and denominator, and geometric mean of n and d were three
of the most popular, the last being my favorite (see TD 1216.4).  Largely
dismissed were metrics based on prime factorization.

Aside from this, two useful concepts discussed and named (thanks to Dave

1. TOLERANCE- Irrational intervals, like those found in ETs, can be consonant
when they are close in size to rational intervals.  A definition of
consonance based on properties of rational numbers (as above) can be adapted
to explain this by including a TOLERANCE function.  Everybody's favorite is
Paul Erlich's Harmonic Entropy, which, along the way, explains one of the
reasons why small-numbered ratios are consonant in the first place.

2. SPAN- When intervals are very small, they can be highly dissonant although
they may be represented by small-numbered ratios.  When intervals are very
large, they may have very low dissonance _and_ consonance.  For example, one
can play almost any notes together 6 octaves apart on a piano and they won't
sound particularly consonant, but they won't clash either.  This runs
contrary to the idea that consonance and dissonance represent opposite ends
of the same spectrum.  No rigorous adjustment for SPAN was worked out,
although normalizing all intervals to the octave between 8/7 and 16/7 is one
expedient workaround.

Bandwidth                                                           8/24/99

>>They may be implied, but this out of tuneness is exactly what got many of
>>us here. If you can accept pitches that extend 17 cents in both directions,
>>you are up to a 34 cent band width. this is so all inclusive as to almost
>>be meaningless.
>Yet many professional singers in the West easily extend 30 cents in either
>direction, giving a 60 cent band width.

There are some who would consider the choir the benchmark, and both fixed JI
and 22 an approximation.

Also, the choir's bandwidth is flexible, and does not necessarily give the
same effect as a fixed deviation from just.

RE: Bandwidth (Paul Erlich)                                         8/28/99

>My statement referred to random, uncontrolled deviations from 12-tET, in the
>context of solo singers accompanied instrumentally.

Are you sure they're random?

>Professional vocal harmonies do tend toward JI intervals.

At a given time, yes, and with far less a bandwidth than 60 cents.

But how do you describe the tuning of a passage?  How do singers balance the
factors of common tones vs. pitch shift?  What scale do singers use when
harmony is not involved?  Many of your posts over the years seem to ignore
the difference between a description of vertical events and of horizontal

In the case of melody (whether it is tied rhythmically to a harmony or not),
why should we take fixed scales and say that performers deviate from them
(unless the deviations are truly random)?  Is it possible that performers
have the ideal scale, to which fixed scales are an approximation?  Are you
sure accuracy of tuning beyond the 3-limit is not important in melody (as
stated in your paper)?

Recent sine-tone stuff                                              9/18/99

>harmonic entropy -- Particularly for tones outside the upper-middle
>register, the lack of harmonics will make it more difficult for the ear to
>discern precisely what frequency ratios are being heard; therefore the
>uncertainty parameter in the harmonic entropy calculations should be
>increased considerably.

I thought the "uncertainty parameter" was based on Goldstein's experiment,
which was done using sine tones... no?

>remember, pitch resolution is poor (I would say in the 50 cent ballpark)
>with sine waves (experiment: play a familiar melody (just one voice) in
>several different tunings using sine waves. Give yourself a blind test if
>possible. Can you hear the difference?).

I admit that I haven't been able to generate anything even close to a sine
tone in my house or apartment.  I'd love to take such a test, and I would be
surprised -- downright astonished -- if I couldn't tell the difference.

>In all, if your carillon is truly producing sine waves, I suspect you will
>find it rather unmusical, a poor demonstration of JI, and probably rather
>annoying to the ear.

This has got to take the non-sequitur cake for the last few issues of digest.
Darren, I doubt you'll wind up with sine tones, considering the parking-lot
horns you'll be playing the thing thru, but even if you do, there isn't a
shred of reason to believe the carillon couldn't be a wonderfully musical and
satisfying instrument.  I can't imagine someone failing to notice it was
alternately tuned.  As for being a poor demonstration of JI, well... so were
most of Partch's instruments.  Utonal chords fun compositionally?  You bet. 
Even without them, the otonal chords will enjoy the common-tone arrangement
of the diamond.

RE: recent sine tone stuff                                          9/20/99

>Yes, and the value of the parameter I like to use (1%) was based on the
>performance in the optimal frequency range, a narrow one in the middle-upper
>register. The parameter rapidly increases outside that range. Most musical
>instuments produce harmonic partials in this range, so the 1% can be assumed
>to apply across a wide range of frequencies. But for sine tones, one gets no
>help from harmonic partials, so the relevant parameter value will usually be
>quite a bit higher.

Sounds wishy-washy to me.  Maybe partials in a complex tone do not help, or
maybe they help a great deal more than simply allowing us to extend the
measurements made in the optimal frequency range to all frequency ranges. 
Paul, this sounds very strange to me.  I would expect that if partials
helped, they would increase resolution but leave the relative resolution at
different frequencies more or less unchanged.  In other words, I would expect
that data about individual partials is thrown out at an early stage in the
listening system; the frequency-resolver probably only gets one wave, which
would carry the extra resolution provided by the partials from an earlier

We need to repeat Goldstein's experiment with complex tones.

RE: Project Retune: my adaptive JI methods                          9/23/99

>4567 is a common tuning for the dominant seventh chord in barbershop
>music, but is more controversial for classical music.
>I haven't been able to get his retuned sequences to work on my computer,

The controversy should disappear when you do; certainly both of these tunings
are viable for the sonata.

>I wouldn't use it except for the augmented sixth chord, where it is
>historically appropriate.

Bach chorales harmonize nicely with 4567 dominants in many spots, speaking by
way of classical theory.

And please acknowledge that this way of speaking is generally at a loss to
explain what is possible when notes can be retuned while they are sounding,
and when modulations can by made by intervals outside the scale.

>First of all, I would like to say that I don't like the idea of using just
>intonation for Mozart and Bach,

Why not?  Mozart is a sucker for it.  Keyboard Bach probably reaches maximum
goodness with extended meantone, but the rest of his output is fair game.

RE: Fokker periodicity blocks (Joe Monzo)                           9/23/99

>I'm interested in studying the historical conceptions of various shapes and
>sizes of periodicity blocks in music all over the world.

I'm with you on that one!

>The most important aspect, it seems to me, is to determine the proper
>lattice metric to portray all the different tunings.

I'm not sure why that would be important.  As far as I can tell the most
important aspect is getting reliable information on how musicians actually
tune their instruments.  Music theory in other countries is no more (and
probably less) reliable than music theory in the West.

>There is no clear consensus on this yet, altho I recall Pauls Erlich and
>Hahn being the most outspoken in favor of the one(s?) they like, and doesn't
>Carl Lumma agree, at least partially?

First, you have to decide what you want to measure.  Psychoacoustic
consonance?  If so, the lattice really isn't of much use, since consonance
has little or nothing to do with factoring.  However, Paul E. does seem to
consider his metric a lattice metric.  He only recognizes factoring by 2's,
so he's got a lattice dimension for every odd factor, to odd-limit infinity.
Because he uses a triangular lattice, every ratio has an octave-equivalent
representation that spans one rung.  So his metric is simply the log of the
odd limit of the ratio.  Why log, I don't know.  The only advantage of using
logs that I ever understood was that it unifies the results of odd and prime
limit factorizations, which shouldn't matter to Paul E, given his stance on
prime numbers in consonance measures.  Last I heard, Paul E. was considering
switching to an octave-specific rectangular lattice; perhaps he can update us
on the pro's and con's of such a thing.

Or, you could measure the modulation distance (Paul H, if you can think of a
better way to explain this...), so to speak, of an interval.  In this
approach, factoring is important.  First, you declare which intervals you
consider consonant, and you give lattice dimensions only to the identities
needed.  Then you count the number of rungs (each of which are defined as
consonant) along the shortest route to the target interval.  In effect, you
are counting the number of common-tone modulations (by consonances) you need
to get to the interval.  Paul Hahn uses this metric in his diameter and
consistency measurements.

RE: Fokker periodicity blocks (Joe Monzo)                           9/24/99

>>Or, you could measure the modulation distance (Paul H, if you can think of
>>a better way to explain this...), so to speak, of an interval.  In this
>>approach, factoring is important.  First, you declare which intervals you
>>consider consonant, and you give lattice dimensions only to the identities
>>needed.  Then you count the number of rungs (each of which are defined as
>>consonant) along the shortest route to the target interval.  In effect, you
>>are counting the number of common-tone modulations (by consonances) you
>>need to get to the interval.  Paul Hahn uses this metric in his diameter
>>and consistency measurements.
>When the overall odd-limit goes to infinity, you get the thing you were
>talking about at the top of this message.

As I posted months ago, and as is appropriate when looking at psychoacoustic
consonance.  This is inappropriate for modulation distance, however, because
we wish to know how certain intervals function within a given system of
consonances.  As you are so fond of pointing out, sensory consonance and
functional consonance are not the same --- the latter evolves with music.
225:224 might be a basic consonance in another 2000 years, but we wish to
know how it functions in the 9-limit systems of today.  By making the odd
limit infinity, we destroy this information.

RE: new lattice diagram on my Robert Johnson page                   9/29/99

>>But to my ears, the vocal in this tune seems firmly 'rooted', melodically,
>>to that 1/1 'D' key.  An argument in favor of this is the fact that the
>>repeated lines where he sings 'my life seems so misery' both occur over G7
>>chords but clearly outline a resolution onto the tonic D7, which is exactly
>>what he plays on the guitar 
>In 12-equal! And yet you'd claim the guitar exerts no influence on his vocal
>intonation, other than determining the 1/1?

Joe, I haven't looked at your Robert Johnson page recently, and Paul, I
vehemently agree that confusion between vertical and horizontal JI has caused
many problems (in fact I have criticized you for it in the past).

The point I wish to make here is that rhythm is a factor that can not be
ignored when considering tuning interaction between parts.  I have heard
vocals that were accompanied by piano and guitar and completely un-influenced
by the fixed instrument's intonation.  Only when the rhythm of the parts is
related do vocals tend to form just intervals with prominent notes in the
accompaniment as Paul suggests.

Old diamonds                                                        10/1/99

; Going over stuff that I might put on the web, I realized this unfinished
; work wasn't going to make it.  I publish it here in case anybody finds it
; interesting...

It all starts with a group of things.  Imagine a group called Moe.  Say there
are x things in it.  Imagine picking the things up and putting them in your
hand.  There are obviously x different hands if you pick up only one of them
at a time.  But how many different ways are there to hold two of them at a
time?  Three at a time?  The answers can be found in Pascal's triangle, by
reading across the row corresponding to x.

Now imagine a set of all the possible hands from Moe.  This is called the
power set of Moe.  How many things does it have in it?  It makes sense to add
up all the numbers in the corresponding row of Pascal's triangle.  It so
happens that this number is 2^x.  If you've ever played a valved brass
instrument, you may have noticed that there are 8 = 2^3 possible ways to
valve the horn, at least on three-valve instruments.

Wilson has called the power set a 0/x thru x/x CPS, or "grand slam".  He
pointed out that it contains all of the CPS's makable by taking any number of
things at a time from any subset of the source set.  For example, the 2|6
[1 3 5 7 9 11] eikosany contains a 2|3 and a 3|3 dekany for each of the six
5-member subsets of [1 3 5 7 9 11].  In this way, each new row of Pascal's
triangle can be seen as restricting the number of possible source sets for
the previous rows.

What Wilson didn't point out is that the power set also contains the diamond
of every scale that can be built by chaining intervals from the source set.
In the case of [1 3 5 7] one such scale would be 1/1, 5/4, 105/64, 15/8.  In
the case of [1/1 3/2 5/3 7/5] one such scale would be 1/1, 5/4, 3/2, 7/4.

What is a diamond?  It is a list of all the intervals in a scale.  In other
words, it's what you get when you rotate the scale's interior intervals
around a common frequency -- what you get if you start each mode of the scale
on 1/1.

Old cross sets                                                      10/1/99

To make a cross set, transpose the scale by each of its own degrees.  Paul E.
noticed that this produces structures with scaling symmetry on the lattice.
Try a triad on the triangular lattice to see what I mean.  I'm not sure about
more than one iteration.  I think I tried a triad of triads and it didn't

Cross sets never have more than ((n^2)/2 + 1) notes, where n is the number of
notes in the seed scale.  This gets fairly economical at higher limits; the
15-limit cross set has only 33 notes.  That means you can have harmonics 8-16
as your scale (if you like harmonic scales as I do), and modulate to every
key with only 33 pitches.  Of course, you can't modulate to every key of
every key, but it isn't too shabby with perfect 15-limit consonance and 33

Constant Structures and periodicity blocks  (Paul E.)               10/3/99

>>Incidentally, by what reasoning did you figure that all periodicity blocks
>>represent each of their acoustic intervals by only one scale position?
>More intuition than reasoning. Can you find any counterexamples?

Howabout the 5-limit periodicity block:

|4 -1|
|0  2| = 8

1/1  135/128  9/8  5/4  45/32  3/2  27/16  15/8

Open problem                                                        10/4/99

"How can one choose unison vectors in linear space so he gets only the MOS's,
and all the MOS's of the generator of the linear space and a given interval
of equivalence?"

I've already shown to my satisfaction that 1D periodicity blocks of the type
Paul Erlich is looking for do not fit the bill.

RE: Open problem                                                    10/5/99

>You're expecting to find some sort of "shortcut" for finding MOSs? I don't
>know why, since MOSs are 1D by nature, and periodicity blocks really only
>show their mathemagic in >1D. Otherwise, they're just the sort of numbers
>you'd be looking at anyway. Or am I misunderstanding?

I'm not looking for a shortcut for finding MOS's, I'm hoping to find higher-D
analogs to MOS's by applying the rule to bigger periodicity blocks.  I didn't
mention this, because it's very speculative; you know Paul, that MOS's are
related to mediants. . . .

RE: Optimum adaptive tuning methods                                 10/8/99

>But as John has to have a program that executes a relatively simple set of
>fixed algorithms, what would one that addresses "root motion by irrational
>and extra-scalar intervals" look like?

First, we need a lookup table that will relate 12-tone chords and just ones.
I can imagine a table based on scalar intervals, but it would likely be quite
a trick to get it to work well, and it wouldn't work at all unless the
composer was paying attention to his spellings (99% of MIDI files out the
window).  So the table would be based on absolute intervals.  For most music
post 1850, a well-working scalar table would be equivalent to an absolute
table for 12 anyway.

Start by taking all dyads, triads, tetrads, and pentads of the set [1 3 5 7 9
15 17 19].  That's less than 210 things.  Tune them each in 12tET.  Since 12
is consistent with respect to this set, there will only be one tempered
version of each just chord.  But 12 is hardly unique with respect to this
set, so sorting by the 12tET side should show multiple just versions for each
tempered chord.  For each tempered chord, rank the just versions inversely by
their MAD from 12 (mean log-frequency deviation from 12tET) times their Farey
limit (the largest number in their harmonic series representation).  Keep
only the top four ranking just chords in each 12tET category.

Now, create a utility which steps through MIDI files, observing note on/off,
timbre, velocity, and channel amplitude information to track the loudness of
MIDI events.  Construct a "previous chord record" that holds up to five note
names and a tuned frequency and loudness for each.  At each tick, consider
the five loudest MIDI events a "chord" and...

1) If the "previous chord record" is empty, tune the current chord with the
highest-ranking just version, write the result to the record, and proceed to
the next tick.

2) Otherwise, if there are common note names between the current chord and
the one in the record,

	a) If there's a tuning option that can duplicate the frequencies for
	all of the common note names, use it, rooting the new chord
	appropriately, over-writing the record, and moving to the next tick.

	b) Otherwise, pick the option that duplicates the most, taking the
	highest-ranking one in case of a tie^, root the new chord to minimize
	the MAD between the frequencies belonging to the common notes^^,
	write the result to the record, and proceed to the next tick.

3) Otherwise, calculate the log-frequency changes between the notes of the
previous chord (in 12tET) and the notes of the current chord (in 12tET).
Apply these changes to the corresponding frequencies in the record and start
over at the current tick.

^In this case, a tie could also be resolved by throwing out the quietest
events, one by one, until one of the other conditions are met.  Trial and
error would probably tell which method works best.

^^Instead of rooting to minimize the change between all pairs of common
tones, it may prove better to hold the loudest pair constant and allow the
others vary the full amount.

Some other items...

1] Everything here assumes octave equivalence.  2] Trial and error may
improve the ranking method.  For example, mean squared deviation may work
better than mean absolute, and the geometric mean of a chord's harmonic
series representation may work better than its Farey limit.  3] I would
probably sneak a few utonal chords, and perhaps the 12141821 tetrad, into
the lookup table, since they are not generated by the method I described.
Still working on a way to measure the dissonance of utonal chords. . . .

Words                                                              10/14/99

One of my favorite things to glimpse is a good word origin.  Most of the
time, it seems, it behooves us to be ignorant that words are symbols.  A high
percentage of speech is actually made of phrases -- many of which even have
literal meanings in contemporary English -- whose meaning has become atomic.
This is a major contributor to the "e-mail bug"; the atomisizing often
depends on intonation.

Besides the e-mail bug, there are one or two other things known to get people
to notice the symbolic nature of words =8^)...  One of them is when words are
new to a context.  To the point...

[Paul Erlich]
>I don't mean to be obstinate, but it depends on what you mean by "fly". The
>decatonic scale, like the ordinary diatonic scale, depends on "puns"
>relative to JI.

Does anybody know who first used the word "pun" in this context?  They
deserve a pat on the back.

RE: the "canon"                                                    10/14/99

>Composers like Bach and Beethoven are put on pedestals for a reason and
>cranks like Partch (who as a composer is not worthy to undo Bach's
>sandlestrap) only show their own crankishness by trying to pull them down.
>(I admire Partch for alot of reasons, but his music criticisim is not one
>of them.)

Dante, I've followed the thread and despite denying the existence of a
cannon, I agree with almost everything you've said.  But...  First off, when
did Partch ever try to tear down Bach or Beethoven?  He called them daring.
He was a classically-trained performer, and according to Gilmore had strong
feelings about how Chopin, for one, should be played.

Second, and most importantly, it is my considered opinion that Partch is to
be listed with Bach and Beethoven among the greatest, say 30, composers in
history.  This would probably still be the case if his music were transcribed
for the piano.  Have you listened to his music?

>And what "equally or more deserving works" from Tchaikovsky's time are there
>not enough recordings of?

Not many that I know of.  There is Nielsen, who was the next generation, but
whose career intersected T's for five years.

The point stands that there are many works more deserving than, say, the
1812, in other periods and even in T's own output, that have been recorded
far less.  Beethoven had trouble getting past Moonlight, much to his chagrin.
How many times have you heard Sonata 28, which fits in the "no better
organization of sound known to man" department?

Tuesday night I saw the Philadelphia orchestra.  They're doing a season of
entirely 20th century works.  A great idea.  What did they play?  The Planets
and Honegger 3.  The Planets is fantastic, and they nailed it, but have you
ever heard anything else by Holst in concert?  They deserve credit for
playing Honegger, but they picked his most famous symphony, and butchered it.

I can hardly get recordings of Honegger's other symphonies.  When the
orchestra plays Cowell and Koechlin (most of which has never been recorded),
I'll call their season innovative.

>To write a "symphony" today, and take Beethoven or Sibelius as your model,
>and want to have it performed by the Philharmonic Society of
>Boredomthroughstagnationsville, is to be a wannabe - a wannabe "classical"
>composer. History will have no qualms about assigning this kind of stuff
>to the dustbin.

How (except maybe when they like Sibelius :) do you get off making this
claim?  Just because a composer wants to write for an orchestra means he's a
wannabe, his stuff going in the historical dustbin?

RE: the "cannon"                                                   10/14/99

[Dan Stearns]
>I too have no problem ascribing Partch Bach and Beethoven the same kinds of
>esteem as composers, but I really don't see the "transcribed for the piano"
>part of your argument.

When I think about a work of music, I often find myself grouping thoughts
into three areas.  They are very much inseparable, but often I think it's
useful to try and separate them anyway -- Notes, Sound, and Concept.

Notes -- Imagine transcribing the work as you listen.  Is that part cool?
Sound -- How do your ears feel?  Is the performance a vehicle or a barrier
for the notes?  Concept -- What's the point?

Partch was a genius of all three.  His concept, the virtue of early man.  His
sound, a custom orchestra, a general attention to intonation, and in many
cases, extended just intonation.  His notes?  Not quite on the level of
Beethoven, but close.  My comment was just addressing the notes part.

>I think that it's pretty safe to say that the vast majority of the admirers
>(or students of one sort or another) of Bach or Beethoven will forever (to
>some degree) scoff & smirk at the form (etc., etc., etc.) of Partch's music
>when these kinds of direct comparisons are invoked - and I say fair enough.

Bleck!  The vast majority of fans of any given style are notorious for
smirking at other styles.  Partch disciples especially.  This is a symptom of
lazy listening.  It is obvious that music goodness depends on the musical
intelligence of the composer, not than the style he's working in.  Until
recently the style he's working in was more or less determined by geography.
So unless you buy into some sort of Bell curve argument for "musical
intelligence", you've only got your laziness to blame for the holes in your
CD collection.  Unless, of course, you listen to music you like, rather than
music you evaluate as good.  Which is fine, so long as you note the
difference; if you listen to anything long enough you'll like it (unless
you're depressed).  Got it?

RE: 53-note midi sequencing                                        10/14/99

>A quick thought Writing music-actually printing ideas as sumbols on paper-
>forces the mind into a detailed reckoning of the material, in the same way
>that a student of English Composition must use the essay form to learn to
>order and present ideas.

I agree, but I don't see the analogy.  Scores and recordings can be viewed as
types of instruments, which, like piano and guitar, dictate certain things
about the music that is written with them.  Both give different and good
results.  When multi-tracking came out, for example, a host of good ideas
came up that nobody would have thought of with paper.

RE: the "cannon"                                                   10/15/99

[Dante Rosati]
>Unless I'm hallucinating, I remember seeing in "Bitter Harvest" writing
>about Bach that was negative.

Bitter Music?  That was the first thing I read after Doty's JI Primer, when I
got into all this business two years ago.  I thought it was a masterpiece.  I
don't remember anything about Bach, and unfortunately I can't check, since my
copy is on loan.

>I dunno, Carl- I admire the guys originality and vision, his inventiveness
>and industry, but the actual music? I played Guitar One in two performances
>of Revelation at the Courthouse Park, and of course have heard other stuff.
>Let's take one of his most frequently performed works- US Highball. Do you
>really consider this "great music" on the same creative level as (forget
>Bach and Beethoven) the best of Lou Harrison, Stravinsky, Shostokovitch etc?

Lou Harrison is great with tea, but let's not mention him with Partch,
Stravinsky, or Shosti.  Essential Partch for me is Oedipus, Revelation, The
Bewitched, Petals, Dreamer.  Oh, I don't know, it's hard!  It's all good (and
I haven't given Delusion a fair shake yet)!

The main problem I see people having with Partch is the timbres.  In fact,
that's the main problem I see in all cross-style listening.  Not that this
applies to you -- maybe we just have different taste.  But have you tried
listening, imagining it all for guitar?  All of those percussion sounds are
notes!  It's a shame that Partch, to date, must be heard through mediocre
performances, bad recordings, and instrument disrepair.

A good friend of mine, who loves prog. rock and bebop, can't get classical.
I think it's because he isn't following the notes.  He just hears sounds --
ever notice that musicians tend to listen, overwhelmingly, to music written
for their instrument?

>Maybe you could recomend what you consider his best piece and I could check
>it out (if I haven't heard it).

Try Dreamer.  Maybe his best short work, and probably his most listenable.
Humor me and listen in the dark.

>T might lack depth sometimes but his melodic invention is second to none.
>Nielsen is not on the same level.

T's melodic invention is quite good, but second to a few.  I agree with that
second bit completely.

>>The point stands that there are many works more deserving than, say, the
>>1812, in other periods and even in T's own output, that have been recorded
>>far less.  Beethoven had trouble getting past Moonlight, much to his
>>chagrin.  How many times have you heard Sonata 28, which fits in the "no
>>better organization of sound known to man" department?
>I think you can walk into any record store and find dozens of recordings of
>all the Beethoven Sonatas. How many do there really need to be?

I wasn't talking about numbers of recordings here.  I was talking about the
number of performances, in the composers' times and today.  So how many times
have you heard #28?

>Sibelius? I remember a time many years ago when I listened to Sibelius'
>Second every day for a couple of months and derived great spiritual
>sustenance from it.

I confess my exposure to Sibelius leaves much to be desired.  So this Second
is recommended?

RE: Free market vs. totalitarianism (John Link)                    10/19/99

>>It certainly wasn't protecting the rights of its citizens.
>>Who said it had to do that?
>Ayn Rand. See, for example, "Capitalism, The Unknown Ideal".

She didn't say it had to, she said it should.  She can hardly claim propriety
on this idea, in fact I can't even think what else government could be for.

The difficulty arises in defining "rights".  Rand's gross oversimplification
of this question leaves her reader with a clear idea of the problem, and no
idea of the solution, although she deserves much credit for writing stories
instead of essays.  I personally feel legislative systems are clearly un-
suited to working out what "rights" are, and would prefer the whole mess done
away with in favor of something like the judicial system (with a little less
catering to an evolving system of precedents).

My point was that government is an emergent behavior- a free spirit.  Nobody
sits down and makes one.  The framers of the US Constitution found that out,
if not while they were alive (with the unwanted appearance of political
parties), then certainly by now.  The whole thing is clearly out of control,
people acting out roles as in a play.  If government improves its longevity
at the cost of it's citizen's rights, what's to stop it?  Certainly not any
piece of paper.

RE: Open problem                                                   10/25/99

[me, Carl Lumma]
>You bet!  Of course any chain-of-fifths tuning can be viewed as a
>periodicity block whose unison vector is the chromatic interval, not just
>the pentatonic scale, and not just the MOS's.

[Paul Erlich]
>Hmm . . . a 6-note chain of fifths has the tritone as unison vector, and I
>would hardly call a tritone a single chromatic step . . .

>Sheesh!  My bad.  I was confusing the note it changed _to_ with the _amount_
>it changed.  Let's see...

Actually, if you define the chromatic interval as the amount of change in the
_note that changes_ when the scale is transposed, then I was correct -- that
amount is always the unison vector.  This is how I would like to define
chromatic interval.  I had posted a chart showing the chromatic interval as
the distance between the new note and the nearest unchanged note...

>However, the nearest one isn't always the one that changes, at least I don't
>think it is.  For example, in the 4-tone case, the change actually jumps a
>scale degree (the chromatic step should be 81:64, or 410 cents in 41tET).
>Can anybody fix the chart?  Or better yet, explain how to transpose a scale
>in Scala without confusing the hell out of yourself?

So I've just fixed the chart- the chromatic interval is the unison vector.
Manuel and Pat came to the rescue with the Scala question...

>Resp. seems to make sense in the help for PIPEDUM, ironically.  But what
>does "different resp. identical" (from the SHOW TRANSPOSE help) mean?

[Manuel Op de Coul]
>It just means the number of identical pitches follows the number of
>different pitches in the output.

That solved, I've moved on to being confused by the wording "degree order"
vs. "any order".  I have convinced myself by playing with it that "degree
order" enforces my preferred definition of chromatic [it does show more than
one note changing when certain connected cyclic scales are transposed by
their generator, which is impossible with the other definition of chromatic.
Indeed, the "any order" column always shows 1 note change when transposing by
the generator.  Where do the >1's in the "degree order" column actually come
from -- Manuel?].

Anyway, I've also convinced myself that MOS's are the scales that have a "1"
in the "degree order" column for a given generator and IE --- that MOS's are
scales whose chromatic changes don't jump scale degrees.  Can anybody prove
it?  Are MOS's the only such scales?

>I'm not looking for a shortcut for finding MOS's, I'm hoping to find
>higher-D analogs to MOS's by applying the rule to bigger periodicity blocks.

[Paul Erlich]
>Aha! How would such analogs be defined?

When they are transposed by any of their unit vectors, no changing notes hop
over unchanging notes.  Currently looking to define this in terms of size of
the unison vectors and 2nds of the scale.  Paul, your search rules out many
of these (only the 2nds built from either end of the chain are important?),
as I noted...

>I've already shown to my satisfaction that 1D periodicity blocks of the type
>Paul Erlich is looking for do not fit the bill.

What would a search for these 2-D MOS analogs be like?

Clearing up TC                                                      11/6/99

Is anybody following this thread?  I think I've got it figured out now.

There are two versions...

Strong: All instances of a pitch in a scale's "Cartesian" cross-set occupy
the same scale-degree position in the row or column in which they fall.

Weak: All pitches appearing in both the original scale and any one of the
transposed scales in the "Cartesian" cross-set occupy the same scale position
in their respective scales.

The question...

If [S -TC-> S*f] and [S -TC-> S*g], must [S -TC-> S*(f-g)]?

Example, S = 1/1 5/4 3/2
If [S -TC-> S*5/4] and [S -TC-> S*3/2], must [S -TC-> S*6/5]?

...asks if these two versions are actually independent.  Here's the
"Cartesian cross-set" of the C major scale in 12tET...

   B   C#  D#  E   F#  G#  A#
   A   B   C#  D   E   F#  G#
   G   A   B   C   D   E   F#
   F   G   A   Bb  C   D   E
   E   F#  G#  A   B   C#  D#
   D   E   F#  G   A   B   C#
   C   D   E   F   G   A   B

...the "original" scale of C major is TC with all the transposed scales in
the set.  However, the F and B transposed versions are not TC with eachother.
So the answer to the question above is "NO!", and the Strong and Weak
versions above really are independent.

What does this mean?  It means there are scales that are Weak TC in some
modes but not in others.  Why?  Because B does get compared to the original
scale when the cross set of the Lydian mode is taken.  Is anybody following

Strong TC is by definition a property of all modes.  It is also equivalent to
CS.  I've said that before...

>>It seems to me, if I've understood correctly how you've defined it, that
>>in most cases it would be most meaninful to consider TC shorthand for "TC
>>by all intervals within the scale itself".
>You mean only those intervals measured up from the root, or measured from
>anywhere?  If you mean the latter, then your suggested definition is
>equivalent to CS (constant structureness).

Here's an outline of my reasoning...

1. CS means all instances of a pitch in a scale's "diamondic" cross-set
occupy the same scale position.

2. Strong TC means all instances of a pitch in a scale's "Cartesian" cross-
set occupy the same scale position.

3. Multiplying a scale by n/d changes its accidentals in the same way (but in
opposite directions) as multiplying the scale by d/n.

4. So making a "Cartesian" cross set makes a "virtual" diamond, which is
equivalent as far as common tones go.

5. I realized that all strictly proper scales are CS.  Paul Erlich realized
that all strictly proper scales are TC.  That seems to add weight to the

Conclusions?  I like Weak TC, because I think it better reflects how this
property works in music.  Although CS now takes on a new level of interest
for me.

The future?  I'd like to learn which of these properties, if any, all MOS's
have.  Strong TC / CS is out, since the diatonic scale in 12tET is MOS.  That
leaves Weak TC.  After that, I'll want to return to lattice TC, and search
for periodicity blocks that have it.

RE: Newbie questions - "modulation"                                11/7/99

>And what if two notes are held from a previous chord.  If they form a
>different interval in the two chords, which is almost always the case, one
>or both have to shift.  Any recommendations?

These are aesthetic questions, and the freedom of choice seems to be what
we've requested.  I recommend playing, and listening.  The recordings listed
at http://lumma.org/music/theory/topten.html offer a good taste of what's
been done.

>Another example if I modulated keys in fifths around the full circle, would
>I accumulate the 2 cent discrepancy all the way around so that I would be 24
>cents off from my previous C when I get back to it, or would I somehow shift
>pitch slightly at convenient places to try and stop the drift?  If shifting
>is the answer, any recommendations for good places or ways to do it?

You could spread it out uniformly by rooting chords in 12tET.  In some
progressions, there are convenient places to insert the shift, rather than
spreading it out by choosing roots from a temperament.  These are usually
spots where the root motion is large (say, on the circle of fifths).  In the
progression A E C F, for example, a jump would probably be least noticed
between E and C.  But in the case of the uniform circle-of-fifths
progression, there's hardly a better solution than equal distribution.  Of
course, you can always leave the drift -- you might like it.

>The only problem is this means the tuning actually becomes quite fluid, with
>intervals always "correct", but no consistent pitches for notes.  Or another
>way to look at it is I'ld always be playing an approximation of a small
>prime limit JI subset of a large note scale (like 53 circle of primes or

The common-tone v. harmony problem is one of the oldest in music theory.
Temperament is a very interesting solution -- the range of temperaments has,
just in the last decade, become well-understood.

One solution from a JI perspective is offered by David Doty, in a recently-
released landmark recording http://www.syntonic-rec.com/ucp.html

I wrote about a solution to the problem from a JI perspective in a series of
posts around the beginning of October.  I've summarized these posts at

You'll also want to check out John deLaubenfels' adaptive tuning page

>2.  I've seen an old article in Electronic Musician talk about playing
>melodies differently than harmonies (ie using different tunings).
>Unfortunately, the article was short, imprecise, and all that, and I haven't
>seen mention of this anywhere else.  I presume he means unacommpanied
>melodies, since any accompanied melodies have harmony?  Any info on what is
>often done here?  I understand the physical basis for using small prime
>limit JI for harmonic situations.  Are there physical basis for melodic
>intervals as opposed to harmonic?

I would say yes.  The melodic basis is proving much harder to discover than
the harmonic.  I believe there are several independent components to the
melodic basis, some but not all of which are mutually exclusive.  And some,
but not all of which are exclusive with the harmonic basis.

One interesting thing I suggest you try melody and harmony coming from
different tunings.

RE: Clearing up TC                                                  11/9/99

>Carl, I'm really trying, but your language is very confusing. Can you please
>demonstrate this:
>>It means there are scales that are Weak TC in some modes but not in others.
>>Why?  Because B does get compared to the original scale when the cross set
>>of the Lydian mode is taken.

Here's the cross-set of the CMaj scale in 12tET...

    B   C#  D#  E   F#  G#  A#
    A   B   C#  D   E   F#  G#
    G   A   B   C   D   E   F#
    F   G   A   Bb  C   D   E
    E   F#  G#  A   B   C#  D#
    D   E   F#  G   A   B   C#
    C   D   E   F   G   A   B

...the question is, if the column on the far left is TC with all other
columns, are all columns necessarily TC with all other columns?  The answer
is no, since the far left column _is_ TC with all the others, and the columns
starting on F and B are not TC with eachother.  So we have two separate
measures, Weak and Strong.  This is trivial, and I gather you understood it.

It is also trivial that the Weak version varies across the modes of a scale,
since the column starting on F would be in the far left if the cross set of
the Lydian mode had been taken instead of the Ionion.  The Scala user can
check this by doing the following...

1. equal 12        ; creates 12tET
2. mos -> 7 -> 4   ; removes "black notes"
3. key 4           ; you're in the ionian mode
4. show transpose  ; observe that the two leftmost columns are the same
5. undo            ; now back to the lydian mode
6. show transpose  ; observe that the two leftmost columns differ

So we have...

1. CS= Are there no ambiguous intervals (in the diamond)?
2. Strong TC= No ambiguous intervals in the cross set?
3. Weak TC= No ambiguous intervals between a mode and its transpositions?

Was there an error in my reasoning that 1 & 2 are the same?

I asked if all MOS's have at least one mode that is Weak TC.  That's the same
as asking if no MOS has at least one ambiguous interval appearance in every
mode.  Last year on November 19th I had asked if any proper scale could have
this property.  I answered the question affirmatively on December 29th, and
the example I gave had Myhill's property...

       C   Eb   Gb
2nds   3    3   *6
3rds  *6   *6    9
4ths  12   12   12

...so it seems not all MOS's have at least one Weak TC mode.  However, this
is a fairly degenerate case -- the generator has not yet spanned the IE.  I
haven't been able to find a "serious" MOS whose modes weren't mostly TC.

Now, could you (Paul Erlich) explain this...

>>When you take a cross set of a scale and itself, you only multiply by the
>>intervals measured up from the root.  So each mode of a scale has a
>>different cross set, right?
>Different only up to a transposition, which shouldn't matter.

RE: Digest Number 388                                              11/11/99

FOUREGUMP@aol.com wrote...

>I have been on this list for approximately two months and I am very
>interested in micotonal music.  While I understand the basic idea, I do not
>understand it in great detail.  What are some books I can read on this
>subject and where can I get them?  Also, what are some good microtonal
>albums for beginners interested in the music?

For recordings -- I have about 100 explicitly microtonal recordings -- I
recommend the following...


As far as books go, I guess it really depends on the angle you're interested
in.  Two I found essential are Harry Partch's _Genesis of a Music_ and David
Doty's _Just Intonation Primer_.  The former is available from Amazon.com for
$15 US; search books for "Partch".  The latter is available from the Just
Intonation Network...


If you're more interested in temperament and/or the extension of common
practice theory, you can't do better than Easley Blackwood's _The Structure
of Recognizable Diatonic Tunings_.  Unfortunately, it's out of print.  I was
able to find it in the New York public library.  An excellent book on
acoustics and tuning in Bill Sethares' _Tuning, Timbre, Spectrum, Scale_.
Search Amazon for "Sethares".

RE: clearing up TC                                                 11/11/99

>That's very strange, but basically you've changed the definition of TC to
>a _relation_ between two absolute pitch sets, and you're saying that F major
>and B major are not TC with one another because the interval between E and
>A#/Bb is a fourth in one and a fifth in the other. Am I following you so

Yes, except that TC has always been a relation between two absolute pitch
sets.  First there was lattice TC, where a set of connected lattice points
could be transposed by any one of the intervals connecting it without any of
the (absolute pitch) common tones between the original and transposed scales
changing scale positions.  Last I remember, we had decided that if you add
connectedness to your search criteria, then lattice TC was basically a weaker
version of your search -- which hardly seemed desirable.

So I proposed a version where the scale -- connected or not -- could be
rooted on any one of its members without having common tones jump scale
positions.  I called this self TC.  There are two versions weak and strong.
Weak means you can take a scale and root it on any of its members without
getting ambiguous intervals between the two scales.  Say you're playing along
in CMaj -- you can switch to FMaj without a hangup.  You can't switch to F#
without a hangup, but you'd never think of switching to F# because there
isn't one in the current scale.  You can switch to D first -- then you'll
have an F# -- but each of the switches will be cool.

Strong TC says, "I'm going to be perverse, and demand that TC hold over more
than single depth of switch.  It has to hold from C thru D to F#.  Because a
listener could still have C in his mind when F# comes around."

>What would be different if you took the cross set of the lydian mode?

A single transposition could turn up an ambiguity.

>>It is also trivial that the Weak version varies across the modes of a
>>scale, since the column starting on F would be in the far left if the
>>cross set of the lydian mode had been taken instead of the ionion.
>But that is not all that would change


>>Different only up to a transposition, which shouldn't matter.
>I already explained that. If you followed that explanation, you'd know that
>the Cartesian square of the C major scale, which you showed above, is
>identical to the Cartesian square of the Bb lydian mode, provided you rotate
>the numbering of the rows and columns

Obviously the relations in the cross set stay the same, which is why Strong
TC is defined for scales, not modes.  So Bb Lydian isn't Strong TC either --
no mode of the major scale is.  Nor is Bb Lydian Weak TC, as Bb Ionion and C
Ionion are.

Hopefully all this is clear by now.  I feel like a bit of an ass -- I've
managed to confuse everyone here, starting with myself -- over something
rather unremarkable.

Clearly the missing ingredient in all these TC's is attention to the
proportion of common tones under transposition.  So basically, what it boils
down to is "Paul, you could narrow your search by insisting that the
periodicity blocks be connected on the triangular lattice, and that
transposition by at least one of the connecting intervals gives a scale with
many common tones."

I'm fairly certain that such a search would find scales with high Rothenberg

RE: More on Fortuin & Peck at Sonic Circuits                       11/26/99

>"A Voice from Iraq" is in 27.35 ET, which, especially on a cent-accurate
>K2000, is the nearly the same as 44-CET.

Every other step of which is Gary Morrison's 88CET.  Which is not surprising,
since his tuning turned up in a similar search.  These tunings join with 41ET
in a series dividing the 3:2 into 8, 16, and 24 equal parts.

Regarding the K2000 -- does the error occur before or after the stacking?  If
before, you can still get closer to 27.35ET than 44CET.

>Interestingly, the above interval matches in 27.35 ET are alternately 3 and
>8 cents off from the ratio values--

Notice that in normal ET's, an interval and it's inversion about the octave
have equal errors in opposite directions.  Since your tuning divides the 3:2
almost exactly, pairs of ratios which sum to 3:2 ought to have this same
property.  Why 11:9 and 27:22 share the same error as 9:7 and 7:6 (and indeed
7:4!), I don't know.  Probably these are all using the same comma...

This tuning (and 27ET) are interesting in that they distinguish the 225:224,
a comma of 8 cents, as one step while the 64/63 of 27 cents vanishes.

RE: Performance announcement                                        12/7/99

>Is there a policy of what is acceptable on the list? I've never received
>Comments from anybody else?

While I usually favor a rather strict, narrow definition of subject matter
for e-mail and usenet discussion, the topic of tuning strikes me as a rather
special one.  The economic and social machinery, the number of fields
involved, in bringing about a conscious evaluation of musical tuning is so
great, that as long as the topic is related to music, I don't see the harm.
One of the greatest things about this list, in my eyes, has been its ability
to foster the tuning _community_.  If I could go and enjoy a concert, and
know that much more about a certain microtonalist's approach to music, and
maybe meet him afterwards, then yes, I'd like to know about it.

Concert announcements are so few on this list, that I don't think anyone
could complain.  OTOH, several listers have complained, and even left the
list, due to a lack of _musical practice_ being discussed here.  Importantly,
I have been to microtonal concerts featuring considerably less microtonality
than the average "normal" concert.

Finally, there is the old, but good, bandwidth is cheap argument.  How much
does it really take to ignore somebody's concert announcement?  When there
are so many microtonal concert listings here that I can't attend all of them,
then I'll make a motion to narrow the subject matter.

John, your post was a little vague as to what microtonal content, if any, it
was going to have.  Next time, just open with a word about that.

RE: Getting rid of the middleman                                   12/11/99

>>Conlon Nancarrow, Conlon Nancarrow, please pick up the white courtesy
>>telephone . . .
>I don't get it. Who is Conlon Nancarrow?

One of the most important composers of the millennium.  A search for "Conlon
Nancarrow" on Google gives the following URL's (among others)...


RE: Perfect pitch, Mr. Pehrson, and kudos                          12/14/99

[Jon Szanto wrote...]
>Anyhow, the researcher making the claims that most anyone could have perfect
>pitch is here in San Diego at the UC. If it is of interest I'll find out how
>to contact her.

According to the article, the main point of her research is that anyone may
be able to learn perfect pitch.  "No duh" would be my response.

The article claims she thinks it can only be learned at an early age.  I
think there's probably an imprinting period for it between the ages of 3 and
6, but like all things, I think it can be learned at any age.

Last time this thread came up, I promised some references on the topic.  Here
they are...

Lola L. Cuddy, "Practice Effects in the Absolute Judgment of Pitch"
Journal of the Acoustical Society of America, vol. 43, #5, 1968.

Paul T. Brady, "Fixed-Scale Mechanism of Absolute Pitch"
JASA, vol. 48, #4, part 2, 1970.

RE: Numerical accuracy conceptions of past music theorists         12/15/99

[Paul Erlich wrote...]
>Whether or not Mr. Woolhouse went "too far", Mr. Hill has failed to address
>Woolhouse's point here. Furthermore, far from overlooking the errors from
>just intonation, Woolhouse sought the best way to reduce them while
>preserving the musical meaning of the notes in the Western tradition.

This would unquestionably involve meantone.  But there are two points I'd
like to make here

[1.]  Free-pitched performances in the western tradition do certainly _not_
perform in temperament, if temperament is a _consistent_ detuning of just
ratios for the purpose of reducing the size of a pitch set.  In the case of
brass quintets and Barbershop quartets, and other a cappella groups,
surprisingly good adaptive JI is achieved by the most renowned performers,
and the emphasis is on vertical sonorities over melodic coherence (JI-related
melodic artifacts can be heard in some of the best performances).

Even in the case of bare melody temperament is not used (or melody with a
relatively quiet and rhythmically un-related accompaniment).  I challenge
anyone to find a performance with melodic fifths consistently tuned 6 cents
flat.  As with vertical sonorities, JI must be applied locally to melody to
make sense (some theorists never seem to tire of pointing out that global
"fixed" JI doesn't make sense harmonically).  The intervals between melodic
notes tend towards ratios of small whole numbers for the same reason that
harmonic intervals do -- the hearing system likes to fit things into a
harmonic series.  The accuracy is far less than with harmonic intervals, at
least because the roughness mechanism is not involved (so melodies produced
on a tempered instrument will sound okay).

On the global scale, there is a desire to simplify the Rothenberg difference
matrix.  Pythagorean chains, MOS, constant structures, and tetrachordality
all simplify the difference matrix.  I reject the idea that only ratios of 3
are important in melodic tuning.  Often, the desire to simplify the DM makes
the global scale definable in terms of a single just interval, and this
interval has historically been the 3:2, because 3:2 is such a strong
consonance.  But chains of any strong consonance could work, and even in
3-based scales, the approximation is only a global one.  Performers often
bend notes locally so that melodic (as well as harmonic) intervals can be
closer to 5:4, 7:4, etc, and in isolated runs one can even hear stuff like

[2.]  As concerns new music, the "problems" of JI would naturally be used to
advantage by the composer.  Music exists, still very much in the western
tradition, which demonstrates this.

[Kraig Grady wrote...]
>I am not quite sure what you mean in the 1st phrase here. It is interesting
>that those composers of notoriety who use different tunings, have use a JI
>system. Partch, Young, Harrison, and Riley.

Let's not forget Wyschnegradsky, Carillo, Haba, Mandelbaum, Blackwood, or

RE: Numerical accuracy conceptions of past music theorists         12/16/99

>And in fact, for ratios of 5 or higher, the accuracy with which ratios of
>_melodic_ intervals are reckoned is so low that factors of "custom" or
>"familiarity" are far more important than the harmonic series.

That's hard to determine.  And what are the driving forces behind custom,
then?  I'd call +/-10 cents a fair estimate of the accuracy of singing the
"base-two under" 7-limit melodic intervals, which is closer than 19-tone
equal temperament.

Far greater accuracy is possible when a drone is present.  Granted, these
would be "harmonic" intervals by our current convention, but... clearly a
melody over a drone is still a melody in some important ways (otherwise we
would not find approximations to the linear series in Indian and Persian

[Reference Bobby McFerrin's "The Voice"]

>Furthermore, if your first sentence above is true, why are minor arpeggios
>at least as common as major arpeggios in non-harmonized music from around
>the world (at least what I've heard)?

You're assuming they approximate a utonal structure?  Stuff I've heard is
usually closer to 7:6 or 19:16.  That may be because of the linear series, or
the harmonic series, but in neither case are subharmonics implicated.  There
is the exception of ethnic flutes, which often play subharmonic series
melodies, and after hearing such, it isn't too hard to sing them...

>>and this interval has historically been the 3:2, because 3:2 is such a
>>strong consonance.  But chains of any strong consonance could work,
>Speaking again strictly of melodic tuning, I fail to see any evidence of

Historical evidence, maybe not (although Wilson may disagree).  But certainly
the melodic validity of Keenan's chain-of-minor-thirds scales -- or the
22-out-of-41 scale you proposed -- is not in dispute, to say nothing of the
myriad of proper MOS's of the 5:4, 7:6, 7:5, and 7:4.

RE: Numerical accuracy conceptions of past music theorists         12/17/99

>>That's hard to determine.  And what are the driving forces behind custom,
>I don't know; join the sociology list.

They are either "frozen accidents" or something that can be discussed on this

>>I'd call +/-10 cents a fair estimate of the accuracy of singing the
>>"base-two under" 7-limit melodic intervals, which is closer than 19-tone
>>equal temperament.
>Come again?

7:4, 5:4, 3:2

>>[Reference Bobby McFerrin's "The Voice"]
>That's a great album -- which track are you thinking of?

Harmonic series segment like things can be heard throughout the album,
especially on track 7 in the "We're in the money" section.

>>You're assuming they approximate a utonal structure?  Stuff I've heard is
>>usually closer to 7:6 or 19:16.  That may be because of the linear series,
>>or the harmonic series, but in neither case are subharmonics implicated.
>>There is the exception of ethnic flutes, which often play subharmonic
>>series melodies, and after hearing such, it isn't too hard to sing them...
>Are you buying into the Schlesinger equally-spaced holes argument? No,
>flutes aren't quite like strings.

What do equal-spaced holes on flutes produce, then?

>I think they could arise on other instruments and even vocally, though; if
>the common overtone happens to coincide with a resonance, a subharmonic
>arpeggio would really stand out.

That's another thing -- perhaps subharmonic structures are important in
melody, assuming that first sentence of mine was true.  But maybe it's true
not because of the virtual pitch mechanism, but because of the spectral
mechanism -- maybe the ear just hears just melodic intervals better because
it is aware of a common harmonic.

>By the way, Carl, you've referred to "linear series" twice now. Are you
>being Wilsonified?

I was Wilsonified long ago.

>>Historical evidence, maybe not (although Wilson may disagree).
>Any examples?

Not that I can remember, but I seem to recall him explaining an ethnic scale
as a chain of something other than 3:2's (was it one of the permutations
articles?).  Kraig?

>>But certainly the melodic validity of Keenan's chain-of-minor-thirds scales
>> -- or the 22-out-of-41 scale you proposed -- is not in dispute,
>I recall much dispute over the latter from yourself, and please give us an
>mp3 of a nice melody with the former.

I disputed the scale only because it's so big -- most melodies would be
subsets of it.  And due to perceptual limitations, even a melody that did use
the entire scale would probably fail to convey a 22-member melodic pitch set.
But it surely contains a wealth of great subsets.

Eeek!  I've been having the worst time getting to my server today.  My ISP is
periodically taken with loosing connection to whole portions of IP space.  At
last, you may try the following, in the proper 8-tone subset of Keenan's

http://lumma.org/01.mp3 - 274K
http://lumma.org/02.mp3 - 634K
http://lumma.org/03.mp3 - 574K
http://lumma.org/04.mp3 - 1.02M

..or, go for all of them, plus 3 more at...

http://lumma.org/8all.zip - 3.37M

JI vs. ET!!!                                                       12/17/99

>>Wilson favorite pastime is pointing out commatic shifts in all types of
>>western singers on tape. Once again Boomsliter and Creel showed this in
>>actually measurements.
>I would like to see more evidence of this.

I'm not aware of any Boomsliter and Creel material that measured the
intonation of recordings.  They did, however, conduct a study where musicians
and non-musicians alike were allowed to tune familiar melodies in any way
they wanted, within the confines of a modified Reed organ.  While the study
was hardly scientific, it is fairly convincing, and I do buy their extended
reference theory, which is not really much of a theory, unfortunately.  It
basically states that many common melodies make use of a parenthesis-closing
type hierarchy, with melodic intervals all being just when related to the
appropriate point in the hierarchy.  The reason it doesn't amount to a theory
is that a method for identifying the hierarchy in a given melody was never
put forth.  Hierarchies were identified in a number of melodies, and shown to
match "preferred tunings" from experiment, but it was never shown that a
given melody has only one such interpretation.

Extended reference falls under what I've recently been calling "local"
melodic JI.  While I do buy into it, and think in principle the problems
mentioned above could be solved, I believe it is only one effect which a
melody may use; plenty of melodies exist which do not use the parenthesis-
closing effect.

It should also be noted that the parenthesis-closing effect does not require
JI.  Its potential simply depends on the number of notes at its disposal.  I
have referred to this as the "weak argument" for microtonality (TD 173.2 and
elsewhere).  But when combined with my "first sentence" (TD 441.15), you get
Extended Reference.

>It would certainly run counter to what I've been saying, but as far as I'm
>concerned any reason to overthrow the hegemony of 12-tET is a good thing.
>Hopefully it would not be to replace it with a hegemony of strict JI.
>Without the puns of temperament most Western keyboard compositions and my
>decatonic stuff would be out the window.

I don't think you have any reason to worry, Paul -- puns are definitely tha
bomb.  Thing is, puns have nothing to do with melody.  At least I've always
considered them a harmonic effect.

I don't believe, in general, that commatic shifts change the melodic identity
of notes.  By increasing the size of the melodic pitch set (or perhaps by
decreasing the certainty with which it is defined), commatic shifts may rob a
melody of clarity.  OTOH, if the shifts are due to extended reference, they
add a level of information to the melody.  But I don't think they effect puns

A pun to me is a forced re-fitting of a dyad or larger-ad in the harmonic
series.  For example, in 12, when modulating from D7 to FM, the dyad AC has
changed from a 7:6 to a 6:5.  So for me, the ability to pun is measured by
the uniqueness of a temperament.

I've always believed that puns add something to music -- and that they take
something away.  They necessarily take away consonance, and also any effect
depending on the "weak argument", since they are only possible in the smaller
temperaments.  It would be very interesting to search for ETs with low
uniqueness yet high consonance.  12 is certainly the king.  After that must
be 15 and 22.

Since I've joined this list, I've been an advocate of JI.  It hasn't been
because I don't believe in puns.  Rather, I'm interested in what may be
gained from greater consonance, and larger pitch sets.

RE: Numerical accuracy conceptions of past music theorists         12/18/99

>>I'd call +/-10 cents a fair estimate of the accuracy of singing the
>>"base-two under" 7-limit melodic intervals, which is closer than 19-tone
>>equal temperament.
>74 is 21 cents off in 19-tone equal temperament.  And the point is?

If I said that 1/4 comma meantone comes within 6 cents, what would the point

>>I was Wilsonified long ago.
>So tell me, why does he see modulus-n as having two primary representations,
>one as a linear series, and one as a high-limit diamond or CPS with added
>notes?  What about representations with intermediate numbers of dimensions?

Why do I feel like I'm being set up?  Extended reference requires a basic set
of intervals be available from several starting points.  You can view each
representation as the most compact pitch set providing the highest number of
starting points as the basic set is varied.  These are only snapshots of what
may happen in a performance -- the performer is likely to mix and match.  Is
there another answer that contradicts something I've said?

>That was the point of it.  Under those qualifications, it would be hard to
>dispute the melodic validity of any large scale with regularities.

True.  And if that was the point of it, have you since changed your mind
about chains of 5:4's?

RE: Reply to Carl Lumma                                            12/20/99

>>If I said that 1/4 comma meantone comes within 6 cents, what would the
>>point be?
>I still don't know.

Duder, just giving a point of reference.

>>Why do I feel like I'm being set up?  Extended reference requires a basic
>>set of intervals be available from several starting points.  You can view
>>each representation as the most compact pitch set providing the highest
>>number of starting points as the basic set is varied.  These are only
>>snapshots of what may happen in a performance -- the performer is likely to
>>mix and match.  Is there another answer that contradicts something I've
>I don't know, but you still haven't answered my question.

If you're trying to make a point, Paul, just do it.  If you don't know the
answer, then tell me a little more about what you're after, and I'll try to
provide it.  The language "So why don't you tell me" seemed slightly odd

>>True.  And if that was the point of it, have you since changed your mind
>>about chains of 5:4's?
>What opinion did I previously have about them?

I don't presume to know your opinion.  But I recall a reserved maybe coming
across.  Which seemed different than the absolutely not I was getting a few
days ago.

RE: re: Reply to Carl Lumma                                        12/21/99

>OK, my point was that concentrating on the two representations -- the linear
>(least compact, lowest dimensionality) and the supplemented CPS/diamond
>(most compact, highest dimensionality) leaves out all kinds of intermediate

I wasn't aware I was ignoring the intermediate ones.  And I wouldn't call the
linear representation anything other than maximally compact.

>OK, I find it hard to comprehend a scale based on 54s rather than 43s
>melodically. I gave a reserved maybe on scales with a tetrachord-like
>structure repeating in 54s rather than 43s. Thing is, when the melody hits
>a near-43 or a near-32, the effect is so powerful it tends to overwhelm
>the 54 structure. It's similar to my view on non-octave based scales.

This makes good sense.  Would the following scale qualify as tetrachordal at
the 5:4?

0 345 386 732 773 1118 1159 1200

RE: re: Reply to Carl Lumma                                        12/21/99

>0 345 386 732 773 1118 1159 1200

Unfortunately, this scale is _wildly_ improper.  The smallest proper MOS
available built from anything like a 5:4 is decatonic.  The closest to 5:4
you can get is 369.2 cents, an error of 17.1 cents.  Fortunately, no other
interval in the scale comes as close to the 5:4.  Unfortunately, 17 cents is
really bad, especially on the flat side of this interval.

I'm not sure how MOS maps to tetrachordality here.

RE: re: Reply to Carl Lumma                                        12/21/99

I wrote...

>Unfortunately, this scale is _wildly_ improprer.  The smallest proper MOS
>available built from anything like a 5:4 is decatonic.  The closest to 5:4
>you can get is 369.2 cents, an error of 17.1 cents.  Fortunately, no other
>interval in the scale comes as close to the 5:4.  Unfortunately, 17 cents is
>really bad, especially on the flat side of this interval.

I just realized this is 10 out of 13tET!  In fact, by Chalmers' formula, the
exact point of non-strict propriety for MOS's occurs when L/s is 2/1 (or
1/2).  So 13tET represents the theoretical best tuning of the 5:4 in this
scale if we want it to be proper.  Also, while the 7-tone scale I posted
earlier is very awkward, this one rocks!  It is far more fluid than even the
8-tone Keenan scale I posted mp3's of a few days back.  I can hear the 5:4's
melodically -- Paul, do you have any way to tune this up, or would you like
an mp3?

RE: re: Reply to Carl Lumma                                        12/21/99

>>I wasn't aware I was ignoring the intermediate ones.
>I think Wilson gives them short shrift, at least in some of his papers.

He depicts some of them on rectangular lattices, and surely the modified
linear series keyboard mappings are to be considered intermediate reps.

>>And I wouldn't call the linear representation anything other than maximally
>It's very stretched out in high-limit space.

Yeah, but there wouldn't be a reason to view them there.  They are maximally
compact when the basic set is shrunk to a single interval.

>>>This makes good sense.  Would the following scale qualify as tetrachordal
>>>at the 5:4?
>>0 345 386 732 773 1118 1159 1200
>Yes, I think it should (though "trichordal" would be more correct).

Is there any MOS that isn't "tetrachordal" by its generator?

>>>I just realized this is 10 out of 13tET!
>> . . .
>>Paul, do you have any way to tune this up, or
>>would you like an mp3?
>Sure -- by the way, this is also 10 out of 26-tET, of course.

I hadn't even thought of that!

>You should try to get all your .mp3s on the Tuning Punks site.

The 8-tone stuff is way too rough for that.  And the CPS demos are, well,
just demos.  I think they're better suited for my website, whenever I get it
up.  Maybe when I get some instruments up and running I'll post some music...

New mp3's                                                          12/21/99

And you thought that last batch was annoying!  As before, I've simply
recorded my doodling.  This time, though, you get the following scales...

* The 10-tone subset of 13-tET discussed earlier.

* Two tetrachordal scales I made up a few months back --- an 8-tone one based
on tetrachords at the 7:5, and a 9-tone one based on tetrachords at 3:2.

* Paul Erlich's Pentachordal Major decatonic, tetrachordal at the 3:2. 

* Balzano's 9-tone subset of 20-tET.  It's not tetrachordal, but what the

At two points in these files, Midi Relay craps out, and you can hear a chord
bend back to 12.  Tell Graham to fix it!

URL = http://lumma.org/chordal.zip

Tetrachordality                                                    12/22/99

>>He depicts some of them on rectangular lattices, and surely the modified
>>linear series keyboard mappings are to be considered intermediate reps.
>Such as?

17-tone Tubulong, genus 3^8 * 5 comes to mind.

>>Is there any MOS that isn't "tetrachordal" by its generator?
>I don't think so, but there are plenty of "tetrachordal" scales that are not

That's what I thought.

>Doesn't bother me. How about the augmented scale, 0 3 4 7 8 11 in 12-tET (or
>0 4 5 9 10 14 in 15-tET or 0 7 9 16 18 25 in 27-tET)? Two closed chains of
>400 cents separated by a 3:2.

Paul, did you see my subsequent post on this?  I first thought you meant MOS
only, but later realized that you meant tetrachordal scales in general.  I'll
check out this scale...

>>* Balzano's 9-tone subset of 20-tET.  It's not tetrachordal, but what the
>Carl, are you using a consistent definition of "tetrachordality" here?

I'm insisting that the tetrachordality occur at a consonance.  Otherwise lots
and lots of scales would be "tetrachordal".  I guess it still would require
some simplification of the interval matrix, but lots of things do that;
tetrachordality is supposed to work because some intervals are "special".

RE: Re: Digest Number 457                                          12/27/99

[Gerald Eskelin wrote...]
>Yes. I'm quite sure that the ear does "temper" such chords "naturally." I
>commonly hear singers make slight pitch modifications in notationally
>"repeated" or "sustained" notes as the fundamental shifts. And, believe me,
>very few of those singers have any notion of numbers, ratios or tuning
>"systems." They simply have sensitive ears.

Jerry, this doesn't sound anything like temperament to me.  It sounds like
the singers you describe are making commatic adjustments consistent with
singing in JI.

RE: Digest Number 459                                              12/28/99

[Marty Hatch wrote...]
>Because I have foregone the right to a grand piano in my office in the
>music department at Cornell, I have been given the opportunity to purchase
>a synthesizer keyboard of my choice. I'd like to get one that has the 88
>touch sensitive keys (so I can still play theory exercises), midi export, a
>nice range of timbres or places for (say, Alesis) cards, etc.,


Are you aware of the microtonal synthesis page?

I can't recommend Alesis keyboards, and I'm pretty sure they are not tuning

If you're willing to put up with 61 keys, there's a great deal right now on
one of the coolest synths ever made, the Kurzweil K2000VP...


Otherwise, the new Korg Triton is a great piece of gear, and you should be
able to get an 88-key version for just under $3000.  Be sure to get the MOSS
upgrade card for the physical modeling synthesis, and presumably the
microtonal capabilities, of the Trinity series (double-check the tuning
capabilities of the Triton w/MOSS before you buy it).

If portability is less of an issue, and the most flexible tuning options are
a priority, then the best way to go would be a separate midi controller and
rackmount synth.

The best midi controllers are by FATAR (I recommend the "semi-weighted"
action, which is what Kurzweil puts in its K2500 and K2600 flagship


Then, teem it up with a K2000VPR or an Emu Proteus 2000.  This setup would
probably give you the best of everything and the lowest price.  The K2000VPR
is only $1500 at Sam Ash...


And the Proteus 2000 is only $800!


Here are links to manufacturer's pages...


I hope this is of some help.

Keenan 10-tone MOS                                                 12/29/99

A while back Dave Keenan posted the results of his search for 7-limit tetrads
in MOS's of 10 or less tones.  His results showed Erlich's decatonics and a
chain of 6:5's to be most promising.  On this list, in the last few months,
these have been examined.  But little mention has been made of his next-most
promising candidate, a 10-tone MOS of generator 125 cents, and interval of
equivalence 2:1.  Here is it's interval matrix...

      2nds 3rds 4ths 5ths 6ths 7ths 8ths 9ths 10ths   7o  7u  5o  5u
   1  125  250  375  450  575  700  825  950  1075   479 369  47
   2  125  250  325  450  575  700  825  950  1075       369      47
   3  125  200  325  450  575  700  825  950  1075                47
   4   75  200  325  450  575  700  825  950  1075                47
   5  125  250  375  500  625  750  875  1000 1125  
   6  125  250  375  500  625  750  875  1000 1075  
   7  125  250  375  500  625  750  875  950  1075   ~49 ~39
   8  125  250  375  500  625  750  825  950  1075   ~49 ~39
   9  125  250  375  500  625  700  825  950  1075   479 ~39  47
  10  125  250  375  500  575  700  825  950  1075   479 369  47

The scale is strictly proper.  To the right, I've listed the 7- and 5-limit
chords that appear in each mode, by the scale degrees they occupy.  We see
that, although this scale contains six 7-limit tetrads (the same number as
Erlich's Pentachordal decatonics), they are not distributed very well (as are
Erlich's).  Particularly, the O- and Utonal versions are never represented by
the same pattern of scale degrees.  I think the scale fails as a 7-limit
generalized diatonic for this reason.  However, if you look at the 5-limit,
you'll notice that the triads are very well spread out, and are tuned with
greater accuracy than in 12-tET.  Unfortunately, 10-tones are a lot to cover
with triads, especially when you've only got 6 of them.  But it's not all
bad, and composing in this scale would certainly be interesting.

Barbershop history                                                 12/29/99

In a recent off-list discussion between Daniel Wolf and I, the history of
Barbershop was briefly explored.  In the process, I dug up a SPEBSQSA
handbook, and found a story that I thought some here might find

"_Was barbershop harmony actually sung in barbershops?_

Certainly -- and on street corners (it was sometimes called "curbstone"
harmony) and at social functions and in parlors.  Its roots are not just the
white, Middle-America of Norman Rockwell's famous painting.  Rather,
barbershop is a "melting pot" product of African-American musical devices,
European hymn-singing culture, and an American tradition of recreational

Immigrants to the new world brought with them a musical repertoire that
included hymns, psalms, and folk songs.  These simple songs were often sung
in four parts with the melody set in the second-lowest voice.  Minstrel shows
of the mid-1800's often consisted of white singers in blackface (and later
black singers themselves) performing songs and sketches based on a
romanticized vision of plantation life.  As the minstrel show was supplanted
by the equally popular vaudeville, the tradition of close-harmony quartets
remained, often as a "four act" combining music with ethnic comedy that would
be scandalous by modern standards.

The "barbershop" style of music is first associated with black southern
quartets of the 1870's, such as "The American Four" and "The Hamtown
Students".  The African influence is particularly notable in the
improvisational nature of the harmonization, and the flexing of melody to
produce harmonies in "swipes" and "snakes".  Black quartets "cracking a
chord" were commonplace at places like Joe Sarpy's Cut Rate Shaving Parlor in
St. Louis, or in Jacksonville, Florida, where, black historian James Weldon
Johnson writes, "every barbershop seemed to have its own quartet".

The first written use of the word "barbershop" when referring to harmonizing
came in 1910, with the publication of the song, "Play That Barbershop
Chord" -- evidence that the term was in common parlance by that time.

_O.C. Cash & Rupert Hall_

While travelling to Kansas City on business, Tulsa tax attorney O.C. Cash
happened to meet fellow Tulsan Rupert Hall in the lobby of the Muehlebach
Hotel.  The men fell to talking and discovered they shared a mutual love of
vocal harmony.  Together they bemoaned the decline of that all-American
institution, the barbershop quartet, and decided to stem that decline.

Signing their names as "Rupert Hall, Royal Keeper of the Minor Keys, and O.C.
Cash, Third Temporary Assistant Vice Chairman," of the "Society for the
Preservation and Propagation of Barber Shop Quartet Singing in the United
States [sic], the two invited their friends to a songfest on the roof garden
of the Tulsa Club, on April 11, 1938.

Twenty-six men attended that first meeting, and returned the following week
with more friends.  About 150 men attended the third meeting, and the grand
sounds from harmony they raised on the rooftop created quite a stir.  A
traffic jam formed outside the hotel.  While police tried to straighten out
the problem, a reporter of the local newspaper heard the singing, sensed a
great story, and joined the meeting.

O.C. Cash bluffed his way through the interview, saying his organization was
national in scope, with branches in St. Louis, Kansas City, and elsewhere. 
He neglected to mention that these "branches" were just a few scattered
friends who enjoyed harmonizing, but knew nothing of Cash's new club.

Cash's flair for publicity, combined with the unusual name (the ridiculous
initials poked fun at the alphabet soup of New Deal programs), made an
irresistible story for the news wire services, which spread it coast to
coast.  Cash's "branches" started receiving puzzling calls from men
interested in joining the barbershop society.  Soon, groups were meeting
throughout North America to sing barbershop harmony.

_The Early Years_

Operating out of living rooms and garages, the Society was an informal affair
in its first years.  Memberships could be had by writing to Cash and Hall and
requesting a certificate, which the founders financed out of their own
pockets.  The first formal convention, held in Tulsa in June of 1939, brought
together 150 men from 17 cities for a weekend of harmonizing and a contest to
crown the "World Champion Quartet" -- a distinction won by the Bartlesville
Barflies of Oklahoma.

The Society grew rapidly in the post-war years, in 1945 doubling in size from
4,000 to 8,200 members in 200 chapters.  By 1957, the Society had grown to
26,000 members in 625 chapters, and an aggressive "Expansion Program" was
initiated to provide educational services and purchase a permanent home for
the Society's administrative staff.  A mansion in Kenosha, Wisconsin, became
the new base of operations.  SPEBSQSA had come of age."

_From the original SPEBSQSA charter_

'In this age of dictators and government control of everything, about the
only privilege guaranteed by the Bill of Rights not in some way supervised or
directed is the art of barbershop quartet singing.  Without a doubt, we still
have the right of peaceable assembly which, we are advised by competent legal
authority, includes quartet singing.

The writers have, for a long time, thought that something should be done to
encourage the enjoyment of this last remaining vestige of human liberty.'

						-- O.C. Cash and Rupert Hall"

RE: 22tet kbd example                                                1/6/00

>The transpositional equivalence these embody (I think) doesn't lend itself
>to navigation as specifically as the 12-tone; in the same manner as
>Reszutek's only relative navigation is embedded so necessitating (again)
>the labelling of keys.

Clark, it is true that the design of the conventional keyboard allows for
easy navigation by touch.  However, I believe transpositional invariance is
a far greater asset, and what's wrong with color-coded keys, or tactile
patterns on the keys (US 5515763)?

Rezsutek's keyboard would probably, in practice, have plenty enough feel
abouts for one to find his way.  It has other problems, though.  Noticeably,
the lack of transpositional invariance.  However, I personally think there's
one thing worth giving up TI for easy access to the scale, and a consistent
fingering pattern for its chords (it seems that TI rules out uniform
fingering for the chords of a generalized diatonic scale...).  However,
Rezsutek's keyboard puts the easiest decatonic on all black keys, which is
bass ackward.

But foremost among my problems with this keyboard is my inability to reach an
octave on it.  According to my calculations, the average octave span would be
28cm if standard key width was assumed.  The good folks over at DS Keyboards
(www.dskeyboards.com) have shown that this width need not be assumed --- the
minimum workable key width turning out an octave span of about 24cm.  Still
too long for your average bear.  Reaching an octave is, in my book, a
requirement for any serious xen keyboard, and so the conclusion you need more
than two ranks of keys!

RE: 22tet kbd example                                               1/10/00

>According to the website, DS only make piano keyboards.  So those findings
>aren't applicable to MIDI keyboards, let alone when they have more than 12
>keys to the octave.  Steve rebuilt a MIDI keyboard, not a piano.

It is true that DS only makes piano keyboards.  But the findings (at least
the ones I've mentioned) are applicable to any keyboard with Halberstadt

>My old PSR keyboard has white keys 1.8 cm center-center.  I found that
>playable, but not convenient.

Too narrow for my fingers.  There'd be too much strain playing runs, having
to ball up my hand.  And, how thick were your blacks?

>When you say "getting stuck between the blacks" does that mean you're
>deliberately playing the top end of the white notes?  This is possible
>with mini-keys, but not easy.  I'd prefer to write music that doesn't
>require such tricks.

Such as music entirely in the key of C Maj?

>With a custom-built keyboard, the blacks could be thickened, and the
>keyboard made playable with a smaller span.

Thickening the relative size of the blacks only makes the getting stuck
problem worse -- or isn't that what you meant?

>But then a custom-built keyboard could have more than two ranks.

Well, that's the thing.

>That's better, you're not talking about pre-requisites for "any serious
>xen keyboard" any more.

Yes I am.  You could spend your life playing wonderful music on a xen
keyboard where you could only reach a 5:3.  But I wouldn't call it a serious
alternative to the Halberstadt keyboard, considering the types of voicings
you'd be giving up.  I wouldn't pay money for a full-sized Halberstadt
anymore either, after playing various scaled-down versions over at DS.  But I
do have a relatively small hand, and I am a bit of a nut.

Xen keyboards, Graham Breed, Mr. Halberstadt                        1/16/00

>The number of notes you can reach with one hand is far more important if
>you're playing live than recording each part to a sequencer.  And having
>interchangeable keyboard mappings means each needn't be as useful on its
>own as if that were all you had.

Graham, please understand that I'm not dissing any plans you may have.  I
strongly believe there's worlds to be done with a retuned halberstadt and a
sequencer -- Brian McLaren has produced hours of such material.  It's just my
choice to spend my time obtaining a keyboard that suits me, and then learning
to play it, rather than learning to play one that doesn't, even though the
latter option could be a fruitful one.  It's just my choice.  It's what
brought me to this list.  In fact, theory for me is just a means to a
keyboard design.

Now, I know I've said that a retuned halberstadt isn't "serious".  That's
based on some very careful thought about what made the Halberstadt so good
for 12-tone tunings in the past 1000 years.  See, I believe the keyboard has
had no small impact on music as we know it -- in fact, I believe it gave
birth to western music.  What I'm saying is, I doubt there will ever be a
serious genre of music based on retuned halberstadts.  If somebody proves me
wrong on that, I certainly wouldn't complain, but...

Like it or not, serious keyboard music is played "live".  The ability for one
person to control a large number of resources in real time with his hands is
not a small deal.  It forces a certain understanding of music.  Serious
keyboards (ones that are up for consideration as a standard) are expected to
do a wide variety of things -- like rehearsing a choir.  You simply cannot
rehearse a choir in JI with a halberstadt.

>A diminished triad contains two minor thirds.  Either could be subminor,
>and with the right keyboard you can try all three versions to see which
>you want.  And do that really easily.

And how would the ability to an octave prohibit this choice?

>Sure, if you've got one.

I'm saying there are good reasons to get one!  That it is worth the effort.
And the expense is not prohibitive!  How could anybody believe such a thing?
Do you drive a car?  Live under a roof?  Use a PC?

>> Caring about what voicings I can reach is arbitrary?
>Insisting on reaching an octave with one hand is arbitrary.

No, it isn't.  What I insist upon, to be clear, is to be able to reach at
least the interval of equivalence with a single hand.  As it is, much
beautiful music would be lost by making the halberstadt bigger so that, say,
Oscar Peterson could reach _only_ an octave.  Yes, it is true that by making
such a strict rule, I imply a sudden change of goodness that doesn't exist.
Yes, I'd probably take a keyboard on which I could only reach a 15:8 if it
appeared on my doorstep.  But every bit you loose, you loose a bit.  The IE
is a good goal to set, because it is the point at which you no longer need
two hands to play every interval in the tuning.

>You can play cool music on a bad instrument.  My contention is that
>extended mappings allow you to do cool things that wouldn't otherwise be
>possible.  And if you're interested in the right scales, it may be worth
>rebuilding a keyboard to make them work better.  So long as that's
>understood, no problem.

Understood, and agreed.

RE: xen keyboards, me, Mr. Halberstadt                              1/18/00

[Graham Breed wrote...]
>An extended mapping for a 2-rank keyboard could take it's place in the
>ensemble or the studio. 

That was Partch's approach.  And he certainly ripped me a new ___ with his
work.  Partch wrote some two-voice bits for the Chromelodeon that deserve to
be written about 1000 years from now.  Like the opening few bars to
Revelation in Courthouse Park.  Right or wrong?

>And some of "what made the Halberstadt so good" wouldn't transfer to a

True!  But all of what made the halberstadt so good (far as I can tell) can
be transferred to multi-rank keyboards.  In its defense, what the Bosanquet
looses is _probably_ made up for by what it gains (transpositional

>Do "retuned halberstadts" include 12-note octave retunings?

I meant non-octave reachable retuned halberstadts there, sorry.

>> Like it or not, serious keyboard music is played "live".
>What a strange assertion.  Is this true of some, most or all such music?

Music written for any instrument is generally considered clever in direct
proportion to how much it extracts from the instrument's resources.  If you
view a N-ORRH (see previous quote) as a melodic instrument, then serious
music for it could just be a melody.  But to me, turning a keyboard into a
melodic instrument is silly.  Of course, N-ORRH's are capable of at least two
voices in all registers.

If the instrument in question is a sequencer hooked up to a N-ORRH, music
made on it may be clever, or not.  But it's no longer keyboard music, it's
sequenced-N-ORRH music.  This has its own challenges and rewards.  But I will
argue that a sequenced-N-ORRH cannot replace a 4- or 5-voice capable
polyphonic keyboard for utility in a genre of music.  I will even argue that
sequencing parts is not as mind-expanding as having to play them at once, in

The choir is just one example.  The use of a Bosanquet or multi-rank keyboard
could change everything.  Orchestras could play Stravinsky in 15-limit JI all
the time, not just under the freakish circumstances that must have existed
the day they recorded Deutsche Grammophon 437-850-2.  The influence that a
keyboard can have, serving as representation of musical materials (if its
layout shares the periods of its tuning, and if what an ensemble might play
in such a tuning can be hinted at by the hands of one person), cannot be

>Hmm.  "Up for consideration as a standard" suggests I wasn't taking your
>"seriously" seriously enough.

You'll have to give me some slack -- I'm trying to explain my entire approach
to music and my life here.  That's not easy.  I can say this: I'm _not_
interested in making a new standard.  I _am_ interested in obtaining for
myself a standard-worthy keyboard instrument.  Make sense?

>There are good reasons to get a Bosanquet keyboard, and reasons why the
>expense is too great.

Such as?

>But even if you were to buy/build one, I wouldn't expect it too be as
>playable as a Halberstadt right away.  It needs time to evolve, along with
>a playing style.

For sure.  Also my playing in general needs some time to evolve. :)

>Now I've got a house, my next priority is to get my own PC, and then Kyma.

I'll trade you that house for my PC!  Looks like we're both after Kyma.  Have
you got Marcus Hobbs' CD yet?

>Keyboards are a secondary issue.  I'll probably go for that Fatar
>semi-weighted job you say is so good: you'd better be right!

I didn't say it was good, I just said it was the best all-around MIDI
keyboard action out there.  You should try different actions to see for
yourself.  The new Kawai stage piano thingy looked soo promising, but I don't
like it.

>The MicroZone is $7500 for 810 keys.  If that were cost-per-key, then the
>128 keys you can actually get from MIDI would only cost $1200 by my
>calculations.  That would be a serious proposition.

There's a lot of things I don't like about the MicroZone.  But it's Erv's
keyboard, so I don't have to like it.  There's absolutely no reason you
couldn't have somebody custom-build you a 5-octave, 48-key/octave, multi-rank
MIDI controller keyboard with reachable octaves for $5000 US.  That's about
1/5 the price of the average grand piano.  If the keyboard was mass produced,
it could be available for $1500 US and still provide a nice profit for
whoever was making it.

>But there's a lot of beautiful music that can't be made without reaching
>inside the semitone.

I think we'd all agree with that.

Extended reference? (was RE: Yes)                                   1/20/00

>>Hmmm... now *there's* something that would make a
>>great tuning analysis project!...
>In the case of Yes or Robert Johnson, I have a (IMO) very important
>suggestion for you. Take several alternative performances of the same piece
>and analyze the tuning. You will then have an idea as to what kind of
>accuracy you can meaningfully ascribe to performer intent and what is due
>to random errors.

Paul, forgive me for butting in here, but if one subscribes to extended
reference, as I believe Joe does (his Robert Johnson analysis and Beethoven
Scherzo fragment both qualify as extended reference interpretations), then
your suggestion is a bit misguided.  Recall that extended reference has no
way of pairing any single tuning to a given structure -- it's just a set of
rules that may be broken or followed by any number of possible tunings.  Two
performances may differ, but each may still be internally consistent.  Of
course one could still learn a lot by comparing different performances of the
same structure!!

The notion that only one set of pitches can satisfy JI in melody seems to me
as ignorant as the harmonic version we're now in the process of debunking
with various "adaptive JI" schemes.  Blasted California school!

RE: xen keyboards, me, Mr. Halberstadt                              1/20/00

>>True!  But all of what made the halberstadt so good (far as I can tell)
>>can be transfered to multi-rank keyboards.  In its defense, what the
>>Bosanquet looses is _probably_ made up for by what it gains
>>(transpositional invariance).
>A Halberstadt is essentially linear, as pitch rises steadily as you go up 
>it, and you have a high degree of freedom perpendicular to this.

You mean you can move your hands closer to or farther from your body as you
play?  That's a keen observation -- it is one of the fatal flaws (IMO) of
Erv's honeycomb design.

>A  multi-rank keyboard needn't spoil that until you use the extra ranks.
>But once you do, it restricts your freedom of finger placement.  This isn't
>a fatal flaw, but it's something that will have to be overcome.

I'm sorry, I don't follow.  Multi-rank keyboards with rectangular keys can
preserve as much or as little perpendicular freedom as you'd like.

>I'm not sure if transpositional invariance is such a big deal.


>Getting more notes playable with one hand is, and anything that minimises
>the complexity is good.


>The inherent simplicity of a linear keyboard is something that will always
>count for Halberstadt.  It all depends on whether or not you want those
>extra notes.

I don't follow the linear thing.  Multirank keyboards can still map pitch in
a "linear" way, just in 2-D instead of 1-D.

The big choice you have with any keyboard is that between transpositional
invariance and what I will call modal invariance.  You can't satisfy both at
once, by definition (unless your basic scale is equal-step, in which case it
doesn't really have modes).  The halberstadt has M.I.  Both major and minor
thirds are thirds, on balance the same width.  This is probably the way to go
for generalized-diatonic scales -- you can play the basic chords without
having to think.  You just get them "under the fingers".  But, you do have to
bother to learn to play in all keys...

>Well, I don't think music should set out to be clever.

That's a good point, and I agree.  But it isn't always bad thing, either.

>As for keyboards as melodic instruments, there are plenty of reasons:
>they're cheap, easily available, easy to play and give full MIDI control.

You can do some cool stuff with a bend wheel and a monophonic patch, to be
sure.  But I'd much rather use (and listen to) a wind controller.

>>If the instrument in question is a sequencer hooked up to a N-ORRH, 
>>music made on it may be clever, or not.  But it's no longer keyboard
>>music,  it's sequenced-N-ORRH music.
>Aaaaaaahhhhhhhhhh!!!!!!!!!  So serious keyboard music is played live
>because, if it isn't live, it isn't keyboard music!

That's right.  [If live means "in realtime".]

>A non-octave reachable retuned Halberstadt won't replace the standard
>article.  But there's so much music -- well, most that uses synthesizers
>-- that doesn't rely on polyphonic playing.


>>The choir is just one example.  The use of a Bosanquet or multi-rank
>>keyboard could change everything.  Orchestras could play Stravinsky in
>>15-limit JI all the time, not just under the freakish circumstances that
>>must have existed the day they recorded Deutsche Grammophon 437-850-2.
>15-limit JI is asking a lot of any two-dimensional keyboard.

Tell that to Erv Wilson!  [Even a plain diamond works quite well -- even
those who don't like Norman Henry's music seldom complain about his skill at
the diamond keyboard.]

>Well, it depends what you'd use it for.  I'm making sequenced music in my
>spare time, and keyboards aren't really that important.  Having a more
>versatile one wouldn't make much, if any, difference to the music.  If
>you're playing live, and need the extra notes, you need the keyboard,
>provided your budget stretches to it.

Just to be sure, I'd like to say again that none of my remarks are meant to
discourage anyone from making sequenced music!  I can't stress that enough!

>That's a tempting offer, but without my house where would I put your PC?
>As for the CD, I'm still confronted by the obstacle of not having a US
>bank account.

Graham, write me off list.  You send pounds, I send disc.

RE: extended reference? (was RE: Yes)                               1/21/00

>How about a nice, formal definition of 'extended reference' for
>my Tuning Dictionary?...

A method of notating a melody in which consecutive pitches are given in
order, by their distance from a previously-tuned note.  The frequency of the
first note is arbitrary, and it serves as the reference point for the second
note, after which either the first or second note may be the reference, and
so on.

Like classical melodic JI, extended reference holds that melodic intervals
are ideally tuned according to ratios of small whole numbers.  But extended
reference differs from the classical application in two important ways:

1. Rather than mapping notes of the scale to a fixed pitch set and performing
modal transpositions on the pitch set, extended reference interprets modal
transposition as a change of reference point (tonic) -- maps scale degrees to
intervals instead of pitches.

2. Rather than map a single just ratio to each scale degree, extended
reference may allow several tunings of a given scalar interval, so long as
all occurrences of that interval are tuned with the same ratio _within a
given reference point_.  [The model borrows from conventional music theory
the idea that harmonic progressions function in a sort of 'parenthesis
checking' hierarchy, where one must first resolve to the local tonic, then to
the next tonic up, and so on...]

<---Throw in references, credit B & C...--->

That tuning method I've been pitching                               1/21/00

>What, for example, do you plan to do when two successive chords
>overlap slightly?  One option, which in fact I do, is to seek out such
>overlaps, evaluate them for duration, and essentially throw away the
>very brief complex chords formed when they don't last longer than some
>preset time, which I typically set to 64 msec.

My procedure only tunes the five loudest notes at a given time; less if there
are less than five unique (after octave equivalence) pitches sounding.

>Do you plan to have any chords actually tuned to 12-tET?


>What is your thought about considering the duration of chords?  If I
>play a chord for a very brief length of time, should it have as much
>pull on tuning as a long sustained chord?

In my version, duration isn't a factor.  But keep in mind it was designed to
be composed for, rather than to retune existing music (although you might get
many interesting results!).  It is for the composer who wants to work in
extended JI, thinking of chord progressions as common-tone paths through
chord space (after Wilson), but who only has provisions for scoring in
12-tone (say he uses MIDI).

>How will you handle drift?  The ol' comma pump, repeated many times?

Ignores drift entirely.  Repeated pump makes the music go flat.

>Will you consider the length of time between chords?  Two conflicting
>chords back-to-back vs. separated by a long silence, say.

Silence is ignored.

>In your scheme, can chords, or the notes within them, interact when
>something else intervenes?

Just the five loudest notes at any given time.  Notes may be retuned as they
sound, if that's the question.

>If there are three successive chords, and only the outer two have a C, say,
>will the two C's consider their relative tuning?

Nope.  In a way, I'm actually after "strict" JI here.  I call it adaptive
because it's getting at JI with only 12-pitches, moving the tonic is all over
the place.  But I'm trying more to access high-limit pitch sets than I am
trying to get JI to conform to meantone logic.

>When you refer to "ticks" in a MIDI file, what does this mean?  I'm
>aware of the MIDI time scheme, but if nothing is changing, there's
>nothing to focus on, yes?

Just the shortest unit of action in the MIDI-verse.  Yes, many ticks will not
feature a change (although not as many as one might think, since event
loudness is sampled at each tick), but the thing is: ain't nothin' gettin'
past this baby.

RE: That tuning method you (Carl L)'ve been pitching                1/22/00

>Mmmm.  My experience has been that, if you don't explicitly go after
>and neutralize those small overlap points, you end up with tuning you
>don't want when the dying notes die.  Real-time schemes don't, of
>course, "know", when a new chord starts, whether the old one is about to
>die or not, which is one of the really tough challenges of real-time
>adaptive JI, as opposed to what I call leisure retuning.

Obviously, I have no idea how my method will sound.  But, keeping in mind
that it re-evaluates the type of chord sounding at every tick, I don't think
it would suffer from the problem you're thinking of.

>So... if the first chord is an augmented triad, or full diminished
>chord, how will it be tuned?  Symmetry in, asymmetry out: a tough

The chords are tuned by way of a lookup table.  There may be several 19-limit
chords that resemble a "full diminished chord" in sound (if you mean a
diminished 7th chord, 10:12:14:17 is one example).

My method tries to squeeze as many consonant relationships out of 12-tone as
possible.  Erlich's method takes the opposite approach, ignoring anything
that isn't a simple 5-limit consonance.  That seems the ideal method for
rendering classical music as it was meant to be heard.  Of course, extended
meantone already does an excellent job, but it makes sense to hide the errors
horizontally rather than vertically, since you can, and the improved fifths
do make a difference.

>It's hard for me to believe you'll be satisfied with that!  Drift
>control is challenging, but, IMHO, all but essential.

The important thing is: drift from what?  Not all music depends on the
vanishing of the syntonic comma.  Progressions are possible that would go
sharp in meantone.

The other thing is, my version doesn't completely ignore drift.  As I said, I
use 12-tet as the ideal in horizontal motion.  How?  1.) In the lookup table,
chords are penalized for their deviation from 12, and 2.) if there aren't any
common tones between adjacent chords, the 12-tet motion is taken as ideal
(see adaptive.txt).

>>Nope.  In a way, I'm actually after "strict" JI here.  I call it
>>adaptive because it's getting at JI with only 12-pitches, moving the
>>tonic is all over the place.  But I'm trying more to access high-limit
>>pitch sets than I am trying to get JI to conform to meantone logic.
>Hmmm, I don't quite follow the whole paragraph.  But listeners like Paul
>E are likely to find mismatches if your method doesn't, I fear.  Of
>course, you may or may not want to get excited about that...

Let me turn you on to Erv Wilson's pitch-space diagrams (if you're not
already hip).  Go to...


Figure 14 depicts the 11-limit Partchian tonality diamond.  The diamond is a
pitch set formed by a group of connected hexads.  In Wilson's chart each
hexad is a pentagon, and each vertex is a note (the 6th note is at the center
of the pentagon).  Otonal hexads point up, utonal hexads point down.  See
them in there?  Notice the progression of hexads around the outside of star
-- each pair of chords is connected by a common dyad.  The point is, there
are progressions which do not require temperament to work!

>As always, I say, go for it, Carl!  None of these challenges, or the
>other ones you face, are insurmountable.

Thanks for your encouragement!

RE: AW.: re: xen keyboards, Graham Breed, etc.                      1/22/00

>Interestingly, European "classical" accordion players (i.e. Teodoro
>Anzelotti and everyone in eastern Europe) prefer triangulated arrays of
>buttons to piano keyboards.  Apparently, the small size of the buttons is
>less of an obstacle than are the greater distances on the piano keyboard.
>Also, the button arrays may have some pitch redundancies to increase

There are a lot of different types of fingering systems for accordions and
their kin.  The important thing to remember is the playing is primarily
melodic.  Even in the left hand, chords can be played with only one button.
Reaching chords and weaving lines fugue-like is less of an issue as it is
with keyboards.  Although I have a recording of my grandfather playing some
wild stuff on his Bandoneon (which, depending on who you ask, is either a
type of concertina or a type of 'button box'). . .  The only comparison of
some of the different systems I'm aware of is at Hans Palm's Accordion


...and I find it telling that he picks the Janko as best.

>The Starr generalized keyboard represents a number of compromises on Erv
>Wilson's part. But I think he was willing to accept them just to finally
>have an instrument of some sort in place after forty-some years of

Oh yeah- I don't mean to knock it.  It will be one of the most powerful
keyboard instruments in the World!  And as I said, the honeycomb thing does
make things more compact, and may actually be conducive to some types of
scale-based playing, which Erv seems to have great interest in (I asked him
about some unreachable chords, and he said he didn't want to play them!).

>The Hackleman-Wilson clavichord keyboard really benefits from the tactile
>qualities of the mixed hardwood keys and the clavichord ranks are more
>steeply elevated than are those on the Starr keyboard.  I find the greater
>elevation useful for orientation, but the very presence of additional ranks
>make the differences in fingering between the hands much less intuitive
>than on a 2-d (or nearly 2-d) keyboard.

I haven't played the clavichord, but I spent a few hours with Michael
Zarkey's 19-tone harpsichord, which I understand has exactly the same
keyboard, scaled up (ahem!) a bit.

I agree that elevation helps orientation, and that 2-D makes the fingering
across ranks easier.  But cross-rank fingering is only a problem on the
elevated version because of the hexagonal keys.  The Janko/Bosanquet setup
has the best of both worlds in this respect.

Two other things I don't like about the MicroZone are...

1. The keys are too small!
2. The action travel is 0.008".  To short!

>However, we just don't know what the learning curve on such a keyboard will
>look like -- certainly the differences between fingering major scales in B
>natural (i.e. l: 4324321 r:12312341) and Bb (l:32143212 r:21231234) on the
>conventional keyboard are quite disorienting (and almost impossible if your
>technique excludes thumbs).

Like JdL says, It's WIDE OPEN!

RE: extended reference?                                             1/23/00

>With the Beethoven fragment, I was consciously *not* seeking small-number
>ratios in many cases, which upon reflection seems to contradict the
>extended-reference idea.  My method here was to tune the 'roots' of the
>chords to small-integer JI, then to tune other notes with commatic
>substitutions of small-integer JI wherever small-integer JI didn't sound
>right to me, and where I imagined Beethoven might want those commatic
>substitutions for harmonic or melodic reasons.
>So is that extended reference or not?

Extended reference is not rigorously defined, by me, Boomsliter & Creel, or
anyone else.  It's just a term for something B. & C. observed.  My definition
tries to put their observation in terms of what's been discussed here over the
last few years.  Let's walk through it.

Say we agree that JI is important in melody.  How should we apply JI in the
following example?


The classical JI school would say [subscribers to this school please forgive
the color that follows], "The diatonic scale, eh?  What intervals does that
involve?"  They write...

1:1  9:8  5:4  4:3  3:2  5:3  15:8

Then they say, "C is the tonic of the above melody!" and write...

1/1 9/8 5/4 4/3 3/2 5/3 15/8
C   D   E   F   G   A   B

...and finally...

1/1  9/8  5/4  9/8  5/4  4/3

The extended reference version is more like, "Diatonic music, eh?  What basic
set of intervals do we need?"

1:1  16:15  9:8  6:5  5:4  4:3  7:5  3:2  8:5  5:3  9:5  15:8

"Now, what's happening in our melody?  The first note is always 1/1..."


"Let's make the 2nd above C a 9:8..."

1/1 9/8
C   D

"And a cracking-good 3rd is 5:4..."

1/1 9/8 5/4
C   D   E

"And we see that the scale degree pattern 1 2 3 is now being repeated on D
[modal transposition].  So D is our new tonic, and the first note is always a
1:1 [D stays 9/8].  Let's make the 2nd above D a 9:8.  That gives..."

1/1 9/8 5/4 9/8 81/64
C   D   E   D   E

"And the third above D should be a 6:5..."

1/1 9/8 5/4 9/8 81/64 27/20
C   D   E   D   E     F

There you have it.  I've used the convention developed on this list -- /'s
for pitches, and :'s for intervals.  Notice the main difference between the
classical JI school and the extended reference school is where the modal
transposition takes place -- on the pitches vs. on the intervals.

[It's important not to hang up on the details of my example.  The "basic set"
actually chosen by the extended reference tuner, the particular just
"diatonic" scale chosen by the classical tuner, and the choice of tonics by
both parties may be entirely different than what I've given.  The methods
themselves have nothing to say about these choices.]

So now maybe you can answer the question.  I would guess that if you were
successful, the Robert Johnson transcription would be extended reference if
the performance was.  The Beethoven?

[P.S.  You can take out the "<--insert references...-->" part of my
definition.  Thanks for inserting the references!!]

Dominant chords in Barbershop?                                       2/7/00

>>I'm not familiar with Forte, or with a good definition of "functional
>>harmony".  But I know what a dominant chord is.  Barbershop music contains
>>plenty of them, often tuned 4:5:6:7.  It also contains tetrads which are
>>not dominant chords, and are tuned 4:5:6:7, and in fact this latter type
>>of chord is at least as plentiful as the former type in the typical
>>Barbershop performance.
>Really? They're not secondary dominants or deceptive dominants? Can you
>give an example?

Barbershop songs modulate primarily by fifths, just as the songs of most
genres.  So if you wanted, you could describe most of the 7th chords as
being dominants of their successors.  But the successor is also a "dominant"
7th, and so on.  Such an analysis wouldn't add much insight to the music,
though.  And there are many progressions which would not succumb to even
this perverse treatment; 4:5:6:7 tetrads are often connected by tritones,
thirds, and half-steps.

Here's a phrase from "Down Our Way", the first Barbershop tune I ever sang
in a quartet...

F7 - Cm7 - E7 - F7
"How  do  you  do?"

The E7 is a typical half-step "passing chord".  Okay, here's a tag called
"Our Last Goodbye"...

Bb7 - G7 - G7 - F#7 - C7 - Dm7 - BbM
"Will this  be  our  last  good  bye?"

Here we have a modulation by a minor third, then a local dominant-tonic
modulation is split into a half-step and a tritone.  Then we return
symmetrically -- the Dm7 subs for the G7, and back to the tonic (by a

Here's a classic tag called "Rainbows in the Sky"...

BbM -- Bb7 - F#7 - C7 - C79 - Gm6 - Am - C7 - F7 - Dm - F7 - Eb6+9 - BbM
"There will  be   rai - ain - bows  in  the  Sky---------y   some   day."

Generalized diatonic scales                                          2/8/00

I've just done a search of 14 different scales, most of them discussed at
some point here in the last year, for "generalized-diatonic" properties.

That could mean a number of things, but the main thing is: given a pattern
of scale degrees, you get _different_ consonant chords in different modes of
the scale.  It turns out that this property is already strong enough to
disqualify most of the scales I looked at, so you don't need to agree with
me on the rest of what else constitutes "generalized diatonicity" to find
this useful.

Here are all the scales that even remotely meet this criterion when the
following chords are considered consonant...

Triads -  4:5:6, 1/(4:5:6), 6:7:9, 5:6:7, 7:9:11, 9:11:15, 9:11:13
Tetrads - 4:5:6:7, 5:6:7:9, 1/(4:5:6:7), 10:12:15:17, 8:10:12:15,
          10:12:15:18, 12:14:18:21, 6:7:9:11, 8:10:11:14

[1.] Standard diatonic scale in 31tet.

  0   5   10   13   18   23   28   31
    5   5    3    5    5    5    3

Scale pattern 1-3-5 yields six 5-limit triads (3 otonal and 3 utonal).
Pattern 1-3-5-7 yields two 8:10:12:15 tetrads and three 10:12:15:18 tetrads.

[2.] Dave Keenan's 125-cent generator, single-chain scale in 29tet.

  0   3   6   9   11   14   17   20   23   26   29
    3   3   3   2    3    3    3    3    3    3

Scale pattern 1-4-7 yields six 5-limit triads (3 otonal and 3 utonal).

[3.] Easley Blackwood's decatonic scale in 15tet.

  0   2   3   5   6   8   9   11   12   14   15
    2   1   2   1   2   1   2    1    2    1

Scale pattern 1-4-7 yields ten 5-limit triads (5 otonal and 5 utonal).
Degrees 1-4-7-10 yield five 8:10:12:15 and five 10:12:15:18 tetrads.

[4.] A decatonic scale I cooked up from two pentatonic chains of 7:4's
rooted a 5:4 apart, tuned in 31tet.

   0   4   6   10   12   16   18   22   25   28   31
     4   2   4    2    4    2    4    3    3    3

Scale pattern 1-4-7 yields four 5-limit triads (2 otonal and 2 utonal).

[5.] A decatonic scale I cooked up from two pentatonic chains of 3:2's
rooted a 7:4 apart and tuned in 31tet.

   0   4   5   10   12   17   18   23   25   30   31
     4   1   5    2    5    1    5    2    5    1

Scale pattern 1-4-7 yields four 5-limit triads (2 otonal and 2 utonal).

[6.] Paul Erlich's pentachordal major decatonic in 22tet.

   0   2   4   7   9   11   13   16   18   20   22
     2   2   3   2   2    2    3    2    2    2

Scale pattern 1-4-7 yields eight 5-limit triads (4 otonal and 4 utonal).
Degrees 1-4-7-9 give three 4:5:6:7 and three 10:12:15:17 tetrads.

Scales that didn't make this list include two of Paul Hahn's nonatonic scales
in 31tet, two tunings of a decatonic MOS proposed by Dave Keenan, 8- and
11-tone versions of the chain-of-minor-thirds scale in 19tet, three tunings
of an 11-tone MOS suggested by Dave Keenan, and a scale by Heinz Bohlen based
on the 5:7:9 triad.  The complete results of the search are downloadable at:


RE: Interval names                                                   2/9/00

>I'm not sure I'm following. What counterexample do you have in mind?

I just meant that the diatonic scale is not 11-limit consistent, so no naming
scheme based on diatonic degrees will be either.

>But each column is now consistent in its own way. One is a list of note
>names, the other a list of interval names.

How are you using consistency here?  The wordy names in your table aren't
even close to consistent with 22.  For example, stacking two of your thirds
would wind one up with, variously, types of fourths, fifths, and sixths.

>I't wouldn't do to have an approximate 4:5 being called a major third in
>most tunings but called a neutral (or submajor, or narrow) third in 22-tET.

There are very good reasons for Paul's position when it comes to diatonic
naming in 22.  The 9:7 and 7:6 are the natural diatonic major and minor
thirds in this tuning, based on the way they fall in the scale.  The o- and
u-tonal 6:7:9 chords function as 1-3-5 triads exactly like the 4:5:6's
function in a meantone diatonic.

Dave, I hope you don't take me wrong on this issue.  When working with the
diatonic scale, diatonic names are appropriate.  When working with the
decatonic scale they are not.

For composers working with the diatonic scale and 11-limit chromatic harmony,
your naming scheme (or something like it) will be essential.  Inconsistency
is just something that composers interested in this setup will have to deal
with.  By the way, diatonic music with high-limit chromatic harmony is one of
the most fertile grounds in tuning theory in my opinion, and I have plans to
explore it in depth one day, when your naming scheme will very likely be
handy to me.

RE: dominant chords in Barbershop                                  2/13/00

>>The subject was not functional harmony, but dominant 7th chords. I said
>>that there are 7th chords in Barbershop that have no teleological
>>interpretation as the 1st, 3rd, 5th, and 7th degrees of a dominant key.
>First of all, go get yourself a copy of Mathieu.

If I was near a library...  No money of mine is going to these guys until
I've seen a lot more.  I've been over this ground a few times already -- as
recently as '96, with my professor at the conservatory of music at Indiana
University.  As it's been presented to me in these previous encounters, I can
only conclude that it's a lot of B.S.  Perhaps I'll state, at some point, how
I've come to this conclusion.

>Secondly, terminolgy may be stinging us again -- the term "dominant 7th" is
>commonly used merely to distinguish the M3, m3, m3 chord from "major 7th",
>"minor 7th", and other seventh chords.

Ouch.  Yes, indeed.  I usually use the term, "maj/min 7th" to speak of
4:5:6:7 chords in 12-tet.  In this thread, I was using dominant chords to
mean 1-3-5-7 chords.  It sounded like you were too, since the context was
functional harmony, etc.

>The term "dominant" is meaningless here, except to remind us where, in the
>diatonic scale, this chord normally appears.

The original statement was "diatonic in function and origin".  I admit this
statement is pretty general, but the way I was reading it, I had to disagree.

Dysfunctional harmony                                               2/13/00

[me, Carl Lumma]
>Perhaps I'll state, at some point, how I've come to this conclusion.

Roughly, I would say that the thing is not falsifiable, and not useful, since
any passage can be given a functional interpretation, and since I'm not aware
of any composer who has used the model (at least not directly) to create
music I enjoy.  I have never seen anything to suggest it reflects any deep
structure in music (as its presentation would imply).  I view it as a simple
descriptive language, which may be reasonably compact when applied to the
music it was based on, but which I find useless for examples like the ones I

Once again, this is something I would find quite reasonable if it was
presented as a descriptive effort, but something which I find quite laughable
when presented as a constructive effort -- with the cock of a chin, no less,
in 4 out of 5 cases (in my experience).

That said, I am prepared to be corrected.  In Berkeley I will once again be
near a library, and I will try to check out some Mathieu.  I will post a full
retraction here if I like it.

RE: Seeking advice on JI live rig                                   2/29/00

>Is anyone familiar with the Justonic program? Perhaps this is old news or
>perhaps it really doesn't fit this thread. I just thought I'd mention it in
>case it is helpful. The URL is 

I bought the software in 97, after I saw its ad in Keyboard magazine.  I
promptly went off to Barbershop "Harmony College", and found that one of my
profs was using the software in his theory class.  When I got back, I
developed some scales with the software, and tweaked some of my midi files
with root-change messages.  Unfortunately, using MIDI cables and a Proteus,
most of the key changes took too long to be usable (they caused artifacts).
I managed to get a few of the slower tunes working by carefully timing the
messages in the midi stream, and by using a software synth.

I corresponded with Thomas Langley about the next version of the software,
which unfortunately never appeared.  When the USB Roland boxes came out,
Thomas wrote me to say that the root changes were groovy on them.  But by
that time, I had decided that Kyma, CSound, and the built-in features of
some synths better met my needs, and that the Justonic software was lost in
Win16 land.

Final analysis: Pitch Palette was over-priced.  The chord-follower was
over-hyped, and for my purposes, useless.  To its credit, it was the only
thing available at the time which would re-tune synths from a PC, and with
the exception of some expensive Kurzweil and late model Emu units, it is
still the only way to get root changes out of the box.

Point of it all                                                      3/9/00

Some readers may wonder what Constant Structures and Propriety are good for.

They both assume that when we hear music, we attempt to assign scale degree
numbers to melodic pitches.  I won't argue about that here; if you don't
think it happens, then propriety and CS aren't for you.  But if you're
interested in getting it to happen or not, then they are for you.

Propriety and CS both measure how easy it is for the listener to
un-ambiguously assign scale degree numbers to pitches as he hears them.  CS
assumes that a listener can recognize intervals by their _specific_ size -- a
3:2 is distinct from a 7:5, and so on.  So CS asks, once you recognize the
interval, will there be any doubt as to what scale degree it is?  If you can
answer "no", your scale is CS.  In the diatonic scale in 12-tet, you can
answer no for all the intervals but the tritone -- it can be a fourth or a
fifth; you need other notes to clarify its position in the scale.

Propriety does the same thing, except it assumes that listeners recognize
intervals by their _relative_ size -- that the listener ranks the intervals
he hears by how big they are.  The actual tuning of the intervals doesn't
matter, so long as their ranks are preserved.  This means the theory can be
tested to see if people perceive a similarity between different tunings of a
scale with the same "rank-order", and if they perceive a difference between
two tunings of a scale with a different rank-order.  Rothenberg has actually
suggested some very cool experiments to test many aspects of his theory,
which are yet to be performed.  Unfortunately, enthno-musicology is simply
not a source of scientific information here.

RE: http://www.intelliscore.net/                                     5/7/00

>Interesting; have you tried out their demo?

Yes, I have (since my last message).  I wasn't surprised, and I'm sure you're
not, to learn that it doesn't really work yet.  I think I could get better
results by spending more time with it, but I also think (admittedly, without
spending that time) that I have an idea of the limits of such improvement.
For certain types of music, with a certain amount of touch-up afterwards, you
could get workable results.  But for music like mine it's just a no-go at
this point.  Ditto for any music of fast tempo, and/or intricate rhythms.
And even in ideal circumstances, I _don't_ buy the 35% saved-time figure.
Touch-up can be very time-consuming.

OTOH, I was just thinking the other day (yesterday, I think!) how far off
something even this good was.  I am shocked and a-ghast.  Now, I'd stop
before I ruling out the possibility that transcribing by ear will be obsolete
in 4-5 years.

RE: http://www.intelliscore.net/                                     5/8/00

>1. When the software guesses wrong, a single mistake often has wide-ranging
>consequences. This suggests that you want to help it along, so that when
>it gets to a place where it isn't sure, you can prevent it from making the

With intelliscore, you have to tap the tempo, and declare the time signature
before the recognition starts.

>2. There are lots of places where the sound is simply ambiguous. You
>likewise want to be able to make decisions about this as part of the first

Well, the recognition works in very nearly realtime on my single P2-400, so
you wouldn't have time on the first pass as it stands.

>One would be to synthesize a score with the extracted pitches, and create a
>second spectrogram, which you could compare by eye with the first.

The software, as it stands, creates a midi file, so it would simply amount to
feeding back the output from that into a second spectrogram.  Trouble is,
synthesized timbres probably look a lot different from real timbres on a
spectrogram at this point.  In fact, at this point, the software cannot tell
the difference between instruments -- its MIDI output is all on a single

>3. There's a lot of variation in the performance of rhythm (and in what
>actually comes out of an instrument) that you want the software to ignore;
>listening, you can often know what's *meant* better than the software
>(though it may be absolutely right about what actually happened).

As it stands, the software is completely ignorant about rhythm.  It seems
there are two general ways to proceed...

>6. In order to correct the transcription, you need to compare the
>transcription to the sound; the software ought to provide tools for doing
>What I'm imagining is something where the sound presented as a spectrogram,
>processed to highlight possible note onsets, etc. Overlaid would be a grid
>showing where pitches (in whatever tuning) would be. The software would
>guess at barlines, and you'd correct them. Likewise for beats within bars.
>Likewise for notes. Once the notes were in place, there'd be various tools
>for comparing the transcription to the original audio.

This would be a great tool.  As far as I'm concerned, it wouldn't even have
to guess the rhythms.  I can do that about as quickly as I can correct them.
The main thing would be the spectrogram display with note overlay, and an
good score entry window below that.  In this setup, the computer simply
handles the absolute and relative pitch skills, which are generally
difficult for humans.

The second way to proceed would be to try and have the computer do
everything, with no touch-up afterwards.  It was this possibility that I
said may be 4 years off.  I can imagine an algorithm, whereby several meter
choices are generated concurrently, and then fit to the sound source with a
tempo map (ala MOTU's FreeStyle).  The tempo maps are then compared, and the
meter choice corresponding to the simplest tempo map wins.


I would also be happy to see these kinds of tools, and I'm sure we won't
have too long to wait.  OTOH, I am bound to ask which is more of an advance
Proliferation of software that can transcribe music as well as humans, or
proliferation of what history and science have shown to be a very _human_
skill throughout our society?  (I said above that absolute and relative
pitch were "difficult" for humans -- what did I mean by that?).

RE: 13 tones equal tempered                                         5/13/00

>Does anyone have some suggestion on what tones go together with 13 tet?

13-tet is a very dissonant tuning, notable for its lack of reasonable 3-limit
ratios (perfect 5ths and 4ths).  If you are looking for some nice
dissonances, simply improvising in this scale will make great results (there
is no need to consciously avoid tonality as in 12-tet).  I gather that you
are looking for consonant relationships, however.

1 = 19/18
2 = 10/9
3 = 7/6
4 = 5/4
5 = wolf 4/3
6 = 11/8
7 = wolf 3/2
8 = wolf 3/2
9 = 8/5 or 13/8
10 = 12/7
11 = 9/5
12 = 15/8
13 = 2/1

Above is one possible just interpretation of 13-tet.  Of these intervals, the
most consonant are probably the 3-step minor third, followed by the 4-step
major third, and the 11-step minor seventh.

As for triads... the best to my ear is 4:5:9 (13-tet degrees 0-4-15).  You
can form a quasi 10:12:15 with degrees 0-3-7 (same pattern as a minor triad
in 12-tet!), and a quasi 12:14:21 with degrees 0-3-11.  All this depends on
the timbre used, naturally.  You'll have to experiment.

One approach would be to really exploit "voicings".  In a tuning this
dissonant, you need to use your octaves!  Don't be afraid to consider 6:7:12
a triad, to consider 6:12:14 a different triad, etc.

>Last week the topic of 13 tones equal tempered scales had three scales
>suggested. I tried them and I like them 3121222 3131212 2141212.

You might also like to try my 10-tone scale 2112112111.  Get a hold of
Xenharmonikon 13, too, if you can.  It has two good articles on 13-tet
harmonic-melodic practice.  Chalmers suggests an 8-tone scale 22122121...

RE: Re: The CPS blues...[combination product sets]                  6/10/00

>>So, Daniel, stellation is only defined for an m-out-of-n CPS if n = m*2,
>No -- sorry, I'm wrong about this.  On page 3 of the "Letter to John
>Chalmers", Wilson indicates five possible solutions for a "stellated
>dekateserany".  (All five are present in the stellated Eikosany.)  It
>appears that the n=2m CPSs will each have single stellations while the
>non-2=2m CPSs  will have m possible stellations.

Wow- I'd like to see that.  There should be exactly one complete stellation
for every CPS, regardless the proportion of m and n.

How many tones would they have?  That's an interesting question.  I don't
suspect the answer would be in Pascal's triangle.  For CPS's with 2:1 m:n, I
think the answer is as simple as t + s(c), where t is the number of tones in
the un-stellated structure, c is the number of un-saturated chords in the
CPS, and s is the number of identities needed to saturate the CPS's basic
chord.  The 14 tones of the stellated hexany make sense here; each of the
hexany's 8 triads needs 1 extra note.  The something is different with the 70
tones of the stellated eikosany, tho; each of its 30 tetrads needs 2 extra
notes, giving 80, but according to Chalmers the stellated 1*3*5*7*9*11 eik.
has 70 notes.  So either 10 notes are redundant due to the composite
9-factor, or 10 points are getting re-used somewhere and my formula is
incorrect (which is unlikely, since the chords of the CPS will already
exploit the maximum number of common-tone relations in prime-factor JI: 2).

For non-2:1-m:n CPS's, the answer is more difficult, since s is not a

RE: Re: The CPS blues...[combination product sets]                  6/12/00

John- I think your 70-note structure is correct, since Scala "shows" the
"locations" of the correct number of chords.

>Anyway, as simple as that formula is, I think you can simplify it further.

I think the formula is elementary, but not simple, since it has three
variables.  So far, it is nothing more than a way to explain how to do the
brute force method that John Chalmers mentions.  I tried when I made the post
to simplify it, but was too sleepy.

>>(which is unlikely, since the chords of the CPS will already exploit the
>>maximum number of common-tone relations in prime-factor JI: 2).
>What's that last lemma, Lumma?

In a factor space built from a list of relatively-prime factors, no two
chords may share more than two points in common.

>>For non-2:1-m:n CPS's, the answer is more difficult, since s is not a
>I still am waiting for Daniel Wolf to reply to my post on that subject so
>that we can have a better sense of what these look like.

Ich auch.  I'm not seeing the existence of multiple complete stellations.
Perhaps Erv was referring to possible incomplete stellations.

>triad out of the minor tetrad. Then come back to the Eikosany. If you want
>to "cheat", look at the factors that make up the notes of the Eikosany (3
>factors for each note) -- each consonant otonal tetrad will have two factors
>in common to all notes (e.g., A*B*C, A*B*D, A*B*E, A*B*F), and each utonal
>tetrad will have two factors absent from all notes (e.g., A*B*C, A*B*D,
>A*C*D, B*C*D).
>Carl -- does that last hint for Joseph help you simplify your formula
>[t + s(c)] any further?

Hrm...  We already know how to find t; my first thought was to find a way to
express c and/or s as something we know from t.  But I don't see it from your
clue yet.

Since I first learned about CPSs, I've wanted a way to find the number of
notes in the basic chord of the CPS (which, in the case of the highly-
symmetrical CPSs, gives s when subtracted from the number of factors, n).

Here's a try for c:

c = tn/(n-s)

8 = 6*4/(4-1)
30 = 20*6/(6-2)
112 = 70*8/(8-3)

...which shows that the number of chords grows more quickly with respect to
the number of factors than with the tonality diamond.

RE: RE: Re: The CPS blues...[combination product sets]              6/12/00

Okay, I'm ready to publish.  Define the following variables:

M: combination count
N: number of factors

Tr: number of tones in the raw CPS, N!/M!(N-M)!
Co: cardinality of the basic otonal chord, 1+(N-M)
Cu: cardinality of the basic utonal chord, 1+M

So: number of tones needed to saturate the basic otonal chord, N-Co
Su: number of tones needed to saturate the basic utonal chord, N-Cu

No: number of basic otonal chords in the CPS, N!/So!(N-So)!
Nu: number of basic utonal chords in the CPS, N!/Su!(N-Su)!
                                                  [Thanks Paul!!]

Ts: number of tones in the stellated CPS, Tr + No(So) + Nu(Su)

Backing out and simplifying:

        N!            N!               N!
Ts=  -------  + -------------- + --------------
     M!(N-M)!   (M-2)!(1+N-M)!   (N-M-2)!(1+M)!

[If the factors were not mutually prime, remove doubled pitches here.]

CPS stuff (was RE: The CPS Blues)                                   6/15/00

[Manuel wrote...]
>If I apply this to the 1)4 tetrany, the outcome is 4+12=16. This is when all
>6 dyads are completed to tetrads.

To utonal tetrads, right.  All the basic otonal chords (a total of 1) are
already complete.

>But I think another way to stellate it is to complete the 4 triads to

There are octahedra in the way.

>Then the number of tones is 4+4=8.

You're counting tones already present in the un-stellated CPS.

>>   N!             N!              N!
>> -------  + -------------- + ------------
>> M!(N-M)!   (M-2)!(1+N-M)!   (N-M-2)!(1+M)!
>The first negative factorial results from dividing by (M-1), which is
>dividing by zero if M=1, so you mustn't do that.

Ah yes.  I found the correct common denominator, but forgot how to distribute
for cubes, and it was very late, so I published the above.

>              3       3
>   N!       (M + (N-M) - N) N!
> -------  + ------------------
> M!(N-M)!     (M+1)!(N-M+1)!

Thanks!  Looks like we have a winner!

RE: Stellation continued                                            6/18/00

[John Chalmers wrote...]
>However, six new incomplete harmonic hexads are created by the addition of
>the first 30 tones above.

Right.  Perhaps we should call the 96-tone structure some sort of "2nd-
iteration stellation". :)  The 1st stellation might be of interest to
somebody, no?

>These are formed by the notes E*E*F and F*F*E for all selections of 2
>factors from the 6 of the generator.

Remembering that the x)6 CPSs mate to form the 6-factor Euler-Fokker genus,
it's no surprise that we run up against Pentadekanies when stellating the
Eikosany.  As above, stellating these adjacent Pentadekanies doesn't really
strike me as the thing to do.

I believe your 96-tone structure is actually a stellated E.F. genus, and I
wonder if your 570-tone is?  The 2)4 is adjacent to the basic, completed 4-
factor chords, and thus requires only a single iteration of stellation to
become a stellated E.F. genus.  The 3)6 must cross a pair of neighboring
m-1)n, m+1)n CPSs to reach the basic hexads, and so it takes two iterations
to become a stellated EFG.  The 4)8 will have more than one pair of
neighbors.  How many "iterations" does your formula perform?

Recent request, 4+ test                                             6/21/00

>When I was much younger (1974 - 1977) I used to go to the Computer Music
>Conferences. I recall meeting one composer, I think his name was Bruce
>Hemmingway, who told me that his goal, and one that he thought all composers
>should share, was to create pieces that would hold a listener's interest for
>more than four minutes. He admitted that he had never done it, and didn't
>think any contemporary composer in the electronic/computer realm had either.
>This "four-minute rule" is one I subscribe to, and I am reminded of some of
>the rather lengthy symphonic pieces that hold my interest, for example, the
>Shostakovich 11th symphony, which, plus or minus some conductors' tempo
>digressions, runs about 65 minutes. So, can someone recommend a piece over
>four minutes that is xenharmonic or xentonal (or both) that holds *your*
>interest? Surely in the years since 1977, someone has written something.

My number-one recommendation is going to be Marcus Hobbs' _From on High_.  If
you like electronic music, you will love it.  I promise.  It may even get you
into groove.


From the category of more 'involved' music, David Doty has no fewer than six
cuts on his album _Uncommon Practice_ which are over four minutes.


Does it have to be electronic?  Paul E. already mentioned Partch and the
Catler Bros. (both of which I whole-heartedly second).  I believe Haverstick
has at least one 4+ cut on _Acoustic Stick_.

How fast do you need to be held?  Michael Harrison and Kraig Grady both have
astonishing long works in JI (see _Ancient Worlds_ and _The Creation of the
Worlds_ respectively).

(E-mail Kraig off-list for info.)

Also in the 'minimal' category, but back to the electronic realm (and with
video!), Bill Alves' recent post should be investigated.

http://www2.hmc.edu/~alves/hiway70.html (one of three!)

Lastly, for instant gratification, and available only in mp3, try Prent
Rodgers' stuff (speaking into the air, Rock10, and Mino9 are all over 4

Hard to believe, but it's all worth a listen.  Bet you didn't bargain for
_that_! :)  Incidentally, I love Shosty 11!

RE: electric Harry                                                   7/4/00

Daniel Wolf wrote...
>In my last posting, I indicated some of the musical problems involved in
>finding a synthesized surrogate for a Chromelodeon but thus far no one has
>suggested any ways to overcome these.

You're right that many stock MIDI controllers (like the DX-7, but the 64+
note poly of most late-model boards should be enough) and wavetable patches
aren't up to the Chromelodeon -- they're not up to most instruments!  But
unlike many instruments, there's nothing particularly hard to synthesize in
the Chromelodeon.  It may be that nobody's ever taken the time...

David Beardsley...
>Maybe a job for physical modeling? - it's worth a try.

The timbre of the instrument succumbs quite readily to additive.  To satisfy
Wolf, and perhaps simulate varying air pressure, it's a matter of mapping
some MIDI control vectors to some envelope and filter stuff, and having
enough polyphony for the job.

As to the general question of the substitution, we may ask which, if not
both, would be objectionable to Partch 1. the substitution of a stock synth
for economic reasons, and 2. substitution of a custom synth setup for
whatever reason.  Most learned colleagues, I humbly suggest that the person
who debates such questions has mis-understood Partch on a very fundamental
level.  If Harry's work lives only as a part of a concert cycle, it's a great
loss.  Partch's work makes for us a much greater challenge -- one we may not
choose to accept, one we may not even hear.  There's no shame either way.
Partch asks nothing of us, but he does challenge us.  And if we claim to
accept the challenge, then it means making art from our souls -- "just like
early man" -- and making life an art in the same spirit.  It's rather
revolutionary, really.  It means new work.  All the time.  And if somebody
performs Partch with a DX-7, let somebody else perform it with a
Chromelodeon.  If somebody transcribes Partch for string quartet, let
somebody else transcribe a string quartet for Partch ensemble (but don't
attend either of those concerts!).

"The world is round." - Gertrude Stein
"The world is wrong." - Harry Partch

RE: basic CPS, stellation, hexany, etc. etc.                         7/5/00

[Monz wrote...]
>The Dictionary is currently undergoing a heavy expansion.  I've asked in the
>past on this List if others could help me with more detailed definitions for
>the Wilson terms, but have only gotten a few enlightening responses.  The
>offer still stands.  (Paul?  Carl?  Kraig?  Manuel?  Daniel Wolf?)

Chris and Joe,

We can take the pitches in a tuning, represent them as dots, and arrange them
in a space so that where any two pitches are separated by a consonant
interval, that interval can be shown as a straight line connecting the two
dots (pitches) involved.

We can define any set of intervals we want as being consonant, and we can
always arrange the dots of any tuning so that _all_ our consonances are shown
as straight lines, with none of the lines crossing, if our space has enough
dimensions.  Sound like Dr. Who yet?  )

Anyway, these spaces are called tonespaces -- usually, we consider the
intervals of just intonation (at some limit) consonant, but really we can use
any intervals we want.

Let's say we use JI.  Consonant intervals are written as fractions in JI.  In
order to keep our lines from crossing, we need one dimension for every number
appearing in a numerator or denominator in our list of consonances.  Usually,
we abide by octave-equivalence, where factors of 2 are ignored, so that a dot
represents all octave copies of its pitch.

One of the simplest and most common tonespaces is the two-factor space of the
5-limit.  Our list of consonances is (1:1, 5:4, 8:5, 3:2, 4:3, 6:5, 5:3).
Ignoring the 2's, we only need the numbers 3 and 5 to write all these
fractions.  So we need two dimensions, x and y.  Adding a 3 to a numerator
will move us + on x, adding a 3 to a denominator will be a - move on x.  Same
for the 5's and y.  So:

    / \     / \    /
   /   \   /   \  /
  /     \ /     \/

Every line on this diagram is in our list of consonances, and no lines cross.
The dots are named with a fraction denoting the _pitch_ in the _tuning_
that's being shown.  In the above case, the tuning is [1/1, 9/8, 5/4, 45/32,
3/2, 15/8].  I wrote those fractions with /'s, to show that they are
_pitches_.  Notice that I wrote the fractions in our list of consonant
intervals with :'s to show they are _intervals_.  That's a convention
developed on this list, to keep intervals straight from pitches.  For
example, starting at 3/2, we move a 5:4 to 15/8.  See?

Now say we wanted to add 15:8 to our list of consonances.  We'd add a
dimension for 15, and then a step from 1/1 to 15/8 would be a single line on
our 15-axis, rather than two lines (one on the 3 and one on the 5).  See?

Now look at this Wilson diagram: http://www.anaphoria.com/dal12.html

It depicts two structures in 6-factor tonespace (1 3 5 7 9 11).  There appear
to be lines crossing, but the lines only cross in 2-D (on the paper).  The
thing on the paper is only a shadow of a 6-dimensional object.  Cool, huh?

Anywho, the beat is that if we desire tunings with a good number of
consonances for a good economy of tones, they look like connected structures
in tonespace.  One type of very highly-connected structure is the tonality
diamond (see Partch -- the figure on the right of the above diagram is the
11-limit diamond).  Another type is the Combination Product Set, or CPS.

To make a CPS tuning, start with your list of consonances, and take, say, two
of them at a time, follow them out on the lattice, and mark a dot where you
stop.  For example, taking our 11-limit consonance list (1 3 5 7 9 11) two
at-a-time, we get...

(1 3)   (3 5)   (5 7)  (7 9)  (9 11)
(1 5)   (3 7)   (5 9)  (7 11)
(1 7)   (3 9)   (5 11)
(1 9)   (3 11)
(1 11)

Order doesn't matter, since moving a 5:4 and then a 3:2 away from 1/1 gets us
to 15/8 the same as moving a 3:2 and then a 5:4.  You can see that all this
moving amounts to multiplying the numbers.  So our pitches are...

3/2   15/8    35/32  63/32   99/64
5/4   21/16   45/32  77/64
7/4   27/16   55/32
9/8   33/32

Notice there's no 1/1!  All those pitches are measured from 1/1, but it's not
in the tuning itself.  That's one cool thing about CPSs -- they have no
central tonality (unlike tonality diamonds).

Note also that while the numbers in these fractions are rather large, the
tuning has a lot of low-numbered consonant relationships.

Lastly, note that we could have taken the factors three at a time (that would
give the figure on the left of the Wilson diagram above), four at a time,
etc.  One at a time gives a tuning the same as the consonant list; six at a
time gives only one pitch.

You're ready to explore CPSs!

RE: limits and ??                                                    7/6/00

>my confusion seems to come from not making a sharp difference between
>factors and roots.  i knew that a ratio like 10/8 was 5-limit because it can
>be factored down to 5/4, but i thought a ratio like 10/9 was also 5-limit
>because 9 can be square-rooted to 3.

No sweat! -- This subject has surfaced before.  If you read my recent post on
CPSs, I touch on the difference between JI tunings (whose pitches are written
as fractions) and JI harmony (where all the intervals in a sonority appear in
a list of consonances).  The terms "just intonation" and "limit" have been
confusingly applied to both these things.

For purposes of discussing consonant intervals, many listers have decided to
write the ratios involved with colons (:), distinguishing _intervals_ from
_pitches in a tuning_ (which they write with slashes (/), following Ellis,
Partch, the American Gamelan folks, et all).  So the JI diatonic scale may be

1/1  9/8  5/4  4/3  3/2  5/3  15/8
C    D    E    F    G    A     B

...and the interval between D and E may be written as 10:9.

For tunings, the "limit" used is often a "prime limit", when the diatonic
scale above is considered 5-limit, since 5 is the largest prime factor needed
to write all the fractions involved.

For harmony, the "limit" used is often "odd", since consonance seems to
change, roughly, with the largest odd factor appearing in any interval in the
chord sounding.  There are good psychoacoustic and practical reasons for
thinking in terms of an odd-limit for harmony; Partch did so.  In we play all
the notes in the above scale at once, the odd limit would be 45, since
there's a 45:32 between F and B.

Prime limit may have come into use for tunings because many tunings are
designed to provide several copies of simple chords (like the diatonic scale
contains triads), and often if you use a prime limit, the factorization will
reveal the harmonic limit of the simple chords which have been transposed
throughout the scale (the triads in the diatonic scale are 5-odd-limit).

More examples:

1/1 and 15/8
prime= 5
odd= 15

1/1 and 5/4 and 15/8
prime= 5
odd= 15

1/1 and 8/5 and 16/15
prime= 5
odd= 15

1/1 and 11/8 and 15/8
prime= 11
odd= 15

As you can see from these examples, either "limit" can be somewhat limiting
for harmony, since a "limit" assumes you'll be using every factor up to the
limit before you increase the limit.  But you may not always want to do this,
and when you don't, it's best to list the factors you consider consonant.  In
the last example, they would be (1, 11, 15) -- 15-limit and 11-limit are both
deceptive here because 7, 5, and 3 are missing.

>so, if limit isn't the right word to refer to these "dimensions", what is?
>is there a term for the infinite 1-dimensional lattice made with powers of
>3?  the 2-dimensional lattice with powers of 3 and 5?  etc etc...

On a lattice, each dimension is usually mapped to a single factor.  If you
include a dimension for every prime factor up to x, then I'd call it a x-
prime-limit lattice.  Same goes for odd-limit lattices, but they do have a
quirk where composite odd numbers will appear in more than once place -- 9
will be one step on the 9-axis, or two steps on the 3 axis, for example.
Normally, this doesn't cause a problem, since our ordinary 3-D space can only
accommodate the 7-limit (which is the same wether odd or prime, since the
first composite odd number is 9).  But on some of Erv Wilson's diagrams, you
can see pitches appearing in multiple places.

New work on Rothenberg's model                                       7/7/00

This paper assumes familiarity with Rothenberg's model of melodic perception.
For those of you who aren't familiar with the model, try the tuning
dictionary, or go to...


...and put message number "4044" in the "jump to" box.  If anything's still
unclear, post your question to the list.


First, let's recall the following exchange, which took place here in March
of this year...

[Dan Stearns wrote...]
>OK, this makes sense to me as well (and I was indeed thinking along these
>lines in the three 3L & 4s examples I gave as well), but what if the
>intervals in question are something like a 17/12 and a 24/17 though?
>Strictly proper but oh so close to not being so... is their some agreed upon
>practical range to draw the line with scales that might recognize these tiny
>commatic differences that would perhaps only trivially distinguish them from
>being strictly proper or just proper?

[Paul Erlich wrote...]
>Dan, much as with consistency, propriety does involve a very sharp boundary,
>and crossing that boundary by a very small amount, while possibly having no
>audible effect, can amount to a change from a strictly proper scale to a
>proper one to an improper one. I was once bothered by the fact that the
>Pythagorean diatonic scale was improper, but it turns out that Rothenberg
>believed that as long as the worst impropriety (in this case, a Pythagorean
>comma) was to small to be perceived (since you're comparing B-f with f-b in
>this case, it would be hard to perceive the difference), the scale is
>essentially perceived as proper. Carl, am I representing Rothenberg
>correctly here?

...I'd like now to clarify the response I gave on March 9th, in a message
titled "barely improper" (egroups message #9175).

First off, yes, Paul, I believe you were representing Rothenberg correctly.
But I think we can go one better than old R here.  With the "show data"
command in Scala 1.6, we see "Lumma stability", followed by "impropriety
factor".  What are they?

Imagine a log-frequency ruler whose total length is the interval of
equivalence ("formal octave") of our periodic scale.  Take all of the unique
intervals in the scale's interval matrix and mark them off on the ruler.
Now, draw line segments on the ruler with colored pencil, using the marks as
endpoints.  Connect all marks belonging to the same scale degree with a
single line, using a different color for each scale degree.  "Lumma
stability" is the portion of the ruler that has no pencil on it -- the
portion of the octave that is not covered by scale degrees.  "Impropriety
factor" is the portion of the ruler that is more than singly covered -- the
part where different colors overlap.  The idea being that when two scale
degrees overlap, the listener will not be able to distinguish them in all
cases -- that's a loss of propriety.  Lumma stability measures how well-
distinguished the non-overlapping degrees are.

In my earlier post, I correctly characterize my measure as being a version
of Rothenberg's, applied to logarithmic pitch space rather than 'scale
degree rank space'.  I say that both versions are useful, begging the
questions "Which is more useful?", "When is one better than the other?", and
so on.

To answer, this paper will contribute "rank standard deviation/range".  This
measure will tell us, for a given scale, how effective "equivalence" is, and
thus how pertinent any of the measures which are "invariants of equivalence"
will be to the given scale.

What is "equivalence"?  It is the process which converts an interval matrix
into a rank-order matrix.  Which measures are "invariants of equivalence"?
Rothenberg-stability, Efficiency, Constant Structures, are all invariant
under equivalence, since they can be defined on the rank-order matrix alone.
But Lumma-stability is _not_, since scales which share the same rank-order
matrix may have different Lumma-stability values.  Therefore, "rank standard
deviation" and range will tell us when my version is better, along with a few
other things...

Rank standard deviation and rank range are designed to measure the complexity
of a scale's interval matrix.  The higher these values, the more difficult it
will be for the listener to construct the matrix, and the more likely it will
be that he uses a matrix he has already learned instead (thus hearing the
scale in question as a re- or mis-tuning of a scale he already knows).  The
measures are easy to calculate.  Again, we need our log-frequency ruler, and
again, we'll take all of the unique intervals in our scale's interval matrix
and mark them off on the ruler.  But this time, instead of involving scale
degrees, we simply find the ratios between all pairs of consecutive values
marked on the ruler.  The measures are then the standard deviation and the
range of the decimal values of these ratios.

For example, take the major 7th chord in 12-tet.  Its interval matrix:

0 4 7 11 12
0 3 7  8 12
0 4 5  9 12
0 1 5  8 12

Unique intervals: 0 1 3 4 5 7 8 9 11 12

Distances between consecutive marks: 1 2 1 1 2 1 1 2 1

Range: 1
(This should be expressed as a fraction of the formal octave, so: 1/12.)

Standard deviation: Well, you get the idea (any measure of central tendency
should work here).

The idea is that since the interval matrix works by ranking intervals by
size, the easiest matrices to use will be ones in which the sizes are most
evenly distributed.  Since for periodic scales the intervals between the
intervals (ratios between consecutive marks on the ruler) must sum to the
formal octave, the mean ratio occurs between every pair of marks in the ideal
case, and the standard deviation will be in direct proportion to the
complexity.  Range should work in a similar way, but may differ by a scaling
factor from the standard deviation?  I'm not sure which is more intuitive.
Perhaps someone with some knowledge of statistics can help me out here.

So there you have it.  I should go over the Rothenberg papers again, to
refresh myself on what he says about the tolerance of perception.  My feeling
was, that since he was working with ETs, and especially since he used a
relatively small ET (12-tET) for testing his measures, he didn't run into
many close calls.  From speaking with him, I got the impression that he
considers equivalence very strong -- scales are gestalts which hold their
character through a great deal of mistuning.  From his point of view, any
instability is undesirable, and scales would be ruled out long before close
calls were an issue.  In other words, Rothenberg is really only interested in
scales for which my rank deviation is very small.

But I, for one, don't frown so much on instability, especially if it isn't
bad in pitch space, and we may arrive at close calls for harmonic reasons.
Rothenberg was interested only in melodic scales -- many of us are interested
in harmony, too, and don't have the luxury of re-tuning scales until they fit
neatly into a rank order matrix.

RE: Rothenberg question                                             7/23/00

Hello, Jason!

>I'm using David Rothenberg's pitch perception calculus in a thesis I'm
>working on and am having a little difficulty with exactly what property of
>scales the efficiency quantity is meant to capture.

I'm a casual observer of Rothenberg's material.  I'll try my best to
communicate my view of things.  I'm doing this from memory, but at the very
least it should be something to chew on. 

>He introduces the idea of efficiency by saying that an efficient musical
>language is one which can form the greatest number of distinct words
>(ordered sequences of tones) given the number of notes in the scale.

Efficiency is defined as the average length of a scale's sufficient sets (in
terms of the length of the entire scale).  The greater this value, the
greater the number of distinct words which contribute _key information_ to
the listener.

>The way I interpret this comment is that, for instance, two augmented
>triads, C E G# and D F# A# (both 444), are not distinct words.

?  Are not distinct words of what scale?  Perhaps you meant "sufficient sets"
instead of "distinct words"?

>The calculation succeeds at what it is clearly meant to do distinguish
>asymmetrical proper scales with high stability such as the diatonic scale,
>from symmetrical scales with higher efficiency (although much less common in
>practice) such as the whole tone and octatonic scales.

One could explain efficiency as a measure of this type of symmetry.  Overtly,
efficiency is a measure of how much information the listener needs to
determine what key (and thus, what mode) of the scale a melody is in.  For
scales where this symmetry is low, the listener may find his place after only
a few scale members are heard.  As this symmetry goes up, more and more
motivic material is required for the listener to locate the frequencies he
hears on the rank-order matrix.  Rothenberg implies that some of the fun of
melody is playing around in this area -- melodies may make puns as to which
key they belong, and so forth.

>This scale contains indistinct 4 note subsets E F# G# c and G# A# c E
>(2244), indistinct 3 note subsets C E A#, E F# A# and F# G# c (246), subsets
>E F# c, G# A# e, and A# c f#, (264) and so on.  But it is because of this
>property of the scale that it recieves a high efficiency rating (it always
>takes 5 notes to distinguish the scale from one of its transpositions),
>relative to the anhemitonic pentonic, which contains more distinct 2, 3 and
>4 note subsets.

I assume you meant "insufficient" instead of "indistinct".

>I must be interpreting "distinct words" incorrectly, in which case the
>question becomes, how should we think of the distinctness of pitch sequences
>as Rothenberg defines it, and what is the value of that way of looking at

Substituting "sufficient" for "distinct", you're pretty much on target.

Importantly, says R., proper scales can participate both in this key-guessing
fun, which results from having lots of insufficient words, and in scale-
degree tracking fun.  But improper scales can only participate in
scale-degree tracking fun, and so will be most desirable when efficiency is
_low_ (so scale-degree tracking can begin as soon as possible).  Proper
scales will be best when efficiency is high, but still functional when it is

I think it's important to keep in mind that with Rothenberg's model, we
address everything in terms of the _entire scale_.  We're used to this kind
of thinking, with 5- and 7-tone scales found the world over, where all the
tones are 'on hand', and the entire scale functions as a unit.  But improper
scales have proper subsets!  I believe that worthwhile composition can also
be done on large, microchromatic scales, where melodies are played locally,
and the sense of a global source set is non-existent.  In fact I believe this
has been done with 12-tET in this century.  Rothenberg's model still applies,
to local events, but the usefulness of the model at this scale (ahem!) is
questionable, IMO.

Let me know if you can make sense out of any of this.

RE: Digest Number 716                                               7/25/00

[Jason Yust]
>>Balzano, in his 1980 paper in Computer Music Journal, showed that the
>>smallest proper set of 12tET to contain all twelve intervals is the
>>diatonic scale.  In fact, I think this is true of any series-of-fifths
>>gerenated scale tuned in the next highest ET in the series 3, 5, 7, 12, 19,

[Christopher Bailey]
>Actually, the smallest subset of 12TET containing all 12 intervals IS the
>C-Db-E-F#, or C-Db-Eb-G set and their inversions.

But they are not proper subsets.

>I think what Balzano was pointing out was a even weirder property of the
>diatonic scale.  If you look at the intervals between: each note of the
>scale, and all of the other notes in the scale; then, you will see that
>each kind of interval appears a unique number of times.

I don't remember him pointing this out, but I wouldn't put it past him --
what possible significance could this property have?

>This helps "efficiency" I guess, as if you get one semitone (say, B-C)
>and then part of the other semitone (F#) then you know what transposition
>of diatonicity you're in (BCDEF#GA).

Sorry, I don't follow.  B-C-F# may be a sufficient set of the diatonic
scale, but I don't see how the unique-number-of-times property helps
lower efficiency.  And since Balzano was interested in duplicating every
property of the diatonic scale he happened to notice, he would want high
efficiency, anyway.

RE: Food for thought                                                7/28/00

>MOS - Rothenberg - Balzano property:
>Has anyone studied this from an information theory perspective?

Rothenberg has; he makes cryptic remarks about it in his papers, including
one spot where he criticizes previous efforts to apply information theory to

"Much confusion has resulted from the application of standard statistical
'redundancy' measures to musical 'messages' without considering the
limitations introduced by the efficiency of the code being used."

>The more unique intervals there are in a scale, the more "information" in
>would contain.

One important feature of the Rothenberg model is that it postulates that
listeners construct reference frames to extract information from music.  So,
we have three places to look: the music, the scale, and the reference frame.
Then, we must ask what we want.  Do we want to maximize the amount of
information in any one of these?  Random tone generation has far more
information than normal music, but most listeners would not prefer it.

RE: wilson mos / rothenberg                                         7/31/00

>>>In fact, I think this is true of any series-of-fifths-gerenated scale
>>>tuned in the next highest ET in the series 3, 5, 7, 12, 19, 31.
>>Eh?  The diatonic scale doesn't contain all of the intervals of 19-tET.
>I could be completely wrong, but I'm thinking of Yasser's series of
>5th-generated scales, where the pentatonic is tuned in 7-tET,  the diatonic
>scale is tuned to 12-tET, the chromatic scale of 12 notes is tuned in
>19-tET, and so on.

Ah- I thought you meant only the diatonic scale.  You meant the 'Yasser
generalized diatonic scale' of the ET in question.

>but always proper but ambiguous in one degree (and it's inversion) when
>tuned in an ET one place higher: as in 11212.  Also, in later case, I
>observed that the scale always exhausts the interval set of the tuning
>system.  I'm not positive, however, whether it's always the proper subset
>of lowest cardnality in the system to do so, but that's the claim I was

I don't think so -- in 19-tET, a 10-tone MOS of generator 17...

0 1 3 5 7 9 11 13 15 17 19

...is proper and exhausts the intervals of the temperament.  Also the 11-tone
MOS of generator 12...

0 1 3 5 6 8 10 12 13 15 17 19

...Both are smaller than the 12-note Yasser chain-of-fifths scale in 19-tET.

>Any generator, as you say, excluding intervals that divide the 8ve by an
>interger in log freq, will produce an infinite number of proper scales.  But
>this disregards the number of tones in these scales.  I'm interested in the
>generator which produces proper scales the most frequently, that is,
>produces x number of proper scales with the fewest number of iterations.

Perhaps the range of a "fifth" is better than most here, but many other
generators do produce proper scales with reasonable number of tones.  And the
fact that the fifth has been a historically popular generator isn't really
explained its utility here, since a given culture usually uses only a single
scale.  Western music is overwhelmingly diatonic, for example.  Why not use
one of the proper MOSs of the minor third?  And, as you pointed out,
Rothenberg's model does not imply that improper scales are "bad".

I think we must attribute the popularity of the fifth to its strong position
in the harmonic series, one way or another.  Paul Erlich's idea of
tetrachordality as a sort of 2nd-order octave equivalence is one good option.

>You also responded to my point that this analysis might explain the use of
>5ths in the generation of scales in many musical cultures, saying that this
>fact is better explained by the concordance of the 3/2.  The stumbling
>block of this latter reasoning (this is not a point of my own invention) is
>that such scales exist in cultures where harmony doesn't exist.

Octaves exist in such cultures as well.

>I would agree: I have found that difference tones (a relatively weak
>phenomenon of harmony) are clearly perceptable in certain melodic
>situations, rapid sequences of high pitched tones.

Yes, but I don't think we need difference tones to explain the 3:2 in melody.

>In any case, a stronger point is that many musical systems using fifth-
>generated scales employ non-harmonic and rapid decay toned idiophones such
>as xylophone-type instruments, where the consonance of the 3/2 is

I'm not so sure.  And in fact, my position would be supported if one could
find evidence that cultures using such instruments subject their fifths to
greater mis-tuning than cultures using instruments with more harmonic

RE: Replies to Banaphshu                                            8/19/00

>>>Absolutely -- the immediate reaction of most Western musicians, after 150
>>>years of 12-tET hegemony, to triads with just thirds is usually that they
>>>sound "dead" or "lifeless".
>>Paul- what is your source on this?
>Lots of reading and anecdotal experience (testing other musicians and
>myself). I'm specifically thinking keyboard instruments using timbres with
>harmonic partials here.

I have seen contradictory findings here.  I've seen studies which conclude
that 'most musicians find just thirds dead, at first', and studies which
conclude that 'most prefer them immediately, and express disbelief that the
"tempered" third is correct'.  I dismiss such studies as unmeaningful -- I
believe the context of the hearing is far stronger than any innate effect, if
there is one.

I myself, exposed to a great variety of music, the piano and trumpet, from an
early age, can remember having both kinds of reactions, at various times,
before becoming fully conscious of the difference between just and tempered

RE: locally concordant tetrads                                      8/28/00

Keenan Pepper wrote...

>This would tend to judge the entropy of the chord as a whole, not a
>particular voicing of the chord. This seemed to ba a problem in the local
>minima of the tetrad graph; some chords didn't make it or got bumped down
>simply beacause they didn't have a voicing open enough. Octave equivalence
>just makes sence.

I disagree.  For dyads octave equivalence can be handy, maybe even roughly
insightful up to the 9-limit.  For triads and higher-ads, it gets messy very
quickly.  Observe how much "voicing" can contribute to a characteristic
sound, even for 5-limit triadic music (let alone jazz!).  Get into higher-
limit JI, and many chords will run into different harmonic-series
representations as you invert them.  Take the 7-limit utonal tetrad.  In
"1st" (Partchian) inversion, (1/1 7/6 7/5 7/4), it doesn't really approximate
any harmonic chord that I know of, but the intervals are all consonant, and
we have a characteristic minor sound.  But in 2nd inversion, the chord
clearly (IMO) approximates a 10:12:15:17 tetrad.

Crunchy chords                                                      8/28/00

Keenan Pepper wrote...

>A good example is the 3-7 square chord, 1/1:21/16:3/2:7/4.  Every interval
>is 7-odd-limit consonant except the 21/16, which is so close to 4/3 as to be
>quite dissonant, giving this chord the melancholy sound common to all
>crunchy chords.

I like this chord quite a bit.  I used to play it on instruments tuned to the
12-of-harmonics-16-through-32 scale (after Denny Genovese and Wendy Carlos),
treating it as a suspended chord.

>1/1:9/8:7/6:3/2:7/4 is also very crunchy because every interval is
>9-odd-limit consonant except the 28/27, a very narrow interval of 63 cents
>or a thirdtone. This chord is recognizable in 12-eq as min7(9), but just
>doesn't have the same bite.

Well said.  This sonority can be heard in various places on Michael
Harrison's excellent _From Ancient Worlds_.

Another example is 1/1 6/5 3/2 7/4 (whose only dissonance is the 35:24).  It
can make a nice microtonal passing chord between 1/1 6/5 3/2 9/5 and
1/1 7/6 3/2 7/4.

Just some of the interesting stuff going on in strict JI -- no temperament
necessary.  In fact, as Keenan pointed out, some of this stuff is lost on the
smaller temperaments (the first example is a no-go even in 22-equal).

Holy smokes. . .                                                    8/31/00

>Some of my latest posts have been geared toward disproving _exactly_ this
>belief. For example, the natural resonances of just intervals lead one most
>naturally to what kind of pentatonic scale? A just one? No.

I would say this grossly depends on how the pentatonic scale is going to be

RE: Holy smokes. . .                                                8/31/30

>OK -- with the most informationless prior about how it's going to be used,
>one finds that tempering it is best. Think of it as the problem of tuning a
>set of harmonic-series-timbre windchimes, or a keyboard that a monkey or
>serialist composer is going to play . . .

I don't see how the minimum-entropy scales are better than, say, harmonics
5-10 under these circumstances.

RE: Holy smokes. . .                                                 9/2/00

>>I don't see how the minimum-entropy scales are better than, say, harmonics
>>5-10 under these circumstances.
>You may not have noticed the line...
>0 268 498 703 885  40.473
>...in my results, but that corresponds to the scale you're talking about.
>Would you like me to show, interval by interval, how this number is arrived
>at, as opposed to how the 39.506 of...
>0 195 390 699 891  39.506
>...is arrived at; or should we step outside this framework and talk about
>its validity?
>I'd like to develop a true odd-limit harmonic entropy measure, but with that
>the general pattern would be the same.  Both lists start with two ratios of
>three.  But then, my scale has one ratio of 3 for every ratio of 5 in your
>scale and one ratio of 5 for every ratio of 7 in your scale.  Finally, both
>scales have 3 intervals that are dissonant in the 7-limit.  So my scale
>clearly wins, and the small about of tempering required doesn't change that

That's as I suspected.  I have no problem with averaging the inversions for
this result, but unless you specify that you intend the scale to convey
different fundamentals with mode changes (per my original complaint), then my
scale must have a lower overall harmonic entropy than yours, since it's
simply a saturated 9-limit otonal chord.  I suggest this would be reflected
by a true higher-adic harmonic entropy measure.

RE: Holy smokes. . .                                                 9/3/00

>>Thanks- that's as I suspected.  I have no problem with averaging the
>>inversions for _this_ result, but unless you specify that you intend the
>>scale to convey different fundamentals with mode changes
>How about just simply chord changes?

That's (more or less) equivalent to what I said.

>I sure prefer chord changes that imply different fundamentals rather than
>the same fundamental over and over again.

Maybe I prefer it too, but the issue was saying so.  I'm big on saying so.
Surely, lots of interesting stuff can happen by subsetting a series over a
fundamental.  I've enjoyed this at various points in Carter Scholz's work.
Denny used to play the resonant filter on the ARP 2500.  Ozric uses the
effect in many synth gurgles.  If music is imitation-speech, then perhaps
this is the most basic type of music, but hardly the least desirable.

>>(per my original complaint), then my scale must have a lower overall
>>harmonic entropy than yours, since it's simply a saturated 9-limit otonal
>>chord.  I suggest this would be reflected by a true higher-adic harmonic
>>entropy measure.
>Undoubtedly. However, if you're playing the scale two notes at a time, the
>diadic measure I've been using is quite appropriate.

I guess that depends on how much stock we decide to place in "echo".  I'm
open to suggestions, but your own examples have taught me, over the years, a
great respect for echo.

Call it persistence of hearing (was RE: Holy smokes. . .)            9/5/00

>>>Undoubtedly. However, if you're playing the scale two notes at a time, the
>>>diadic measure I've been using is quite appropriate.
>>I guess that depends on how much stock we decide to place in "echo".  I'm
>>open to suggestions, but your own examples have taught me, over the years,
>>a great respect for echo.
>You mean examples of my guitar playing?  Well, I think of what I do with
>delay more in terms of counterpoint (as in a round) rather than creating one
>big chord with all the notes in the scale.

Sorry for not making myself clear, but I meant a kind of psychoacoustic
"echo" -- musical context, if you will.  I suspect the common-fundamental,
chord-like quality of the harmonic series segment would prevail over many
duets tuned to it.  But I see what you're saying Paul, and I'm all for it.  I
just wanted readers to know that your views are part of a large,
self-consistent model of music-making -- one that reflects, in places, the
type of music you're interested in...  Bottom line: I don't believe the
39-point scale is any lower in harmonic entropy than the 40-point one, until
we assume multiple fundamentals, chord changes, yada.  I don't think the duet
restriction will do it (cough).

Incidentally, I've had many blissful listening moments hearing delay as
counterpoint -- but I almost always find it nasty by the time it starts
making a chord out of the scale in a melodic passage.  By "your examples", I
meant theory-ones about approximations and such -- one involving the 8:5
comes to mind.

RE: diatonicity                                                      9/6/00

Hello all, Jacky, Paul,

>Could you explain the "action between the latter two" in laymens terms?
>When you say "results in diatonicity", am I understanding correctly that it
>is the interaction between the Rothenberg property and "partial positions"
>(perception of overtones in the timbre being used), is what results in the
>listener's perception of a scale's "diatonicity"?

Correct, except for the definition of "partial positions", and that's my
fault, for writing gibberish.  I meant one level higher than timbre here, a
sort of meta-timbre.  The perception of the position of the fundamental of a
melody note in the harmonic series of the fundamentals of the other notes.
Kapeesh?  Idea is, you can simply play a harmonic series over a drone and get
a good, though primitive, sort of tune.

>Very interesting!  Do explain more.

It isn't all that impressive, since I basically _define_ diatonicity with it.
Nevertheless, very few scales satisfy the definition, and I like the ones
that do.  Here are all the ones I'm aware of:

o Standard meantone pentatonic
o Standard meantone diatonic
o Standard meantone harmonic minor
o Rimski-Korsakov octatonic, 12-tET (0 1 3 4 6 7 9 10)
o Blackwood decatonic, 15-tET (0 1 3 4 6 7 9 10 12 13)
o Erlich pentachordal decatonic, 22-tET (0 2 4 7 9 11 13 16 18 20)
o Erlich symmetrical decatonic, 22-tET (0 2 4 7 9 11 13 15 18 20)
o Keenan decatonic MOS, 29-tET (0 3 6 9 11 14 17 20 23 26)
o Lumma decatonic, 31-tET (0 4 6 10 12 16 18 22 25 28)
o Erlich double-diatonic, 26-tET (0 2 4 6 8 10 12 13 15 17 19 21 23 25)

The basic theme is that any time your brain/ear can follow something, there's
potential to exploit it for melody.  Let's re-name and define the three
things now:

  #       NAME                          METRIC 
  01......Pitch tracking................Miller limit
  02......Scalar interval tracking......Rothenberg propriety
  03......Partial tracking..............Odd-limit

These conflict with each other in spots, and diatonicity is the compromise
that results -- that's what I meant by "action between them".  I can get
diatonicity from 02 and 03, or from 01 and 03.  Let's do 02 and 03 first,
since that's what you asked about.
___________________________________________________ __  __   ___    ___

     I.  Diatonicity = Scalar interval tracking + Partial tracking

We already have "partial position" tracking down, and that gives us harmonic
series segments.

Rothenberg propriety is a metric for an _interval ranking_ process.  R. says
we can measure intervals between pitches, and recognize interval patterns by
the order of the sizes of the intervals, ignoring the absolute sizes of the
intervals involved.  This allows the modal transposition of themes.  It's
how, says R., we can recognize the tune Happy Birthday, when played in the
natural minor.

6-12 is the highest complete octave in the harmonic series that's proper.
After that, a compromise must be made (some harmonics must be left out).
This is the beginning of the "action between the two" I was talking about.

Further, we want to transpose our theme in harmonic numbers as well as scale
degree distances.  Aside from the small amount of natural repetitiveness in
the harmonic series (i.e. 6:9 = 3:2), we'll need to tweak our harmonic series
scales so that different instances of a given scalar interval can function as
a single harmonic interval.  Poof, diatonicity!

Now, let's look at 01 and 03, and get the same result.
___________________________________________________ __  __   ___    ___

        II.    Diatonicity = Pitch tracking + Partial tracking

Again, let's start by assuming partial tracking (I like starting with partial
tracking because I think we get the ability from listening to speech).  This
time, though, we'll add pitch tracking instead of scalar interval tracking...

>A little help with the Miller Limit would be of great help.  Got a paper,
>post or web page on that one?

George Miller was a cognitive psychologist who wrote a famous paper on short-
term memory and perception.  Basically, it says that people can, on average,
hold up to about 7 things at a time in their short-term memory.  What caused
the stir was that this number should be the same for many different types of
things.  We'll apply it to pitches.  The URL for the paper has been posted
here a few times...


...my original post on the subject was sent to the Mills server.  I can re-
post it if anybody would like.

So Miller basically says we can remember and work with up to 7, give or take,
pitches at a time.  So now, rather than just following a note in a harmonic
series, we can take a set of pitches, and assign each of them a position in
our imaginary harmonic series.  This opens the door for the symmetries of
common-tone modulation -- holding a pitch constant while changing its
position in a harmonic series.  For example, with V-I in C-major, we have G
going from 1 to 3.  You can even do more than one pitch at a time if you go
from major to minor, or by using certain forms of temperament.

So, aside from the natural repetitiveness in the harmonic series (i.e. 6:9 =
3:2, etc.), we'll need to adjust our harmonic series segments to allow
pitches to change their harmonic functions.  Poof, diatonicity.

One thing about saying folks can only keep track of 7 things at a time is
that you have to define "things".  Miller allows a "thing" to be a group of
related things.  He calls grouping to squeeze more out of memory "chunking".
Interestingly, both octave-equivalence (symmetry at the 2:1) and
tetrachordality (symmetry at the 3:2) can be seen as chunking the pitch set
to reduce the burden on the listener's short term memory.

Sorry this post is so kludgy, but I'm pressed for time.  Please post
questions and comments.

RE: M. Schulter's Idea, the electronic Bosanquet keyboard            9/6/00

>>The Wilson Uath-108 (MicroZone) 810-key microtonal MIDI keyboard 7500.00
>Well folks, at $7500.00 I think I'll continue learning creative ways to use
>the old "5 black-7 white", my midi wind and percussion controllers to get
>the job done.

That's all fine and good, but considering what Mr. Starr has done, I think
$7500 is a fine price, and I can't understand why everybody around here seems
to have such a hard time with it.  If you can do it for less, I'd like to see

>I find Margo's idea's about using 2 midi manuals to be much more cost

Not to mention completely limiting.  Organs and harpsichords have had
multiple manuals for hundreds of years, yet I know of no serious microtonal
work ever done in this manner.  Can you play chords across manuals 1-handed,
for instance?  What if I split your favorite piano keyboard into separate 7-
and 5-tone manuals.  How would you like that?

>and besides your not in danger of losing years of acquired keyboard skills
>with this method.

What kind of skill is it that can you loose by learning a new one?  I'd like
to know.  This rampant fear of new keyboards is pure mind fungus (and not the
good kind, either!).

>Has anyone on this list actually played these Starr Labs boards, or better
>yet do you know of anyone that has composed music on one that may be able to
>tell us what the transition to this Bosanquet keyboard was like? Monz? What
>was it like for you when you played it? I've seen you play keyboards before,
>and you've got skills - how'd it feel to you? Surely there would be a bit of
>a learning curve - and especially when it would come to setting up mappings.

I've had the chance to play only one generalized keyboard, and that only
briefly, and I didn't like it, because it had Wilson's hexagonal keys (as
does the Uath), and I prefer rectangular keys.  However, the harpsichord I
played had tiered ranks, which is great for rectangular but bad for
hexagonal.  I have reason to believe the planar layout of the Uath suits its
hexagonal keys far better.

The other key thing about the Uath is the action.  When I tried the "mute"
prototype in 1998, the action travel was .008".  Which is a great idea, with
the polyphonic aftertouch Starr has planned.  His aftertouch mechanism is
simply awesome.  But if I understood the web site, poly touch isn't available
just yet.  I wouldn't get the board without it.

>Got to admit, it does look really cool!! If only money were no object.

Do you play keyboard?  Have you looked at the price of pianos lately?  Yes,
it's expensive.  No, I can't afford it today either.  But musical instruments
are expensive, and one day, I just may buy it.

RE: re: "stability" and "efficiency"                                9/10/00

>The recent threads regarding "propriety" made me realize once again that
>both "stability" and "efficiency" are missing from Joe Monzo's "Definitions
>of tuning terms"... I think fleshing these terms out would help give the
>"propriety" definition some much needed additional focus. Anyone care to
>take a crack at some suitably layman-like definitions for these?

Manuel just posted an excellent definition of stability.  Here's a stab at

If a listener hears random samples of a pitch set tuned to a given scale,
then the _Rothenberg efficiency_ of the scale is the time it takes her, on
average, to determine which scale position each pitch represents.

...Stability measures how easy it is for a listener to determine which
interval class an interval belongs to.  Efficiency measures how _difficult_
it is for her to determine what scale position a pitch represents.

RE: new webpage, JI modulation                                      9/10/00

>OK, Monz -- I'm glad you asked me to do this myself, because the Vicentino
>idea doesn't really work for chords like that C-G-A chord . . .

What a drag!  This looks to be a serious limitation of the Vicentino method.

>I get it -- the D note in the D chord is not held over into the G chord. So,
>with G as 1/1, you're retuning the sequence as:
>4/3 ---- 4/3 5/4 10/9-10/9 1/1
>5/3 16/9 1/1 --- 1/1 50/27 5/4
>    10/9 1/1 5/4 4/3 40/27 1/1
>Which is the same as Monz' second example.

Yup, this looks to be the optimum solution, as far as a ground for 5-limit
JI.  Is anybody actually unhappy with this version?  I like it.

>Guys, this problem would be a lot more interesting if the D sustained into
>the final G chord.

In this case, Partch's 3rd version isn't bad, or we may choose to distribute
the comma as much as possible.  The optimum distribution is easy, since the
sequence is short and there isn't anything common bigger than a monad...

|          |          |         |         |         |         |         |
              A (10/9)                       A (10/9)  A (9/8)            
                                                       D (3/2)   D (3/2)  
    C (4/3)   C (4/3)    C (4/3)             C (4/3)                      
                                   B (5/4)                       B (5/4)  
                         G (1/1)   G (1/1)   G (1/1)             G (1/1)  
                                                       F# (15/8)          
              F (16/9)                                                    
    E (5/3)                                                               
|          |          |         |         |         |         |         |
    +/- 0.0     +3.1      +6.1      +9.2      +12.3     -6.1      -3.1

The numbers across the bottom indicate the tuning change, in cents, to be
applied to the pitches in their respective columns.  For example, the pitches
of the d-major chord would be, in cents, D=695.9, F#=1082.2, and A=197.8.

RE: essential listening                                             9/13/00

Something else I would recommend.  Not a top-ten microtonal recording, but
the person who likes brass and/or early music must not be without...

The Art of Baroque Trumpet, Vol 2.
Niklas Eklund, baroque trumpet
Knut Johannessen, organ
Naxos 8.553593

Niki is one of the greatest artists playing the trumpet today.  The album
features music by Fantini, Viviani, Frescobaldi, Sweelinck, Pezel, and the
first "period" recordings of tunes by Prentzl and the German composer Lowe
von Eisenach.  Good 5-limit just intonation is often achieved, and the organ
is tuned in "1/4 syntonic comma meantone".  Importantly, the performances are
in the same good spirit as this sublime music.  And it's Naxos, which means
you get good to excellent production for around $5 US.

The only Sweelinck tune is, rather predictably, the Chromatic Fantasia.
Overplayed, this is still an awesome tune, and if you've never heard it in
meantone, you're in for a treat.

Then comes track 11, _Toccata settima_ by Michelangelo Rossi, which is my
current pick for best showcase of meantone as a "microtonal" tuning.  And
this is truly an astounding work, meantone or no.  Sounds like Bach's
_Little Harmonic Labyrinth_ a 100 years ahead of schedule.

I never believed all that stuff about valveless horns sounding sweeter, but
it's apparently true.

Quote from liner notes, natural trumpet, etc.                       9/14/00

From the liner notes of Naxos 8.553593...

The natural trumpet itself, a trumpet without valves, is restricted to the
notes of the harmonic series, the so-called partials.  Some of these notes
are impure and cause problems with intonation, especially if the trumpet is
played together with an instrument of fixed pitch, like the organ.  The 11th
harmonic is too high for F' and too low for F#"; the 13th is too low for A"
and is nearer to G#"; the 7th and 14th harmonics are slightly too low for
Bb' and Bb", respectively.  In mean-tone temperament, however, the impure
notes are less conspicuous and an experienced performer can reduce these
problems by his embouchure.  This is also easier on an old trumpet because of
irregularities in the tubing.  The matter is complicated by modern replicas,
with even tubing which makes the pitch of every harmonic more stable.  As the
harmonics in the upper register lie very close together, it is difficult to
achieve secure attack.  With the modern valve trumpet lower harmonics are
used, making the attack more secure.  The modern trumpet, with its six valve
combinations, is, in fact, a combination of seven natural trumpets.  On the
other hand the valve trumpet needs more wind than the natural trumpet, the
player will, therefore, have to be prepared to adjust his technique to
achieve a secure attack with the natural instrument.  In order to help the
modern performer shift between the valve and natural trumpet a finger-hole
system was devised in 1960 by Otto Steinkopf.  By opening one hole all even-
numbered harmonics are excluded and by opening another all the odd ones.  In
this way, by opening one of the holes the nearest harmonic can be excluded,
giving greater security of attack. . . Trumpets with such holes may be call
Baroque trumpets, to distinguish them from pure natural trumpets.

...In the first years of the 17th century trumpet parts were confined to a
limited number of notes, as, for example, in the 5-part trumpet _Toccata_ in
Monteverdi's _Orfeo_.  The melodic part ... consisted only of the notes C",
D", E", F", and G", and less often A".  Fantini broke with this tradition and
developed a use of the solo trumpet in its entire range, from the 2nd to the
16th harmonic, from C to C'".

RE: Schenker                                                        9/18/00

>   The smallest interval in tonal music is the _second_ (2nd), an interval
>   which encompasses only two notes.  This is the interval we find between
>   one scale degree and the next - for example, Bb to C in the Bb major
>   scale.  The 2nd has two sizes, a fact easily verified by looking at a
>   piano keyboard.
>That's it.  At no point is the harmonic meaning (or near-meaning) of any
>interval discussed.  No tuning system is discussed (how could it be, when
>the theoretical harmonic basis of tuning is omitted?).

John- you're right that no book on music theory should be without a
discussion on the psycho-acoustic basis of harmony.  However, here, the
author may not be committing a crime by assuming a connection between scale
degrees and acoustic intervals, since in the case of the meantone diatonic
scale it exists.  Though incomplete without a discussion of tuning, theory in
terms of scale degrees may be the most useful kind for the practitioner of
common-practice-style music -- if I had to have one without the other, I'd
probably take the scale-based theory.

But if this kind of thing gets your goat, as it did mine after 13 years of
music education (including a year of composition, theory, and piano at the
Conservatory of Music at Indiana University) without one mention of tuning,
you'd probably like Doty's JI Primer as much as I did.  It totally busts
these Schenker types (with apologies to Schenker for the term).

>Here's Forte on parallel fifths (p. 54)
>   Parallel 5ths are avoided because the 5th formed by scale degree 1 and 5
>   is the primary harmonic interval, the interval which divides the scale
>   and thus defines the key.  The direct succession of two 5ths raises doubt
>   concerning the key.

An important point, in addition to the one Paul mentioned.

>   The prohibition of parallel 5ths is more than a pedantic dictum.  It is
>   an important negative principle which is responsible for many harmonic
>   and melodic features of tonal music ...  Without the limitation placed on
>   parallel 5ths and 8ves the art of tonal music would not have developed
>   the elaborate and intricate forms which have given it such a unique
>   position.

He's dreaming.  As Paul pointed out, this is a reversal of history. Self-
important music theory at its best.

>Doubt concerning home key???  That seems bizarre the distraction, if any,
>caused by parallel fifths seems to my ear no greater than things that DO
>happen and are encouraged by these rules.

Though the existence of parallel fifths alone is hardly enough, sections of
tonal music harmonized in parallel fifths can easily run into problems, since
this kind of music often relies on a steady pace of tonal motion, and as
Rothenberg points out, the diatonic scale is highly efficient, so it takes
some work to achieve this.  Take an hour, and set a hymn to 4-part harmony in
three different ways, and you'll see for yourself immediately.  Paul Erlich
presents a reasonable version of all of this in his paper, with
characteristic dissonances, and so on.  Rothenberg's sufficient sets are
another approach.

>The art could not have developed?  It would not have developed in exactly
>the same way, to be sure, but failing to prohibit something does not mean
>that it must appear in every composition!

Of course.  I would also like to point out that, with mastery of the tonal
style, harmonizing in parallel fifths is fine.  The rule that Forte so highly
recommends is, in fact, a rule given to beginners, since learning to leave
enough anchor points in a progression requires some subtlety.  Beethoven
didn't have a problem with it, though.  Or, you can skip the anchor points,
and return to a more modal style, as in Jazz (harmonizing in parallel fifths
is almost standard in some voicings).

If these rules of music theory really did dictate the behavior of composers,
rather than describing it as Paul and I suggest, then perhaps they could
also take credit for setting up a hegemony for great music to break!  (This
may sound funny, but if you read your liner notes, you'll find nothing but
praise for rule-breaking from Forte's colleagues.  With apologies to Forte
for the term.)

>Anyway, the parallel fifths thing is O.T., I suppose, and the idea of
>getting the book WAS to learn the sometimes strange rules under which some
>music (music I often like, to be sure!) was written, whether or not I agree
>with those rules.  Trouble is, I have a hard time getting motivated to read
>the book when so little of it resonates with me.

John, I would recommend you F all that.  Get yourself some basic tools -- a
polyphonic instrument, do a little ear training every time you warm up, a
good score-entry package (paper ain't bad), and play (literally).

Microtonality and new music?                                        9/21/00

>As far as finding a totally new music through microtonal investigations, I
>think thats a bit like an artist deciding to paint in a totally new color.

I think the better analogy is that we are learning to paint with a much
larger palette, rather than just a single new color.

RE: the key to Wilson's diagrams                                    9/21/00

Joseph Pehrson wrote,

>The objections are nonsense.  Anything that prohibits someone from learning
>something is nonsense...

Joe, I can't disagree more.  Plato said all virtue is knowledge, and I agree
(though I'll ask what _isn't_ knowledge, from that POV).  But you seem to be
implying that all knowledge is virtue, which I think is nonsense.

RE: the key to Wilson's diagrams                                    9/22/00

Introductory note:

      Anytime you start to criticize criticizing, you're setting yourself up
      for trouble (naturally, this note is no exception).  This can even get
      to the point where the problem is actually complaining about the
      problem.  I think we saw that today.  What's more damaging to the list:
      a few angry flames, or 200 messages about what constitutes an
      appropriate message?  My usual response is silence, but I thought I'd
      add this intro in case it might help out in the future.  I think the
      list description at our egroups page (which was recently quoted
      on-list) is all we should need to determine appropriate material.  I
      would add that anytime you see a message subject in caps, you should
      adjust the case if you want to reply.  Now, back to the subject of
      Wilson's papers...

Joseph Pehrson wrote...

>It doesn't ruin the original in *ANY* way... especially if provided by
>somebody competant like Monzo... and perhaps with the explanatory materials
>reviewed by Wilson himself...

If reviewed by Wilson himself, then fine.

>>>The objections are nonsense.  Anything that prohibits someone from 
>>>learning something is nonsense...
>>Joe, I can't disagree more.  Plato said all virtue is knowledge, and I
>>agree (though I'll ask what _isn't_ knowledge, from that POV).  But you
>>seem to be implying that all knowledge is virtue, which I think is
>Well then, Carl... what would you prefer... ignorance and obstructions to

Try again.  That's denying the antecedent, and attacking the person.

Joseph- as the conceptual movement tried to show, the presentation of a work
has great influence on how the work functions.  Wilson's papers aren't just
papers, they're the part of Wilson's work he's decided to give us, in the way
he's decided to give them to us.  Monz hit the nail on the head:

>There's one thing I'm really amazed at  there's absolutely *nothing*
>preventing anyone else here from learning Wilson's work the same way I did,
>by painstakingly redoing the originals in a way that's easier for them to
>understand - not for public consumption, but just for their own personal
>benefit.  Of course, it does require...  a lot of   W O R K  !!!!
>I guess that's the problem...

That's not the problem, that's the point!  I'm the first guy to get up and
say I only understand a fraction of Wilson's stuff.  If I work at it, and
think I can help explain it to somebody else, I will.  But I _won't_ mark up
Wilson's papers and publish the result without his explicit consent.

RE: The passion of the list                                         9/22/00

Very inspiring post, Neil.  I'd love to share a few words on the topic.

I got into music because it seems so cool- it's only there (like all art I
guess) because it's beautiful.  It's a way for humans to share what we find
beautiful.  It seems unlike the other arts, though (and I'm not the first to
point this out), in that it takes sound to a whole new level on this earth.
A visual artist may show us something that's never been seen (like the
fantasy of a Roger Dean painting) or even a new way to see (like a Picasso or
Van Gough), but it never really seems to compare to the fantastic visual
detail around us.  Computer graphics are getting there, I guess (if you've
ever seen the movie Antz).  Architecture gives us a new way to live -- an
artificial environment -- but on balance, I've always preferred the woods
(where my waste is recycled and the food and air are fresher, and, the latest
science informs us, more healthful).

Music, on the other hand, is definitely at the top of its game.  There are
those who may prefer the sound of a running stream, or a traffic jam, but I'm
not one of them.  To me the greatest sound on this earth is the sound of
human voices singing in just intonation.  I like to imagine that animals like
dogs and cats experience pure bliss when they hear human music.  I even
played a concert about this, called _Chorus in the Cello_.  I used the cello
register of my piano, which can sound a lot like a tenor voice, and tried to
play what a human choir might sound like if you couldn't understand the
words.  There are many stories of angelic choirs.  For an agnostic like me,
angelic choirs are human ones.

I'm allergic to cats, and I don't really like them.  But when I lived in
Berkeley and had no piano to play, and my neighbor let me use his Steinway, I
had to put up with his cat.  But I soon grew to love that cat, because when I
played, it stretched out on top of the piano, on its back, and hung its head
over the fall board, and looked right at me with its blue eyes.  It would
stay there for hours while I played.  I would try to imagine what it must be
like to hear that, while feeling it vibrate all around and through you.  And
if I ever wanted the cat to leave, I would simply switch to harpsichord (they
_hate_ it!).  :)

For me, music is one of the richest things in life.  Take Beethoven's late
string quartets, a good Bach fugue, or some extended YES tunes.  I've also
heard "ethnic" music from Africa, India, and Persia that I simply couldn't

The kind of thing I'm saying seems to be supported, somewhat, by recent
experiments where listening to "complex" music (Mozart is only one example)
temporarily improves scores on spatial reasoning tests.  I first learned of
those experiments from an article Erv Wilson gave me.  He's into this aspect
of music, too.

I guess what I'm saying is, I see music as one of the best things about being
human, and thus as one of the best things to be involved with as a human, to
improve humanity if such is possible.  Take Beethoven's Ode to Joy -- a
celebration of the fact that all humans experience, from time to time, the
feeling of joy, and isn't that great?  Sure, the work is over played, but if
I can live 150 years after it was written and still get that out of it, then

I got into tuning because it seems like the natural concern of a good
musician.  From my earliest memory, I can remember wondering what gave
Barbershop quartets and boys' choirs that special sound.  When I found
out about just intonation, I was shocked.  I can still remember the first
time I heard the familiar Barbershop, not as a timbre, but as a locked
7-limit chord.  I was _really_ shocked.  I can still remember the first time
Denny played his 15-limit slide guitar for me.  I was blown away.  I knew I
was on to something, but that I'd have to learn as much as I could if I was
going to synthesize these new materials into good music in my lifetime.

You said, Neil, that theory didn't have anything to do with making good
music, and I can see what you're saying.  But it only applies to the guy
who's had a 19-tone guitar dropped in his lap -- or, for that matter, a
12-tone one!  I've made outstanding music on the top of an empty can of
Pavich organically-grown Jumbo Thompson seedless raisins... but I think I
prefer polyphonic instruments that approximate just intonation (with a
raisin-can holding down the beat, of course!).

A 19-tone guitar may be about the same as a 12-tone one, but consider the
piano.  The white keys make a strong statement.  They suggest there's
something special about the diatonic scale.  I'm a keyboardist in search of
an instrument, and it will be expensive.  I want to choose carefully.  That's
why I'm on this list, trying to understand all this stuff.  I still believe
that the raw materials of 11-limit JI are enough to keep me occupied making
awesome music for many years, but I'm becoming convinced that organizing
these materials a bit before I start will be worth my time.  When I get
there, we'll have to jam!

RE: Good 5-limit scale generators                                   9/30/00

>Single chain:                        No. gene-  Notes in  m/n
>           No.                       rators in  smallest  Gen. is
>           triads  Min      Min      interval   proper    approx
>Generator  in 8    7-limit  7-limit  2  4  5    MOS with  m steps
>(+-1c)     notes   RMS err  MA err.  3  5  6    >4 notes  of n-tET
>317c  m3  *   4     1.0c     1.4c    6  5  1    15        5/19
>380c  M3      6     4.6c     6.0c    5  1  4    16 or 19  6/19, 13/41

Groovy, Dave!

>Most of them suck melodically, as the second-last column shows, and 2 or 4
>triads in 8 notes isn't really harmonically good.

Above, I see 4 and 6 triads in 8 notes, which can be very good!  The 317c
scale, while not MOS and not very stable, is proper.  As for the 380c one...

The latest version of my diatonicity rules may be found at...

...I think very highly of them.  In fact, there are only two properties I
would call very useful which don't appear on the chart -- tetrachordality and
efficiency.  These are two properties that our traditional diatonic scale
has, but which I don't think are needed for or related to what I'd call
diatonic effects.

Anyway, you'll notice that I list Rothenberg stability _or_ chord coverage as
sufficient for interval coding.  I haven't checked yet, but with six triads
in eight notes, the 380c scale has probably got the coverage thing down.

>>>It only has 2 major and 2 minor triads.
>>Yeah -- a really good amount for a 7-tone scale! 
>I assume you're being facetious.

I don't think he was.  Anyway, the scale looks cool to me.

RE: Good 5-limit scale generators                                   10/2/00

>I was mistaken in calling it a MOS, but it _is_ proper.  Note that the
>generator here can be either a fifth or a major third.  They end up as the
>same 8 note scale.  In 12-tET it is 12121212.  Outside of 12-tET the minor
>thirds must stay at 300 cents but you are free to narrow the fifths to
>improve the major thirds.

Aha!  Thanks.  So we can describe the octatonic scale as two chains of minor
thirds a fifth apart.  Any improvement over 12-tET is going to be very small.

>Here's the 9 note proper scale with 12 triads, based on 3 chains of fifths
>(or minor thirds) 1/3 octave apart. In 12-tET it's 121121121. The major
>thirds are stuck at 400 cents but you can widen the fifths to improve the
>minor thirds. If you lose an outside note you lose 3 triads. Lose a second
>note on the same side and you only lose another 2 triads, leaving 7 triads
>in 7 notes.

Wild.  Unfortunately, neither the major or minor versions of this 7-note
structure are anywhere near proper, and their triads do not occur on a
regular pattern of scale steps.

>Do these symmetrical 8 and 9 note scales have standard names?

I am not familiar with the 9-note, but the 8-note one is commonly called the
"octatonic" scale.  Messiaen is known for using it and writing about it, but
Stravinsky also used it extensively, and I believe Rimski-Korsakov was the
first to use it.

RE: Good 5-limit scale generators                                   10/2/00

>>Single chain:                        No. gene-  Notes in  m/n
>>           No.                       rators in  smallest  Gen. is
>>           triads  Min      Min      interval   proper    approx
>>Generator  in 8    7-limit  7-limit  2  4  5    MOS with  m steps
>>(+-1c)     notes   RMS err  MA err.  3  5  6    >4 notes  of n-tET
>>317c  m3  *   4     1.0c     1.4c    6  5  1    15        5/19
>>380c  M3      6     4.6c     6.0c    5  1  4    16 or 19  6/19, 13/41

Dave Keenan!  I just realized that the 8-tone 317-cent scale above is my 8-
tone subset of your 11-tone chain-of-minor-thirds scale from last year!  Four
triads, on 1-3-5!  I don't think I ever realized that!!

RE: Good 5-limit scale generators                                   10/3/00

[Dan Stearns wrote...]
>I don't see why you can't look at these as MOS scales... I do!  Periodicity
>is periodicity with MOS structures no?

[Paul Erlich wrote...]
>Erv Wilson invented the term MOS to refer to scales with a _single_
>generator, and it's a very strange term . . . if you're after a different
>meaning, then why not just coin a different term?

Good point (be creative, Dan!).  Also, other authors since Wilson have
recognized Myhill's property as a sufficient and necessary condition for MOS;
no multi-chain scales have Myhill's property.

>>Wait a minute, Dan... are you doing non-octave MOSs?  They are still MOS,
>>but it isn't fair to use an octave to talk about them (i.e. describe them
>>as the superposition of two octave-periodic chains).
>Right. The generalized "Golden scale" algorithm I use only defines the
>periodic space in which the MOS pattern exists:

Eeep!  You're way over my head here, Dan.  I'm not really quick with math,
and I have very little time to spend on tuning issues (I can barely track the
threads as it is now).  Is there a chance you could explain what you're up to
and why in language for simple folk?

I'll do us the favor of showing how a non-octave MOS can be viewed as a
superposition of multiple, octave-based MOSs.  Take the wholetone scale in
12-tET (0 2 4 6 8 10 12).  It doesn't have Myhill's property, so it isn't an
MOS.  But it can be described as the superposition of (0 2 4 6 8 12) and
(0 4 6 8 10 12), both of which have Myhill's property.  Now, call 1300 cents
the interval of equivalence.  The set (0 2 4 6 8 10 13) _is_ a single MOS
here.  Did I make a mistake?

In short, a set is MOS if and only if it has Myhill's property at the
interval of equivalence specified; scales can be MOS at a given IE, but not
at another.

RE: Amateur question                                                10/5/00

>Judith -- the poster wanted to have much more than 12 notes per octave. The
>harpsichord would not work for this purpose, _unless_ you had multiple (n)
>manuals, in which case you could get 12*n notes per octave --- right?

Paul- it isn't the number of manuals, but the number of scales inside that
counts.  On many multiple-manual harpsichords, the manuals may be assigned
in various ways to what's inside by way of lever.  The same goes for (less
common) single-manual, multi-scale instruments.  The trouble is that the
different scales are traditionally used to change the timbre of the
instrument, and so the microtonal user will have to put up with interleaved
timbres.  D'oh!

RE: Amateur question                                                10/6/00

>Carl -- the original poster wanted to be able to perform quarter-tone runs,
>etc. If you have a single manual, how fast would you have to shake the lever
>back and forth to perform a quarter-tone run? :)

I guess that depends... I'm pretty fast.  Actually, I think such a lever
would be great.  Michael Harrison has done impressive things with a pedal,
and the "lever" on a harpsichord is often in the form of a pedal.  In fact,
since harpsichords already have this kind of pedal, they are arguably better-
suited to this technique than pianos, with which Harrison was left to
modifying the una corda mechanism.  It would be great to have a harpsichord
designed with two identical scales, with jacks to switch by pedal, scaled for
24 out-of 31-tET.

The root-controller idea takes on new scope with electronics, and the ability
to root on all twelve of the top keys, rather than onto only two.  I have
often wondered how easy it might be to take a midi controller -- say, a
pedalboard -- and use it to control the roots on a normal keyboard up top.
Note-ons from the pedal could be sorted by note number -- perhaps by a CAL
applet, which would then apply the appropriate pitch bend to all the notes
from the keys.

RE: Amateur question                                                10/6/00

>>It would be great to have a harpsichord designed with two identical scales,
>>with jacks to switch by pedal, scaled for 24 out-of 31-tET.
>But any chords would have to come from one of two sets of 12, right?  That
>doesn't seem very useful.

Are you kidding?  You could play most things through Beethoven in meantone,
for starters.  Yes, this is more of a consonance-added approach than a
xenharmonic one, but for some of us, 12-tET just doesn't cut it for 5-limit
on acoustic instruments.  And Harrison showed that 24 tones of 7-limit JI
are interesting.  It isn't hard to subset the 7-limit in such a way that all
complete tetrads fall on one manual at a time.  Yes, this approach is
ultimately going to be just a subset of Margo's double-manual suggestion, but
I think it's a great deal more intuitive at the keyboard, and more than
promising enough to justify the construction of a few harpsichords.

RE: Amateur question                                                10/7/00

>I don't get it. Wouldn't many keys require some notes from one set of 12,
>and other notes from the other set of 12, at the same time?

You can get 12 non-manual-spanning diatonic keys from 18 tones by tuning one
manual to...

                \  / \  / \  / \  / \  / \  / \  / \  / \
                 \/   \/   \/   \/   \/   \/   \/   \/   \

...and the other to...

                \  / \  / \  / \  / \  / \  / \  / \  / \
                 \/   \/   \/   \/   \/   \/   \/   \/   \

If you don't overlap, you can still get 12 non-spanning keys, plus six
spanning keys, although I'd probably prefer to overlap and get continuity.

Interestingly, Michael Zarkey has fitted his historic harpsichord with a 19-
tone generalized keyboard that allows one to do chromatic runs without a
lever.  But you do get the timbre-switching I mentioned, which is absolutely

10 triads, 10 notes                                                10/13/00

A while back Dave Keenan discussed the 10-tone 7/22 MOS...

0  1  6  7  8 13 14 15 16 21 22
1  5  1  1  5  1  1  1  5  1  

...It has ten triads (5 major, 5 minor)!  We can improve the tuning without
loosing any chords by taking it from 41-tET...

0  2 11 13 15 24 26 28 30 39 41
2  9  2  2  9  2  2  2  9  2  

...Both versions are wildly un-proper, but the 41-tET version is slightly
more stable, at least in log-freq. space.

Recently, Dave posted this...

>Single chain                         No. gene-  Notes in  m/n
>           No.                       rators in  smallest  Gen. is
>           triads  Min      Min      interval   proper    approx
>Generator  in 8    7-limit  7-limit  2  4  5    MOS with  m steps
>(+-1c)     notes   RMS err  MA err.  3  5  6    >4 notes  of n-tET
>380c  M3      6     4.6c     6.0c    5  1  4    16 or 19  6/19, 13/41

...which is 8-out-of the scale above.  Its chord/note ratio is lower, and the
chords no longer fall on a regular pattern of scale steps.

But by omitting 1 note, we have another MOS...

0 11 13 24 26 37 39 41
11 2 11  2 11  2  2

...with 4 triads, falling on a regular pattern of scale steps.  Like the 10-
note version, the pattern is different for the major and minor chords, which
means we can not consider these scales generalized diatonics, at least by my
book.  But they are interesting.

Also like the 10-note and 8-note versions, this 7-note one is wildly
improper.  But, as I told Dave, I believe chord coverage can at least
partially replace propriety in many kinds of music.  The 10-tone version is
obviously covered, with its 1:1 chord/note ratio.  I wrote...

>I haven't checked yet, but with six triads in eight notes, the 380c scale
>has probably got the coverage thing down.

Indeed it does, although, as my rules state, for coverage to work in place of
propriety, the triads must fall on a regular scale-step pattern, which they
do not in the 8-note scale.  They do in the 7-note, but only 6 of its 7 tones
are covered by its 4 triads.

In short... On paper, I don't like the 8-note version, but I do like the 10-
note one, and the 7-note one is somewhere in between.  Now, to play with 'em!

RE: 10 triads, 10 notes                                            10/15/00

>The actual choice of the number of notes for a scale should be based also
>on melodic criteria and playability criteria as Carl has done.  Myhill's
>property (or MOS) (only two sizes of scale step) seems to be an almost
>essential requirement for good behaviour in these regards.

I disagree, and I see you do now, too.

>But N chains spaced 1/N of an octave apart will result in an non-Myhill,
>non-MOS scale, though it still may have only two step sizes.  There's an
>"official" name for scales with two step sizes -- I forget what it is.
>Let's keep forgetting it.

:) Good call.  But by "two step sizes", do you mean two sizes of 2nd??

>I think two-step-size is way more important than Myhill's/MOS.

I don't think any of it is important, per se.  They do simplify things,
greatly, but I don't believe in any _sudden_ change in goodness from three
step sizes to two (as opposed to four to three).  If I had to guess, I'd say
the best measure for this business was Rothenberg's mean variety.

RE: 10 triads, 10 notes                                            10/15/00

>>:) Good call. But by "two step sizes", do you mean two sizes of 2nd??
>Er. Yes. What else could "step" mean in this context?

Well, in our recent discussion of Myhill's property, we used "step sizes" to
mean all generic intervals (not just 2nds).

>>>I think two-step-size is way more important than Myhill's/MOS.
>>I don't think any of it is important, per se. They do simplify things,
>>greatly, but I don't believe in any _sudden_ change in goodness from three
>>step sizes to two (as opposed to four to three).
>Ok. I can go along with that.  Particularly since one may have two step
>sizes that differ by only a few cents.

Sure, but I think it goes deeper.  Basically, I think it's one of those false
positives... just like prime-limit for chords.  Many times, two types of
seconds will result in a simple matrix entirely, and in these cases I believe
it's the fact the _entire_ matrix is simple that counts; nothing special
about 2nds.  To check this, try a two-2nds-sizes scale where the 2nds are un-
evenly distributed (thus, causing havoc on the rest of the matrix), and see
what you think of the scale.  I've tried this, and concluded that stuff like
maximal evenness is suspiciously tack-on.

>>If I had to guess, I'd say the best measure for this business was
>>Rothenberg's mean variety.
>I don't recall that one. Is it easy to explain?

You'll be delighted.  It's the average number of sizes per generic interval,
excluding the interval of equivalence.  So for Myhill scales, the mean
variety is always 2.

RE: the sound of a distant acoustic horn...                        10/19/00

Kraig Grady wrote,

>One of the problems of doing music on a computer is that one will not
>develop skills like one does on an instrument unless you see typing 80
>words a minute a technique.

Does that mean that using a MIDI controller with a synthesizer is not
computer music?

>First you have to  convince me that electronic music is "improving" do
>modern synth./computers sound better than old moogs?

Sound better (!?), who would disagree?  Even if all you want are annoying
lead timbres, modern digital synths can do everything and far more than old
analog machines could do...  Not to mention timbres which are suitable for
rendering polyphonic music tastefully.  The master of this with analog
machines was Wendy Carlos, and she proved that far more could be done on
modern equipment with albums like _Digital Moonscapes_!  Marcus Hobbs' _From
on High_, which is one of the most important achievements in the history of
computer music, wouldn't have been possible with anything less than Kyma,
which is a very recent development.

>Electronic music changes and offers more control, but what you have to do to
>your brain to control more is think about things that are already a given on
>an acoustic instrument.

I understand and agree with what you are saying, and I've argued this here
many times in the past.  But that's a problem of controllers, not of timbre
generation.  There are plenty of things to control with current digital
timbres -- all we need are sensible controllers, and these, as I hope to
prove, are not a hard thing.

>I have more f..king power than any of these things -)

Sounds like issues to me.  It probably won't comfort you to know that you may
live to see "machines" more intelligent than any human.

I've got issues too.  For example, we all participate in this medium,
generally considered a type of "advanced" communication, which, the idea
often seems to go, is a very recent phenomenon, and which we are therefore
very privileged to use, living recently as we do.  But what does "advanced
communication" really mean?  I think it's easy to confuse the medium, which
I could see calling "advanced", with the communication, which I consider
rather primitive.  I believe that humans communicate ideally in small troops,
when fed with fresh food, when basing their report for complex work by
sharing simple work like farming, when speaking face to face, when able to
spend their time communicating instead of sitting behind the wheel of a car,
when breathing fresh air...

RE: the sound of a distant acoustic horn...                        10/19/00

>what does it say that computers work best when they borrow "controllers"
>from traditional instruments

Well... I would say the idea of a keyboard comes naturally from our
physiology, with the hands and fingers and all.  It has been transplanted
across several technologies... organs, harpsichords, pianos... what's so
different about synthesizers?

>> Sound better (!?), who would disagree?
>There is something about Hendrix i just can't get by. The electronic
>modification of acoustic sources still sounds better to my ear than pure
>synth sounds.

Agreed.  But I thought it was analog synths (like Moogs) vs. digital synths.
In that case, the modern gear must win, and win big.

>haven't heard his latest so can't say.  

?  Chalmers played it for me on my way up to see you in '98.  You _must_ get
a hold of it!

>take a flute starting low playing a set of quick note ending on a tone that
>is taken up by a violin harmonic. With instruments we hear two instruments,
>with electronics we have only a slight tweak of a sound.

For $3500 you can get a basic Kyma setup, which will enable you to model an
entire orchestra, with as many independent "instruments" as you like.

>When i first started, electronics weren't even a good possibility, and at
>the time those who where interested in that direction had to wait.

Absolutely.  Even today the sound of acoustic chamber ensembles is my
favorite.  As you know, I really appreciate what you've done in this area.

But I'm starting now, and as far as the waiting goes, things are reversed.
It now costs _less_ to do a digital orchestra, in whatever tuning I like,
than it would to build and maintain even a small percussion ensemble, as you
have done... and the sound is good enough to jump for joy.

RE: "warm fuzzy feelings"                                          10/20/00

>>I believe in mathematics, "elegance" is a real component of what makes a
>>good proof, or at least an aesthetically attractive proof.
>Science is often about aesthetics because the simplest explanation is
>usually the most beautiful, the best epistemologically (leading to the most
>insight), _and_ the most likely to be correct.

The last two are the same, and the first is a behavior-creating emotion we
evolved because being correct is good for survival.

>For a piece of aesthetically beautiful tuning theory, see Balzano's paper
>_The Group-theoretic Description of 12-Fold and Microtonal Pitch Systems_.
>This has got to be the most elegant and aesthetically attractive proposal of
>a new tuning system that I've ever seen. In fact, it's so well-constructed
>that some of our finest minds have been convinced by it. But does that make
>it in any way a "proof" or "scientific"? It almost appears to be, but I've
>spent some time on the list exposing its holes, and ultimately it's a house
>of cards that comes tumbling to the table.

Perhaps I never understood Balzano's paper, Paul, but I'm going to have to
disagree.  He makes an awful lot of assumptions... and when did you ever
expose any holes?  The only argument I can remember you making was on the
failure of the scheme to produce consonant chords -- something which Balzano
lists as one of his goals in his introduction.

RE: Dekany in 4D (Excel)                                           10/24/00

>I think it is an exageration to say that any CPS has no preferred tonal
>center. All consonant dyads are not equally consonant and otonal chords are
>not equal to utonal.

True, but there is more to tonality than consonance.  In fact I think it's an
abuse of the term to call CPSs atonal.  I think what has been meant in the
past, with comparisons to the diamond, etc., is the way the common tones are
distributed between the chords.

>But still it's an extraordinary acheivement to get as close to atonal as
>this, in JI.

Not especially.  It's quite easy to use the diamond atonally.  Or, one could
use a bunch of chords distant from eachother on the lattice.  The cool thing
about CPS is the number of consonant chords you can use, from a small number
of notes, without boring the hell out of people by having them always hear a
single common frequency as in the diamond.

>I think that complete atonality can only be obtained by using all the notes
>from some equal temperament (equal division of the octave).

I disagree, even if we take atonality to mean a lack of consonance.  One
central theme of the list seems to be that the brain is hard-wired for JI...
it will always hear some intervals as more consonant than others, ET or

RE: non-centricity of CPS/Pythagorean Hexany piece                 10/29/00

David Finnamore wrote,

>So what do other composers seek to accomplish with CPS structures?  Kraig?

Actually, I've never composed strictly within a CPS, at least not
intentionally (other than a few demos which I'll post shortly).

One thing I'm very interested in as a composer is leaving diatonicity and
exploring a style where (often simple) melodies are derived from non-
diatonic common-tone modulations, and subject to heavy variation (with the
interesting catch that the borders between variations are intentionally
blurred -- but that's another topic for another list).  Inspired by late
Beethoven, YES, and Jazz, and much 20th-century music, this is the direction
I've taken with 12-tET so far.  Anywho, I think CPSs are a great way to
familiarize oneself with the common-tone progressions of a given limit, and
I like the way their incomplete chords help thin out higher-limit utonal
sonorities.  For these reasons, I plan on spending some quality time with
the pentadekany and eikosany at some point.  Eventually, however, I imagine
I'd work in free-JI (and perhaps even extend common-tone possibilities
somewhat with temperament), but armed with my CPS experience.

Of course, I'm also interested in new types of diatonicity (it's not
diatonicity I'm bored of... just 5-limit 7-tone diatonicity!) like Paul
Erlich's decatonic scales in 22-tET, or Easley Blackwood's decatonic in 15-
tET (his 15-tone etude is one of my favorite microtonal tunes).  To that end,
the hexany and dekany might prove to be good theoretical tools for locating
"generalized diatonic" scales.

BTW, I like your hexany piece too!  And your contribution to the tape swap
was way cool, as I've said here before!

RE: my paper                                                       11/12/00

>Can anyone direct me to pertinent sources, on or off-line?

Heya Brian!

The de facto microtonal home page is John Starrett's...


...from there, you can probably find all these links, and more...


(This is the de facto place for Partch on the web.  Check out the "Show us
your instruments" section under "The Art Institute of Gualala" heading.  If
you haven't already, get your hands on Harry's book, _Genesis of a Muisc_.)


(Ivor Darreg was another experimental instrument builder, and Sonic Arts
carries on his work.)


(Ivor published many articles in _Xenharmonikon_, along with many other
interesting authors.  This home page has TOCs and errata, and a link to Frog
Peak music, which can furnish the back issues.)


(Speaking of experimental musical instruments...)


(The microtonal synthesis page is a phenomenal resource, showing the tuning
capabilities of nearly every synth made.)


(Dante Rosati retuned a guitar in Just Intonation, and made some great music
with it.)


(Harold Fortuin created a nifty microtonal keyboard instrument, and made some
cool music with it.)


(Kraig Grady has built an acoustic orchestra to play tunings designed by Erv
Wilson.  His music is unreal... literally.)


(Last, but certainly not least, John deLaubenfels has realized the world's
first practical adaptive tuning solution.)

...Good Luck!

Microthon '00                                                      11/14/00

Hi everybody, just a few notes on our recent Microthon in Manhattan...

Unfortunately, I missed everything prior to Pete Rose's _Medieval Nights_,
which, according to the program, consisted of...

Don Conreaux in _Outdoor Gonging_
David Eggar's _Four on the Floor_
Wim Hoogewerf's performance of _Preludes_ and _Mysteries_.
Pagano and Beardsley's _Ataraxya_
Joe Maneri's _Sharafuddin bin Yaya Maneri il Mulk_
Giacinto Scelsi's _L'ame Ailee_
Erik Nauman's _Refridgeration Segments_
Ron Kozak and Scott Lee's _Jazz_
John Negri's _Blue Guitar_

...so I can't comment on these, other than to say I was really looking
forward to Wim's and Pagano & Beardsley's segments, and I'm sorry I missed
them.  Also, I've heard nothing but praise for the Scelsi, and I regret
missing that, too.

As far as what I did see, Pete Rose's recorder playing was both refreshing
and impressive.  Marc Jones' _Mostly Electric String Quartet_ was very
interesting, but lost something due to the fact that only one voice was
performed live (the rest supplied from tape).  The highlight of the first
half for me was Erlich and Sarkissian's segment -- especially their third

Likewise, a long dinner kept me out of Lou Harrison's _Sinfony in Free
Style_, which I had been looking forward to (damn!), and also Judith Conrad's
performance of the _Chromatic Fantasy & Fugue in Dmin_.  What I did see in
the second half waxed a bit long, IMO, but Wim's premiere of Joseph Pehrson's
_Just in Time_ was cute, and Reinhard & Pierce's Wyschnegradsky segment was
*perfect*.  To close, Alejandro Sanchez tickled the Dinarra to some Jazz
standards, which was most enjoyable, if it did suffer from a poor
amplification setup.

I gather recent talk of Judith Conrad's performance was about the Bach, since
_Ill Tempered_ seemed to come off without a hitch.  Throughout the concert,
the basketball was annoying, but the acoustics and general atmosphere of the
venue were excellent.  I can understand the classical performer's quarrels
with the way the event is organized, compared to a normal "concert", but
given the sheer variety of musical styles being presented, and our usual
microtonal situation, Reinhard's event is nothing short of amazing.

I will say that I'm not sure I approved of the Partch, even though I have
been highly critical of the "Partch purists" on this list in the past... In
general, I feel the event could probably be more successful if Reinhard's
AFMM ensemble did fewer numbers, making the show shorter.

Greater-than 13-limit JI?                                         11/15/00

Dave Keenan wrote...

>Thanks for a good laugh David. It does rather point up the ridiculousness
>of claiming that some piece is greater-than 15 limit JI. Who could tell?
>Even 13 is pushing things.

I believe this statement is at best ambiguous.  Dave, if you mean to say what
limit we begin to encounter significant tolerance effects, I would say 7.

At what limit does increasing the limit add no new 'independent' ratios?  For
practical purposes, in a tonal setting, I would say 19 (in certain settings,
like using complete series segments in an almost timbre-like way, maybe much

I'll grant you that 19 adds only one 'independent' ratio as far as I can tell
-- 19:16.  But then, how do we define "independent"?  In theory, every ratio
falls under a part of every other ratio's distribution...

I can live without 19, but I can't live without 17 -- it's part of some
gorgeous chords, including my favorite "diminished 7th".  As for 13, I can
tune it by ear very quickly and to a great degree of accuracy, if I am
allowed reference pitches at 12 and 14, and I can tune each of those from a
single reference pitch -- any one you'd like from about G2 to G5.

JI fudge factor                                                    12/14/00

>For the record, note that I agree pretty much with Dave Keenan's restriction
>to "within +-0.5 cents" for a true JI label.  I've always contended that
>differences of 1 cent or less are negligible under most circumstances,
>excluding only such particular cases as La Monte Young sound installations.

I know I may make some enemies saying so, but I seriously doubt that anyone
could tell the difference if the tones of the dream house were randomly
fudged by 0.5 cents.

While changes greater than a cent may make a noticeable difference to those
who are very familiar with the dream house, I seriously doubt they would come
at the expense of any affect in the composition.  The dream house is full of
beating combination tones; I'm not really sure that JI is all that important
to it.  There, I said it.

>Show me any Barbershop Quartet singer, or any singer that can deliberately
>sing "within +-0.5 cents" of Just Intonation, and sustain this degree of
>"mechanical" accuracy over the course of entire compositions, and I will dub
>him "Man Machine".

Agreed.  And further, how do we intend to measure this sort of thing?  Even
if there was a way to get spectral data that accurate, the process of
assigning fundamentals to the parts, in order to measure their "accuracy"
with respect to JI, isn't determined to within +/- 0.5 cents by any existing

>In light of the above challenge, can it truly be shown that Barbershop
>Quartet singing is really the ironclad and unshakeable proof we need of the
>definition of JI? To clarify my point here: If indeed a "beatlessness" is
>perceived in the chords of this music, and yet we are saying also that there
>must be "within +-0.5 cents" accuracy for it to truly be considered JI, then
>which am I to understand is correct; an audible quality of beatlessness, or
>the "within +-0.5 cents" rendered accuracy of the performance?

Obviously the former.  Accuracy of barbershop?  Their singers are capable of
sustained chords tuned beatless to the limit of the timbres.  See my post,
"Barbershop Spectrogram", circa Dec. '98 (the files for the post are temp.
off line, but I will provide them upon request).

>The only question is whether +-5 cents qualifies as "true JI" vs.
>"quasi-JI".  Yes?

No!  That question is meaningless.  Shame on you all for entertaining it
under the guise of good scholarship.  If anyone disagrees, then he should
provide the criteria that were used to arrive at these numbers, and why a
binary distinction around them is warranted.

Further, given the multiplicity of criteria sets, with respect to physics
(limit, timbre, volume), effect (beating, periodicity), and affect
(noticeable difference, annoying difference, etc.), he should explain why the
particular criteria set should be the basis for the definition of such a
basic and general term as "just intonation".

I must be crazy.

Recent Partch                                                      12/15/00

>I have only ever heard one very short piece by Partch (I forget its name,
>something about a waterfall? Carl Lumma?)

_The Waterfall_, from Eleven Intrusions.

>and I certainly couldn't tell if it contained Just simultaneities or not,
>due to the previously mentioned rapidly decaying inharmonic timbres and
>rapid progress.

Listen again!  The adapted guitar (I think) plays clear just chords (which
the "waterfall" marimba arpeggios retrace).  BTW, howabout the change on

>I have however examined some of Partch's tunings (reproduced from his
>mathematical description of them), and given the freedom to sustain their
>chords and use harmonic timbres, I am quite prepared to agree that they are

The chromelodeon parts are usually the best for hearing the JI.  I like the
2-part harmony in the beginning of _Revelation_.  Thanks, Daniel, for the
excerpt from _Ring Around the Moon_.

RE: Beauty in the Beast                                            12/18/00

>>BTW- Easley Blackwood's 12 microtonal etudes (13-tET uo to 24-tET) is also
>>full of suprising flavours despite the cheesy sound of the old synth he

[David Beardsley]
>I find nothing cheesy about the synth.  Analog is so rich and warm.

I don't know if I'd call them warm, but I certainly agree with David that
they aren't cheesy.  I consider _Etudes_ a masterpiece of synthesis as well
as of composition.  As Easley explained on the phone, many of the timbres
were painstakingly chosen for each tuning, to maximize the concordance.
Problem was, he had to plan very carefully every spot he wanted a given
timbre, and lay down all of them before using another timbre, as exactly
re-creating a patch on the synth was impossible.

Fare-well                                                          12/22/00

Ladies and Gentlemen, it's been a grand ruckus!  Along the way I've learned
unspeakable things from places men fear to tread.  If I managed to make some
sort of contribution in the process, so much the better.  Many of you have
become close friends, as far as long-distance semi-professional acquaintance
allows...  We've made arguments, but hopefully, friendly and productive ones.
I'd like to stay subscribed forever, but I figure 3.5 years is a good run...

I'm moving to Berkeley again, to work for Keyspan (www.keyspan.com).  If
y'all need a Firewire adapter, drop me a line!  The line is...


...my clumma@nni.com address, fiction of my ISP here in Pennsylvania, will be
defunct very shortly.

I reserve the right to, at some future time, drop by.  In the meantone, I
wish you all productive musings, and good music!  I will un-subscribe on
Sunday, Dec. 24th.  Blessed solstice, all.