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Date: Mon, 21 May 2001.
Subject: Friendly guide to what we were talking about here, part 1
I want to say hello and welcome to Marc Jones (who happened to play in
the same "key" in 32 as I did in 22 on the AFMM radio broadcast); to
Gary Morrison, a fine mind whose voice had been regrettably absent for a
while; and to all the silent people out there who are crazy enough to
stay subscribed.
Undoubtedly, many people are lost in the lingo (I'm imagining myself
day-trading) that comes up in the technical tuning discussions, and so
stop reading those threads after a while. So I thought I'd try to
outline the basic thinking for those who don't care to get mathematical,
and sum up what we've learned since that hasn't been done yet.
There is a long history of theorists whose work anticipates these ideas,
but for convenience I'll stick to just four: Partch and Wilson, Euler
and Fokker. I invite Kraig Grady to correct any errors in what follows
and provide an alternative viewpoint should he so desire.
Somewhere between Partch's "One-Footed Bride" graph and his numerical
theories, Partch enunciated a theory of consonance. It seems to be the
perfect way of applying psychoacoustics to the problem of intervals in
an octave-repeating tuning system. That is, if trying to decide what
set of pitches you want, given that they will repeat every 2:1 octave,
Partch gives you the measure of consonance to use.
Partch's conception of "tonality" was that one pitch was at the center
of all the consonances, i.e. it should be accompanied by all the pitches
consonant with it. This concept, married to his consonance model, bears
child in the form of the Tonality Diamond scales.
Erv Wilson showed how Partch's scales could be geometrically depicted,
with the tonic lying in the center of a fantastically symmetrical shape.
Erv's constructions relied on a concept called the lattice: every pitch
in a JI tuning system can be mapped to a point in space (sometimes
higher-dimensional space), such that its nearest neigbors are the
pitches most consonant with it.
Erv Wilson also created new scales which were the perfect theoretical
complement to Partch's Tonality Diamonds. Rather than containing one
central pitch, the new scales put every note in the scale in an equal
position, each sharing an equal number of consonances with its
neighbors. Yet there is no "space" in the lattice into which any
central (or more central) pitch could be inserted. He called these
scales "CPS scales". Very mathematical.
Getting back to Partch, Partch realized that his Tonality Diamond,
though symmetrically centered in the lattice, had some melodic "holes"
in it when ordered as an ascending or descending scale. Thus he added
some extra pitches to his scale, which partook of about as many strong
consonances as possible in the expanded scale.
Wilson, designing keyboards and instruments to play his CPS scales,
found it similarly logical to fill the melodic holes in his scale with
extra, consonant pitches. He thought of this process largely in terms
of a closed cycle of fifths, mainly the consonant 3:2 fifths, but with
some of the fifths "mutated" to be more complex JI ratios, in order to
accomodate the CPS scale. He used the same process when designing
keyboards for Partch's scale, and doubtlessly others of which I am
unaware.
Now we bring in one more theorist, Fokker. Fokker's main realization
was a simple but elusive one. Fokker studied the JI lattice. He
showed that by defining two or three or four (depending on the number of
dimensions in the lattice) small JI intervals to be "unisons", one could
divide the lattice into congruent "chunks" each of which, as a scale:
has a lot of consonant intervals (by virtue of it being a chunk); is
melodically pretty smooth; provides a "unison" to each of the infinity
of pitches in the lattice; tiles with "unison" copies of itself
indefinitely to fill the lattice. He called these scales PBs.
Fokker created just a few PBs of his own. But we've found that most of
Wilson's scales, including his CPS scales, and nearly enough Partch's
scale, actually _are_ PBs! Undoubtedly, Wilson understood the basic PB
insight in many different mathematical forms. For example, the cycle of
fifths in Wilson's keyboard designs can be visualized as a straight line
through the infinite lattice, along the direction representing the 3:2
interval. Meanwhile, the lattice is divided into PB chunks (chunky
peanut butter) sliced along the directions of the "unisons". Each time
the line of fifths crosses the boundary from one PB to the next, it must
"mutate" by a "unison" in order to compensate for the "unison" pitch
shift between one PB and its neighbor. Eventually, it passes through a
note in a distant PB that is the "image" of the beginning note in the
beginning PB, and the cycle is closed.
Another insight of Wilson had to do with another way of slicing the
infinite lattice, not into "unisonious" pieces but into pieces aligned
along the directions of the "prime" consonances (those relating a
fundamental and an overtone). This construction was found by Euler
centuries ago, and he called each piece a "genus". Fokker extended the
list of "primes", much like Partch did, and the higher-dimensional
"genus" became known as an EFG (Euler-Fokker genus). Wilson found that
each of his "families" of CPS scales, suitably "housed", would fit with
no gaps into an EFG. Thus the infinite lattice could be seen as an
infinite number of CPS scales, fitting together with no gaps. Or
alternatively, as Carl Lumma (I think) pointed out, as an infinite
number of Tonality Diamonds, with some CPSs filling the gaps.
So a PB, being a chunk of the lattice, has to contain a Tonality
Diamond, a CPS scale, some of both, or parts of both. Regardless, there
will generally be lots of consonant intervals. Partch (guided more by
judgment than by mathematics) and Wilson carefully constructed each of
their PBs to carefully fit around one Tonality Diamond or around one CPS
or one EFG, with as few extra notes as possible.