Date: Mon, 25 Aug 2003.
Subject: Re: n-tet tunings by ear
> I'd also like to discuss some musical and structural
> possibilities inherent in these temperaments!!!
well, i guess you've come to the right place! here's a
"bingo card" for 28-equal, showing the 5-limit consonances that
can be formed with each pitch as its immediate neighbors...
http://groups.yahoo.com/group/tuning/files/perlich/28.gif
28-equal is one i've discussed early and often -- it's got nice
octatonic (diminished) scales, and you can modulate them around a
circle of 7-equal fifths if you want. 28-equal is not only a member
of the "diminished" family, it's also a member of the "negri" family
(which involves 9- and 10-note scales, all of whose steps except one
are equal), as you can see on the first graph here...
http://sonic-arts.org/dict/eqtemp.htm
the data for this graph is here...
http://tinyurl.com/4xuu
the ninth row of this table, which begins "major diesis", corresponds
to the diminished, 'octatonic' system of constructing scales and
chords. as you can see, the optimal period and generator for this
scale are [300, 94.1] -- meaning the scale repeats itself every 300
cents, 4 times per octave, and that the generator for new pitches is
about 94.1 cents. in 28-equal this would be approximated by 85.7
cents, while in 12-equal it's 100 cents.
as you can also see in that row, the the first three primes 2, 3, and
5 are approximated in this system as [4,0] [6,1] and [9,1] in terms
of the generator. this means that the prime 2 is approximated by...
4*300 + 0*94.1 cents = 1200 cents, or in 28-equal,
4*300 + 0*85.7 cents = 1200 cents;
the prime 3 is approximated by
6*300 + 1*94.1 cents = 1894.1 cents, or in 28-equal,
6*300 + 1*85.7 cents = 1885.7 cents;
the prime 5 is approximated by
9*300 + 1*94.1 cents = 2794.1 cents, or in 28-equal,
9*300 + 1*85.7 cents = 2785.7 cents.
the approximations other consonant intervals like 6:5 can be found by
vector addition: the represenation of 6:5 is the representation of 2
plus the represenation of 3 minus the representation of 5, since 6/5
= 2*3/5. the represenation of 6:5 is therefore...
[4,0] + [6,1] - [9,1] = [1,0], so the approximation to 6:5 is
always 300 cents in the diminished, 'octatonic' system.
the 8-tone-per octave scale created by applying the period an
infinite number of times and the generator once, is called the
octatonic or diminished scale and is probably familiar to you from
its 12-equal incarnation. trace it out on the 28-equal bingo card to
see that it contains 4 major triads and 4 minor triads per octave.
notice that these chords wouldn't all be possible if the bingo card
didn't repeat itself at the vector shown as "major diesis" on the
second graph back at...
http://sonic-arts.org/dict/eqtemp.htm
this is the reason the relevant row in the table is designated "major
diesis" (which is the name of the interval formed by the vector in
question on the just intonation bingo card, ratio 648:625).
the thirteenth row of the table corresponds to the "negri" system.
negri actually used the system within a 19-equal tuning, but it's
perfectly applicable (though with not as much harmonic purity) in
28-equal. see if you can run through the above analyses with the
negri system, which the table associates with a period and generator
of [1200, 126.238272] and, in terms of which, the primes 2, 3, and 5
are associated with [1,0], [2,-4], and [2,3] respectively. trace out
9- and 10-note scales on the 28-equal bingo card, and convince yourself
that all the chords wouldn't be simultaneously possible if not for
the 28-equal bingo card repeating itself at the vector shown as
"(negri)" on the second graph back at...
http://sonic-arts.org/dict/eqtemp.htm
...a vector which corresponds to the nameless 16875:16384 interval
in just intonation.
all these analyses could be carried out in the 7-limit as well, but
all the charts and diagrams would need an additional dimension, and
it's hard to show complicated 3-dimensional structures in 2
dimensions. it's been done on a case-by-case basis for some simple
cases, though.
hopefully this makes sense, you can have fun looking at other equal
temperaments and systems, and hopefully make some neat music from it
all!