truncating a lattice at a periodicity other than the octave

Here's another example of some of the interesting results that can
occur by truncating a lattice at a periodicity other than the octave.

This example was inspired by a bit I happened upon that mentioned
Olivier Messiaen's use of scales with "a concept of limited
transposability". One of the examples -- along with the 6-tone
"whole-tone", the 8-tone "diminished", and the 9-tone so-called
"Tcherepnin scale" -- was the 10-tone C Db D Eb F F# G Ab A B scale.
This is a rotation of the 8s2L symmetric decatonic; a familiar scale
to those that frequents list as the 22-tET, 0 2 4 7 9 11 13 16 18 20
decatonic has been proposed and championed by Paul Erlich in his
"Tuning, Tonality, and Twenty-Two-Tone Temperament" as a sort of
generalized major scale that treats the 4:5:6:7 as the basic
consonance.

If we allow that "P" = 1:2^(1/2), the 8s2L can be seen as a single
generator scale whose sL index is 1/4 0/1; by "sL index" I mean two
adjacent fractions whose denominators indicate the amount of small and
large steps in a given two-stepsize scale (the numerators indicate the
extreme boundaries of a given indexes generator).

If we think of this generator as a generalized fifth, I'll call it "Q"
after Paul's paper, then we could say that Q = 3/2^(3/2). This would
give the following one-dimensional, single generator chain:

588-90-192-294-396-498-(600/0)-102-204-306-408-510-12

This results in a sort of pseudo Pythagorean comma of ~12� at
2^(19/2):729.

Taking this to two dimensions would give the following decatonic
lattice with periodicity at the half-octave.

5/4----------15/2^(7/2)----45/32
/ \ / \ /
/ \ / \ /
/ \ / \ /
/ \ / \ /
/ \ / \ /
/ \ / \ /
1/1-----------3/2^(3/2)-----9/8

As 1:1 = 1:2^(2/1), this results in a sort of pseudo syntonic comma of
~10� at 45:2^(11/2).

Piggybacking this mapping results in a static symmetrical major
JI/temperament hybrid:

0 102 204 386 488 600 702 804 986 1088 1200
0 102 284 386 498 600 702 884 986 1098 1200
0 182 284 396 498 600 782 884 996 1098 1200
0 102 214 316 418 600 702 814 916 1018 1200
0 112 214 316 498 600 712 814 916 1098 1200

Running the first mapping:

5/4----------15/sqrt(2)*8
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
1/1-----------3/sqrt(2)*2---9/8

consecutively with the following:

5/sqrt(2)*3---5/4
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
4/3-----------1/1-----------3/sqrt(2)*2

would result in an ssLsssLsss JI/temperament hybrid of:

0 102 204 386 488 600 702 884 986 1098 1200
0 102 284 386 498 600 782 884 996 1098 1200
0 182 284 396 498 680 782 894 996 1098 1200
0 102 214 316 498 600 712 814 916 1018 1200
0 112 214 396 498 610 712 814 916 1098 1200
0 102 284 386 498 600 702 804 986 1088 1200
0 182 284 396 498 600 702 884 986 1098 1200
0 102 214 316 418 520 702 804 916 1018 1200
0 112 214 316 418 600 702 814 916 1098 1200
0 102 204 306 488 590 702 804 986 1088 1200

With the odd mix of just 3s and 5s, and irrationals nearer to 17s than
they are to 7s, this curious hybrid would probably find a better sonic
match if it were to be embedded in some of the higher consistency ETs
of the 1/4 0/1 seeded tree -- like 58 and 80-tET for example -- than
it would in 22-tET.

--Dan Stearns

Dan Stearns wrote,

>This example was inspired by a bit I happened upon that mentioned
>Olivier Messiaen's use of scales with "a concept of limited
>transposability". One of the examples -- along with the 6-tone
>"whole-tone", the 8-tone "diminished", and the 9-tone so-called
>"Tcherepnin scale" -- was the 10-tone C Db D Eb F F# G Ab A B scale.
>This is a rotation of the 8s2L symmetric decatonic; a familiar scale
>to those that frequents list as the 22-tET, 0 2 4 7 9 11 13 16 18 20
>decatonic has been proposed and championed by Paul Erlich in his
>"Tuning, Tonality, and Twenty-Two-Tone Temperament" as a sort of
>generalized major scale that treats the 4:5:6:7 as the basic
>consonance.

Well, the scale 0 2 4 7 9 11 13 16 18 20 quote is C Db D Eb F F# G A Ab B,
not the Messaien scale, but the one I do kind of champion in the paper, the
pentchordal decatonic scale.

However, I now like the symmetrical decatonic just as much (if you heard my
first piece at the Microthon you know why).