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some tetrachordality results

🔗Carl Lumma <carl@lumma.org>

12/2/2001 6:53:16 PM

All;

I've implemented in scheme some of the stuff regarding the
generalized tetrachordality measure that went around a
while back.

My procedure finds the mean absolute deviation of a scale
from its transposition at 702 cents, first in scale order,
then in any order. Specifically, I do a brute force note-
to-note compare with all rotations of the notes of the
transposed scale in the former case, permutations in the
latter case, and return the minimum for each (source available).
Scales in ()s are degrees of 12-tET. Values are mean cents
deviation, rounded to the nearest cent.

Pentatonic Scale
(0 2 5 7 9) = 21, 21
1/1 9/8 4/3 3/2 27/16 = 18, 18

Diatonic Scale
(0 2 4 5 7 9 11) = 16, 16
1/1 9/8 5/4 4/3 3/2 5/3 15/8 = 16, 16

Diminished chord
(0 3 6 9) = 102, 102

Wholetone scale
(0 2 4 6 8 10) = 102, 102

Diminished scale
(0 2 3 5 6 8 9 11) = 50, 50
(0 1 3 4 6 7 9 10) = 102, 102

Minor scales w/'gypsy' tetrachord
(0 1 4 5 7 8 11) = 44, 44
(0 1 4 5 7 8 10) = 44, 44
(0 1 4 5 7 9 10) = 44, 44

The issues I'd like to bring to your attention are:

() Anybody care to compare results, or see
any obvious bugs? Debugging always appreciated.

() Different modes of a scale will give different
results, as with the diminished scale above. I
consider all "rotations", but I do not normalize
the rotations to their tonics to get modes. Maybe
I should? IOW, would we think omnitetrachordality
could be used find stable modes of a scale?

() Notice that the allowing permutations (2nd set of
values) never lowers the score. So I either have a
bug, or I don't need to fuss with permutations.
Normally, we could lower the meandev. between these
two...

(1 2 3 10 11 12)
(1 10 2 11 3 12)

By allowing permutations, vs. enforcing neighboring
order. However, maybe because I'm restricting myself
to transpositions, this sort of situation never arises.
Sound reasonable?

-Carl