melodic coordinate systems

>From reading the list, it looks like I'm not the only person
using harmonic axes. However, I haven't seen any mention of
the full algebraic treatment I apply to them. I will outline
some of my ideas here, and someone can tell me whether they
are original and, if not, how they are notated.

First a column matrix needs to be defined to correspond to
harmonic axes:

oct (log(2))
H ----- (log(3))
log(2) (log(5))

H can be generalised to have as many dimensions as the matrix
you are multiplying it by. Using this, a fifth is (-1 1)H, a
major third (-2 0 1)H and a subminor seventh (-2 0 0 1)H.

H is useful for defining harmonies. For melodies, though,
it is more convenient to use the smallest intervals that
occur in a scale. For a diatonic scale with comma shifts,
these will be:

(t) ( 1 -2 1)
(s) ( 4 -1 -1) H
(p) (-4 4 -1)

Where t is a minor tone, s a diatonic semitone and p a
syntonic comma. These intervals completely define a 3-D JI,
and so the matrix can be inverted:

( 5 2 3) (t)
H ( 8 3 5) (s)
(12 4 7) (p)

This will be true of any three intervals whose defining
matrix has a determinant of +/- 1.

A fifth can now be written:

( 5 2 3)(t) (t)
(-1 1 0)H (-1 1 0)( 8 3 5)(s) (3 1 2)(s)
(12 4 7)(p) (p)


Other coordinate systems can be defined from t, s and p or
directly from H. One I find useful is:

(r5) (-3 -1 2 0) (12 7 3 0)(r5)
(q5) ( 7 0 -3 0)H H (19 11 5 0)(q5)
(p5) (-4 4 -1 0) (28 16 7 0)(p5)
(p7) ( 1 2 -3 1) (34 19 8 1)(p7)

You can add commas for as many harmonic dimensions as you
like. Equally tempered scales can then be defined
according to the number of steps in these intervals:

nTET r5 q5 p5 p7

12 1 0 0 0
15 1 0 1 0
19 1 1 0 0
22 1 1 1 1
31 2 1 0 0
34 2 1 1 0
41 2 2 1 1
46 3 1 1 0
53 3 2 1 1

This is the clearest representation I know of. From the
defining matrix, nTET 12*r5 + 7*q5 + 3*p5.

That'll do for the time being.

Graham

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