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Euclidean reduced 9-note scales

🔗genewardsmith <genewardsmith@juno.com>

4/18/2002 12:49:59 AM

Here are the Euclidean reductions for h9 in the 5 and 7 limits:

5-limit: [1, 16/15, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 15/8]

This is pretty much of a classic--a rhombic block.

7-limit: [1, 15/14, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 28/15]

This, with steps of size 16/15 and 15/14, as well as 9/8 and
28/25, fairly cries aloud to be tempered via 225/224~1. If we do that of course both scales become the same.

I looked at all of the permutations of the 72-et version of this scale, and the original form turned out to be the clear winner:

[0, 7, 19, 23, 30, 42, 49, 53, 65]

Here are the 5,7, and 9-limit characteristic polynomials:

x^9-16*x^7-16*x^6+57*x^5+84*x^4-34*x^3-84*x^2-8*x+16
x^9-21*x^7-28*x^6+65*x^5+100*x^4-71*x^3-116*x^2+26*x+44
x^9-29*x^7-80*x^6-39*x^5+70*x^4+52*x^3-16*x^2-9*x+2

We have 40 9-limit triads here, which looks pretty good!

Here is a runner-up scale:

[0, 7, 19, 23, 30, 42, 49, 53, 60]

x^9-16*x^7-16*x^6+57*x^5+86*x^4-27*x^3-80*x^2-18*x+6
x^9-19*x^7-20*x^6+67*x^5+76*x^4-79*x^3-84*x^2+30*x+28
x^9-27*x^7-64*x^6+3*x^5+104*x^4+37*x^3-40*x^2-14*x