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Genesis Minus as a block

🔗genewardsmith <genewardsmith@juno.com>

2/9/2002 1:33:47 AM

I undertook a proceedure to determine if "Genesis Minus", by which I mean Genesis less 11/10 and 20/11, is a block by my understanding of what a block is. That understanding is that a block is epimorphic and convex, where by "convex" I mean it is convexly closed: every lattice point contained in the convex hull of the octave equivalence classes of the scale are already in the scale. Equivalently, a block is epimorphic and has the property that there exists a norm such that no point can be added to the block without increasing its diameter. It seems Genesis Minus is a block by this definition (I hope Joe notices I said the word "definition".)

I downloaded and ran the "qhull" program, which gave me a set of inequalities defining the convex hull, which I converted into a vector space norm. By this norm, Genesis Minus is within a radius of one of the unison, whereas the nearest comma seems to be 385/384 at a distance of three. The Voroni cells of the lattice of 41-et commas using this distance measure give the appropriate tiling of the 4D space.

Here are the distances from unity of the tones of Genesis Minus according to this distance measure:

1 0
81/80 1.
33/32 1.
21/20 1.
16/15 1.
12/11 1.
10/9 1.
9/8 .5196155385
8/7 1.
7/6 1.
32/27 1.
6/5 1.
11/9 1.
5/4 1.
14/11 1.
9/7 1.
21/16 1.
4/3 .0392310770
27/20 1.
11/8 1.
7/5 1.
10/7 1.
16/11 1.
40/27 1.
3/2 .0392310770
32/21 1.
14/9 1.
11/7 1.
8/5 1.
18/11 1.
5/3 1.
27/16 1.
12/7 1.
7/4 1.
16/9 .5196155385
9/5 1.
11/6 1.
15/8 1.
40/21 1.
64/33 1.
160/81 1.

I'll run some more computations; computing this distance measure is a little slow.