[tuning-math] Geometric comma measures
- From: Gene W Smith <genewardsmith@xxxxxxxx>
- To: tuning-math@xxxxxxxxxxxxxxx
- Date: Wed, 17 Jul 2002 18:20:14 -0700
If we want to consider temperaments of codimension one, which is to say,
ones using a single comma, we need a way of measuring comma complexity,
and then of putting that together with the rms value of the comma. If we
use a size hueristic in place of the rms value, we then want a reasonable
way of putting together complexity (in particular, geometric complexity)
with size to get a comma goodness measure.
One way to do this is to appeal to Baker's theorem, which implies that if
L(q) is a Euclidean metric on the p-limit group (turning it into a
lattice), then good(q) = -ln(ln(q)/ln(L(q)) is bounded above, so there
are infinite sets of commas with
good(q) > A for a suitable choice of A.
Here is a list of all 7-limit intervals of size less than 50 cents,
within a radius of 10 of the unison, and such that
good(q) > 2.6:
[36/35, 49/48, 50/49, 64/63, 81/80, 245/243, 126/125, 4000/3969,
1728/1715, 1029/1024, 225/224, 10976/10935, 3136/3125, 5120/5103,
6144/6125, 65625/65536, 32805/32768, 2401/2400, 4375/4374]
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