RE: [tuning] Re: EXPERIMENT: Xeno-Gothic, 17-tet, and "near-7-lim it"

Margo wrote,

>Thus the normal 17-tet intervals have a kinship to the so-called
>"near-7-limit" intervals of 24-note Xeno-Gothic, but one importance
>difference for our testing purposes: they are much further from
>7-based ratios than the ~3.80 cents of the septimal comma:

You mean septimal schisma?

> 17-tet interval cents 7-based ratio cents difference
> --------------- ------- ------------- ------- ----------
> M2 (3 steps) ~211.76 8:7 ~231.17 ~19.41
> m3 (4 steps) ~282.35 7:6 ~266.87 ~15.48
> M3 (6 steps) ~423.53 9:7 ~435.08 ~11.55
> M6 (13 steps) ~917.65 12:7 ~933.13 ~15.48
> m7 (14 steps) ~988.24 7:4 ~968.83 ~19.41

The corresponding table for 22-tET, used in a Pythagorean manner:

22-tet interval cents 7-based ratio cents difference
--------------- ------- ------------- ------- ----------
M2 (4 steps) ~218.18 8:7 ~231.17 ~12.99
m3 (5 steps) ~272.73 7:6 ~266.87 ~5.86
M3 (8 steps) ~436.36 9:7 ~435.08 ~1.28
M6 (17 steps) ~927.27 12:7 ~933.13 ~5.86
m7 (18 steps) ~981.82 7:4 ~968.83 ~12.99

Related to these better approximations is the fact that 17-tET is not even
consistent in the 7-limit (or 9-limit), while 22-tET is.

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