RE: Consistency generalized

Paul Hahn,

Your generalization is quite correct and produces the desired result of a
more stringent criterion. The musical meaning of integer levels of
consistency higher than 1, namely, the importance of matching just
intonation in combinations of n-limit intervals, I find dubious. So level 2
consistency seems like an arbitrary requirement, as level 2.1 consistency
and level 1.9 consistency would lead to results which are slightly different
but just as meaningful musically.

One thing I looked at long ago was what we would now call level 3
consistency, since 12-tET is just barely level 3 consistent at the 5-limit.
It seemed reasonable that an equal temperament could be "fuzzified" so that
it covered one-third of frequency space, while two-thirds remained
out-of-tune with respect to the equal temperament. The simplest qualifiers
are:


3-limit: 5-tET
5-limit: 12-tET
7-limit: 31-tET
9-limit: 171-tET
11-limit: 342-tET (=2*171)

My Commodore 64 couldn't do the higher-limit calculations within a
reasonable period of time, but these days it's just a few seconds:

13-limit: 5585-tET (=5*1117)
15-limit: 5585-tET
17-limit: 16808-tET (=22*764)
19-limit: 20203-tET (=89*227)

-Paul Erlich


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