Consistency

Daniel Wolf wrote,

>A resolution of 1.56 cents. So that a 3/2 is approximated by an offset of 1
>unit, but a 9/8 by 3 units. So for the TX81Z the best 9/8 is not equal to
>the sum of 3/2s.

In other words, 768-equal is not consistent within the 9-limit (although
7-limit is fine). A much simpler example is 15-equal. The chord 0 9 18 is
fine as a 3/2*3/2, but 0 18 is not the best 9/4 (0 17 is closer). So
stacking 3-limit chords poses no difficulties in 15-equal, and the same
holds through the 7-limit, but full 9-limit harmony per se is impossible, in
the sense of taking the best approximations to just intervals and expecting
them to be as related to one another as the just intervals themselves.

I wonder, in Brian's scheme of mapping real harmonic series to equal
temperaments in the quest for "better" timbres, what happens when you reach
a harmonic that is inconsistent with the temperament, e.g. the 11th harmonic
in 12-equal?


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