Reply to John Chalmers

I think you are confusing the concepts of "consistency" and "articulation"
(or uniqueness). For example, 31tet does not uniquely articulate the
11-limit, since 11:10 and 12:11 are both represented by 4 steps. However,
31tet consistently represents the 11-limit, since given any 11-limit triad,
the sum of the best approximations to the inner intervals will be the best
approximation to the outer interval. In equal tunings with a small number of
notes, consistency is usually a less stringent criterion, while for large
numbers of notes, consistency will be the more stringent criterion. The
point of the consistency criterion is partially that consonance calculations
looking at all consonant intervals individually may not tell you anything
about consonant chords. Consistency with respect to triads, however, does
imply consistency with respect to tetrads, pentads, etc.

(By 11-limit triad I mean a triad made up of three just intervals, each of
which has a numerator and denominator not higher than 11 (aside from factors
of 2).)


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