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First of all, I left out any comments to see what people would say. The
most important thing I left out was that local maxima and minima have
limited relevance unless your music uses continuous sweeps of the
interval spectrum. I have always held this as a (very mild) criticism of
some of Sethares's arguments. It only takes a tiny change in the
harmonic entropy function (say, a change of 1 in N) to convert a local
maximum into a local minimum or vice versa. The value of the function
need change only very little at any given interval, but the just ratios
will tend to be near these local extrema. The values of the function are
more important, however these are dependent on whether the allowed
fractions in the analysis are defined to have numerator less than N,
denominator less than N, num. + den. < 2N, etc. The choice of one of
these rules is a difficult one, but the local extrema, I think, should
be independent of this choice, which is why I only reported those.

>My immediate medievalist reaction: shouldn't 16:9 be listed here as a
>basic ratio at 996 cents? Somehow my Pythagorean predilections find it
>curious that this interval should be treated mainly as a point of
>maximum ambiguity between the 5-prime and 7-prime standards, rather
>than a 3-prime standard in its own right.

As you may recall, I don't believe in prime standards for consonance,
only odd standards. This analysis did not allow for any octave
equivalence efffects, so evens are just as good ass odds and integer
standards reign. 16 is a very high integer.

>Also, this interval seems
>very basic to me since it is derived from two pure 4:3 fourths.

I agree that it can be so derived in a musical context. However,
presented on its own, the ear has NO WAY of analysing the 16:9 into two
4:3s. This kind of "derivation" has no _acoustical_ relevance to the
interval on its own. Try tuning two strings a minor seventh apart and
see what ratios you are drawn to.

>I'm curious if this result for 16:9 might say anything about the
>Pythagorean m7 (which often gets used prominently in 13th-14th century
>music, and which I tend to agree with some medieval theorists has a
>certain degree of "compatibility" or even "concord")

In medieval times how sure could you be that your m7 wasn't a 7:4 or a
9:5, unless it was built up from two 4:3s, in which case my previous
caveat applies? Anyway, these three minor seventh differ relatively
little in the value of the entropy function.

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🔗Eduardo Sabat-Garibaldi <esabat@...>

🔗"Paul H. Erlich" <PErlich@...>