Reply to David Finnamore

I thank you for your well-thought out comments on multidimensionality.
>From a compositional perspective, the points you raise are all very
relevant. But from a purely psychoacoustic perspective, it is unlikely
that making exceptions to the prevailing odd-limit by allowing large
powers of a small prime makes much sense. Using Kameoka & Kuriagawa, for
example, there is no sense in which 27/16 will be more consonant that
5/3 for any listener. It may be more familiar due to melodic properties
of the Pythagorean scale, but in terms of harmonic consonance, either
the listener is sensitive to the 5-limit and will prefer 5/3 or the
listener is not sensitive to the 5-limit and will simply find both
intervals dissonant. The point (in the "evolution" toward greater
dissonance) at which 9-limit intervals become consonant is somewhere
between the point at which 7-limit intervals become consonant and the
point at which 11-limit intervals become consonant. As for higher-limit
intervals, the conditions under which they are important depend more on
voicing to acheive certain virtual root and combination tone effects.

The assumption of exact octave equivalence, though it has some
psychoacoustic basis, should be taken more as a conveniece than an exact
truth. In some sense, it is not the odd-limit but the integer-limit that
really correlates with psychoacoustic consonance. However, in the
traditional approach of pondering octave-repeating scales, the
assumption of octave equivalence allows you the vast simplification of
only considering what goes on within one octave.

David, you may be interested to look into Georg Hajdu's paper in
Interface, he cites some mysterious formula that gives a measure of the
"complexity" of an integer; primality is a factor so that 7 is
considered more complex than 9. As I have touched on above, I doubt such
an approach is valid, but if you disagree with me, maybe you can find
something useful in his formula.

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I had written:
>>>>>>>>
Dan had responded:
>A look at the Sonatas will show you 3 in C Major, 3 in F Major, 2 in Bb
>Major, 1 in Eb Major, 1 in G Major, 3 in D Major, 1 in A Major, 1 in a
>minor, and the Sonata and Fantasie in c minor, exactly the extent of the
>meantone keys. Most of the Sonatas feature chromaticism, the spelling
>consistant with meantone set to Eb to A. I personally find the chromaticism
>in the Fantasien to be especially vivid in meantone. My difficulty with
>claims for a temperament other that meantone for Mozart lies chiefly in
>the absence of music in other keys.

Well, the key a piece is written in is only one of the tonalities
introduced. By "chromaticism" I meant not only the use of chromatic
non-harmonic tones, but also the frequent use of modulations to foreign
keys. A quick glance at my copy of the c minor Fantasia K.475 shows what I
would consider modulations from C minor to Bb minor to Db major to B major
all within the first 11 bars, including a big fat F#653 chord. That would
sound vivid indeed in meantone!

Bill

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^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
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