To Gary, Paul E, Re: non-octave scales

> under certain strict conditions (in
> which the partial structure of the tones is more or less irrelevant),
> powers of 3 (and possibly of 5) do replicate the smilarity condition
> eliceted by powers of 2 in tonal or modal polyphony.

Aha! I'm working my way through Bill Sethares' manuscript at the moment, but
I would definitely be interested in reading that later. Thanks for the
offer/suggestion.

I can vouch for the meaningfulness of that suggestion in two regards: First,
I have briefly worked with ... "squashing" for lack of a better word ...
traditional music (melodies alone mostly, but not entirely) to a tuning of 12
equal steps per 3:2 instead of 12 equal steps 2:1. After listening to that for
a while, I found that 3:2 started to have a mild duplicating effect much like
the octave.

This of course harkens back to the debate not long ago about whether or not
you can hear primes or not. After he did his first rendition of his taped
lecture-demo "Introduction to Nontraditional Harmony", Dave Hill changed his
belief from the idea that each prime provides a new harmonic feeling, to his
belief that each ODD number provides a new feeling. As I think I mentioned, he
concluded that that was probably true because powers of two give a feeling of
duplication, thus ruling out multiples of 2 from having truly fundamental
effects upon harmony. Everything left though, he concluded, should have truly
fundamental effects. My immediate response to that was, "why do you conclude
that there are no other fundamental effects associated with other primes other
than 2's duplication effect? Why should we rule out the possibility that powers
of three or of five have a distinctive effects of their own?"

The second regard in which I can relate to your conclusion that other
intervals can have moderate duplicating effects, is with regard to my search in
88CET for pitch relationships that can function for note-doubling. I described
my conclusions in my series on 88CET, but the short version is that that is - to
a large extent anyway - a matter of the doubling interval needing only to be
more "bland" than the other intervals. For example in a comparatively
harmonically simple chord like a major triad, you can't get a doubling effect
from fifths. Our ears perceive the result as a basically different harmony,
like a seventh or ninth chord. But if you start with a more complex and intense
harmony, like a 7:9:11 chord, doubling a note by a much more easy-going,
simpler-sounding harmony like a fifth (or even a major tenth, perhaps
surprisingly) doesn't seem to have much effect on the overall character of the
chord.


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