Mathematical explanation of consonance

Most of us who do a lot of work in
just-intonation would probably agree
pretty much with Polansky's "low primes
vs. few exponents is the basic idea".

But as Polansky and Erlich point out,
the issue is *much* more complicated
than a simplistic number theory will
allow. How does one explain the fact
that so many musicians who are less
"well tuned" than us will find 12-equal
"3rd"s to be consonant, for instance?
Most of us on this List wouldn't say
that, but lots of other listeners would.

Modern psychoacoustic research has found
many surprising results in connection with
proving or disproving old acoustical theories,
some of which have been very well-established
for a long time. Much more work still needs
to be done, and much of what I read about
this modern work calls into question some
of my own theories, and leads me to take a
more humble stance, ` la Partch in his
section on "resolutions", that I'm stating
not facts or rules, but rather, just my own
observations and speculations.

I think this is an eminently good topic for
discussion in this list. I've focussed on
studying historical tunings and theories,
and my own knowledge of modern research is
not as complete as I'd like it to be. It
would be good to have some of the main avenues
of this research succintly described, so that
in this particular forum different approaches
in the research could be brought together and
elaborated.

- Joe Monzo
monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html

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Sometime ago, I reported the following :

"Pitch bend, I checked with the range at 1-semitone and
found no clear pattern in the data values that caused each successive
pitch change.
etc"

Well, I just came upon this in Yamha's XG Guidebook (available on their
web site) :
"XG-compatible instruments follow all GM guidelines as described above,
and are also capable of receiving Polyphonic Key Pressure.
The shift curve used for pitch bend is defined as linear by cents,
ensuring pitch bend compatibility between XG instruments.
The XG Specification also states that Pitch bend can be used to affect
rhythm
channels as well as melody voices."

So, I'm somewhat annoyed/confused, since the anomaly means it is
not possible to enter precise pitch bend values into a sequencer without
using ye old frequency meter.

Then again, I suppose this to be a problem with just about any
commercial sample-playback device. Can anyone confirm my suspicion
that a synth that is actually a synth, synthesizing it's own waveforms
(i.e., FM, PM, Additive, etc.) is much more reliable at delivering
more stable/precise pitches.

Later,
Drew

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