RE: 3 emails with Martin Braun

Hi Daniel -- e-mail 1 never came through, but thanks.

(H.E. List: this concerns some e-mails forwarded to me from Daniel Wolf,
containing conversations with Martin Braun.)

Yes, between 1kHz and 4kHz, it's been found that harmonics higher than 7 can
contribute to pitch perception. Most musical timbres in a medium to high
register have audible harmonics in this pitch range range -- and the only
thing stopping one from "understanding" arbitrarily high harmonic
relationships is the _resolution_ with which the central pitch processor
receives the pitch information about these harmonics. At the optimal point
in this range (about 3kHz), Goldstein found, among around 10 listeners,
central pitch processor accuracy implying a resolution ranging from 0.6% to
1.2%.

Even if one can only process pairs of harmonics between 260¢ and 520¢, the
dyadic harmonic entropy calculation remains essentially unchanged within
that range, since only ratios in and immediately bordering that range will
be considered. A visit to
/harmonic_entropy/files/dyadic/ and a download
of het006_16.zip will show that within this range, an best-case listener
(standard deviation 0.6%) will be able to perceive, according to the
harmonic entropy model, local minima at not only 4/3, 5/4, and 6/5, but also
9/7, and even a wee little one at 11/9.

Braun wrote,

> Harmonic chords generate several low-frequency periodicities in the
auditory
> system. The combination of these effects contributes to the specific
quality
> of a chord perception.

. . . and an entropy function on the relative saliences of these
periodicities can model one component of discordance.

. . . this doesn't mean that even more complex ratios can't be recognized
and tuned by ear, by eliminaiting beating. I've tuned 17/13 by ear by
eliminating beating. It's only that beating is an additional component of
discordance, beyond that modeled by harmonic entropy.

I wrote,

>Even if one can only process pairs of harmonics between 260¢ and 520¢, the
dyadic harmonic >entropy calculation remains essentially unchanged within
that range, since only ratios in and >immediately bordering that range will
be considered. A visit to
>/harmonic_entropy/files/dyadic/ and a download
of het006_16.zip >will show that within this range, an best-case listener
(standard deviation 0.6%) will be able >to perceive, according to the
harmonic entropy model, local minima at not only 4/3, 5/4, and >6/5, but
also 9/7, and even a wee little one at 11/9.

And 7/6 is in there too, of course!

P.S. For the Farey-based (rather than Tenney-based) version of this graph,
see /harmonic_entropy/files/perlich/ent_006.jpg -- it's
the same shape but with an overall downward slope.