RE: [tuning] Answer to Carl Lumma on min7

Hi Pierre,

Although there is a possible psychoacoustical justification for subharmonic
chords in the common overtone (guide tone), your "inverse differential" is,
I'm afraid, pure fantasy. The ear produces combination tones of many
different orders (e.g, 2a-b, 5a-3b, etc.) whose loudnesses as a (nonlinear)
function of a and b have been very well measured (see, for example, Plomp's
_Aspects of Tone Sensation_. For the most part, these causes of these
combination tones are very well understood, resulting from the ear's
nonlinear response to loud input stimuli (see, for example, the Feynman
Lectures in Physics to see a derivation of how combination tones result from
nonlinear response curves). However, there is no phenomenon (analogous or
otherwise) that would produce the "inverse differential" tones.

Pierre, following up on Wim's remarks -- you should use your own ears,
rather than just awaiting the consensus of others, to inform your
theoretical speculations. To what extent have you done so?

-Paul