RE: More on spacing

>(1) Your modification of my timbre restriction to "generic harmonic timbres
>played at a reasonably loud volume" is fine. My own practical work has been
>with sine waves, however, (Rayna synthesizer) and I probably should have
>identified the subharmonic by lowest (theoretical) summation tone. (Your
>phrase has the virtue of sparing me a summation tone controversy; but the
>use of (theoretical) summation tones has been useful to me in the past in
>describing aspects of a significant repertoire of real music (Wagner,
>Debussy, Varese, Partch, Young, among others).

The effect of a subharmonic complex of sine waves seems to me to be
undistinguished in any respect. The combination tones will fill the entire
pitch spectrum, leading to lots of roughness and very little tonalness (or
tonal fusion), while the combination tones of a harmonic complex of sine
waves will form a single harmonic series, a very distinguished and
digestible sensation. It is only with the addition of harmonic spectra that
subharmonic chords become distinguished in any psychoacoustical way from
their immediate neighbors in n-1 dimensional pitch space.

>(2) I am interested in evaluating as many algorithms as possible. I am
>especially interested in the possibility of a sequential processing of
>intervals, from most simple to most complex, allowing for re-evaluation of
>the complex with the addition of each new interval.

Could you elaborate on this idea?

>(3) I have deliberately avoided the issue of temperaments which contain
>_puns_ of just intervals (being hearable as approximations of two or more
>distinctive rational intervals), because that is an issue best determined
>by diachronic context.

In a compositional sense, your characterization may be useful, but in a
purely psychoacoustical sense, a chord played will evoke certain
interpretations in terms of the harmonic series. This is what my algorithm
tells you, and the results are most interesting when applied to
temperaments. Are you still interested in evaluating this algorithm?

>(I must add my disagreement to your statement that
>an interpretation of a pitch complex as 10/12/15 or 16/19/32 (and, I may
>add, 54/64/81) makes "no musical difference". In terms of diachronic
>relationships, these difference are profound. Consider, for example, a
>progression from a tonic minor triad to the Major triad on the flat sixth
>degree (i to bVI, or c minor to Ab Major): each of the above
>interpretations of the minor triad suggests an alternative tuning for the
>second chord, and each
>progression is part of a very different scalar environment (resp. triadic
>just, extended just or harmonic, and pythagorean), each of which implies
>very different possibilities for going on further (try it for yourself:
>continue the progression by descending fifths to _return_ to the _tonic_.))

Of course, the only one that will work on paper is the pythagorean, but in
equal temperament, you can do it, too! The ear doesn't continue its harmonic
series math from one chord to the next; it doesn't know that 19 is not
factorable by 3 or whatever. The ear doesn't follow the calculations through
and say, "Ah, although the tonic is the same pitch as before, it is now
represented by a different number, so something must have been wrong with my
calculations -- let me go back and correct them." The issue of "puns" cannot
be avoided, since even a pythagorean minor triad will be heard, under
certain circumstances, as a 16:19:24, and never as 54:64:81 -- the ear's
central pitch processor simply doesn't go that far. 16:19:24 which is
subltely more tonal (rooted) a sound than 10:12:15, and that does make a
musical difference (I didn't say "no musical difference" -- indeed I
postulated a stylistic change in Western music that could be attributable to
this very difference). The reason the difference is subtle is that the main
impression made by these minor triads on the central pitch processor is that
of a perfect fifth with an added non-harmonic note. Of these minor triads,
the 10:12:15 may minimize roughness, but that is a seperable issue, I
believe. This is what I meant by interpretation -- not what you put down on
paper, but how the brain's central pitch processor understands the sound.

>(4) From this pitch spacing perspective, the "optimal frequency range" may
>be more relevantly identified with the perceived fundamental of a complex
>when mapped onto a harmonic series. The most profound musical conditions
>seem to be determined by the lower limit to _pitch_ perception. When a
>fundamental falls below this boundary, perception is made more difficult
>(although work with standing waves played for very long durations suggests
>that it is not impossible when additional _listening_ techniques (the sense
>of touch, for example) are called into play),

Actually, these are two seperate concepts, and a register-specific theory
would have to take _both_ into account. My algorithm is indeed derived by
first identifying this lower limit, and then allowing it to approach zero
(since the algorithm is to be register-unspecific). The pitch resolution of
the central pitch processor as a function of frequency would also have to be
taken into account by a register-specific theory. If the audible partials of
a chord fall into the optimal frequency range (the range where the pitch
resolution is greatest), the chord will be interpreted most accurately --
otherwise, the ear will resort to interpretations in terms of lower members
of the harmonic series.

>(5) My comment on emotional effect should not be taken too literally. I am
>aware of research on "happiness" and low electrical activity in the brain;
>the calculation hypothesis is my own; ascribing quantum effects to the
>infintessimal is not without controversy.

I didn't take your quantum business seriously; controversy is not the word.


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