More on spacing

Responding to Paul�s comments:

(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).

(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.

(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. (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_.))


(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),

(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.

Daniel Wolf

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