Pitch spacings

I am currently drafting a fairly detailed text (for Xenharmonikon,
possibly, but certainly for my own composition cogitation) about the
spacing (or voicing) of pitch materials. This subject has only been treated
in passing in harmony and orchestration texts. I will outline some of the
main points, in hope of getting some feedback:

Assumptions (big ones, I know, but useful in staking out the territory):
(a) heard materials are mapped by the listener onto a just structure. (b)
we are listening to an orchestra of sine waves (thus, the timbres of real
instruments are not considered). (c) we have unlimited sample lengths
(thus, perception of interval differences within the critical band is
possible and harmonic rhythm is not an immediate concern).

(1) General characterization of pitch spacing: (a) harmonically (larger
intervals at the bottom, smaller at top), (b) neutrally (no preponderence
of intervals) or (c) subharmonically. A characterization as harmonic
follows when a pitch complex is analyzable at a lower position in a
harmonic series than in a subharmonic series; a subharmonic
characterization follow a contrary analysis; a neutral characterization is
inconclusive.

(2) Specification of pitch complex: (a) harmonic sonorities by fundamental,
(b) neutral by central frequency - or ambiguously, between a fundamental
and a guiding tone, (c) subharmonic by guiding tone (the lowest pitch
common to the spectra of all pitches in the complex).

(3) Evaluation of relative complexity: lowest mapping of the complex onto a
harmonic series. (eg Major triad maps onto 4,5,6, minor onto 10,12,15 (or
possibly 6,7,9)) divided by frequency of the fundamental in that mapping.
(A method of calculating in greater detail needs to be established;
possible models are in Chalmers, Divisions of the Tetrachord).

(4) Cohensiveness: "gaps" in the series seem to increase instability or
ambiguity in perception. (eg the interval 3/2 is easier to perceive than
the interval 3/1). (Why is it easier to tune superparticulars?)
Suggestion: gaps increase the uncertainty of the harmonic/subharmonic
analysis.

Comment: I am encouraged in making a fundamental distinction between
harmonic and subharmonic materials by recent brain research identifying the
emotion of happiness with lowered electrical activity in the brain. Whether
this is caused by eating chocolate or the processing of small-number ratios
seems all the same. It makes sense that the harmonic series, marching
happily onward up to the infinite is much easier to process than the
subharmonic, going inexorably to the quantum depths of the infinitesimal.
Could this be part of the mechanism though which we class the Major triad
as "happy" and the minor as "sad".


Daniel Wolf
Ludwig-Landmann-Str. 84 B
60488 Frankfurt Germany
+49 69 764307
http://ourworld.compuserve.com/homepages/DJWOLF_MATERIAL




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