our McMaster representative

I find this subject interesting, yet perhaps not too appropriate for this
list except in its categorization as "specialized allotonal musical
instruments".
Dr. Rapoport: have you seen the Neanderthal bone flute?

-Adam

_________________
Adam B. Silverman
153 Cold Spring Street; A3
New Haven, CT 06511
(203) 782-1765

abs22@pantheon.yale.edu



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Bill Sethares, in the manuscript for his up-coming book, describes a detailed
study of levels of discord. I think the study was conducted by Plomp and Levelt
(or something like that - I don't have the reference in front of me).

But anyway, two factors strike me as especially significant about Bill's
approach:
1. It builds from experimental results that strike me as hard to believe, but
I'm 90% confident are due to the fact that the experimenters used
sinewaves (their "dissonance" curve looked somewhat like the integral of
a typical bell-curve if you can imagine that), and
2. Bill uses this seemingly unrealistic set of experimental results with
sinewaves, to form what is almost certainly the most complete, accurate,
and practical approach to the meaning of "consonance" and "dissonance"
(the latter more accurately being called "discord").
Along the first lines - surprising conclusions of the indepth experimental study
- it concluded that minor ninths and major sevenths are about as consonant as
octaves. I find that difficult to swallow even with sinewaves. I have a
preconceived notion that major sevenths and minor ninths have to be
comparatively discordant in ANY timbre. I have to confess that, based upon his
accompanying tape, it could well be that my preconceived notion in that regard
is at least partly false.

But anyway, I'm about 98% certain that what I'll find as I read further, is
that he'll attribute the discord of major sevenths and minor ninths absolutely
100% to a "rough" (fast) beat between the second harmonic of the lower tone with
the fundamental of the higher tone. That suggests that half-wave symmetrical
tones (no even harmonics) will not produce discordant M7s and m9s, which again
strikes me as counterintuitive.

But the undeniable fact of the matter is that Bill's techniques and ideas
absolutely DO work. That in turn seems to make it hard to deny that what I
described above is indeed a case of preconceived notions.


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Ooops, I just realized that the integral of a bell curve is all wrong. Plomp
and Levelt's curve actually looks like the negative of the derivative of a bell
curve, if you can picture that.

Whatever ...


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Incidentally, Plomp and Levelt did find that even with sine waves, in-tune
octaves were more consonant than their out-of-tune neighbors. This has been
attributed to second-order beating, which can be thought of as beating
between the lower tone's summation tone with itself and the upper tone.
However, at a low enough volume, second-order beating should disappear. I
vaguely remember reading that it doesn't disappear, which was taken as
evidence for periodicity pitch detection.

Plomp and Levelt's main results need to be taken seriously in any theory of
consonance. K&K (two Japanese workers -- I can't remember their names)
calculated consonance curves based on Plomp and Levelt's results. K&K
assumed some typical harmonic series, and assumed (I think) that total
roughness is the sum of the individual sine-wave roughnesses calculated
according to P&L's results. They derived the usual consonance/dissonance
relations by defining consonant intervals as local minima on the graph.
Clearly, with inharmonic timbres, none of the usual consonant intervals
would appear (except the unison), and a whole new set would arise. This is
the basis of Sethares' work, I believe.

However, there is an additional component to consonance that is a somewhat
separable issue. It goes under many names, such as "spectral fusion,"
"rootedness," etc., and is the degree to which the entire set of sine waves,
belonging to more than one complex tone, approximates a single harmonic
series. The harmonic series is privileged in our auditory system as the
information is "compressed" into a single pitch (the fundamental, even if it
is physically absent) and timbre (which represents the relative amplitudes
of the harmonics). For instruments with harmonic timbres, this phenomenon
explains the difference in consonance between "otonal" and "utonal" chords,
whose roughness (if equal to the sum of the roughnesses of the individual
intervals) is identical, but whose consonance is not. For minor vs. major
triads, this may be arguable, but for complete 11-limit hexads, it's pretty
easy to hear that the utonal chord is less consonant. This is because the
otonal chord forms a single harmonic series, while the utonal chord can only
be considered a harmonic series if four- and five-digit harmonic numbers are
allowed. Tones with completely inharmonic spectra may be formed into chords
that have the same degree of roughness as an "otonal" or "utonal" chord of
harmonic spectra, but this other component of consonance will favor both of
the chords with harmonic spectra over the inharmonic one. This is because
any pair of tones, even in the utonal chord, forms a harmonic relationship,
and, particularly for the perfect fifth, the set of harmonics resulting from
the pair of tones forms a single harmonic series (in the perfect fifth, this
harmonic series is 2/3 complete).


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The two factors contributing to consonance, roughness and tonalness, are
often associated with Helmholtz and Rameau, respectively. See John R.
Pierce's book, "The Science of Musical Sound" for a candid discussion.
Pierce expected (and Sethares' theory would conclude) that a piece of music
would retain its character if all intervals, including those between
partials, were stretched by a fixed factor. He found to his surprise that
this was not the case.


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