RE: a definition of sensory dissonance

To recap some of my earlier postings and e-mails to J. Kukula:

I do believe that the place and periodicity mechanisms are both at play.
I also believe that Plomp's model gives a fine account of the
place-related component of dissonance, which I like to call roughness.
Combination tones complicate the matter but with a knowledge of the
amplitudes and frequencies of all combination tone components, Plomp's
algorithm can be still applied. But a phenomenon called "virtual pitch"
or "fundamental tracking" is central to Parncutt's treatment of
dissonance and does represent, I believe, an additional factor besides
critical band roughness. This phenomenon is clearly distinct from the
combination tone phenomenon, but it may have a lot to do with
periodicity mechanisms. There is a very strong propensity for the ear to
try to fit what it hears into one or a small number of harmonic series,
and the fundamentals of these series, even if not physically present,
are either heard outright, or provide a more subtle sense of overall
pitch known to musicians as the "root". As a component of consonance,
the ease with which the ear/brain system can resolve the fundamental is
known as "tonalness." I have proposed a concept called "relative
harmonic entropy" to model this component of dissonance. The harmonic
entropy is based on the concept that the critical band represents a
certain degree of uncertainty in the perception of pitch, and for any
"true" interval, the auditory system will perceive a range of intervals
spanning a number of simple-integer ratios. Simple-integer ratios come
into the picture because if the heard tones are to be understood as
harmonic overtones of some missing fundamental or root, they must form a
simple-integer ratio with one another. The range is a sort of
probability distribution, and a certain amount of probability is
associated with each of the simple-integer ratios.

One way of modeling this is with a Farey series and its mediants. The
Farey series of order n is simply the set all the ratios of numbers not
exceeding n, and the mediant between two consecutive fractions in a
Farey series is the sum of the numerators over the sum of the
denominators (this definition has many mathematical and acoustical
justifications). The simpler-integer ratios take up a lot of room,
defined as the interval between the mediant below and the mediant above,
in interval space, and so are associated with large "slices" of the
probability distribution, while the more complex ratios are more crowded
and therefore are associated with smaller "slices." Now the harmonic
entropy is defined, just like in information theory, as the sum over all
ratios of a certain function of the probability associated with that
ratio. The function is x*log(x). (See an information theory text to find
out why.) When the true interval is near a simple-integer ratio, there
will be one large probability and many much smaller ones. When the true
interval is far from any simple-integer ratios, many more complex ratios
will all have roughly equal probabilities. The entropy function will
come out quite small in the former case, and quite large in the latter
case. In the case of 700 cents, 3/2 will have far more probabilty than
any other ratio, and the harmonic entropy is nearly minimal. In the case
of 300 cents, 6/5 will have the largest probability in most cases, but
7/6, 13/11, and 19/16 will all have non-negligible amounts of
probability, so the harmonic entropy is moderate. In the case of 100
cents, 15/14, 16/15, 17/16, 18/17, 19/18, 20/19, and 1/1 will all have
significant probability, and the harmonic entropy is nearly maximal.

In terms of the periodicity model, we can imagine a process which
samples the signal for random periods of time (with some probability
distribution that is large for very short times and vanishes for long
enough times) and in each period, counts the cycles of each pitch to
come up with a ratio (or equivalently, to come up with a fundamental
frequency, of which the heard note will be harmonic overtones and
therefore possess a small-integer ratio by implication). Note that
harmonic partials within the heard tones are irrelevant because the
cycles here need not be sinusoidal for the counting to occur.

If logs to the base 2 are used in the definition above, the entropy
measures the expected amount of information, in bits, needed in an
optimal code to communicate the ratio being heard. So the entropy really
measures, in a sense, "cognitive dissonance." Now the exact probability
distribution of sampling times, or the order of Farey series one should
use, is something that may be difficult to determine. However, as the
order of the Farey series is increased more and more, the entropy curve
(defined as a function of interval width) continues rising but stops
changing shape (I have observed this numerically but not proved it
mathematically). In the limit of a Farey series of order infinity, one
should find a smooth "relative entropy" curve that gives a good
approximation of the ups and downs of the entropy curve for any
reasonably large finite order. These curves look remarkably like many of
the Helmholtz/Plomp curves that were derived from completely different
assumptions, and though they are meant to represent a completely
different component of dissonance, they lead to the same conclusions for
intervals of tones with some appropriate overtone structure.

However, when three or more notes are involved, the two components of
dissonance can have quite different behavior. Consider Partch's "otonal"
and "utonal" chords. Adding higher identities to both chords increases
the roughness of both by the same amount. But while the periodicity of
the otonal chords will be unchanged or perhaps multiplied by small
powers of two, the periodicity of the utonal chords increases
dramatically. Thus the process of counting will not be significantly
complicated, and may even be aided, by adding higher identities to the
otonal chords, while in the utonal case the likelihood of counting the
same relative numbers of cycles in each sampling period becomes very
small, and thus the entropy becomes very large. So the high-limit utonal
chords, though just as much minima of roughness as the corresponding
otonal chords, are almost impossible to assign a fundamental frequency
to and are therefore not minima of harmonic entropy.

It is often possible for the brain to look for periodicities among some
components of the signal and dismiss the rest as "noise." This is why
the root of a major triad does not appear to change when the third is
decreased from 5/4 through 11/9 to 6/5 and the chord becomes a minor
triad; although the minor triad can be understood as 10:12:15, these
numbers are already too high for the entropy of the entire signal to be
low enough to compete with the low entropy of the perfect fifth alone
(10:15 = 2:3); even the major third alone (12:15 = 4:5) is stronger and
can dominate if the "third" is in the bass. In the otonal case, looking
at any subset of the notes present (except 9:3, etc.) will lead to a
periodicity which is octave-equivalent to, if not identical to, that of
the entire chord, so various combinations of components of the signal
effect a reinforcement of the tonalness of the overall chord. How to
weigh the various subsets' contributions to the probabilities of
particular fundamentals in an overall analysis is unclear. Even without
the consideration of subsets, there appears to be no mathematical theory
of ratios of three of more numbers analogous to Farey theory, and no
easy way to create one. Unlike roughness, tonalness is not merely
concerned with pairwise interactions of tones but three-way and higher
interactions as well. A mathematical model for it is out of my grasp at
the moment.

It's amazing I get any work done around here . . .


SMTPOriginator: tuning@eartha.mills.edu
From: Carl Lumma
Subject: Erlich's Theory
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