On harmonic entropy

Harmonic entropy is the simplest possible model of consonance.
Nothing simpler is possible. It asks the question, "how confused is
my brain when it hears an interval?" It assumes only one parameter in
answering this question.

Our _brain_ determines what pitch we'll hear when we listen to a
sound. It does so by trying to match the frequencies in the sound's
spectrum (timbre) with a harmonic series. The pitch we hear is high
or low depending on whether the frequency of the fundamental of the
best-fit harmonic series is high or low. The pitch corresponding to
the fundamental itself need not be physically present in the sound.
Sometimes, the meaning of "best-fit" will not be clear and we'll hear
more than one pitch. This happens when several tones are playing
together, or when the spectrum of the instrument is highly inharmonic.

Entropy is a mathematical measure of disorder, or confusion. For a
dyad, consisting of two tones which are sine waves or have harmonic
spectra, one can immediately understand the behavior of the harmonic
entropy function. The brain's attempt to fit the stimulus to a
harmonic series is quite unambiguous when the ratio between the
frequencies is a simple one, such as 2:1 or 3:2. More complex ratios,
or irrational ones far enough from any simple one, and the limited
resolution with which the brain receives frequency information makes
it harder for it to be sure about how to fit the stimulus into a
harmonic series. The resolution mentioned is parameterized by the
variable s. A computer program is used to calculate the entropy for
every possible interval (in, say, 1¢ increments). The set of
potential "fitting" ratios is chosen to be large enough (by going
high enough in the harmonic series) so that further enlargements of
the set cease to affect the basic shape of the harmonic entropy curve.

You can see some examples at

/tuning-math/files/dyadic/het01_16.j
pg

/tuning-math/files/dyadic/t2_015_13p
2877.jpg (the "weighting" referred to here is not a weighting of
anything in the model but merely refers to a computational shortcut
used).

Considering ratios to be different if they are not in lowest terms
(appropriate, for example, if we assume 6:3 might be interpreted as
the sixth and third harmonics, rather than simply as a 2:1 ratio)
leads to this slightly different appearance:
/tuning-math/files/dyadic/t3_01_13p2
877.jpg

Certain chords of three or more notes blend so well that it sounds
like fewer notes are playing than there actually are. We hear
a "root" which is kind of the overall pitch of the chord, and the
most stable bass note for it. The harmonic entropy of these chords
(which is not a function of the harmonic entropies of the intervals
in the chord) is low.

Our non-laboratory experiments on the harmonic entropy list seem to
conclusively show that the dissonance of a chord can't be even close
to a function of the dissonances of the constituent intervals. For
example, everyone put the 4:5:6:7 chord near the top of their ranking
of 36 recorded tetrads from least to most dissonant, while everyone
put 1/7:1/6:1/5:1/4 much lower. These two chords have the same
intervals. Therefore, it seems to be the case that dissonance
measures which are functions of dyadic (intervallic) dissonance
account for, at best, a relatively small portion of the dissonance of
chords. Such measures include those of Plomp and Levelt, Kameoka and
Kuriyagawa, and Sethares.

If you're interested in discussing further, please join the harmonic
entropy group:

/harmonic_entropy

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