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:

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