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Article: Harmonic Entropy

By Paul Erlich
  Note from Editor: We are thankful to Paul Erlich for his permission to reproduce this article and explanations, as also for his permission to reproduce the Harmonic entropy graph.

Harmonic entropy is the simplest possible model of consonance. 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.

Harmonic entropy graph

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.

Further clarifications about the units used in the graph on the X- and Y-axes, etc.:

Nats are a unit of entropy or information content. More familiar, from computer science, are bits. To convert from nats to bits, multiply be log2e = 1.442 695.

Cents are a logarithmic measure of musical interval size. For two frequencies p and q, the interval in cents between them is defined as 1200*log2(p/q).

n*d<10000 -- 10000 is the 'seed limit'. This means that the harmonic entropy is calculated by assuming that our central pitch processor could ideally recognize any ratio n/d such that n*d is less than 10000. This is chosen arbitrarily. Increasing this number has very little effect on the shape of the harmonic entropy curve, but increases the overall entropy level.

s=1.2% means that the resolution with which auditory information is relayed to our central pitch processor is assumed to be 1.2%. This is essentially the only free parameter in the harmonic entropy model. The appropriate value for s will depend on timbre, register, duration, the particular listener in question, and other factors. 1.2% is a generic value that seems to represent many typical situations well. The other graph used 0.6% which represents very fine hearing conditions.

1/sqrt(n*d) weighting: It turns out that the "width" of each ratio n/d in interval space -- the amount of "room" it has between neighboring ratios within the 'seed' set -- is approximately proportional to 1/sqrt(n*d). The approximation is very good when n and d are small, but breaks down as n*d approaches the 'seed limit' -- in this case, 10000. 1/sqrt(n*d) weighting simply means that rather than using the actual "widths", the 1/sqrt(n*d) proportionality is assumed to hold exactly, no matter how close n*d gets to 10000. This has only a small effect on the harmonic entropy curve, but makes it a lot less sensitive to the actual choice of 'seed limit'. Artifacts that result from the specific choice of 10000 get washed out by this method.




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