Harmonic entropy: Difference between revisions

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'''Harmonic entropy''' ('''HE''') is a simple model to quantify the extent to which musical chords align with the harmonic series, and thus tend to partly "fuse" into the perception of a single sound with a complex timbre and '''virtual fundamental''' pitch. It was invented by Paul Erlich and developed extensively on the Yahoo! tuning and harmonic_entropy lists, and draws from prior research by Parncutt and Terhardt. Various later contributions to the model have been made by Steve Martin, Mike Battaglia, Keenan Pepper, and others.

Note: the terms dyad, triad and tetrad usually refer to chord with 2, 3 or 4 [[Pitch class|pitch classes]]. But in this discussion they refer to chords with 2, 3 or 4 <u>pitches</u>. Thus C-E-G-C is a tetrad not a triad.

== Background ==

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The general idea of Harmonic Entropy is to first develop a discrete probability distribution quantifying how strongly an arbitrary incoming dyad "matches" every element in a set of basis rational intervals, and then seeing how evenly distributed the resulting probabilities are. If the distribution for some dyad is spread out very evenly, such that there is no clear "victor" basis interval that dominates the distribution, the dyad is considered to be more discordant; on the other extreme, if the distribution tends to concentrate on one or a small set of dyads, the dyad is considered to be more concordant.

A clear mathematical way of quantifying this "dispersion" is via the [~~http~~://en.wikipedia.org/wiki/Entropy_(information_theory) Shannon entropy] of the probability distribution, which can be thought of as describing the "uncertainty" in the distribution. A distribution which has a very high probability of picking one outcome has low entropy and is not very uncertain, whereas a distribution which has the probability spread out on many outcomes is highly uncertain and has a high entropy.

A clear mathematical way of quantifying this "dispersion" is via the [https://en.wikipedia.org/wiki/Entropy_(information_theory) Shannon entropy] of the probability distribution, which can be thought of as describing the "uncertainty" in the distribution. A distribution which has a very high probability of picking one outcome has low entropy and is not very uncertain, whereas a distribution which has the probability spread out on many outcomes is highly uncertain and has a high entropy.

=== Definitions ===

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== Harmonic Rényi Entropy ==

An extension to the base Harmonic Entropy model, proposed by Mike Battaglia, is to generalize the use of [~~http~~://en.wikipedia.org/wiki/Entropy_(information_theory) Shannon entropy] by replacing it instead with [~~http~~://en.wikipedia.org/wiki/R%C3%A9nyi_entropy Rényi entropy], a [~~http~~://en.wikipedia.org/wiki/Q-analog q-analog] of Shannon's original entropy. This can be thought of as adding a second parameter, called <math>a</math>, to the model, reflecting how "intelligent" the brain's "decoding" process is when determining the most likely JI interpretation of an ambiguous interval.

An extension to the base Harmonic Entropy model, proposed by Mike Battaglia, is to generalize the use of [https://en.wikipedia.org/wiki/Entropy_(information_theory) Shannon entropy] by replacing it instead with [https://en.wikipedia.org/wiki/R%C3%A9nyi_entropy Rényi entropy], a [https://en.wikipedia.org/wiki/Q-analog q-analog] of Shannon's original entropy. This can be thought of as adding a second parameter, called <math>a</math>, to the model, reflecting how "intelligent" the brain's "decoding" process is when determining the most likely JI interpretation of an ambiguous interval.

=== Definitions and Background ===

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 '''Harmonic entropy''' ('''HE''') is a simple model to quantify the extent to which musical chords align with the harmonic series, and thus tend to partly "fuse" into the perception of a single sound with a complex timbre and '''virtual fundamental''' pitch. It was invented by Paul Erlich and developed extensively on the Yahoo! tuning and harmonic_entropy lists, and draws from prior research by Parncutt and Terhardt. Various later contributions to the model have been made by Steve Martin, Mike Battaglia, Keenan Pepper, and others.
+Note: the terms dyad, triad and tetrad usually refer to chord with 2, 3 or 4 [[Pitch class|pitch classes]]. But in this discussion they refer to chords with 2, 3 or 4 <u>pitches</u>. Thus C-E-G-C is a tetrad not a triad.
 == Background ==
@@ Line 35: / Line 37: @@
 The general idea of Harmonic Entropy is to first develop a discrete probability distribution quantifying how strongly an arbitrary incoming dyad "matches" every element in a set of basis rational intervals, and then seeing how evenly distributed the resulting probabilities are. If the distribution for some dyad is spread out very evenly, such that there is no clear "victor" basis interval that dominates the distribution, the dyad is considered to be more discordant; on the other extreme, if the distribution tends to concentrate on one or a small set of dyads, the dyad is considered to be more concordant.
-A clear mathematical way of quantifying this "dispersion" is via the [http://en.wikipedia.org/wiki/Entropy_(information_theory) Shannon entropy] of the probability distribution, which can be thought of as describing the "uncertainty" in the distribution. A distribution which has a very high probability of picking one outcome has low entropy and is not very uncertain, whereas a distribution which has the probability spread out on many outcomes is highly uncertain and has a high entropy.
+A clear mathematical way of quantifying this "dispersion" is via the [https://en.wikipedia.org/wiki/Entropy_(information_theory) Shannon entropy] of the probability distribution, which can be thought of as describing the "uncertainty" in the distribution. A distribution which has a very high probability of picking one outcome has low entropy and is not very uncertain, whereas a distribution which has the probability spread out on many outcomes is highly uncertain and has a high entropy.
 === Definitions ===
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 == Harmonic Rényi Entropy ==
-An extension to the base Harmonic Entropy model, proposed by Mike Battaglia, is to generalize the use of [http://en.wikipedia.org/wiki/Entropy_(information_theory) Shannon entropy] by replacing it instead with [http://en.wikipedia.org/wiki/R%C3%A9nyi_entropy Rényi entropy], a [http://en.wikipedia.org/wiki/Q-analog q-analog] of Shannon's original entropy. This can be thought of as adding a second parameter, called <math>a</math>, to the model, reflecting how "intelligent" the brain's "decoding" process is when determining the most likely JI interpretation of an ambiguous interval.
+An extension to the base Harmonic Entropy model, proposed by Mike Battaglia, is to generalize the use of [https://en.wikipedia.org/wiki/Entropy_(information_theory) Shannon entropy] by replacing it instead with [https://en.wikipedia.org/wiki/R%C3%A9nyi_entropy Rényi entropy], a [https://en.wikipedia.org/wiki/Q-analog q-analog] of Shannon's original entropy. This can be thought of as adding a second parameter, called <math>a</math>, to the model, reflecting how "intelligent" the brain's "decoding" process is when determining the most likely JI interpretation of an ambiguous interval.
 === Definitions and Background ===