Harmonic entropy: Difference between revisions
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'''Harmonic Entropy''', sometimes abbreviated as "HE", is a simple model to quantify the extent to which musical chords exhibit various psychoacoustic effects, lumped together in a single construct called psychoacoustic '''concordance'''. It was invented by Paul Erlich and developed extensively on the Yahoo! tuning and harmonic_entropy lists. Various later contributions to the model have been made by Steve Martin, Mike Battaglia, Keenan Pepper, and others. | '''Harmonic Entropy''', sometimes abbreviated as "HE", is a simple model to quantify the extent to which musical chords exhibit various psychoacoustic effects, lumped together in a single construct called psychoacoustic '''concordance'''. It was invented by Paul Erlich and developed extensively on the Yahoo! tuning and harmonic_entropy lists. Various later contributions to the model have been made by Steve Martin, Mike Battaglia, Keenan Pepper, and others. | ||
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So for example, if we want to express the probability that the incoming dyad "400 cents" is perceived as the JI basis interval "5/4," we would write that as the conditional probability | So for example, if we want to express the probability that the incoming dyad "400 cents" is perceived as the JI basis interval "5/4," we would write that as the conditional probability | ||
<math>\newcommand{\cent}{\text{¢}}</math> | |||
<math>P(J=5/4|C=400\cent)</math> | <math>P(J=5/4|C=400\cent)</math> | ||
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Given a spreading function and set of basis rationals, there are two different procedures commonly used to assign probabilities to each rational. The first, the '''domain-integral approach''', works for arbitrary nowhere dense sets of rationals without any further free parameters. The second, the '''complexity-normalization approach''', has nice mathematical properties which sometimes make it easier to compute and which may lead to generalizations to infinite sets of rationals which are sometimes dense in the reals. It is conjectured that there are certain important limiting situations where the two converge; both are described in detail below. | Given a spreading function and set of basis rationals, there are two different procedures commonly used to assign probabilities to each rational. The first, the '''domain-integral approach''', works for arbitrary nowhere dense sets of rationals without any further free parameters. The second, the '''complexity-normalization approach''', has nice mathematical properties which sometimes make it easier to compute and which may lead to generalizations to infinite sets of rationals which are sometimes dense in the reals. It is conjectured that there are certain important limiting situations where the two converge; both are described in detail below. | ||
=Domain-Integral Probabilities= | ==Domain-Integral Probabilities== | ||
For sets of JI basis rationals which are nowhere dense, and in particular for a finite set of basis rationals, the log-frequency spectrum can be divided up into '''domains''' assigned to each ratio. Each ratio is assigned a domain with lower bound equal to the mediant of itself and its nearest lower neighbor, and likewise with upper bound equal to the mediant of itself and its nearest upper neighbor. If no such neighbor exists, <math>\pm \infty</math> is used instead. Mathematically, this can be represented via the following expression: | For sets of JI basis rationals which are nowhere dense, and in particular for a finite set of basis rationals, the log-frequency spectrum can be divided up into '''domains''' assigned to each ratio. Each ratio is assigned a domain with lower bound equal to the mediant of itself and its nearest lower neighbor, and likewise with upper bound equal to the mediant of itself and its nearest upper neighbor. If no such neighbor exists, <math>\pm \infty</math> is used instead. Mathematically, this can be represented via the following expression: | ||
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In the case where the set of basis rationals consists of a finite set bounded by Tenney or Weil height, the resulting set of widths is conjectured to have interesting mathematical properties, leading to mathematically nice conceptual simplifications of the model. These simplifications are explained below. | In the case where the set of basis rationals consists of a finite set bounded by Tenney or Weil height, the resulting set of widths is conjectured to have interesting mathematical properties, leading to mathematically nice conceptual simplifications of the model. These simplifications are explained below. | ||
=Complexity-Normalization Probabilities= | ==Complexity-Normalization Probabilities== | ||
It has been noted empirically by Paul Erlich that, given all those rationals with Tenney height under some cutoff <math>N</math> as a basis set, that the domain widths for rationals sufficiently far from the cutoff seem to be proportional to <math>\frac{1}{\sqrt{nd}}</math>. | It has been noted empirically by Paul Erlich that, given all those rationals with Tenney height under some cutoff <math>N</math> as a basis set, that the domain widths for rationals sufficiently far from the cutoff seem to be proportional to <math>\frac{1}{\sqrt{nd}}</math>. | ||