Harmonic entropy: Difference between revisions

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'''Harmonic entropy''' ('''HE''') is a simple model to quantify the extent to which musical [[chord]]s 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.

'''Harmonic entropy''' ('''HE''') is a simple model to quantify the extent to which musical [[chord]]s 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.

A simple way to state this, is: harmonic entropy measures degree of certainty in the perception of the (virtual) [[root]].

Harmonic entropy 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.

An interactive harmonic entropy graph can be found in [[Scale Workshop]] version 3 in the Analysis tab.

Note: the terms dyad, triad and tetrad usually refer to chord with 2, 3, or 4 [[pitch class]]es. But in this discussion they refer to chords with 2, 3, or 4 ''pitches''. Thus {{dash|C, E, G, C}} is a tetrad instead of a triad.

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Lastly, you will note that for the special value {{nowrap|''w'' {{=}} 0.5}}, corresponding to the usual <math>\sqrt{nd}</math> weighting, we end up dividing by the term ζ(1). This is the only pole in the zeta function, so we wind up dividing by infinity, making the entire function zero, as pointed out by Martin Gough. However, as we can get arbitrarily close to {{nowrap|''w'' {{=}} 0.5}} and still exhibit the behavior that the unreduced and reduced functions are scaled versions of one another, we can simply use the unreduced version of exp-UHE for {{nowrap|''w'' {{=}} 0.5}} and consider it equivalent to reduced exp-UHE in the limit.

~~== Todo ==~~

~~There are a number of things that need to be added to this article. Below are listed some for reference:~~

* 3HE, both for finite HE and for {{nowrap|''N'' → ∞}}

* Write-up of fast computation for infinite zeta-UHE, perhaps with a zeta table

* Addition of many more pictures

== References ==

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* [https://yahootuninggroupsultimatebackup.github.io/harmonic_entropy Harmonic entropy group on Yahoo] (archive)

* [http://www.mikebattagliamusic.com/HE-JS/HE.html Harmonic entropy graph calculator (JavaScript)]

== See also ==

* [[Harmonic entropy of just intervals]]

* [[Low harmonic entropy linear temperaments]]

* [[User:Sintel/Validation of common consonance measures]] — an informal empirical study which calls into question the effectiveness of harmonic entropy for explaining [[consonance]]

== Todo ==

{{todo|inline=1|complete section|text=Add 3HE, both for finite HE and for ''N'' → ∞.}}

{{todo|inline=1|complete section|add table|text=Write-up of fast computation for infinite zeta-UHE, perhaps with a zeta table.}}

{{todo|inline=1|add illustration|text=Addition of many more pictures.}}

[[Category:Terms]]

@@ Line 1: / Line 1: @@
-'''Harmonic entropy''' ('''HE''') is a simple model to quantify the extent to which musical [[chord]]s 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.
+'''Harmonic entropy''' ('''HE''') is a simple model to quantify the extent to which musical [[chord]]s 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.
+A simple way to state this, is: harmonic entropy measures degree of certainty in the perception of the (virtual) [[root]].
+Harmonic entropy 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.
+An interactive harmonic entropy graph can be found in [[Scale Workshop]] version 3 in the Analysis tab.
 Note: the terms dyad, triad and tetrad usually refer to chord with 2, 3, or 4 [[pitch class]]es. But in this discussion they refer to chords with 2, 3, or 4 ''pitches''. Thus {{dash|C, E, G, C}} is a tetrad instead of a triad.
@@ Line 787: / Line 793: @@
 Lastly, you will note that for the special value {{nowrap|''w'' {{=}} 0.5}}, corresponding to the usual <math>\sqrt{nd}</math> weighting, we end up dividing by the term ζ(1). This is the only pole in the zeta function, so we wind up dividing by infinity, making the entire function zero, as pointed out by Martin Gough. However, as we can get arbitrarily close to {{nowrap|''w'' {{=}} 0.5}} and still exhibit the behavior that the unreduced and reduced functions are scaled versions of one another, we can simply use the unreduced version of exp-UHE for {{nowrap|''w'' {{=}} 0.5}} and consider it equivalent to reduced exp-UHE in the limit.
-== Todo ==
-There are a number of things that need to be added to this article. Below are listed some for reference:
-* 3HE, both for finite HE and for {{nowrap|''N'' → ∞}}
-* Write-up of fast computation for infinite zeta-UHE, perhaps with a zeta table
-* Addition of many more pictures
 == References ==
@@ Line 801: / Line 800: @@
 * [https://yahootuninggroupsultimatebackup.github.io/harmonic_entropy Harmonic entropy group on Yahoo] (archive)
 * [http://www.mikebattagliamusic.com/HE-JS/HE.html Harmonic entropy graph calculator (JavaScript)]
+== See also ==
+* [[Harmonic entropy of just intervals]]
+* [[Low harmonic entropy linear temperaments]]
+* [[User:Sintel/Validation of common consonance measures]] — an informal empirical study which calls into question the effectiveness of harmonic entropy for explaining [[consonance]]
+== Todo ==
+{{todo|inline=1|complete section|text=Add 3HE, both for finite HE and for ''N'' → ∞.}}
+{{todo|inline=1|complete section|add table|text=Write-up of fast computation for infinite zeta-UHE, perhaps with a zeta table.}}
+{{todo|inline=1|add illustration|text=Addition of many more pictures.}}
 [[Category:Terms]]