Harmonic entropy

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=Introduction= 
**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.

=Background= 
The general workings of the human auditory system lead to a plethora of well-documented and sonically interesting phenomena that can occur when a musical chord is played:
* The perception of partial **timbral fusion** of the chord into one complex sound
* The appearance of a **virtual fundamental** pitch in the bass
* Timbral **beatlessness**, compared to mistunings of the chord in the surrounding area
* The appearance of a quick fluttering effect sometimes known as **periodicity buzz**

All of these effects share two common characteristics for chords played with harmonic timbres:
* <span style="line-height: 1.5;">they tend to appear most strongly for those chords with large subsets that correspond to simple chunks of the harmonic series</span>
* <span style="line-height: 1.5;">the effects produced exhibit some degree of tolerance for mistuning</span>

As the presence or absence of these effects tend to appear in tandem and are highly correlated with one another, we can speak of a general notion of the psychoacoustic **concordance** of an interval - the degree to which effects such as the above will appear when an arbitrary musical chord is played. Additionally, chords which are very inharmonic often exhibit a quality known as psychoacoustic **discordance**.

While psychoacoustic concordance is not a feature universal to all styles of music, it has been utilized significantly in Western music in the study of intonation. For instance, flexible-pitch ensembles operating within 12-EDO, such as barbershop quartets and string ensembles, will often adjust intonationally from the underlying 12-EDO reference to maximize the concordance of individual chords. Indeed, the entire history of Western tuning theory -- from meantone temperament, to the various Baroque well-temperaments, to 12-EDO itself, to the modern [[@xenharmonic/Regular Temperaments|theory of regular temperament]] -- can be seen as an attempt to reason mathematically about how to generate manageable tuning systems that will maximize concordance and minimize discordance.

The Harmonic Entropy model is a simple way of quantifying how much an arbitrary chord will exhibit psychoacoustic concordance.

Concordance has often been confused with actual musical consonance, an unfortunate fact made more common by the<span style="line-height: 1.5;"> psychoacoustics literature under the unfortunate name </span>**<span style="line-height: 1.5;">sensory consonance</span>**<span style="line-height: 1.5;">, most often used to refer to phenomena related to roughness and beatlessness specifically. This is not to be confused with the more familiar construct of tonal stability, typically just called "consonance" in Western common practice music theory and sometimes clarified as "musical consonance" in the music cognition literature. To make matters worse, the literature has also at times referred to concordance -- and not tonal stability -- as </span>**<span style="line-height: 1.5;">tonal consonance</span>**<span style="line-height: 1.5;">, often referring to phenomena related to virtual pitch integration, creating a complete terminological mess. As a result, the term "consonance" has been completely avoided in this article.</span>

=Model= 
The original Harmonic Entropy model limited itself to working with dyads. More recently, work by Steve Martin and others has extended this basic idea to higher-cardinality chords. This article will concern itself with dyads, as the dyadic case is still the most well-developed, and many of the ideas extend naturally to larger chords without need for much exposition.

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 is via <span style="line-height: 1.5;">the [[@http://en.wikipedia.org/wiki/Entropy_(information_theory)|Shannon entropy]] of the probability distribution:</span>

[[math]]
H(d) = -\sum_{b} p_d(b) \log p_d(b)
[[math]]

where H(d) is the Shannon entropy of the dyad d, the b are all of the basis rationals in the set, and the p<span style="font-size: 90%; vertical-align: sub;">d</span>(b) is the probability assigned to basis rational b given an input dyad of d. This is the Harmonic Entropy of the dyad d.

In order to systematically assign a probability distribution to this dyad, we first start by defining a **spreading function** that dictates how the dyad is "smeared" out in log-frequency space, representing how the auditory system allows for some tolerance for mistuning. The typical choice that we will assume here for a spreading function is a Gaussian distribution, with mean set to be centered around the incoming dyad, and standard deviation **s** typically taken as a free parameter in the system.

A fairly typical choice of settings for a basic dyadic HE model would be:
* The set of basis rationals is the set of all intervals bounded by some maximum Tenney height, with the bound typically notated as **N** and set to at least 10000
* The spreading function is typically a Gaussian distribution with s=17 cents, or about a frequency deviation of 1% either way

Other spreading functions have also been explored, such as the use of the "Vos function" of a·exp(b|x|) rather than the Gaussian distribution. We will assume the Gaussian distribution as the spreading function for the remainder of this article, so that the spreading function for a dyad d can be written as follows:

[[math]]
s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}
[[math]]

where //s// becomes the standard deviation of the Gaussian, being an ASCII-friendly version of the more familiar symbol σ for representing the standard deviation. We assume here that the variable x represents cents, and will adopt this convention for the remainder of the article.

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= 
For sets of basis rationals which are nowhere dense, the log-frequency spectrum can be divided up into **domains** assigned to the basis rationals. Each rational 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, ±∞ is used instead. Mathematically, this can be represented via the following expression:

[[math]]
p_d(b) = \int_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx
[[math]]

where s<span style="vertical-align: sub;">d</span>(x) is the spreading function associated with d<span style="line-height: 1.5;">, b</span><span style="line-height: 1.5; vertical-align: sub;">l</span><span style="line-height: 1.5;"> and b</span><span style="line-height: 1.5; vertical-align: sub;">u</span><span style="line-height: 1.5;"> are the domain lower and upper bounds associated with basis rational b, and ¢(f) = 1200·log2(f), or the "cents" function converting frequency ratios to cents. Normally, b</span><span style="line-height: 1.5; vertical-align: sub;">l </span><span style="line-height: 1.5;">is set equal to the mediant of b and its nearest lower neighbor (if it exists), or -∞ if not; likewise with b</span><span style="line-height: 1.5; vertical-align: sub;">u</span><span style="line-height: 1.5;">.</span>

This process can be summarized by the following picture, taken from [[@http://sethares.engr.wisc.edu/paperspdf/HarmonicEntropy.pdf|William Sethares' paper on Harmonic Entropy]]:

[[image:http://i.imgur.com/aQlqRXz.png]]
Note the difference in terminology here - in this example, the f<span style="font-size: 90%; vertical-align: sub;">j+n</span> are the basis rationals, the r<span style="font-size: 12px; vertical-align: sub;">j+n</span> are the domains for each basis rational, and the bounds for each domain are the mediants between each f<span style="font-size: 12px; vertical-align: sub;">j+n </span>and its nearest neighbor. The probability assigned to each basis rational is then the area under the spreading function curve for each rational's domain. The entropy of this probability distribution is then the Harmonic Entropy for that dyad.

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= 
It has been noted empirically by Paul Erlich that, given all those rationals with Tenney height under some cutoff N as a basis set, that the domain widths for rationals sufficiently far from the cutoff seem to be proportional to 1/sqrt(nd). While it's still an open conjecture that this pattern holds for arbitrarily large N, the assumption is sometimes made that this is the case, and hence that for these basis rational sets, 1/sqrt(nd) "approximations" to the width are sufficient to estimate domain-integral Harmonic Entropy. This modifies the expression for the p<span style="font-size: 12px; vertical-align: sub;">d</span>(b) as follows, noting that for the moment the "probabilities" won't sum to 1:

[[math]]
p_d(b) = \frac{s_d(\cent(b))}{\sqrt{n_b \cdot d_b}}
[[math]]

where n<span style="vertical-align: sub;">b</span> and d<span style="vertical-align: sub;">b</span> are the numerator and denominator, respectively, of basis rational b. Again, the set of basis rationals here is assumed to be all of those rationals of Tenney Height ≤ N for some N.

A similar observation for the use of Weil-bounded subsets of the rationals suggests domain widths of 1/max(n,d), yielding instead the following formula:

[[math]]
p_d(b) = \frac{s_d(\cent(b))}{\max(n_b,d_b)}
[[math]]

where this time the set of basis rationals is assumed to be all of those of Weil Height ≤ N for some N.

In both cases, the general approach is the same: the value of the spreading function, taken at the value of cents(b), is divided by some sort of "complexity" function representing how much weight is given to that rational number. While the two complexity functions considered thus far were derived empirically by observing the asymptotic behavior of various height-bounded subsets of the rationals, we can generalize this for arbitrary basis sets of rationals and arbitrary complexities as follows:

[[math]]
p_d(b) = \frac{s_d(\cent(b))}{\|b\|}
[[math]]

where ||b|| denotes a complexity function mapping from rational numbers to reals.

Note that, in general, the probabilities in this "distribution" don't sum to 1, making it not a probability distribution at all and invalidating the use of the Shannon Entropy. To rectify this, the distribution is normalized so that the probabilities do sum to 1:

[[math]]
\hat{p}_d(b) = \frac{p_d(b)}{\sum_b p_d(b)}
[[math]]

The p̂<span style="vertical-align: sub;">d</span>(b) are then used directly to compute the entropy.

This alteration is useful because it hypothetically makes it possible to consider the Harmonic Entropy of sets of rationals which are dense in the reals, or the entire set ℚ, although the best way to do this is a subject of ongoing research.

=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 q-analog of Shannon's original entropy. The **Harmonic Rényi Entropy of order a** of an incoming dyad can be defined as follows:

[[math]]
H(d) = \frac{1}{1-a} \log \sum_b p_d(b)^a
[[math]]

where the p_d(b) are the probabilities assigned by dyad d to each basis rational b. It is noteworthy that Rényi entropy converges to Shannon entropy in the limit as a→1, a fact which can be verified using L'Hôpital's rule as found [[@http://www.sonycsl.co.jp/person/nielsen/Note-HopitalRuleShannonRenyiTsallis.pdf|here]].

Musically speaking, the parameter //**a**// can be interpreted as the extent to which the rational-matching process for incoming dyads is considered to be an intelligent, active process by which the auditory system actively analyzes the probability distribution and seeks out the "victor" rational with the greatest probability. Some psychoacoustic effects naturally fit into this paradigm, such as the virtual pitch integration process, which actually does attempt to find a single victor when matching incoming chords with chunks of the harmonic series. Other psychoacoustic effects, such as that of beatlessness, may instead be better viewed as "dumb" processes whereby nothing in particular is being "chosen," but where a more uniform distribution of matching rational numbers for a dyad simply generates a more discordant sonic effect. Increasing values of //a// correspond to a model that is more "active" in this way.

This interpretation comes from the use of the Rényi entropy in cryptography as a measure of the strength of a cryptographic code in the face of an intelligent attacker, an application for which Shannon entropy has long been known to be insufficient as described in [[@http://users.cis.fiu.edu/~smithg/papers/qest11.pdf|this paper]] and [[@http://www.ietf.org/rfc/rfc4086.txt|this RFC]]. More precisely, the Rényi entropy of order ∞, also called the **min-entropy**, is used to measure the strength of the randomness used to define a cryptographic secret against a "worst-case" attacker who has complete knowledge of the probability distribution from which cryptographic secrets are drawn. In a musical context, by considering the incoming dyad as analogous to a cryptographic code which is attempting to be "cracked" by an intelligent auditory system, we can consider that the analogous "worst-case attacker" would be a "best-case auditory system" which has complete awareness of the probability distribution for any incoming dyad and actively chooses the strongest rational.

The use of a=∞ min-entropy would reflect this view of an "intelligent" auditory system that can analyze the incoming probability distribution and always pick one of the matching rationals that holds a plurality. In contrast, the use of a=1 Shannon entropy reflects a much "dumber" process which performs no such analysis; at best, if we want to shoehorn this into the paradigm where something is being "chosen," it can be thought of as the auditory system <span style="line-height: 1.5;">selecting a rational at random </span>from the distribution. Intermediate values of a can be seen as interpolating between these two cases.

Certain values of //a// reduce to simpler expressions and have special names:

===a=0: Harmonic Hartley Entropy=== 
[[math]]
H(d) = \log |R|
[[math]]
where |R| is the cardinality of the set of basis rationals. This assumes, in essence, an "infinitely dumb" auditory system which can do no better than picking a rational number from a uniform distribution completely at random. All dyads have the same Harmonic Hartley Entropy.

===a=1: Harmonic Shannon Entropy (Harmonic Entropy)=== 
[[math]]
H(d) = -\sum_{b} p_d(b) \log p_d(b)
[[math]]
This is Paul's original Harmonic Entropy. This can be thought of as an auditory system which simply selects a rational at random from the incoming distribution, weighted via the distribution itself.

===a=2: Harmonic Collision Entropy=== 
[[math]]
H(d) = -\log \sum_b p_d(b)^2 = -log (P_d = Q_d)
[[math]]
where P<span style="font-size: 90%; vertical-align: sub;">d</span> and Q<span style="font-size: 90%; vertical-align: sub;">d</span> are independent and identically distributed random variables corresponding to the same dyad, and the collision entropy is the same as the negative log of the probability that the two variables produce the same outcome.

===a=∞: Harmonic Min-Entropy=== 
[[math]]
-\log \max_b p_d(b)
[[math]]
This is the min-entropy, which simply takes the negative log of the largest probability in the distribution. This can be thought of as representing the "strength" of the incoming dyad from being "deciphered" by a "best-case" auditory system.

=References= 
[[http://www.webcitation.org/60qOlJVFS|Paul Erlich article]]
[[http://sethares.engr.wisc.edu/paperspdf/HarmonicEntropy.pdf|William Sethares article]]
[[http://www.tonalsoft.com/enc/h/harmonic-entropy.aspx|Harmonic entropy (TonalSoft encyclopedia)]]
[[http://launch.groups.yahoo.com/group/harmonic_entropy/|Harmonic entropy group on Yahoo]]
[[@http://www.mikebattagliamusic.com/HE-JS/HE.html|Harmonic entropy graph calculator (JavaScript)]]

Original HTML content:

<html><head><title>Harmonic Entropy</title></head><body><!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:12 -->Introduction</h1>
 <strong>Harmonic Entropy</strong>, sometimes abbreviated as &quot;HE&quot;, is a simple model to quantify the extent to which musical chords exhibit various psychoacoustic effects, lumped together in a single construct called psychoacoustic <strong>concordance</strong>. 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc1"><a name="Background"></a><!-- ws:end:WikiTextHeadingRule:14 -->Background</h1>
 The general workings of the human auditory system lead to a plethora of well-documented and sonically interesting phenomena that can occur when a musical chord is played:<br />
<ul><li>The perception of partial <strong>timbral fusion</strong> of the chord into one complex sound</li><li>The appearance of a <strong>virtual fundamental</strong> pitch in the bass</li><li>Timbral <strong>beatlessness</strong>, compared to mistunings of the chord in the surrounding area</li><li>The appearance of a quick fluttering effect sometimes known as <strong>periodicity buzz</strong></li></ul><br />
All of these effects share two common characteristics for chords played with harmonic timbres:<br />
<ul><li><span style="line-height: 1.5;">they tend to appear most strongly for those chords with large subsets that correspond to simple chunks of the harmonic series</span></li><li><span style="line-height: 1.5;">the effects produced exhibit some degree of tolerance for mistuning</span></li></ul><br />
As the presence or absence of these effects tend to appear in tandem and are highly correlated with one another, we can speak of a general notion of the psychoacoustic <strong>concordance</strong> of an interval - the degree to which effects such as the above will appear when an arbitrary musical chord is played. Additionally, chords which are very inharmonic often exhibit a quality known as psychoacoustic <strong>discordance</strong>.<br />
<br />
While psychoacoustic concordance is not a feature universal to all styles of music, it has been utilized significantly in Western music in the study of intonation. For instance, flexible-pitch ensembles operating within 12-EDO, such as barbershop quartets and string ensembles, will often adjust intonationally from the underlying 12-EDO reference to maximize the concordance of individual chords. Indeed, the entire history of Western tuning theory -- from meantone temperament, to the various Baroque well-temperaments, to 12-EDO itself, to the modern <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Regular%20Temperaments" target="_blank">theory of regular temperament</a> -- can be seen as an attempt to reason mathematically about how to generate manageable tuning systems that will maximize concordance and minimize discordance.<br />
<br />
The Harmonic Entropy model is a simple way of quantifying how much an arbitrary chord will exhibit psychoacoustic concordance.<br />
<br />
Concordance has often been confused with actual musical consonance, an unfortunate fact made more common by the<span style="line-height: 1.5;"> psychoacoustics literature under the unfortunate name </span><strong><span style="line-height: 1.5;">sensory consonance</span></strong><span style="line-height: 1.5;">, most often used to refer to phenomena related to roughness and beatlessness specifically. This is not to be confused with the more familiar construct of tonal stability, typically just called &quot;consonance&quot; in Western common practice music theory and sometimes clarified as &quot;musical consonance&quot; in the music cognition literature. To make matters worse, the literature has also at times referred to concordance -- and not tonal stability -- as </span><strong><span style="line-height: 1.5;">tonal consonance</span></strong><span style="line-height: 1.5;">, often referring to phenomena related to virtual pitch integration, creating a complete terminological mess. As a result, the term &quot;consonance&quot; has been completely avoided in this article.</span><br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc2"><a name="Model"></a><!-- ws:end:WikiTextHeadingRule:16 -->Model</h1>
 The original Harmonic Entropy model limited itself to working with dyads. More recently, work by Steve Martin and others has extended this basic idea to higher-cardinality chords. This article will concern itself with dyads, as the dyadic case is still the most well-developed, and many of the ideas extend naturally to larger chords without need for much exposition.<br />
<br />
The general idea of Harmonic Entropy is to first develop a discrete probability distribution quantifying how strongly an arbitrary incoming dyad &quot;matches&quot; 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 &quot;victor&quot; 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 is via <span style="line-height: 1.5;">the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Entropy_(information_theory)" rel="nofollow" target="_blank">Shannon entropy</a> of the probability distribution:</span><br />
<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]&lt;br/&gt;
H(d) = -\sum_{b} p_d(b) \log p_d(b)&lt;br/&gt;[[math]]
 --><script type="math/tex">H(d) = -\sum_{b} p_d(b) \log p_d(b)</script><!-- ws:end:WikiTextMathRule:0 --><br />
<br />
where H(d) is the Shannon entropy of the dyad d, the b are all of the basis rationals in the set, and the p<span style="font-size: 90%; vertical-align: sub;">d</span>(b) is the probability assigned to basis rational b given an input dyad of d. This is the Harmonic Entropy of the dyad d.<br />
<br />
In order to systematically assign a probability distribution to this dyad, we first start by defining a <strong>spreading function</strong> that dictates how the dyad is &quot;smeared&quot; out in log-frequency space, representing how the auditory system allows for some tolerance for mistuning. The typical choice that we will assume here for a spreading function is a Gaussian distribution, with mean set to be centered around the incoming dyad, and standard deviation <strong>s</strong> typically taken as a free parameter in the system.<br />
<br />
A fairly typical choice of settings for a basic dyadic HE model would be:<br />
<ul><li>The set of basis rationals is the set of all intervals bounded by some maximum Tenney height, with the bound typically notated as <strong>N</strong> and set to at least 10000</li><li>The spreading function is typically a Gaussian distribution with s=17 cents, or about a frequency deviation of 1% either way</li></ul><br />
Other spreading functions have also been explored, such as the use of the &quot;Vos function&quot; of a·exp(b|x|) rather than the Gaussian distribution. We will assume the Gaussian distribution as the spreading function for the remainder of this article, so that the spreading function for a dyad d can be written as follows:<br />
<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]&lt;br/&gt;
s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}&lt;br/&gt;[[math]]
 --><script type="math/tex">s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}</script><!-- ws:end:WikiTextMathRule:1 --><br />
<br />
where <em>s</em> becomes the standard deviation of the Gaussian, being an ASCII-friendly version of the more familiar symbol σ for representing the standard deviation. We assume here that the variable x represents cents, and will adopt this convention for the remainder of the article.<br />
<br />
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 <strong>domain-integral approach</strong>, works for arbitrary nowhere dense sets of rationals without any further free parameters. The second, the <strong>complexity-normalization approach</strong>, 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc3"><a name="Domain-integral Probabilities"></a><!-- ws:end:WikiTextHeadingRule:18 -->Domain-integral Probabilities</h1>
 For sets of basis rationals which are nowhere dense, the log-frequency spectrum can be divided up into <strong>domains</strong> assigned to the basis rationals. Each rational 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, ±∞ is used instead. Mathematically, this can be represented via the following expression:<br />
<br />
<!-- ws:start:WikiTextMathRule:2:
[[math]]&lt;br/&gt;
p_d(b) = \int_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx&lt;br/&gt;[[math]]
 --><script type="math/tex">p_d(b) = \int_{\cent(b_l)}^{\cent(b_u)} s_d(x) dx</script><!-- ws:end:WikiTextMathRule:2 --><br />
<br />
where s<span style="vertical-align: sub;">d</span>(x) is the spreading function associated with d<span style="line-height: 1.5;">, b</span><span style="line-height: 1.5; vertical-align: sub;">l</span><span style="line-height: 1.5;"> and b</span><span style="line-height: 1.5; vertical-align: sub;">u</span><span style="line-height: 1.5;"> are the domain lower and upper bounds associated with basis rational b, and ¢(f) = 1200·log2(f), or the &quot;cents&quot; function converting frequency ratios to cents. Normally, b</span><span style="line-height: 1.5; vertical-align: sub;">l </span><span style="line-height: 1.5;">is set equal to the mediant of b and its nearest lower neighbor (if it exists), or -∞ if not; likewise with b</span><span style="line-height: 1.5; vertical-align: sub;">u</span><span style="line-height: 1.5;">.</span><br />
<br />
This process can be summarized by the following picture, taken from <a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/HarmonicEntropy.pdf" rel="nofollow" target="_blank">William Sethares' paper on Harmonic Entropy</a>:<br />
<br />
<!-- ws:start:WikiTextRemoteImageRule:56:&lt;img src=&quot;http://i.imgur.com/aQlqRXz.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://i.imgur.com/aQlqRXz.png" alt="external image aQlqRXz.png" title="external image aQlqRXz.png" /><!-- ws:end:WikiTextRemoteImageRule:56 --><br />
Note the difference in terminology here - in this example, the f<span style="font-size: 90%; vertical-align: sub;">j+n</span> are the basis rationals, the r<span style="font-size: 12px; vertical-align: sub;">j+n</span> are the domains for each basis rational, and the bounds for each domain are the mediants between each f<span style="font-size: 12px; vertical-align: sub;">j+n </span>and its nearest neighbor. The probability assigned to each basis rational is then the area under the spreading function curve for each rational's domain. The entropy of this probability distribution is then the Harmonic Entropy for that dyad.<br />
<br />
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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc4"><a name="Complexity-normalization Probabilities"></a><!-- ws:end:WikiTextHeadingRule:20 -->Complexity-normalization Probabilities</h1>
 It has been noted empirically by Paul Erlich that, given all those rationals with Tenney height under some cutoff N as a basis set, that the domain widths for rationals sufficiently far from the cutoff seem to be proportional to 1/sqrt(nd). While it's still an open conjecture that this pattern holds for arbitrarily large N, the assumption is sometimes made that this is the case, and hence that for these basis rational sets, 1/sqrt(nd) &quot;approximations&quot; to the width are sufficient to estimate domain-integral Harmonic Entropy. This modifies the expression for the p<span style="font-size: 12px; vertical-align: sub;">d</span>(b) as follows, noting that for the moment the &quot;probabilities&quot; won't sum to 1:<br />
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p_d(b) = \frac{s_d(\cent(b))}{\sqrt{n_b \cdot d_b}}&lt;br/&gt;[[math]]
 --><script type="math/tex">p_d(b) = \frac{s_d(\cent(b))}{\sqrt{n_b \cdot d_b}}</script><!-- ws:end:WikiTextMathRule:3 --><br />
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where n<span style="vertical-align: sub;">b</span> and d<span style="vertical-align: sub;">b</span> are the numerator and denominator, respectively, of basis rational b. Again, the set of basis rationals here is assumed to be all of those rationals of Tenney Height ≤ N for some N.<br />
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A similar observation for the use of Weil-bounded subsets of the rationals suggests domain widths of 1/max(n,d), yielding instead the following formula:<br />
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p_d(b) = \frac{s_d(\cent(b))}{\max(n_b,d_b)}&lt;br/&gt;[[math]]
 --><script type="math/tex">p_d(b) = \frac{s_d(\cent(b))}{\max(n_b,d_b)}</script><!-- ws:end:WikiTextMathRule:4 --><br />
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where this time the set of basis rationals is assumed to be all of those of Weil Height ≤ N for some N.<br />
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In both cases, the general approach is the same: the value of the spreading function, taken at the value of cents(b), is divided by some sort of &quot;complexity&quot; function representing how much weight is given to that rational number. While the two complexity functions considered thus far were derived empirically by observing the asymptotic behavior of various height-bounded subsets of the rationals, we can generalize this for arbitrary basis sets of rationals and arbitrary complexities as follows:<br />
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[[math]]&lt;br/&gt;
p_d(b) = \frac{s_d(\cent(b))}{\|b\|}&lt;br/&gt;[[math]]
 --><script type="math/tex">p_d(b) = \frac{s_d(\cent(b))}{\|b\|}</script><!-- ws:end:WikiTextMathRule:5 --><br />
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where ||b|| denotes a complexity function mapping from rational numbers to reals.<br />
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Note that, in general, the probabilities in this &quot;distribution&quot; don't sum to 1, making it not a probability distribution at all and invalidating the use of the Shannon Entropy. To rectify this, the distribution is normalized so that the probabilities do sum to 1:<br />
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\hat{p}_d(b) = \frac{p_d(b)}{\sum_b p_d(b)}&lt;br/&gt;[[math]]
 --><script type="math/tex">\hat{p}_d(b) = \frac{p_d(b)}{\sum_b p_d(b)}</script><!-- ws:end:WikiTextMathRule:6 --><br />
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The p̂<span style="vertical-align: sub;">d</span>(b) are then used directly to compute the entropy.<br />
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This alteration is useful because it hypothetically makes it possible to consider the Harmonic Entropy of sets of rationals which are dense in the reals, or the entire set ℚ, although the best way to do this is a subject of ongoing research.<br />
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<!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc5"><a name="Harmonic Rényi Entropy"></a><!-- ws:end:WikiTextHeadingRule:22 -->Harmonic Rényi Entropy</h1>
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An extension to the base Harmonic Entropy model, proposed by Mike Battaglia, is to generalize the use of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Entropy_(information_theory)" rel="nofollow" target="_blank">Shannon entropy</a> by replacing it instead with <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/R%C3%A9nyi_entropy" rel="nofollow" target="_blank">Rényi entropy</a>, a q-analog of Shannon's original entropy. The <strong>Harmonic Rényi Entropy of order a</strong> of an incoming dyad can be defined as follows:<br />
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H(d) = \frac{1}{1-a} \log \sum_b p_d(b)^a&lt;br/&gt;[[math]]
 --><script type="math/tex">H(d) = \frac{1}{1-a} \log \sum_b p_d(b)^a</script><!-- ws:end:WikiTextMathRule:7 --><br />
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where the p_d(b) are the probabilities assigned by dyad d to each basis rational b. It is noteworthy that Rényi entropy converges to Shannon entropy in the limit as a→1, a fact which can be verified using L'Hôpital's rule as found <a class="wiki_link_ext" href="http://www.sonycsl.co.jp/person/nielsen/Note-HopitalRuleShannonRenyiTsallis.pdf" rel="nofollow" target="_blank">here</a>.<br />
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Musically speaking, the parameter <em><strong>a</strong></em> can be interpreted as the extent to which the rational-matching process for incoming dyads is considered to be an intelligent, active process by which the auditory system actively analyzes the probability distribution and seeks out the &quot;victor&quot; rational with the greatest probability. Some psychoacoustic effects naturally fit into this paradigm, such as the virtual pitch integration process, which actually does attempt to find a single victor when matching incoming chords with chunks of the harmonic series. Other psychoacoustic effects, such as that of beatlessness, may instead be better viewed as &quot;dumb&quot; processes whereby nothing in particular is being &quot;chosen,&quot; but where a more uniform distribution of matching rational numbers for a dyad simply generates a more discordant sonic effect. Increasing values of <em>a</em> correspond to a model that is more &quot;active&quot; in this way.<br />
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This interpretation comes from the use of the Rényi entropy in cryptography as a measure of the strength of a cryptographic code in the face of an intelligent attacker, an application for which Shannon entropy has long been known to be insufficient as described in <a class="wiki_link_ext" href="http://users.cis.fiu.edu/~smithg/papers/qest11.pdf" rel="nofollow" target="_blank">this paper</a> and <a class="wiki_link_ext" href="http://www.ietf.org/rfc/rfc4086.txt" rel="nofollow" target="_blank">this RFC</a>. More precisely, the Rényi entropy of order ∞, also called the <strong>min-entropy</strong>, is used to measure the strength of the randomness used to define a cryptographic secret against a &quot;worst-case&quot; attacker who has complete knowledge of the probability distribution from which cryptographic secrets are drawn. In a musical context, by considering the incoming dyad as analogous to a cryptographic code which is attempting to be &quot;cracked&quot; by an intelligent auditory system, we can consider that the analogous &quot;worst-case attacker&quot; would be a &quot;best-case auditory system&quot; which has complete awareness of the probability distribution for any incoming dyad and actively chooses the strongest rational.<br />
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The use of a=∞ min-entropy would reflect this view of an &quot;intelligent&quot; auditory system that can analyze the incoming probability distribution and always pick one of the matching rationals that holds a plurality. In contrast, the use of a=1 Shannon entropy reflects a much &quot;dumber&quot; process which performs no such analysis; at best, if we want to shoehorn this into the paradigm where something is being &quot;chosen,&quot; it can be thought of as the auditory system <span style="line-height: 1.5;">selecting a rational at random </span>from the distribution. Intermediate values of a can be seen as interpolating between these two cases.<br />
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Certain values of <em>a</em> reduce to simpler expressions and have special names:<br />
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<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc6"><a name="Harmonic Rényi Entropy--a=0: Harmonic Hartley Entropy"></a><!-- ws:end:WikiTextHeadingRule:24 -->a=0: Harmonic Hartley Entropy</h3>
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H(d) = \log |R|&lt;br/&gt;[[math]]
 --><script type="math/tex">H(d) = \log |R|</script><!-- ws:end:WikiTextMathRule:8 --><br />
where |R| is the cardinality of the set of basis rationals. This assumes, in essence, an &quot;infinitely dumb&quot; auditory system which can do no better than picking a rational number from a uniform distribution completely at random. All dyads have the same Harmonic Hartley Entropy.<br />
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<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc7"><a name="Harmonic Rényi Entropy--a=1: Harmonic Shannon Entropy (Harmonic Entropy)"></a><!-- ws:end:WikiTextHeadingRule:26 -->a=1: Harmonic Shannon Entropy (Harmonic Entropy)</h3>
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H(d) = -\sum_{b} p_d(b) \log p_d(b)&lt;br/&gt;[[math]]
 --><script type="math/tex">H(d) = -\sum_{b} p_d(b) \log p_d(b)</script><!-- ws:end:WikiTextMathRule:9 --><br />
This is Paul's original Harmonic Entropy. This can be thought of as an auditory system which simply selects a rational at random from the incoming distribution, weighted via the distribution itself.<br />
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<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc8"><a name="Harmonic Rényi Entropy--a=2: Harmonic Collision Entropy"></a><!-- ws:end:WikiTextHeadingRule:28 -->a=2: Harmonic Collision Entropy</h3>
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H(d) = -\log \sum_b p_d(b)^2 = -log (P_d = Q_d)&lt;br/&gt;[[math]]
 --><script type="math/tex">H(d) = -\log \sum_b p_d(b)^2 = -log (P_d = Q_d)</script><!-- ws:end:WikiTextMathRule:10 --><br />
where P<span style="font-size: 90%; vertical-align: sub;">d</span> and Q<span style="font-size: 90%; vertical-align: sub;">d</span> are independent and identically distributed random variables corresponding to the same dyad, and the collision entropy is the same as the negative log of the probability that the two variables produce the same outcome.<br />
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<!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc9"><a name="Harmonic Rényi Entropy--a=∞: Harmonic Min-Entropy"></a><!-- ws:end:WikiTextHeadingRule:30 -->a=∞: Harmonic Min-Entropy</h3>
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-\log \max_b p_d(b)&lt;br/&gt;[[math]]
 --><script type="math/tex">-\log \max_b p_d(b)</script><!-- ws:end:WikiTextMathRule:11 --><br />
This is the min-entropy, which simply takes the negative log of the largest probability in the distribution. This can be thought of as representing the &quot;strength&quot; of the incoming dyad from being &quot;deciphered&quot; by a &quot;best-case&quot; auditory system.<br />
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<!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc10"><a name="References"></a><!-- ws:end:WikiTextHeadingRule:32 -->References</h1>
 <a class="wiki_link_ext" href="http://www.webcitation.org/60qOlJVFS" rel="nofollow">Paul Erlich article</a><br />
<a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/HarmonicEntropy.pdf" rel="nofollow">William Sethares article</a><br />
<a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/h/harmonic-entropy.aspx" rel="nofollow">Harmonic entropy (TonalSoft encyclopedia)</a><br />
<a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/harmonic_entropy/" rel="nofollow">Harmonic entropy group on Yahoo</a><br />
<a class="wiki_link_ext" href="http://www.mikebattagliamusic.com/HE-JS/HE.html" rel="nofollow" target="_blank">Harmonic entropy graph calculator (JavaScript)</a></body></html>