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Mention interactive HE graph in SW3
 
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{{Legacy}}
'''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.  
'''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 {{dash|C, E, G, C}} is a tetrad instead of a triad.
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.


== Background ==
== Background ==
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There has been much research specifically on the musical implications critical band effects in the literature (e.g. Sethares's work), which are perhaps the psychoacoustic phenomena that readers are most familiar with. However, the modern xenharmonic community has displayed immense interest in exploring the other effects mentioned above as well, which have proven extremely important to the development of modern xenharmonic music.
There has been much research specifically on the musical implications critical band effects in the literature (e.g. Sethares's work), which are perhaps the psychoacoustic phenomena that readers are most familiar with. However, the modern xenharmonic community has displayed immense interest in exploring the other effects mentioned above as well, which have proven extremely important to the development of modern xenharmonic music.


These effects sometimes behave differently, and do not always appear strictly in tandem with one another. For instance, Paul Erlich has noted that most models for beatlessness measure 10:12:15 and 4:5:6 as being identical, whereas the latter exhibits more timbral fusion and a more salient virtual fundamental than the former. However, it is useful to have a term that refers to the general presence of these types of effects. The term "consonance" has sometimes been used for this; however, there are many other meanings of consonance which may not be psychoacoustic in nature. Thus, we instead speak of a general notion of psychoacoustic '''concordance''' - the degree to which effects such as the above will appear when an arbitrary musical interval or chord is played - as well and psychoacoustic '''discordance'''. Timbral fusion, the appearance of virtual fundamentals, beatlessness, and periodicity buzz, can all be thought of as different aspects of psychoacoustic concordance.
These effects sometimes behave differently, and do not always appear strictly in tandem with one another. For instance, Paul Erlich has noted that most models for beatlessness measure 10:12:15 and 4:5:6 as being identical, whereas the latter exhibits more timbral fusion and a more salient virtual fundamental than the former. However, it is useful to have a term that refers to the general presence of these types of effects. The term "consonance" has sometimes been used for this; however, there are many other meanings of consonance which may not be psychoacoustic in nature. Thus, we instead speak of a general notion of psychoacoustic '''concordance''' – the degree to which effects such as the above will appear when an arbitrary musical interval or chord is played – as well and psychoacoustic '''discordance'''. Timbral fusion, the appearance of virtual fundamentals, beatlessness, and periodicity buzz, can all be thought of as different aspects of psychoacoustic concordance.


Harmonic Entropy was originally intended to measure, in particular, the "virtual fundamental" aspect of psychoacoustic concordance, being modeled on J. Goldstein's [https://asa.scitation.org/doi/10.1121/1.1914448 1973 paper] "An optimum processor theory for the central formation of the pitch of complex tones." It can also be thought of as an elaboration on similar research by Terhardt, Parncutt and others, which addresses some of the shortcomings suggested by Erlich in prior models. The model basically asks how "confused your brain is," in Erlich's words, when trying to match the incoming sound to that of one single harmonic timbre played on a missing fundamental.
Harmonic entropy was originally intended to measure, in particular, the "virtual fundamental" aspect of psychoacoustic concordance, being modeled on J. Goldstein's [https://asa.scitation.org/doi/10.1121/1.1914448 1973 paper] "An optimum processor theory for the central formation of the pitch of complex tones." It can also be thought of as an elaboration on similar research by Terhardt, Parncutt and others, which addresses some of the shortcomings suggested by Erlich in prior models. The model basically asks how "confused your brain is," in Erlich's words, when trying to match the incoming sound to that of one single harmonic timbre played on a missing fundamental.


For dyads, the basic harmonic entropy model is fairly simple: it places the dyad we are trying to measure amidst a backdrop of JI candidates. Then, it uses a point-spread function to determine the relative strengths of the match to each, which are then normalized and treated as probabilities. The "entropy" of the resulting probability distribution is a way to measure how closely this distribution tends to focus on one possibility, rather than being spread out among a set of equally-likely possibilities. If there is only one clear choice of dyad which far exceeds all others in probability, the entropy will be lower. If, on the other hand, there are many equally-likely probabilities, the entropy will be higher. The basic harmonic entropy model can also be extended to modeling triads, tetrads, and so on; the standard way to do so is to simply look at the incoming triad's match to a set of candidate JI triads, and likewise with tetrads, and etc.
For dyads, the basic harmonic entropy model is fairly simple: it places the dyad we are trying to measure amidst a backdrop of JI candidates. Then, it uses a point-spread function to determine the relative strengths of the match to each, which are then normalized and treated as probabilities. The "entropy" of the resulting probability distribution is a way to measure how closely this distribution tends to focus on one possibility, rather than being spread out among a set of equally-likely possibilities. If there is only one clear choice of dyad which far exceeds all others in probability, the entropy will be lower. If, on the other hand, there are many equally-likely probabilities, the entropy will be higher. The basic harmonic entropy model can also be extended to modeling triads, tetrads, and so on; the standard way to do so is to simply look at the incoming triad's match to a set of candidate JI triads, and likewise with tetrads, and etc.
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In recent years, it has become clearer that the model can also be very useful in modeling other types of concordance as well, particularly for dyads, where the same model does a very good job in also predicting beatlessness, periodicity buzz, and so on. In particular, Erlich has often suggested the same model, perhaps with slightly different parameters, can also be useful to measure how easy it is to tune a dyad by ear on an instrument such as a guitar, or how much of a sense of being "locked-in" the dyad gives as it is tuned more closely to JI. This may be less related to the perception of virtual fundamentals than it is to beatlessness and so on.
In recent years, it has become clearer that the model can also be very useful in modeling other types of concordance as well, particularly for dyads, where the same model does a very good job in also predicting beatlessness, periodicity buzz, and so on. In particular, Erlich has often suggested the same model, perhaps with slightly different parameters, can also be useful to measure how easy it is to tune a dyad by ear on an instrument such as a guitar, or how much of a sense of being "locked-in" the dyad gives as it is tuned more closely to JI. This may be less related to the perception of virtual fundamentals than it is to beatlessness and so on.


However, it should be noted that the various aspects of psychoacoustic concordance tend to diverge quite strongly in their behavior for larger chords, and thus, when modeling different aspects of psychoacoustic concordance, different ways of generalizing the dyadic model to higher-cardinality chords may be appropriate. In particular, when modelling beatlessness, Erlich has suggested instead looking only at the entropies of the pairwise dyadic subsets of the chord, so that the major and minor chords would be ranked equal in beatlessness, whereas they would not be ranked equal in their ability to produce a clear virtual fundamental (the major chord would be much stronger and lower in entropy).
However, it should be noted that the various aspects of psychoacoustic concordance tend to diverge quite strongly in their behavior for larger chords, and thus, when modeling different aspects of psychoacoustic concordance, different ways of generalizing the dyadic model to higher-cardinality chords may be appropriate. In particular, when modeling beatlessness, Erlich has suggested instead looking only at the entropies of the pairwise dyadic subsets of the chord, so that the major and minor chords would be ranked equal in beatlessness, whereas they would not be ranked equal in their ability to produce a clear virtual fundamental (the major chord would be much stronger and lower in entropy).


=== Concordance vs. actual consonance ===
=== Concordance vs. actual consonance ===
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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 [[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. Consonance and dissonance, on the other hand, is a much more general phenomenon which can even exist in music which is predominantly monophonic and uses no chords at all.
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 [[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. Consonance and dissonance, on the other hand, is a much more general phenomenon which can even exist in music which is predominantly monophonic and uses no chords at all.


== Basic Model: Shannon Entropy ==
== Basic model: Shannon entropy ==
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 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.
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.
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An extension to the base Harmonic Entropy model, proposed by Mike Battaglia, is to generalize the use of {{w|Entropy (information_ theory)|Shannon entropy}} by replacing it instead with {{w|Rényi entropy}}, a {{w|q-analog|''q''-analog}} of Shannon's original entropy. This can be thought of as adding a second parameter, called ''a'', 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 {{w|Entropy (information_ theory)|Shannon entropy}} by replacing it instead with {{w|Rényi entropy}}, a {{w|q-analog|''q''-analog}} of Shannon's original entropy. This can be thought of as adding a second parameter, called ''a'', 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 ===
=== Definitions and background ===
The '''Harmonic Rényi entropy of order ''a''''' of an incoming dyad can be defined as follows:
The '''Harmonic Rényi entropy of order ''a''''' of an incoming dyad can be defined as follows:


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This enables us to speak cognizantly of the harmonic entropy of an interval as measured against ''all'' rational numbers.
This enables us to speak cognizantly of the harmonic entropy of an interval as measured against ''all'' rational numbers.


=== Background: Unnormalized entropy ===
=== Background: unnormalized entropy ===
Our derivation only analytically continues the entropy function for the "unnormalized" set of probabilities, which we previously wrote as ''Q''(''j''|''c''). For this definition to be philosophically perfect, we would want to analytically continue the entropy function for the normalized sense of probabilities, previously written as ''P''(''j''|''c'').
Our derivation only analytically continues the entropy function for the "unnormalized" set of probabilities, which we previously wrote as ''Q''(''j''|''c''). For this definition to be philosophically perfect, we would want to analytically continue the entropy function for the normalized sense of probabilities, previously written as ''P''(''j''|''c'').


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Lastly, it so happens that it will be much easier to understand our analytic continuation if we look at the exponential of the UHE times ({{nowrap|1 − a}}), rather than the UHE itself. The reasons for this will become clear later. If we do so, we get
Lastly, it so happens that it will be much easier to understand our analytic continuation if we look at the exponential of the UHE times ({{nowrap|1 − ''a''}}), rather than the UHE itself. The reasons for this will become clear later. If we do so, we get


$$\displaystyle \exp((1-a) \text{UHE}_a(c)) = \left( S^a \ast K^a \right)(-c)$$
$$\displaystyle \exp((1-a) \text{UHE}_a(c)) = \left( S^a \ast K^a \right)(-c)$$
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Note that this function is simply a monotonic transformation of the original, and so preserves the exact same concordance ranking on all intervals.
Note that this function is simply a monotonic transformation of the original, and so preserves the exact same concordance ranking on all intervals.


==== Analytic Continuation of the Convolution Kernel ====
==== Analytic continuation of the convolution kernel ====
The definition for ''K'' is:
The definition for ''K'' is:


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Finally, it is noteworthy that for {{nowrap|''a'' &gt; 2}}, we end up looking at slices of the zeta function for which {{nowrap|Re(''z'') &gt; 1}}. This is where our original unnormalized HE function should converge as {{nowrap|''N'' → ∞}}, corresponding to the region where the Riemann zeta function Dirichlet series converges. For these values of ''a'', the exp-UHE ''is'' positive. So, we can take the log again and look at the usual UHE. This can be useful for plotting, since exp-UHE tends to "flatten" out the curve for high values of ''a'', whereas taking the log accentuates the minima and maxima (and more closely resembles the usual HRE).
Finally, it is noteworthy that for {{nowrap|''a'' &gt; 2}}, we end up looking at slices of the zeta function for which {{nowrap|Re(''z'') &gt; 1}}. This is where our original unnormalized HE function should converge as {{nowrap|''N'' → ∞}}, corresponding to the region where the Riemann zeta function Dirichlet series converges. For these values of ''a'', the exp-UHE ''is'' positive. So, we can take the log again and look at the usual UHE. This can be useful for plotting, since exp-UHE tends to "flatten" out the curve for high values of ''a'', whereas taking the log accentuates the minima and maxima (and more closely resembles the usual HRE).


=== Interpretation as a New Free Parameter: the Weighting Exponent ===
=== Interpretation as a new free parameter: the weighting exponent ===
In our original derivation of the analytic continuation, we temporarily changed the weighting for rationals from (''nd'')<sup>0.5</sup> to some other (''nd'')<sup>''w''</sup>, with {{nowrap|''w'' &gt; 1}}, for the sake of obtaining a series that converges. We then changed the exponent back to 0.5.
In our original derivation of the analytic continuation, we temporarily changed the weighting for rationals from (''nd'')<sup>0.5</sup> to some other (''nd'')<sup>''w''</sup>, with {{nowrap|''w'' &gt; 1}}, for the sake of obtaining a series that converges. We then changed the exponent back to 0.5.


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$$\displaystyle |\zeta(w+i t)|^2 = \left[ \sum_{j_c \in \mathbb{N}^+} \frac{1}{{j_c}^{2w}} \right] \cdot \left[ \sum_{j \in \mathbb{Q}} \frac{e^{i  t \log (\frac{j_{n'}}{j_{d'}})}}{(j_{n'} j_{d'})^{w}} \right]$$
$$\displaystyle |\zeta(w+i t)|^2 = \left[ \sum_{j_c \in \mathbb{N}^+} \frac{1}{{j_c}^{2w}} \right] \cdot \left[ \sum_{j \in \mathbb{Q}} \frac{e^{i  t \log (\frac{j_{n'}}{j_{d'}})}}{(j_{n'} j_{d'})^{w}} \right]$$


where the left summation now has {{nowrap|''j''<sub>''c''</sub> ∈ ℕ{{mpp}}}}, the set of strictly positive rational numbers, and the right summation now has {{nowrap|''j'' ∈ ℚ}} the set of reduced rationals. Note again that the product above yields all unreduced rationals, thanks to the ''j''<sub>''c''</sub>.
where the left summation now has {{nowrap|''j''<sub>''c''</sub> ∈ ℕ{{+}}}}, the set of strictly positive rational numbers, and the right summation now has {{nowrap|''j'' ∈ ℚ}} the set of reduced rationals. Note again that the product above yields all unreduced rationals, thanks to the ''j''<sub>''c''</sub>.


Now, note that that left series is, itself, just another Dirichlet series that converges to the zeta function. We have
Now, note that that left series is, itself, just another Dirichlet series that converges to the zeta function. We have
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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.
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.
== To Do ==
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 ==
== References ==
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* [https://yahootuninggroupsultimatebackup.github.io/harmonic_entropy Harmonic entropy group on Yahoo] (archive)
* [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)]
* [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]]
[[Category:Terms]]
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[[Category:Consonance and dissonance]]
[[Category:Consonance and dissonance]]
[[Category:Harmonic entropy| ]] <!-- main article -->
[[Category:Harmonic entropy| ]] <!-- main article -->
[[Category:Book-style articles]]
[[Category:Essays]]
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