Harmonic entropy - Revision history

Fredg999: Text replacement - "Riemann zeta function" to "Riemann zeta function"

2026-07-16T18:37:21Z

Text replacement - "Riemann zeta function" to "Riemann zeta function"

← Older revision		Revision as of 18:37, 16 July 2026
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	The only technical caveat is that we use the HE of the "unnormalized" probability distribution. However, in the large limit of ''N'', this appears to agree closely with the usual HE. We go into more detail below about this.		The only technical caveat is that we use the HE of the "unnormalized" probability distribution. However, in the large limit of ''N'', this appears to agree closely with the usual HE. We go into more detail below about this.

	Our basic approach is: rather than weighting intervals by (''nd'')<sup>0.5</sup>, we choose a different exponent, such as (''nd'')<sup>2</sup>. For an exponent which is large enough (we will show that it must be greater than 1), HE does indeed converge as {{nowrap\|''N'' → ∞}}, and we show that this yields an expression related to the [[~~The Riemann zeta function and tuning\|~~Riemann zeta function]]. We can then use the analytic continuation of the zeta function to obtain an analytically continued curve for the (''nd'')<sup>0.5</sup> weighting, which we then show empirically does indeed appear to be what HE converges on for large values of ''N''.		Our basic approach is: rather than weighting intervals by (''nd'')<sup>0.5</sup>, we choose a different exponent, such as (''nd'')<sup>2</sup>. For an exponent which is large enough (we will show that it must be greater than 1), HE does indeed converge as {{nowrap\|''N'' → ∞}}, and we show that this yields an expression related to the [[Riemann zeta function]]. We can then use the analytic continuation of the zeta function to obtain an analytically continued curve for the (''nd'')<sup>0.5</sup> weighting, which we then show empirically does indeed appear to be what HE converges on for large values of ''N''.

	In short, what we will show is that the Fourier Transform of this unnormalized harmonic Shannon entropy is given by		In short, what we will show is that the Fourier Transform of this unnormalized harmonic Shannon entropy is given by
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	It would be nice to show the exact relationship of unnormalized entropy to the normalized entropy in the limit of large ''N'', and whether the two converge to be exactly equal (perhaps given some miniscule adjustment in ''s'' or ''a''). However, we will leave this for future research, as well as the question of how to do an exact derivation of normalized HE.		It would be nice to show the exact relationship of unnormalized entropy to the normalized entropy in the limit of large ''N'', and whether the two converge to be exactly equal (perhaps given some miniscule adjustment in ''s'' or ''a''). However, we will leave this for future research, as well as the question of how to do an exact derivation of normalized HE.

	For now, we will start with a derivation of the unnormalized entropy for {{nowrap\|''N'' {{=}} ∞}}, as an interesting function worthy of study in its own right—not only because it looks exactly like HE, but because it leads to an expression for unnormalized HE in terms of the [[~~The Riemann zeta function and tuning\|~~Riemann zeta function]].		For now, we will start with a derivation of the unnormalized entropy for {{nowrap\|''N'' {{=}} ∞}}, as an interesting function worthy of study in its own right—not only because it looks exactly like HE, but because it leads to an expression for unnormalized HE in terms of the [[Riemann zeta function]].

	=== Derivation ===		=== Derivation ===

ArrowHead294: update shortcut to something easier to type

2026-02-08T14:50:16Z

update shortcut to something easier to type

← Older revision		Revision as of 14:50, 8 February 2026
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	$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in J} \frac{e^{i t \log (j_n/j_d)}}{(j_n \cdot j_d)^{w}}$$		$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in J} \frac{e^{i t \log (j_n/j_d)}}{(j_n \cdot j_d)^{w}}$$

	Now, suppose we want to analytically continue this so that the set ''J'' is the set of all reduced rational numbers. We can first do so by starting again with unreduced rationals, but expressing each rational not as {{sfrac\|''n''\|''d''}}, but rather as {{nowrap\|{{sfrac\|''n''{{´}}\|''d''{{-´}}}} · {{sfrac\|''c''\|''c''}}}}, where ''n''{{´}} and ''d''{{-´}} are coprime, and ''c'' is the GCD of both. For example, we would express {{sfrac\|6\|4}} as {{nowrap\|{{sfrac\|3\|2}} · {{sfrac\|2\|2}}}}. Doing so, and assuming that we denote the set of unreduced rationals by ''U'', we get the following equivalent expression of the same convolution kernel above:		Now, suppose we want to analytically continue this so that the set ''J'' is the set of all reduced rational numbers. We can first do so by starting again with unreduced rationals, but expressing each rational not as {{sfrac\|''n''\|''d''}}, but rather as {{nowrap\|{{sfrac\|''n''{{``}}\|''d''{{-`}}}} · {{sfrac\|''c''\|''c''}}}}, where ''n''{{``}} and ''d''{{-`}} are coprime, and ''c'' is the GCD of both. For example, we would express {{sfrac\|6\|4}} as {{nowrap\|{{sfrac\|3\|2}} · {{sfrac\|2\|2}}}}. Doing so, and assuming that we denote the set of unreduced rationals by ''U'', we get the following equivalent expression of the same convolution kernel above:

	$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in \mathbb{U}} \frac{e^{i t \log (\frac{j_c j_{n'}}{j_c j_{d'}})}}{(j_c j_{n'} \cdot j_c j_{d'})^{w}} = \|\zeta(w+i t)\|^2$$		$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in \mathbb{U}} \frac{e^{i t \log (\frac{j_c j_{n'}}{j_c j_{d'}})}}{(j_c j_{n'} \cdot j_c j_{d'})^{w}} = \|\zeta(w+i t)\|^2$$

ArrowHead294: Update shortcut

2026-01-26T17:15:03Z

Update shortcut

← Older revision		Revision as of 17:15, 26 January 2026
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	$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in J} \frac{e^{i t \log (j_n/j_d)}}{(j_n \cdot j_d)^{w}}$$		$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in J} \frac{e^{i t \log (j_n/j_d)}}{(j_n \cdot j_d)^{w}}$$

	Now, suppose we want to analytically continue this so that the set ''J'' is the set of all reduced rational numbers. We can first do so by starting again with unreduced rationals, but expressing each rational not as {{sfrac\|''n''\|''d''}}, but rather as {{nowrap\|{{sfrac\|''n''{{,}}\|''d''{{-,}}}} · {{sfrac\|''c''\|''c''}}}}, where ''n''{{,}} and ''d''{{-,}} are coprime, and ''c'' is the GCD of both. For example, we would express {{sfrac\|6\|4}} as {{nowrap\|{{sfrac\|3\|2}} · {{sfrac\|2\|2}}}}. Doing so, and assuming that we denote the set of unreduced rationals by ''U'', we get the following equivalent expression of the same convolution kernel above:		Now, suppose we want to analytically continue this so that the set ''J'' is the set of all reduced rational numbers. We can first do so by starting again with unreduced rationals, but expressing each rational not as {{sfrac\|''n''\|''d''}}, but rather as {{nowrap\|{{sfrac\|''n''{{´}}\|''d''{{-´}}}} · {{sfrac\|''c''\|''c''}}}}, where ''n''{{´}} and ''d''{{-´}} are coprime, and ''c'' is the GCD of both. For example, we would express {{sfrac\|6\|4}} as {{nowrap\|{{sfrac\|3\|2}} · {{sfrac\|2\|2}}}}. Doing so, and assuming that we denote the set of unreduced rationals by ''U'', we get the following equivalent expression of the same convolution kernel above:

	$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in \mathbb{U}} \frac{e^{i t \log (\frac{j_c j_{n'}}{j_c j_{d'}})}}{(j_c j_{n'} \cdot j_c j_{d'})^{w}} = \|\zeta(w+i t)\|^2$$		$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in \mathbb{U}} \frac{e^{i t \log (\frac{j_c j_{n'}}{j_c j_{d'}})}}{(j_c j_{n'} \cdot j_c j_{d'})^{w}} = \|\zeta(w+i t)\|^2$$

ArrowHead294: Use actual prime symbol

2026-01-26T02:13:38Z

Use actual prime symbol

← Older revision		Revision as of 02:13, 26 January 2026
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	$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in J} \frac{e^{i t \log (j_n/j_d)}}{(j_n \cdot j_d)^{w}}$$		$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in J} \frac{e^{i t \log (j_n/j_d)}}{(j_n \cdot j_d)^{w}}$$

	Now, suppose we want to analytically continue this so that the set ''J'' is the set of all reduced rational numbers. We can first do so by starting again with unreduced rationals, but expressing each rational not as {{sfrac\|''n''\|''d''}}, but rather as {{nowrap\|{{sfrac\|''n''{{'}}\|''d''{{`}}}} · {{sfrac\|''c''\|''c''}}}}, where ''n''{{'}} and ''d''{{`}} are coprime, and ''c'' is the ~~gcd~~ of both. For example, we would express {{sfrac\|6\|4}} as {{nowrap\|{{sfrac\|3\|2}} · {{sfrac\|2\|2}}}}. Doing so, and assuming that we denote the set of unreduced rationals by ''U'', we get the following equivalent expression of the same convolution kernel above:		Now, suppose we want to analytically continue this so that the set ''J'' is the set of all reduced rational numbers. We can first do so by starting again with unreduced rationals, but expressing each rational not as {{sfrac\|''n''\|''d''}}, but rather as {{nowrap\|{{sfrac\|''n''{{,}}\|''d''{{-,}}}} · {{sfrac\|''c''\|''c''}}}}, where ''n''{{,}} and ''d''{{-,}} are coprime, and ''c'' is the GCD of both. For example, we would express {{sfrac\|6\|4}} as {{nowrap\|{{sfrac\|3\|2}} · {{sfrac\|2\|2}}}}. Doing so, and assuming that we denote the set of unreduced rationals by ''U'', we get the following equivalent expression of the same convolution kernel above:

	$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in \mathbb{U}} \frac{e^{i t \log (\frac{j_c j_{n'}}{j_c j_{d'}})}}{(j_c j_{n'} \cdot j_c j_{d'})^{w}} = \|\zeta(w+i t)\|^2$$		$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in \mathbb{U}} \frac{e^{i t \log (\frac{j_c j_{n'}}{j_c j_{d'}})}}{(j_c j_{n'} \cdot j_c j_{d'})^{w}} = \|\zeta(w+i t)\|^2$$

ArrowHead294 at 21:39, 15 January 2026

2026-01-15T21:39:01Z

← Older revision		Revision as of 21:39, 15 January 2026
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	$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in J} \frac{e^{i t \log (j_n/j_d)}}{(j_n \cdot j_d)^{w}}$$		$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in J} \frac{e^{i t \log (j_n/j_d)}}{(j_n \cdot j_d)^{w}}$$

	Now, suppose we want to analytically continue this so that the set ''J'' is the set of all reduced rational numbers. We can first do so by starting again with unreduced rationals, but expressing each rational not as {{sfrac\|''n''\|''d''}}, but rather as {{nowrap\|{{sfrac\|''n''{{'}}\|''d''{{-'}}}} · {{sfrac\|''c''\|''c''}}}}, where ''n''{{'}} and ''d''{{-'}} are coprime, and ''c'' is the gcd of both. For example, we would express {{sfrac\|6\|4}} as {{nowrap\|{{sfrac\|3\|2}} · {{sfrac\|2\|2}}}}. Doing so, and assuming that we denote the set of unreduced rationals by ''U'', we get the following equivalent expression of the same convolution kernel above:		Now, suppose we want to analytically continue this so that the set ''J'' is the set of all reduced rational numbers. We can first do so by starting again with unreduced rationals, but expressing each rational not as {{sfrac\|''n''\|''d''}}, but rather as {{nowrap\|{{sfrac\|''n''{{'}}\|''d''{{`}}}} · {{sfrac\|''c''\|''c''}}}}, where ''n''{{'}} and ''d''{{`}} are coprime, and ''c'' is the gcd of both. For example, we would express {{sfrac\|6\|4}} as {{nowrap\|{{sfrac\|3\|2}} · {{sfrac\|2\|2}}}}. Doing so, and assuming that we denote the set of unreduced rationals by ''U'', we get the following equivalent expression of the same convolution kernel above:

	$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in \mathbb{U}} \frac{e^{i t \log (\frac{j_c j_{n'}}{j_c j_{d'}})}}{(j_c j_{n'} \cdot j_c j_{d'})^{w}} = \|\zeta(w+i t)\|^2$$		$$\displaystyle \mathcal{F}\left\{K(n)\right\}(t) = \sum_{j \in \mathbb{U}} \frac{e^{i t \log (\frac{j_c j_{n'}}{j_c j_{d'}})}}{(j_c j_{n'} \cdot j_c j_{d'})^{w}} = \|\zeta(w+i t)\|^2$$

Facade: /* Accidentally typed 12 before instead of 17 */

2025-11-26T20:28:41Z

Accidentally typed 12 before instead of 17

← Older revision		Revision as of 20:28, 26 November 2025
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	If we discretize this to an integer array of cents, give S a standard deviation of 12 cents, and represent all delta functions in K as a vertical line of height 1, we can visualize S and K like so:
			If we discretize this to an integer array of cents, give S a standard deviation of 17 cents, and represent all delta functions in K as a vertical line of height 1, we can visualize S and K like so:

	[[File:S function.png\|alt=S function with a standard deviation of 17 cents\|frameless]][[File:K function.png\|alt=Visualization of K(c) on 1201 samples for intervals of up to numerator/denominator of 200\|frameless]]		[[File:S function.png\|alt=S function with a standard deviation of 17 cents\|frameless]][[File:K function.png\|alt=Visualization of K(c) on 1201 samples for intervals of up to numerator/denominator of 200\|frameless]]

Facade: /* Extra matplotlib images for visualization and python code example */

2025-11-26T20:27:30Z

Extra matplotlib images for visualization and python code example

@@ Line 290: / Line 290: @@
 We note that the left factor in the convolution product is always the same ''S''(−''c''), which is not dependent on ''j'' in any way. Since convolution distributes over addition, we can factor the ''S'' out of the summation to obtain
-$$\displaystyle \psi(c) = \left[S \ast \left(\sum_{j \in J} \frac{\delta_{-\cent(j)}}{\|j\|}\right)\right](-c)$$
+<nowiki>$$\displaystyle \psi(c) = \left[S \ast \left(\sum_{j \in J} \frac{\delta_{-\cent(j)}}{\|j\|}\right)\right](-c)$$</nowiki>
 We can clean up this notation by defining the auxiliary distribution ''K'':
-$$\displaystyle K(c) = \sum_{j \in J} \frac{\delta_{-\cent(j)}}{\|j\|}$$
+<nowiki>$$\displaystyle K(c) = \sum_{j \in J} \frac{\delta_{-\cent(j)}}{\|j\|}$$</nowiki>
@@ Line 301: / Line 303: @@
 $$\displaystyle \psi(c) = \left[S \ast K\right](-c)$$
 ==== Convolution product for ρ<sub>a</sub>(''c'') ====
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 We have succeeded in representing harmonic Rényi entropy in simple terms of two convolution products, each of which can be computed in {{nowrap|''O''(''N'' log ''N'')}} time.
 == Extending HE to ''N'' {{=}} ∞: zeta-HE ==

Fredg999: Mention interactive HE graph in SW3

2025-08-04T19:06:27Z

Mention interactive HE graph in SW3

← Older revision		Revision as of 19:06, 4 August 2025
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	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.		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.		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.

BudjarnLambeth: /* See also */ add link to Sintel's page

2025-06-16T02:27:49Z

BudjarnLambeth: Add a sentence to introduction (the sentence was written by sintel on the XA Discord when explaining HE to me. It wasn't written by me.)

2025-05-05T11:41:33Z

Add a sentence to introduction (the sentence was written by sintel on the XA Discord when explaining HE to me. It wasn't written by me.)

← Older revision		Revision as of 11:41, 5 May 2025
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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.

	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.		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.

← Older revision		Revision as of 02:27, 16 June 2025
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	* [[Harmonic entropy of just intervals]]		* [[Harmonic entropy of just intervals]]
	* [[Low harmonic entropy linear temperaments]]		* [[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 ==