Wilson norm - Revision history

FloraC: /* External links */ change of name

2025-12-11T13:35:14Z

External links: change of name

← Older revision		Revision as of 13:35, 11 December 2025
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	== External links ==		== External links ==
	* [https://github.com/FloraCanou/temperament_evaluator/wiki/Tenney%E2%80%93Wilson~~-parametric~~-norm~~-profile~~ Github \| ''Tenney–Wilson ~~parametric~~ norm ~~profile~~'' · FloraCanou/temperament_evaluator Wiki] – a unified, parametrized Tenney–Wilson norm used by [[Flora Canou]]'s [https://github.com/FloraCanou/temperament_evaluator Temperament Evaluator].		* [https://github.com/FloraCanou/temperament_evaluator/wiki/Parametric-Tenney%E2%80%93Wilson-norm Github \| ''Parametric Tenney–Wilson norm'' · FloraCanou/temperament_evaluator Wiki] – a unified, parametrized Tenney–Wilson norm used by [[Flora Canou]]'s [https://github.com/FloraCanou/temperament_evaluator Temperament Evaluator].

	== References ==		== References ==

FloraC: Add my research as an external link for now as I'm figuring out how to integrate it here

2025-11-20T15:33:02Z

Add my research as an external link for now as I'm figuring out how to integrate it here

← Older revision		Revision as of 15:33, 20 November 2025
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	* 98/97: 113 (97-limit)		* 98/97: 113 (97-limit)
	* 83/82: 126 (83-limit)		* 83/82: 126 (83-limit)

			== External links ==
			* [https://github.com/FloraCanou/temperament_evaluator/wiki/Tenney%E2%80%93Wilson-parametric-norm-profile Github \| ''Tenney–Wilson parametric norm profile'' · FloraCanou/temperament_evaluator Wiki] – a unified, parametrized Tenney–Wilson norm used by [[Flora Canou]]'s [https://github.com/FloraCanou/temperament_evaluator Temperament Evaluator].

	== References ==		== References ==

FloraC: Move relevant info here (see talk)

2025-11-01T11:57:14Z

Move relevant info here (see talk)

← Older revision		Revision as of 11:57, 1 November 2025
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	The '''Wilson norm''', otherwise known as '''Wilson height''', is a measure of [[complexity]] for [[just interval]]s, similar to the [[Tenney norm]], but has some very beneficial properties that make it an excellent metric to look at.		The '''Wilson norm''', otherwise known as '''Wilson height''', is a measure of [[complexity]] for [[just interval]]s, similar to the [[Tenney norm]], but has some very beneficial properties that make it an excellent metric to look at.

	If ''p''/''q'' is a positive rational number reduced to its lowest terms, then the Wilson norm is the [http://mathworld.wolfram.com/SumofPrimeFactors.html sum of prime factors with repetition] of the number ''pq'', counting multiplicity. This function is often written sopfr(''pq'').		If ''p''/''q'' is a positive rational number reduced to its lowest terms, then the Wilson norm is the [http://mathworld.wolfram.com/SumofPrimeFactors.html sum of prime factors with repetition] of the number ''pq'', counting multiplicity. This function is often written sopfr(''pq''). This is called ''Wilson's complexity'' in [[John Chalmers]]'s ''Divisions of the Tetrachord''<ref>[http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf ''Division of the Tetrachord''], page 55. John Chalmers. </ref>

	Note that we have {{nowrap\| sopfr(''pq'') {{=}} sopfr(''p'') + sopfr(''q'') }}, similar to {{w\|logarithm}} – as a result, this function is sometimes even referred to as the "integer logarithm". So, equivalently, we can define the Wilson norm of a rational number ''p''/''q'' as the Wilson norm of ''p'', plus the Wilson norm of ''q''.		Note that we have {{nowrap\| sopfr(''pq'') {{=}} sopfr(''p'') + sopfr(''q'') }}, similar to {{w\|logarithm}} – as a result, this function is sometimes even referred to as the "integer logarithm". So, equivalently, we can define the Wilson norm of a rational number ''p''/''q'' as the Wilson norm of ''p'', plus the Wilson norm of ''q''.
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	* 98/97: 113 (97-limit)		* 98/97: 113 (97-limit)
	* 83/82: 126 (83-limit)		* 83/82: 126 (83-limit)

			== References ==

	[[Category:Interval complexity measures]]		[[Category:Interval complexity measures]]
	[[Category:Consonance and dissonance]]		[[Category:Consonance and dissonance]]

FloraC: FloraC moved page Wilson height to Wilson norm over redirect: Follow Tenney norm

2025-09-03T02:24:22Z

FloraC moved page Wilson height to Wilson norm over redirect: Follow Tenney norm

← Older revision	Revision as of 02:24, 3 September 2025
(No difference)

FloraC: Height -> norm, following Tenney norm, and move the norm definition up

2025-09-03T02:24:03Z

Height -> norm, following Tenney norm, and move the norm definition up

Show changes

FloraC: Cleanup

2025-09-03T02:14:18Z

Cleanup

Show changes

Lériendil: capitalization

2025-06-11T23:40:04Z

capitalization

← Older revision		Revision as of 23:40, 11 June 2025
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	# The interpretation of Wilson height for individual JI intervals is less direct, but can be thought of as a composite measure of the interval's psychoacoustic complexity and its prime limit. One way to think of it is that it measures how well the interval "could potentially fit" into simple JI chords with simple subsets. For instance, 15/8 fits into 8:10:15, 8:12:15, 8:10:12:15, each of which has simple subsets such as 2:3, 4:5, 4:5:6, etc. It has a Wilson height of 14. In comparison, 13/6 does not have quite as many simple-subset triads and tetrads that it can fit into, and has a Wilson height of 18.		# The interpretation of Wilson height for individual JI intervals is less direct, but can be thought of as a composite measure of the interval's psychoacoustic complexity and its prime limit. One way to think of it is that it measures how well the interval "could potentially fit" into simple JI chords with simple subsets. For instance, 15/8 fits into 8:10:15, 8:12:15, 8:10:12:15, each of which has simple subsets such as 2:3, 4:5, 4:5:6, etc. It has a Wilson height of 14. In comparison, 13/6 does not have quite as many simple-subset triads and tetrads that it can fit into, and has a Wilson height of 18.

	== L1 ~~Norm~~ on ~~Monzos~~ ==		== L1 norm on monzos ==

	The Wilson height has a nice, simple definition as a norm on monzos, which we can call the '''Wilson norm'''. It is given by		The Wilson height has a nice, simple definition as a norm on monzos, which we can call the '''Wilson norm'''. It is given by
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	We can likewise keep the <math>p</math> and <math>1/p</math> weighting, but change things so that we have a weighted <math>\ell_2</math> norm instead of a weighted <math>\ell_1</math>. We can call this the '''Wilson-Euclidean''' norm, and likewise use it to create metrics similar to the [[Tenney-Euclidean_metrics]], including a Wilson-weighted version of the [[Cangwu_badness]].		We can likewise keep the <math>p</math> and <math>1/p</math> weighting, but change things so that we have a weighted <math>\ell_2</math> norm instead of a weighted <math>\ell_1</math>. We can call this the '''Wilson-Euclidean''' norm, and likewise use it to create metrics similar to the [[Tenney-Euclidean_metrics]], including a Wilson-weighted version of the [[Cangwu_badness]].

	== Wilson ~~Height~~ and Tenney ~~Height~~: ~~A Psychoacoustic Comparison~~ ==		== Wilson height and Tenney height: a psychoacoustic comparison ==

	Note that the Wilson height doesn't really correspond directly to the psychoacoustic concordance, perhaps thought of as "crunchiness," of a simple JI interval or chord in the same way that the Tenney height does.		Note that the Wilson height doesn't really correspond directly to the psychoacoustic concordance, perhaps thought of as "crunchiness," of a simple JI interval or chord in the same way that the Tenney height does.
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	Lastly, while we will not get into the weeds of measuring the "learnedness" of some listener here - which is clearly extremely subjective and dependent on musical context - it would be a rather interesting hypothesis to see if some listener's musical perception of JI intervals moves from the Tenney Height toward the Wilson Height given additional ear training in JI, or exposure to a style of JI music that frequently uses simple upper structure ratios and modulations, etc. Or, if it moves back towards the Tenney Height if one embraces a style of perhaps more "spectralist" music in which one mashes huge harmonic series chords without caring much about simple JI subchord relationships.		Lastly, while we will not get into the weeds of measuring the "learnedness" of some listener here - which is clearly extremely subjective and dependent on musical context - it would be a rather interesting hypothesis to see if some listener's musical perception of JI intervals moves from the Tenney Height toward the Wilson Height given additional ear training in JI, or exposure to a style of JI music that frequently uses simple upper structure ratios and modulations, etc. Or, if it moves back towards the Tenney Height if one embraces a style of perhaps more "spectralist" music in which one mashes huge harmonic series chords without caring much about simple JI subchord relationships.

			== Superparticular ratios ==
	== Superparticular ~~Ratios~~ ==

	Perhaps the most immediate use of the Wilson height is in subgroup temperament comma searches. To illustrate this, below is a list of superparticular ratios from 2/1 to 100/99, ranked by this "Wilson height," as well as a note about the smallest prime limit each ratio fits into.		Perhaps the most immediate use of the Wilson height is in subgroup temperament comma searches. To illustrate this, below is a list of superparticular ratios from 2/1 to 100/99, ranked by this "Wilson height," as well as a note about the smallest prime limit each ratio fits into.

Lériendil at 23:39, 11 June 2025

2025-06-11T23:39:27Z

← Older revision		Revision as of 23:39, 11 June 2025
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	[[File:WilsonHeightIntegerLimit.png\|300px\|thumb\|right\|Diagram by Lériendil showing the Wilson height (vertical axis) versus integer limit (horizontal axis) of simple intervals.]]The '''Wilson height''' is a different way of weighting rational numbers than the [[Tenney height]], but has some very beneficial properties that make it an excellent metric to look at.		[[File:WilsonHeightIntegerLimit.png\|300px\|thumb\|right\|Diagram by Lériendil showing the Wilson height (vertical axis) versus [[integer limit]] (horizontal axis) of simple intervals.]]The '''Wilson height''' is a different way of weighting rational numbers than the [[Tenney height]], but has some very beneficial properties that make it an excellent metric to look at.

	If p/q is a positive rational number reduced to its lowest terms, then the Wilson height is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of the number p*q, counting multiplicity. This function is often written <math>\text{sopfr}(pq)</math>.		If p/q is a positive rational number reduced to its lowest terms, then the Wilson height is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of the number p*q, counting multiplicity. This function is often written <math>\text{sopfr}(pq)</math>.

BudjarnLambeth at 02:43, 26 May 2025

2025-05-26T02:43:59Z

← Older revision		Revision as of 02:43, 26 May 2025
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	[[File:WilsonHeightIntegerLimit.png\|~~200px~~\|thumb\|right\|Diagram by Lériendil showing the Wilson height (vertical axis) versus integer limit (horizontal axis) of simple intervals.]]The '''Wilson height''' is a different way of weighting rational numbers than the [[Tenney height]], but has some very beneficial properties that make it an excellent metric to look at.		[[File:WilsonHeightIntegerLimit.png\|300px\|thumb\|right\|Diagram by Lériendil showing the Wilson height (vertical axis) versus integer limit (horizontal axis) of simple intervals.]]The '''Wilson height''' is a different way of weighting rational numbers than the [[Tenney height]], but has some very beneficial properties that make it an excellent metric to look at.

	If p/q is a positive rational number reduced to its lowest terms, then the Wilson height is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of the number p*q, counting multiplicity. This function is often written <math>\text{sopfr}(pq)</math>.		If p/q is a positive rational number reduced to its lowest terms, then the Wilson height is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of the number p*q, counting multiplicity. This function is often written <math>\text{sopfr}(pq)</math>.

BudjarnLambeth at 02:42, 26 May 2025

2025-05-26T02:42:04Z

← Older revision		Revision as of 02:42, 26 May 2025
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	[[File:WilsonHeightIntegerLimit.png\|200px\|thumb\|right\|Diagram by Lériendil showing the Wilson height (vertical axis) versus integer limit (horizontal axis) of simple intervals.]] The '''Wilson height''' is a different way of weighting rational numbers than the [[Tenney height]], but has some very beneficial properties that make it an excellent metric to look at.		[[File:WilsonHeightIntegerLimit.png\|200px\|thumb\|right\|Diagram by Lériendil showing the Wilson height (vertical axis) versus integer limit (horizontal axis) of simple intervals.]]The '''Wilson height''' is a different way of weighting rational numbers than the [[Tenney height]], but has some very beneficial properties that make it an excellent metric to look at.

	If p/q is a positive rational number reduced to its lowest terms, then the Wilson height is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of the number p*q, counting multiplicity. This function is often written <math>\text{sopfr}(pq)</math>.		If p/q is a positive rational number reduced to its lowest terms, then the Wilson height is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of the number p*q, counting multiplicity. This function is often written <math>\text{sopfr}(pq)</math>.