Height - Revision history

FloraC: - duplicate info

2025-11-01T11:57:34Z

- duplicate info

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	Where \|\|''q''\|\|<sub>T1</sub> is the [[Generalized Tenney norms and Tp interval space #The Tenney Norm (T1 norm)\|tenney norm]] of ''q'' in monzo form, and v<sub>''p''</sub>(''q'') is the [[Wikipedia: P-adic order\|''p''-adic valuation]] of ''q''.		Where \|\|''q''\|\|<sub>T1</sub> is the [[Generalized Tenney norms and Tp interval space #The Tenney Norm (T1 norm)\|tenney norm]] of ''q'' in monzo form, and v<sub>''p''</sub>(''q'') is the [[Wikipedia: P-adic order\|''p''-adic valuation]] of ''q''.

	The function sopfr (''nd'') is the [https://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of ''n''·''d''. Equivalently, this is the L<sub>1</sub> norm on monzos, but where each prime is weighted by ''p'' rather than log (''p''). 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>.		The function sopfr (''nd'') is the [https://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of ''n''·''d''. Equivalently, this is the L<sub>1</sub> norm on monzos, but where each prime is weighted by ''p'' rather than log (''p'').

	Some useful identities:		Some useful identities:

Sintel: equivalence section: no need to talk about p-adic valuations here

2025-04-26T13:04:02Z

equivalence section: no need to talk about p-adic valuations here

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	Exponentiation and logarithm are such functions commonly used for converting a height between arithmetic and logarithmic scales.		Exponentiation and logarithm are such functions commonly used for converting a height between arithmetic and logarithmic scales.

	A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''q''<sub>1</sub> and ''q''<sub>2</sub> are considered equivalent if ~~the following is true:~~		A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''q''<sub>1</sub> and ''q''<sub>2</sub> are considered equivalent if they differ only by factors of 2.
			We can also consider other equivalences. For example, we can assume tritave equivalence by ignoring factors of 3.
	~~<math>\displaystyle~~ 2~~^{-v_2 \left( {q_1} \right)} q_1 = 2^{-v_2 \left( {q_2} \right)} q_2</math>~~

	~~where v<sub>''p''</sub>(''q'') is the [[Wikipedia: P-adic valuation\|''p''-adic valuation]] of ''q''~~.

	~~Or equivalently, if ''n'' has any integer solutions:~~

	~~<math>\displaystyle q_1 = 2^n q_2</math>~~

	~~If the above condition is met~~, we ~~may then establish the following~~ equivalence ~~relation:~~

	~~<math>\displaystyle q_1 \equiv q_2</math>~~

	~~By changing the base~~ of ~~the exponent to a value other than 2, you can set up completely different equivalence relations. Replacing the 2 with a~~ 3 ~~yields tritave-equivalence, for example~~.

	== Height versus norm ==		== Height versus norm ==

Roeesi: /* Examples of height functions */ Moved from the general complexity article

2024-12-03T18:29:24Z

Examples of height functions: Moved from the general complexity article

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	Height functions can also be put on the points of [http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html projective varieties]. Since [[abstract regular temperament]]s can be identified with rational points on [[Wikipedia: Grassmannian\|Grassmann varieties]], complexity measures of regular temperaments are also height functions.		Height functions can also be put on the points of [http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html projective varieties]. Since [[abstract regular temperament]]s can be identified with rational points on [[Wikipedia: Grassmannian\|Grassmann varieties]], complexity measures of regular temperaments are also height functions.

			See [[Dave Keenan & Douglas Blumeyer's guide to RTT/Alternative complexities]] for an extensive discussion of heights and semi-heights used in regular temperament theory.

	== History ==		== History ==

FloraC: Recategorize

2023-11-26T11:12:19Z

Recategorize

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	[[Category:Math]]		[[Category:Math]]
	[[Category:Interval complexity ~~measure\|~~ ]] ~~<!-- main article -->~~		[[Category:Interval complexity measures]]

FloraC: Recategorize

2023-08-29T13:45:03Z

Recategorize

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	<references/>		<references/>

	~~[[Category:Height\| ]] <!-- main article -->~~
	[[Category:Math]]		[[Category:Math]]
			[[Category:Interval complexity measure\| ]] <!-- main article -->

FloraC: +section comparing height and norm

2023-08-29T06:57:47Z

+section comparing height and norm

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	<math>\displaystyle H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</math>		<math>\displaystyle H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</math>

	Exponentiation and logarithm are such functions commonly used for converting a height between ~~the~~ arithmetic and logarithmic ~~scale~~.		Exponentiation and logarithm are such functions commonly used for converting a height between arithmetic and logarithmic scales.

	A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''q''<sub>1</sub> and ''q''<sub>2</sub> are considered equivalent if the following is true:		A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''q''<sub>1</sub> and ''q''<sub>2</sub> are considered equivalent if the following is true:
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	By changing the base of the exponent to a value other than 2, you can set up completely different equivalence relations. Replacing the 2 with a 3 yields tritave-equivalence, for example.		By changing the base of the exponent to a value other than 2, you can set up completely different equivalence relations. Replacing the 2 with a 3 yields tritave-equivalence, for example.

			== Height versus norm ==
			Height functions are applied to ratios, whereas norms are measurements on interval lattices [[wikipedia: embedding\|embedded]] in [[wikipedia: Normed vector space\|normed vector spaces]]. Some height functions are essentially norms, and they are numerically equal. For example, the [[Tenney height]] is also the Tenney norm.

			However, not all height functions are norms, and not all norms are height functions. The [[Benedetti height]] is not a norm, since it does not satisfy the condition of absolute homogeneity. The [[taxicab distance]] is not a height, since there can be infinitely many intervals below a given bound.

	== Examples of height functions ==		== Examples of height functions ==

Xenllium at 03:44, 1 June 2023

2023-06-01T03:44:33Z

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	The '''height''' is a mathematical tool to measure the [[complexity]] of [[JI]] [[interval]]s.		The '''height''' is a mathematical tool to measure the [[complexity]] of [[JI]] [[interval]]s.

	~~__TOC__~~

	== Definition ==		== Definition ==

	A '''height''' is a function on members of an algebraically defined object which maps elements to real numbers, yielding a type of complexity measurement (see [[Wikipedia: Height function]]). For example we can assign each element of the positive rational numbers a height, and hence a complexity. While there is no consensus on the restrictions of a height, we will attempt to create a definition for positive rational numbers which is practical for musical purposes.		A '''height''' is a function on members of an algebraically defined object which maps elements to real numbers, yielding a type of complexity measurement (see [[Wikipedia: Height function]]). For example we can assign each element of the positive rational numbers a height, and hence a complexity. While there is no consensus on the restrictions of a height, we will attempt to create a definition for positive rational numbers which is practical for musical purposes.

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	== Examples of height functions ==		== Examples of height functions ==

	{\| class="wikitable"		{\| class="wikitable"
	! Name		! Name
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	\| Height		\| Height
	\| <math>\text{sopfr}(n d)</math>		\| <math>\text{sopfr}(n d)</math>
	\| <math>2^{\large{\text{sopfr}(~~n d~~)}}</math>		\| <math>2^{\large{\text{sopfr}(q)}}</math>
	\| <math>\text{sopfr}(q)</math>		\| <math>\text{sopfr}(q)</math>
	\|-		\|-
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	== History ==		== History ==

	The concept of height was introduced to xenharmonics by [[Gene Ward Smith]] in 2001<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_31418#31488 Yahoo! Tuning Group \| ''Super Particular Stepsize'']</ref>; it comes from the mathematical field of number theory (for more information, see [[Wikipedia: Height function]]). It is not to be confused with the musical notion of [[Wikipedia: Pitch (music) #Theories of pitch perception\|''pitch height'' (as opposed to ''pitch chroma'')]]<ref>Though it has also been used to refer to the size of an interval in cents. On page 23 of [https://www.plainsound.org/pdfs/JC&ToH.pdf ''John Cage and the Theor of Harmony''], Tenney writes: "The one-dimensional continuum of pitch-height (i.e. 'pitch' as ordinarily defined)", and graphs it ''as opposed to'' his concept of "harmonic distance", which was ironically the first measurement named by Gene Ward Smith as a "height": "Tenney height".</ref>.		The concept of height was introduced to xenharmonics by [[Gene Ward Smith]] in 2001<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_31418#31488 Yahoo! Tuning Group \| ''Super Particular Stepsize'']</ref>; it comes from the mathematical field of number theory (for more information, see [[Wikipedia: Height function]]). It is not to be confused with the musical notion of [[Wikipedia: Pitch (music) #Theories of pitch perception\|''pitch height'' (as opposed to ''pitch chroma'')]]<ref>Though it has also been used to refer to the size of an interval in cents. On page 23 of [https://www.plainsound.org/pdfs/JC&ToH.pdf ''John Cage and the Theor of Harmony''], Tenney writes: "The one-dimensional continuum of pitch-height (i.e. 'pitch' as ordinarily defined)", and graphs it ''as opposed to'' his concept of "harmonic distance", which was ironically the first measurement named by Gene Ward Smith as a "height": "Tenney height".</ref>.

	== See also ==		== See also ==

	* [[Commas by taxicab distance]]		* [[Commas by taxicab distance]]
			* [[Harmonic entropy]]

	== References ==		== References ==

Fredg999 category edits: Removing from Category:Theory using Cat-a-lot

2023-03-02T03:53:56Z

Removing from Category:Theory using Cat-a-lot

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	<references/>		<references/>

	~~[[Category:Theory]]~~
	[[Category:Height\| ]] <!-- main article -->		[[Category:Height\| ]] <!-- main article -->
	[[Category:Math]]		[[Category:Math]]

Fredg999 category edits: Removing from Category:Measure using Cat-a-lot

2023-02-26T06:34:33Z

Removing from Category:Measure using Cat-a-lot

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	[[Category:Height\| ]] <!-- main article -->		[[Category:Height\| ]] <!-- main article -->
	[[Category:Math]]		[[Category:Math]]
	~~[[Category:Measure]]~~

FloraC: Explain what this "equivalence class" would commonly be

2023-02-13T14:45:00Z

Explain what this "equivalence class" would commonly be

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	<math>\displaystyle H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</math>		<math>\displaystyle H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</math>

			Exponentiation and logarithm are such functions commonly used for converting a height between the arithmetic and logarithmic scale.

	A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''q''<sub>1</sub> and ''q''<sub>2</sub> are considered equivalent if the following is true:		A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''q''<sub>1</sub> and ''q''<sub>2</sub> are considered equivalent if the following is true: