Vals and tuning space - Revision history

Sintel: /* Example */ fix broken latex

2025-04-26T15:22:01Z

Example: fix broken latex

← Older revision		Revision as of 15:22, 26 April 2025
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	<math>\displaystyle		<math>\displaystyle
	\left<31 \; \frac{49}{\log_2(3)} \; \frac{72}{\log_2(5)} \; \frac{87}{\log_2(7)}\right\|		\left< 31 \; \frac{49}{\log_2(3)} \; \frac{72}{\log_2(5)} \; \frac{87}{\log_2(7)}\right\|
	~~%original was <31 49/log2(3) 72/log2(5) 87/log2(7)\|~~</math>		</math>

	which is approximately {{val\| 31.000 30.916 31.009 30.990 }}. The standard Euclidean norm would then be the square root of the sum of squares of this vector, which is approximately sqrt (3838.694), or 61.957. To use the RMS we divide that by sqrt (4) = 2, giving 30.976 for the TE norm. Note that the TE norm for this val is approximately 31; any val closely approximating JI is expected to have the TE norm close to its division of the octave.		which is approximately {{val\| 31.000 30.916 31.009 30.990 }}. The standard Euclidean norm would then be the square root of the sum of squares of this vector, which is approximately sqrt (3838.694), or 61.957. To use the RMS we divide that by sqrt (4) = 2, giving 30.976 for the TE norm. Note that the TE norm for this val is approximately 31; any val closely approximating JI is expected to have the TE norm close to its division of the octave.

Sintel: -legacy

2025-03-29T18:13:10Z

-legacy

← Older revision		Revision as of 18:13, 29 March 2025
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	}}		}}
	{{Expert\|Val}}		{{Expert\|Val}}
	~~{{Legacy}}~~
	A '''val''' "maps" [[just intonation]] to a certain number of steps in a chain of [[generator]]s; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … ''a''<sub>''k''</sub> }}, where the numbers ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … are the number of steps along the chain that the first ''k'' [[prime]]s are mapped to. This can be generalized so that ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.		A '''val''' "maps" [[just intonation]] to a certain number of steps in a chain of [[generator]]s; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … ''a''<sub>''k''</sub> }}, where the numbers ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … are the number of steps along the chain that the first ''k'' [[prime]]s are mapped to. This can be generalized so that ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.

Lériendil at 16:53, 17 February 2025

2025-02-17T16:53:23Z

← Older revision		Revision as of 16:53, 17 February 2025
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	}}		}}
	{{Expert\|Val}}		{{Expert\|Val}}
	{{Legacy~~\|Vals_and_tuning_space~~}}		{{Legacy}}
	A '''val''' "maps" [[just intonation]] to a certain number of steps in a chain of [[generator]]s; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … ''a''<sub>''k''</sub> }}, where the numbers ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … are the number of steps along the chain that the first ''k'' [[prime]]s are mapped to. This can be generalized so that ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.		A '''val''' "maps" [[just intonation]] to a certain number of steps in a chain of [[generator]]s; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … ''a''<sub>''k''</sub> }}, where the numbers ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … are the number of steps along the chain that the first ''k'' [[prime]]s are mapped to. This can be generalized so that ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.

Lériendil: deploying new cat

2025-02-17T16:07:13Z

deploying new cat

← Older revision		Revision as of 16:07, 17 February 2025
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	}}		}}
	{{Expert\|Val}}		{{Expert\|Val}}
			{{Legacy\|Vals_and_tuning_space}}
	A '''val''' "maps" [[just intonation]] to a certain number of steps in a chain of [[generator]]s; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … ''a''<sub>''k''</sub> }}, where the numbers ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … are the number of steps along the chain that the first ''k'' [[prime]]s are mapped to. This can be generalized so that ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.		A '''val''' "maps" [[just intonation]] to a certain number of steps in a chain of [[generator]]s; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … ''a''<sub>''k''</sub> }}, where the numbers ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … are the number of steps along the chain that the first ''k'' [[prime]]s are mapped to. This can be generalized so that ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.

Lériendil: Changed protection level for "Vals and tuning space" ([Edit=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)))

2025-02-17T16:00:20Z

Changed protection level for "Vals and tuning space" ([Edit=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:00, 17 February 2025
(No difference)

FloraC: The first part of the definition can be promoted to the intro np

2024-09-03T08:01:57Z

The first part of the definition can be promoted to the intro np

← Older revision		Revision as of 08:01, 3 September 2024
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	{{Expert\|Val}}		{{Expert\|Val}}

	~~== Definition ==~~		A '''val''' "maps" [[just intonation]] to a certain number of steps in a chain of [[generator]]s; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … ''a''<sub>''k''</sub> }}, where the numbers ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … are the number of steps along the chain that the first ''k'' [[prime]]s are mapped to. This can be generalized so that ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.
	A [[val]] "maps" just intonation to a certain number of steps in a chain of ~~generators~~; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … ''a''<sub>''k''</sub> }}, where the numbers ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … are the number of steps along the chain that the first ''k'' ~~primes~~ are mapped to. This can be generalized so that ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.

	A ''rank-r'' temperament has ''r'' generators, and thus is defined by ''r'' vals. In the usual coordinates for the [[~~Harmonic~~ limit\|''p''-limit]], the set of generators are the first ''k'' prime numbers and the set of vals for a ''p''-limit temperament gives you the coordinates for each prime harmonic in the ''p''-limit. For example, all 5-limit rank-1 temperaments, or [[equal temperament]]s, will be defined by a val {{val\| ''a'' ''b'' ''c'' }}, where ''a'' is the number of generators it takes to reach the 2nd harmonic (2/1), ''b'' is the number of generators to reach the 3rd harmonic (3/1), and ''c'' is the number of generators it takes to reach the 5th harmonic (5/1). All 5-limit rank-2 temperaments are defined by two vals: {{monzo\| {{val\| ''a''<sub>1</sub> ''b''<sub>1</sub> ''c''<sub>1</sub> }}, {{val\| ''a''<sub>2</sub> ''b''<sub>2</sub> ''c''<sub>2</sub> }} }}. Now, we locate the 2nd harmonic (2/1) with the 2-dimensional coordinates (''a''<sub>1</sub>, ''a''<sub>2</sub>), sometimes written as {{monzo\| ''a''<sub>1</sub> ''a''<sub>2</sub> }}, meaning go up ''a''<sub>1</sub> of the first generator, and up ''a''<sub>2</sub> of the 2nd generator, to reach 2/1. Similarly, the 3rd harmonic and 5th harmonic will be reached by {{monzo\| ''b''<sub>1</sub> ''b''<sub>2</sub> }} and {{monzo\| ''c''<sub>1</sub> ''c''<sub>2</sub> }} respectively.		A ''rank-r'' temperament has ''r'' generators, and thus is defined by ''r'' vals. In the usual coordinates for the [[harmonic limit\|''p''-limit]], the set of generators are the first ''k'' prime numbers and the set of vals for a ''p''-limit temperament gives you the coordinates for each prime harmonic in the ''p''-limit. For example, all 5-limit rank-1 temperaments, or [[equal temperament]]s, will be defined by a val {{val\| ''a'' ''b'' ''c'' }}, where ''a'' is the number of generators it takes to reach the 2nd harmonic (2/1), ''b'' is the number of generators to reach the 3rd harmonic (3/1), and ''c'' is the number of generators it takes to reach the 5th harmonic (5/1). All 5-limit rank-2 temperaments are defined by two vals: {{monzo\| {{val\| ''a''<sub>1</sub> ''b''<sub>1</sub> ''c''<sub>1</sub> }}, {{val\| ''a''<sub>2</sub> ''b''<sub>2</sub> ''c''<sub>2</sub> }} }}. Now, we locate the 2nd harmonic (2/1) with the 2-dimensional coordinates (''a''<sub>1</sub>, ''a''<sub>2</sub>), sometimes written as {{monzo\| ''a''<sub>1</sub> ''a''<sub>2</sub> }}, meaning go up ''a''<sub>1</sub> of the first generator, and up ''a''<sub>2</sub> of the 2nd generator, to reach 2/1. Similarly, the 3rd harmonic and 5th harmonic will be reached by {{monzo\| ''b''<sub>1</sub> ''b''<sub>2</sub> }} and {{monzo\| ''c''<sub>1</sub> ''c''<sub>2</sub> }} respectively.

	As an example, consider meantone temperament, where 81/80 vanishes. Meantone can be considered a 5-limit rank-2 temperament, defined by the two-val mapping {{monzo\|{{val\| 1 1 0 }}, {{val\| 0 1 4 }}}}. This tells us just about everything we need to know about how the 5-limit is mapped in meantone: since 2/1 is mapped to {{monzo\| 1 0 }}, that tells us that the first generator ''is'' a 2/1, and since 3/1 is mapped to {{monzo\| 1 1 }}, that tells us that the 2nd generator is a 3/2; then, since 5/1 is mapped to {{monzo\| 0 4 }}, aka four 3/2's up, that tells us that 81/64 (which is (3/2)<sup>4</sup>) equals 5/1 (which is 80/64). Since 81/64 is equated with 80/64 here, that tells us that 81/80 is tempered out! Thus it is possible to derive from the mapping the approximate size of the two generators, the commas that are tempered out, and roughly the complexity of the temperament (the number of notes of the temperament we need to reach all the prime harmonics in the ''p''-limit). This makes the val an extremely compact and useful bit of notation for describing regular temperaments, since we can readily find where all of the primes are mapped along the temperament's chain of generators essentially at a glance.		As an example, consider [[meantone]] temperament, where [[81/80]] vanishes. Meantone can be considered a [[5-limit]] rank-2 temperament, defined by the two-val mapping {{monzo\|{{val\| 1 1 0 }}, {{val\| 0 1 4 }}}}. This tells us just about everything we need to know about how the 5-limit is mapped in meantone: since 2/1 is mapped to {{monzo\| 1 0 }}, that tells us that the first generator ''is'' a 2/1, and since 3/1 is mapped to {{monzo\| 1 1 }}, that tells us that the 2nd generator is a 3/2; then, since 5/1 is mapped to {{monzo\| 0 4 }}, aka four 3/2's up, that tells us that 81/64 (which is (3/2)<sup>4</sup>) equals 5/1 (which is 80/64). Since 81/64 is equated with 80/64 here, that tells us that 81/80 is [[tempering out\|tempered out]]! Thus it is possible to derive from the mapping the approximate size of the two generators, the commas that are tempered out, and roughly the complexity of the temperament (the number of notes of the temperament we need to reach all the prime harmonics in the ''p''-limit). This makes the val an extremely compact and useful bit of notation for describing regular temperaments, since we can readily find where all of the primes are mapped along the temperament's chain of generators essentially at a glance.

	Whenever one of the generators of a temperament is a 2/1 the key information is carried by the other vals, assuming octave equivalence (i.e. 3/1 = 3/2 = 6/1 etc.). Thus the essential character of 5-limit meantone is defined by a single val (the one for the 3/2 generator), written {{val\| 0 1 4 }}.		Whenever one of the generators of a temperament is a 2/1 the key information is carried by the other vals, assuming [[octave equivalence]] (i.e. 3/1 = 3/2 = 6/1 etc.). Thus the essential character of 5-limit meantone is defined by a single val (the one for the 3/2 generator), written {{val\| 0 1 4 }}.

	=== Definition for mathematicians ===		== Definition for mathematicians ==
	The ''p''-limit [[~~Monzos~~ and interval space\|monzos]] M form a free abelian group, or ℤ-module, of finite rank π (''p''), which is the number of primes up to and including ''p''. The [http://planetmath.org/encyclopedia/DualModule.html dual ℤ-module] M* is [[Wikipedia: Group isomorphism\|isomorphic]] to M, but not in a canonical way. Hence it, the group (Z-module) of '''vals''', is also a free abelian group of rank π (''p''). Just as monzos are often written as [http://mathworld.wolfram.com/Ket.html kets], vals are typically written as [http://mathworld.wolfram.com/Bra.html bras]. Vals are homomorphisms from a subgroup of finite rank of ℚ*, the abelian group of the positive rational numbers under multiplication, to the integers ℤ. The number theorist [[Yves Hellegouarch]] seems to have been the first to write about them, under the name "degrees".		The ''p''-limit [[monzos and interval space\|monzos]] M form a free abelian group, or ℤ-module, of finite rank π (''p''), which is the number of primes up to and including ''p''. The [http://planetmath.org/encyclopedia/DualModule.html dual ℤ-module] M* is [[Wikipedia: Group isomorphism\|isomorphic]] to M, but not in a canonical way. Hence it, the group (Z-module) of '''vals''', is also a free abelian group of rank π (''p''). Just as monzos are often written as [http://mathworld.wolfram.com/Ket.html kets], vals are typically written as [http://mathworld.wolfram.com/Bra.html bras]. Vals are {{w\|group homomorphism\|homomorphisms}} from a subgroup of finite rank of ℚ*, the abelian group of the positive rational numbers under multiplication, to the integers ℤ. The number theorist [[Yves Hellegouarch]] seems to have been the first to write about them, under the name "degrees".

	== Vals and monzos ==		== Vals and monzos ==
	If V is a val and M is a monzo of the same rank, then the [http://mathworld.wolfram.com/AngleBracket.html angle bracket], written ⟨V\|M⟩ (or occasionally V(M)), is the result of applying the ~~[[Wikipedia: Group~~ homomorphism~~\|homomorphism]]~~ V to M. For example, if V = {{val\| 12 19 28 34 }} and M = {{monzo\| -5 2 2 -1 }} then ⟨V\|M⟩ equals 12×(-5) + 19×2 + 28×2 - 34 = 0.		If ''V'' is a val and ''M'' is a monzo of the same rank, then the [http://mathworld.wolfram.com/AngleBracket.html angle bracket], written ⟨''V''\|''M''⟩ (or occasionally ''V''(''M'')), is the result of applying the homomorphism ''V'' to ''M''. For example, if ''V'' = {{val\| 12 19 28 34 }} and ''M'' = {{monzo\| -5 2 2 -1 }} then ⟨''V''\|''M''⟩ equals 12×(-5) + 19×2 + 28×2 - 34 = 0.

	This tells us that in septimal 12 equal, represented by V, the interval 225/224, represented by M, is mapped to 0, which represents 1. Hence, 225/224 vanishes in septimal 12 equal; it is in the [http://mathworld.wolfram.com/GroupKernel.html kernel] of V. One should note in particular that the coordinates of V represent where the successive primes 2, 3, 5 and 7 are mapped.		This tells us that in septimal 12 equal, represented by ''V'', the interval 225/224, represented by ''M'', is mapped to 0, which represents 1. Hence, 225/224 vanishes in septimal 12 equal; it is in the [http://mathworld.wolfram.com/GroupKernel.html kernel] of ''V''. One should note in particular that the coordinates of ''V'' represent where the successive primes 2, 3, 5 and 7 are mapped.

	By ~~[[Wikipedia: Embedding~~\|embedding]] the monzos into a suitable vector space, norms may be placed on the monzos in various ways, turning them into [http://mathworld.wolfram.com/PointLattice.html lattices] in a vector space. Given a vector space norm on a space of ket vectors, the [http://mathworld.wolfram.com/DualNormedSpace.html dual vector space norm] on the space of bra vectors is defined as the least quantity \|\|V\|\| making		By {{w\|embedding}} the monzos into a suitable vector space, norms may be placed on the monzos in various ways, turning them into [http://mathworld.wolfram.com/PointLattice.html lattices] in a vector space. Given a vector space norm on a space of ket vectors, the [http://mathworld.wolfram.com/DualNormedSpace.html dual vector space norm] on the space of bra vectors is defined as the least quantity ‖''V''‖ making

	<math>\displaystyle		<math>\displaystyle
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	</math>		</math>

	to be always true. The dual of the [http://mathworld.wolfram.com/L1-Norm.html ''L''<sup>1</sup> norm] is the [http://mathworld.wolfram.com/L-Infinity-Norm.html ''L''-infinity norm], and the dual space of Tenney interval space is Tenney tuning space. The embedding of monzos into a real normed vector space automatically induces a dual embedding of vals into a corresponding normed vector space, tuning space, in which vals are lattice points. The dual norm to the ''L''<sup>2</sup> norm is the ''L''<sup>2</sup> norm, and the dual space to Tenney-Euclidean interval space is ''Tenney-Euclidean tuning space''. The Euclidean norm on a val V is given by		to be always true. The dual of the [http://mathworld.wolfram.com/L1-Norm.html ''L''<sup>1</sup> norm] is the [http://mathworld.wolfram.com/L-Infinity-Norm.html ''L''-infinity norm], and the dual space of Tenney interval space is Tenney tuning space. The embedding of monzos into a real normed vector space automatically induces a dual embedding of vals into a corresponding normed vector space, tuning space, in which vals are lattice points. The dual norm to the ''L''<sup>2</sup> norm is the ''L''<sup>2</sup> norm, and the dual space to Tenney-Euclidean interval space is ''Tenney-Euclidean tuning space''. The Euclidean norm on a val ''V'' is given by

	<math>\displaystyle		<math>\displaystyle
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	</math>		</math>

			It useful to renormalize to the RMS ({{w\|root mean square}}) instead, which requires dividing the above by sqrt (''n''), where ''n'' = π (''p'') is the number of primes up to ''p''. This is the [[Tenney-Euclidean metrics\|Tenney-Euclidean norm]], or TE norm.

			It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or [[JIP]], which in weighted coordinates is ''J'' = {{val\| 1 1 1 … 1 }}. It has the property that if ''M'' is a monzo in weighted coordinates, then ⟨''J''\|''M''⟩ or ''J'' (''M'') if you prefer, is exactly the log base two of the interval ''M'' represents, hence the name. In unweighted coordinates, ''J'' = {{val\| 1 log<sub>2</sub> (3) … log<sub>2</sub> (''p'')}}, and applied to a monzo this gives the log base two of the corresponding interval.
	It useful to renormalize to the RMS (root mean square) instead, which requires dividing the above by sqrt (''n''), where ''n'' = π (''p'') is the number of primes up to ''p''. This is the [[Tenney-Euclidean metrics\|Tenney-Euclidean norm]], or TE norm.

	It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or [[JIP]], which in weighted coordinates is J = {{val\| 1 1 1 … 1 }}. It has the property that if M is a monzo in weighted coordinates, then ⟨J\|M⟩ or J (M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, J = {{val\| 1 log<sub>2</sub> (3) … log<sub>2</sub> (''p'')}}, and applied to a monzo this gives the log base two of the corresponding interval.

	== Example ==		== Example ==
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	[[Category:Tuning space]]		[[Category:Tuning space]]
	[[Category:Val]]		[[Category:Val]]

	~~{{Todo\| intro }}~~

FloraC: Update Jp translation; mark as expert page; style

2024-04-20T08:43:31Z

Update Jp translation; mark as expert page; style

← Older revision		Revision as of 08:43, 20 April 2024
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	{{interwiki		{{interwiki
	\| de = Val		\| de = Val
	\| en = Vals and ~~Tuning Space~~		\| en = Vals and tuning space
	\| es =		\| es =
	\| ja = ~~ヴァルと音程空間~~		\| ja = ヴァルと調律空間
	}}		}}
			{{Expert\|Val}}

	~~__FORCETOC__~~
	== Definition ==		== Definition ==
	A [[val]] "maps" just intonation to a certain number of steps in a chain of generators; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … ''a''<sub>k</sub> }}, where the numbers ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … are the number of steps along the chain that the first ''k'' primes are mapped to. This can be generalized so that ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.		A [[val]] "maps" just intonation to a certain number of steps in a chain of generators; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … ''a''<sub>''k''</sub> }}, where the numbers ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … are the number of steps along the chain that the first ''k'' primes are mapped to. This can be generalized so that ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.

	A ''rank-r'' temperament has ''r'' generators, and thus is defined by ''r'' vals. In the usual coordinates for the [[Harmonic limit\|''p''-limit]], the set of generators are the first ''k'' prime numbers and the set of vals for a ''p''-limit temperament gives you the coordinates for each prime harmonic in the ''p''-limit. For example, all 5-limit rank-1 temperaments, or [[equal temperament]]s, will be defined by a val {{val\| ''a'' ''b'' ''c'' }}, where ''a'' is the number of generators it takes to reach the 2nd harmonic (2/1), ''b'' is the number of generators to reach the 3rd harmonic (3/1), and ''c'' is the number of generators it takes to reach the 5th harmonic (5/1). All 5-limit rank-2 temperaments are defined by two vals: {{monzo\| {{val\| ''a''<sub>1</sub> ''b''<sub>1</sub> ''c''<sub>1</sub> }}, {{val\| ''a''<sub>2</sub> ''b''<sub>2</sub> ''c''<sub>2</sub> }} }}. Now, we locate the 2nd harmonic (2/1) with the 2-dimensional coordinates (''a''<sub>1</sub>, ''a''<sub>2</sub>), sometimes written as {{monzo\| ''a''<sub>1</sub> ''a''<sub>2</sub> }}, meaning go up ''a''<sub>1</sub> of the first generator, and up ''a''<sub>2</sub> of the 2nd generator, to reach 2/1. Similarly, the 3rd harmonic and 5th harmonic will be reached by {{monzo\| ''b''<sub>1</sub> ''b''<sub>2</sub> }} and {{monzo\| ''c''<sub>1</sub> ''c''<sub>2</sub> }} respectively.		A ''rank-r'' temperament has ''r'' generators, and thus is defined by ''r'' vals. In the usual coordinates for the [[Harmonic limit\|''p''-limit]], the set of generators are the first ''k'' prime numbers and the set of vals for a ''p''-limit temperament gives you the coordinates for each prime harmonic in the ''p''-limit. For example, all 5-limit rank-1 temperaments, or [[equal temperament]]s, will be defined by a val {{val\| ''a'' ''b'' ''c'' }}, where ''a'' is the number of generators it takes to reach the 2nd harmonic (2/1), ''b'' is the number of generators to reach the 3rd harmonic (3/1), and ''c'' is the number of generators it takes to reach the 5th harmonic (5/1). All 5-limit rank-2 temperaments are defined by two vals: {{monzo\| {{val\| ''a''<sub>1</sub> ''b''<sub>1</sub> ''c''<sub>1</sub> }}, {{val\| ''a''<sub>2</sub> ''b''<sub>2</sub> ''c''<sub>2</sub> }} }}. Now, we locate the 2nd harmonic (2/1) with the 2-dimensional coordinates (''a''<sub>1</sub>, ''a''<sub>2</sub>), sometimes written as {{monzo\| ''a''<sub>1</sub> ''a''<sub>2</sub> }}, meaning go up ''a''<sub>1</sub> of the first generator, and up ''a''<sub>2</sub> of the 2nd generator, to reach 2/1. Similarly, the 3rd harmonic and 5th harmonic will be reached by {{monzo\| ''b''<sub>1</sub> ''b''<sub>2</sub> }} and {{monzo\| ''c''<sub>1</sub> ''c''<sub>2</sub> }} respectively.

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2022-03-13T14:50:43Z

Style, improve readability, and internalize Wikipedia links

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Mike Battaglia: Protected "Vals and tuning space" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

2021-11-15T03:03:12Z

Protected "Vals and tuning space" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

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2021-11-08T05:08:53Z

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← Older revision		Revision as of 05:08, 8 November 2021
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	__FORCETOC__		__FORCETOC__
	== Definition ==		== Definition ==
	A val "maps" just intonation to a certain number of steps in a chain of generators; by putting vals together we can define the mapping of a [[Regular_Temperaments\|regular temperament]] and thereby define the temperament. A val is written in the form {{val\|a1 a2 a3 ... ak}}, where the numbers a1 a2 a3 ... are the number of steps along the chain that the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any JI basis is mapped to, whereas a JI basis for a [[Just_intonation_subgroups\|just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.		A [[val\|val]] "maps" just intonation to a certain number of steps in a chain of generators; by putting vals together we can define the mapping of a [[Regular_Temperaments\|regular temperament]] and thereby define the temperament. A val is written in the form {{val\|a1 a2 a3 ... ak}}, where the numbers a1 a2 a3 ... are the number of steps along the chain that the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any JI basis is mapped to, whereas a JI basis for a [[Just_intonation_subgroups\|just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.

	A ''rank r'' temperament has r generators, and thus is defined by r vals. In the usual coordinates for the [[Harmonic_Limit\|p-limit]], the set of generators are the first k prime numbers and the set of vals for a p-limit temperament gives you the coordinates for each prime harmonic in the p-limit. For example, all 5-limit rank-1 temperaments, or [[Equal\|equal temperaments]], will be defined by a val {{val\|a b c}}, where a is the number of generators it takes to reach the 2nd harmonic (2/1), b is the number of generators to reach the 3rd harmonic (3/1), and c is the number of generators it takes to reach the 5th harmonic (5/1). All 5-limit rank-2 temperaments are defined by two vals: {{monzo\|{{val\|a1 b1 c1}}, {{val\|a2 b2 c2}}}}. Now, we locate the 2nd harmonic (2/1) with the 2-dimensional coordinates (a1, a2), sometimes written as {{monzo\|a1 a2}}, meaning go up a1 of the first generator, and up a2 of the 2nd generator, to reach 2/1. Similarly, the 3rd harmonic and 5th harmonic will be reached by {{monzo\|b1 b2}} and {{monzo\|c1 c2}} respectively.		A ''rank r'' temperament has r generators, and thus is defined by r vals. In the usual coordinates for the [[Harmonic_Limit\|p-limit]], the set of generators are the first k prime numbers and the set of vals for a p-limit temperament gives you the coordinates for each prime harmonic in the p-limit. For example, all 5-limit rank-1 temperaments, or [[Equal\|equal temperaments]], will be defined by a val {{val\|a b c}}, where a is the number of generators it takes to reach the 2nd harmonic (2/1), b is the number of generators to reach the 3rd harmonic (3/1), and c is the number of generators it takes to reach the 5th harmonic (5/1). All 5-limit rank-2 temperaments are defined by two vals: {{monzo\|{{val\|a1 b1 c1}}, {{val\|a2 b2 c2}}}}. Now, we locate the 2nd harmonic (2/1) with the 2-dimensional coordinates (a1, a2), sometimes written as {{monzo\|a1 a2}}, meaning go up a1 of the first generator, and up a2 of the 2nd generator, to reach 2/1. Similarly, the 3rd harmonic and 5th harmonic will be reached by {{monzo\|b1 b2}} and {{monzo\|c1 c2}} respectively.
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	== Example ==		== Example ==
	The rank-1 [[7-limit\|7-limit]] ~~patent~~ [[~~val\|~~val]] corresponding to [[31edo\|31edo]] is {{val\|31 49 72 87}}. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes		The rank-1 [[7-limit\|7-limit]] [[patent val]] corresponding to [[31edo\|31edo]] is {{val\|31 49 72 87}}. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes

	<math>\displaystyle		<math>\displaystyle