Val - Revision history

Godtone: Undo revision and remove technical usage of term "inconsistent" that causes confusion, let me know what you think is most confusing and what you feel is most valuable of what i am reverting and we can brainstorm the section

2026-07-19T06:16:37Z

Undo revision and remove technical usage of term "inconsistent" that causes confusion, let me know what you think is most confusing and what you feel is most valuable of what i am reverting and we can brainstorm the section

← Older revision		Revision as of 06:16, 19 July 2026
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	For a more mathematically intensive introduction to vals, see [[Vals and tuning space]]. For the characterization of higher-rank temperaments, see [[Mapping]].		For a more mathematically intensive introduction to vals, see [[Vals and tuning space]]. For the characterization of higher-rank temperaments, see [[Mapping]].

	== Relationship with (equal) temperaments ==		== Relationship with equal temperaments ==
	{{Todo\|inline=1\| improve readability }}		{{Todo\|inline=1\| improve readability }}

	~~Despite having~~ no contradictions, ~~stacking~~ the ~~tempered intervals of~~ the ~~val will inevitably cause error to accumulate, when compared to~~ the ~~JI counterpart that is supposed to be represented~~. This ~~is because temperaments temper~~ out an infinite set of ~~commas~~, ~~which can be derived from~~ a ~~select set~~ of simple/musically relevant commas that are ~~all nullified~~ in the val.		The guarantee that there are no contradictions comes with an interesting feature: somehow, you have managed to approximate JI in an ''internally-consistent'' way despite the fact that the approximations get worse the more you combine the errors. This corresponds to [[tempering out]] an infinite set of [[comma]]s, though there is a finite number of simple/musically relevant commas; that set is simply the set of all intervals that are mapped to 0 steps ([[1/1]]) by the val. This explains where the additional "structure" went – if there are two or more primes, then you need to specify two or more integers in the exponents of the prime factorization (i.e. in the monzo). So we have lost information and structure by simplifying everything to a single integer coordinate; exactly the information that corresponds to ''equating'' any two intervals whose difference is one of the commas tempered, so we have found a precise sense in which we can equate two nearby intervals that are not actually equal – by mapping according to a val that maps the difference to zero. In fact, you do not have to use an [[edo]] tuning as you could use multiple vals ''simultaneously'' to map a single interval if you want to preserve more of the information in JI rather than just increasing the size of the edo; this corresponds to [[regular temperaments]] generally rather than just the 1-dimensional ("rank-1") case that vals correspond to. Therefore, a val specifies a rank-1 temperament a.k.a. an equal temperament.

	~~All temperaments compromise JI by reducing the number of primes used, so for instance, 5-limit requires 2,3,5 to represent any pitch. If a 5-limit comma is tempered out~~, ~~the structure~~ is ~~collapsed, and error is introduced to compensate for something that was not~~ a ~~unison now being one. In mathematical terms, this is equivalent to making one of the basis vectors~~ of ~~JI linearly dependent.~~		Furthermore, there is actually a lot of applications of vals and monzos that are not necessarily about approximating things in edos or even regular temperaments for that matter, discussed in [[#Applications]], though all of them do still use the idea of the ''mapping'' provided by the val, so really, a val is a ''mapping'' from JI to the numbers with certain properties.

	When tempering out enough commas, JI is collapsed onto a quantized line; an equal temperament or rank-1 tuning. This is where vals come into play. Each of the primes is determined by a certain number of quanta, corresponding to octave divisions (edosteps) in [[EDO\|edos]], tritave divisions in [[EDT\|edts]], et cetera. There are many applications of vals and monzos ~~disjoint from RTT~~, discussed in [[#Applications]], though all of them still ~~treat vals as providing~~ ''~~mappings~~'' from JI to the numbers, with ~~constraints~~.

	== Patent val and generalized patent val ==		== Patent val and generalized patent val ==

Eufalesio: /* Relationship with equal temperaments */ Improve readability

2026-05-08T13:17:21Z

Relationship with equal temperaments: Improve readability

← Older revision		Revision as of 13:17, 8 May 2026
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	For a more mathematically intensive introduction to vals, see [[Vals and tuning space]]. For the characterization of higher-rank temperaments, see [[Mapping]].		For a more mathematically intensive introduction to vals, see [[Vals and tuning space]]. For the characterization of higher-rank temperaments, see [[Mapping]].

	== Relationship with equal temperaments ==		== Relationship with (equal) temperaments ==
	{{Todo\|inline=1\| improve readability }}		{{Todo\|inline=1\| improve readability }}

	~~The guarantee that there are~~ no contradictions ~~comes with an interesting feature: somehow~~, ~~you have managed~~ to ~~approximate~~ JI ~~in an internally-consistent way despite the fact~~ that ~~the approximations get worse the more you combine the errors so can get arbitrarily inconsistent~~. This ~~corresponds to [[tempering~~ out]] an infinite set of ~~[[comma]]s~~, ~~though there is~~ a ~~finite number~~ of simple/musically relevant commas; that ~~set is simply the set of~~ all ~~intervals that are mapped to 0 steps ([[1/1]]) by the val. This explains where the additional "structure" went – if there are two or more primes, then you need to specify two or more integers~~ in the exponents of the prime factorization (i.e. in the monzo). So we have lost information and structure by simplifying everything to a single integer coordinate; exactly the information that corresponds to ''equating'' any two intervals whose difference is one of the commas tempered, so we have found a precise sense in which we can equate two nearby intervals that are not actually equal – by mapping according to a val that maps the difference to zero. In fact, you do not have to use an [[edo]] tuning as you could use multiple vals ''simultaneously'' to map a single interval if you want to preserve more of the information in JI rather than just increasing the size of the edo; this corresponds to [[regular temperaments]] generally rather than just the 1-dimensional ("rank-1") case that vals correspond to. Therefore, a val ~~specifies a rank-1 temperament a.k.a. an equal temperament~~.		Despite having no contradictions, stacking the tempered intervals of the val will inevitably cause error to accumulate, when compared to the JI counterpart that is supposed to be represented. This is because temperaments temper out an infinite set of commas, which can be derived from a select set of simple/musically relevant commas that are all nullified in the val.

	~~Furthermore~~, ~~there~~ is ~~actually~~ a ~~lot~~ of applications of vals and monzos ~~that are not necessarily about approximating things in edos or even regular temperaments for that matter~~, discussed in [[#Applications]], though all of them do still ~~use the idea of the~~ ''~~mapping'' provided by the val, so really, a val is a ''mapping~~'' from JI to the numbers with ~~certain properties~~.		All temperaments compromise JI by reducing the number of primes used, so for instance, 5-limit requires 2,3,5 to represent any pitch. If a 5-limit comma is tempered out, the structure is collapsed, and error is introduced to compensate for something that was not a unison now being one. In mathematical terms, this is equivalent to making one of the basis vectors of JI linearly dependent.

			When tempering out enough commas, JI is collapsed onto a quantized line; an equal temperament or rank-1 tuning. This is where vals come into play. Each of the primes is determined by a certain number of quanta, corresponding to octave divisions (edosteps) in [[EDO\|edos]], tritave divisions in [[EDT\|edts]], et cetera. There are many applications of vals and monzos disjoint from RTT, discussed in [[#Applications]], though all of them still treat vals as providing ''mappings'' from JI to the numbers, with constraints.

	== Patent val and generalized patent val ==		== Patent val and generalized patent val ==

Godtone: this "Mathematical definition" seems to restate fairly trivial information in a verbose and technical way. the only real information given is linearity, which is already mentioned elsewhere

2026-05-08T02:15:44Z

this "Mathematical definition" seems to restate fairly trivial information in a verbose and technical way. the only real information given is linearity, which is already mentioned elsewhere

← Older revision		Revision as of 02:15, 8 May 2026
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	Furthermore, there is actually a lot of applications of vals and monzos that are not necessarily about approximating things in edos or even regular temperaments for that matter, discussed in [[#Applications]], though all of them do still use the idea of the ''mapping'' provided by the val, so really, a val is a ''mapping'' from JI to the numbers with certain properties.		Furthermore, there is actually a lot of applications of vals and monzos that are not necessarily about approximating things in edos or even regular temperaments for that matter, discussed in [[#Applications]], though all of them do still use the idea of the ''mapping'' provided by the val, so really, a val is a ''mapping'' from JI to the numbers with certain properties.

	~~== Mathematical definition ==~~
	Mathematically, a val is a type of function that inputs a {{w\|rational number}} (ratio) and outputs a number of steps that represents what interval of the edo we use to approximate that frequency ratio. If ''a''/''b'' is our ratio and ''k'' is the output of the function, then the interval* of ''N''-edo is 2<sup>''k''/''N''</sup> if we assume a pure octave tuning which is often written with the backslash notation.

	It is not just any such function though; it is a function with a special property called {{w\|linearity}} that allows our arithmetic to be internally consistent (having an internal logic) in the way described above; here ''internally consistent'' is meant in the English sense, so should not be confused with [[consistency]] in the aforediscussed technical sense. The most obvious use of a val (the one discussed in the example) is to algorithmically determine JI interpretations of intervals in edo, which is called using the edo as an equal temperament or rank-1 temperament, where ''rank-1'' means that it corresponds to a 1-dimensional grid of notes related by the same (usually {{w\|irrational number\|irrational}}) frequency ratios.

	~~Also note that in practice vals are ''very far'' from just any list of positive integers; rather, they are generally equal to or one off from the lists of integers that correspond to a ''patent val''.~~

	== Patent val and generalized patent val ==		== Patent val and generalized patent val ==

FloraC: Rework. See talk

2026-05-07T21:04:35Z

Rework. See talk

← Older revision		Revision as of 21:04, 7 May 2026
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	{{~~interwiki~~		{{Interwiki
			\| en = Val
	\| de = Val		\| de = Val
	~~\| en = Val~~
	\| es =		\| es =
	\| ja = ヴァル		\| ja = ヴァル
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	A [[val]] – short for ''valuation'' – is like an algorithm or procedure for finding out how to represent intervals of [[just intonation\|just intonation (JI)]] with the pitches of an [[equal tuning]] such as an [[edo]]. They are typically written using the notation {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>4</sub> ''a''<sub>5</sub> ''a''<sub>6</sub> … }}, where ''a''<sub>''i''</sub> are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, are represented in edosteps, up to some [[harmonic limit\|prime limit]].		A [[val]] – short for ''valuation'' – is like an algorithm or procedure for finding out how to represent intervals of [[just intonation\|just intonation (JI)]] with the pitches of an [[equal tuning]] such as an [[edo]]. They are typically written using the notation {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>4</sub> ''a''<sub>5</sub> ''a''<sub>6</sub> … }}, where ''a''<sub>''i''</sub> are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, are represented in edosteps, up to some [[harmonic limit\|prime limit]].

	The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.		The val is one of the fundamental concepts in [[regular temperament theory]]. The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.

	== Motivation ==		== Motivation ==
	One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2. More generally, combining the approximations in an edo does not necessarily give you the same result as multiplying their ratios first and then using the direct approximation of that in the edo. ~~When~~ this ~~happens~~, we ~~say that the arithmetic is~~ ''~~inconsistent~~''. ~~Therefore when this does not happen~~, we ~~say~~ that the ~~result~~ is [[~~consistent~~]].		One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2. More generally, combining the approximations in an edo does not necessarily give you the same result as multiplying their ratios first and then using the direct approximation of that in the edo, so direct approximations of chords are not guaranteed to exist.

			Rather than giving up and saying that we cannot use this particular harmony in this particular edo, it turns out we ''can'' if we look at interval approximation in a different way.

			In direct approximation, we are treating the approximations as isolated, unrelated free variables, but as we see, ''two'' intervals on top of each other form a triad with ''three'' component intervals that cannot be altered individually. That is why it is important to recognize the fact that intervals like 3/2, 5/4 and 6/5 are related to each other: by stacking 5/4 and 6/5, 3/2 is found; by removing 6/5 from 3/2, 5/4 is found; and by removing 5/4 from 3/2, 6/5 is found. It follows that for the ''three'' intervals in the [[5-odd-limit]], there are ''two'' free variables. If we know any two of them, the third can always be derived (even though it might not be the closest approximation).

	~~Unfortunately~~, ~~it is not possible~~ to ~~fix inconsistency, except~~ by ~~avoiding this particular harmony in this particular edo~~, ~~but rather than giving up and saying that~~ we ~~cannot use it~~, ~~it turns out~~ we ''~~can~~'' if we are willing to allow one or more of these ratios to use an alternative approximation, especially considering that we probably do not mind using the second-best approximation in more complex intervals if we can guarantee that the arithmetic never fails us. A val will show us how to do that.		To take this idea further, we notice that the infinitely many intervals of JI can be reduced to a few representatives from which the rest can be derived by some form of combination, so we only need to keep track of the steps of the representatives. Usually, we choose the steps of the [[prime interval\|prime harmonics]], by which we mean each interval with frequency ratio ''p''/1 where ''p'' is a {{w\|prime number}}.

	== Definition ==		== Definition ==
	A val is a list of numbers that shows the approximation of each prime harmonic used in an edo in terms of steps~~, where by ''prime harmonic'' we mean each frequency ratio ''p''/1 where ''p'' is a {{w\|prime number}}~~. This list of integers by convention corresponds to all primes up to some largest prime (the [[limit]]) so that we can tell what number represents the ''mapping'' of what prime by its place in the list. First place is prime 2's mapping (a.k.a. the edo), second place is prime 3's mapping, third place is prime 5's mapping, fourth is prime 7's, etc.		A val is a list of numbers that shows the approximation of each prime harmonic used in an edo in terms of steps. This list of integers by convention corresponds to all primes up to some largest prime (the [[limit]]) so that we can tell what number represents the ''mapping'' of what prime by its place in the list. First place is prime 2's mapping (a.k.a. the edo), second place is prime 3's mapping, third place is prime 5's mapping, fourth is prime 7's, etc.

	The val is used to ~~understand~~ the edo's approximations to ratios involving those primes, like 2 × 5 / 3 / 3 = [[10/9]] for primes {2, 3, 5}. This list does not have to be the closest approximation for each prime, but it usually is. Thus a val is essentially just a list of numbers that we are interpreting as having a certain meaning.		The val can be used to compute the edo's approximations to ratios involving those primes, like 2 × 5 / 3 / 3 = [[10/9]] for primes {2, 3, 5}. This list does not have to be the closest approximation for each prime, but it usually is. Thus a val is essentially just a list of numbers that we are interpreting as having a certain meaning.

	== Examples ==		== Examples ==

FloraC: Undo. See talk

2026-05-07T20:37:07Z

Undo. See talk

← Older revision		Revision as of 20:37, 7 May 2026
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	{{Beginner\|Vals and tuning space}}		{{Beginner\|Vals and tuning space}}

	A [[val]] – short for ''valuation'' – is like an algorithm or procedure for finding out how to represent intervals ~~(aka. [[frequency ratio]]s)~~ of [[just intonation\|just intonation (JI)]] with the pitches of an [[equal tuning]] such as an [[edo]]. They are typically written using the notation {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>4</sub> ''a''<sub>5</sub> ''a''<sub>6</sub> … }}, where ''a''<sub>''i''</sub> are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, are represented in edosteps, up to some [[harmonic limit\|prime limit]].		A [[val]] – short for ''valuation'' – is like an algorithm or procedure for finding out how to represent intervals of [[just intonation\|just intonation (JI)]] with the pitches of an [[equal tuning]] such as an [[edo]]. They are typically written using the notation {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>4</sub> ''a''<sub>5</sub> ''a''<sub>6</sub> … }}, where ''a''<sub>''i''</sub> are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, are represented in edosteps, up to some [[harmonic limit\|prime limit]].

	The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.		The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.

	== Motivation ==		== Motivation ==
	One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2, though practically this is usually true for tunings of interest, so the realistic examples appear for more complex cases like (the direct approximations of) [[~]][[9/8]] and [[~]][[5/4]] combined not being equal to the direct approximation of [[45/32]] (= 9/8 * 5/4), as discussed in the 26edo example below. More generally, combining the approximations in an edo does not necessarily give you the same result as multiplying their ratios first and then using the direct approximation of that in the edo. When this happens, we say that the arithmetic is ''inconsistent''. Therefore when this does not happen, we say that the result is [[consistent]].		One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2. More generally, combining the approximations in an edo does not necessarily give you the same result as multiplying their ratios first and then using the direct approximation of that in the edo. When this happens, we say that the arithmetic is ''inconsistent''. Therefore when this does not happen, we say that the result is [[consistent]].

	Unfortunately, it is not possible to fix inconsistency, except by avoiding this particular harmony in this particular edo, but rather than giving up and saying that we cannot use it, it turns out we ''can'' if we are willing to allow one or more of these ratios to use an alternative approximation, especially considering that we probably do not mind using the second-best approximation in more complex intervals if we can guarantee that the arithmetic never fails us. A val will show us how to do that.		Unfortunately, it is not possible to fix inconsistency, except by avoiding this particular harmony in this particular edo, but rather than giving up and saying that we cannot use it, it turns out we ''can'' if we are willing to allow one or more of these ratios to use an alternative approximation, especially considering that we probably do not mind using the second-best approximation in more complex intervals if we can guarantee that the arithmetic never fails us. A val will show us how to do that.

	== Definition ==		== Definition ==

Godtone: the example discussed should be mentioned to help the reader connect more dots. also, "frequency ratios" is two words that are so self-evident that they clearly explain the concept as contrasted to "interval" where even experts get confused for negative intervals

2026-05-07T19:58:08Z

the example discussed should be mentioned to help the reader connect more dots. also, "frequency ratios" is two words that are so self-evident that they clearly explain the concept as contrasted to "interval" where even experts get confused for negative intervals

← Older revision		Revision as of 19:58, 7 May 2026
Line 7:		Line 7:
	{{Beginner\|Vals and tuning space}}		{{Beginner\|Vals and tuning space}}

	A [[val]] – short for ''valuation'' – is like an algorithm or procedure for finding out how to represent intervals of [[just intonation\|just intonation (JI)]] with the pitches of an [[equal tuning]] such as an [[edo]]. They are typically written using the notation {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>4</sub> ''a''<sub>5</sub> ''a''<sub>6</sub> … }}, where ''a''<sub>''i''</sub> are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, are represented in edosteps, up to some [[harmonic limit\|prime limit]].		A [[val]] – short for ''valuation'' – is like an algorithm or procedure for finding out how to represent intervals (aka. [[frequency ratio]]s) of [[just intonation\|just intonation (JI)]] with the pitches of an [[equal tuning]] such as an [[edo]]. They are typically written using the notation {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>4</sub> ''a''<sub>5</sub> ''a''<sub>6</sub> … }}, where ''a''<sub>''i''</sub> are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, are represented in edosteps, up to some [[harmonic limit\|prime limit]].

	The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.		The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.

	== Motivation ==		== Motivation ==
	One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2. More generally, combining the approximations in an edo does not necessarily give you the same result as multiplying their ratios first and then using the direct approximation of that in the edo. When this happens, we say that the arithmetic is ''inconsistent''. Therefore when this does not happen, we say that the result is [[consistent]].		One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2, though practically this is usually true for tunings of interest, so the realistic examples appear for more complex cases like (the direct approximations of) [[~]][[9/8]] and [[~]][[5/4]] combined not being equal to the direct approximation of [[45/32]] (= 9/8 * 5/4), as discussed in the 26edo example below. More generally, combining the approximations in an edo does not necessarily give you the same result as multiplying their ratios first and then using the direct approximation of that in the edo. When this happens, we say that the arithmetic is ''inconsistent''. Therefore when this does not happen, we say that the result is [[consistent]].

	Unfortunately, it is not possible to fix inconsistency, except by avoiding this particular harmony in this particular edo, but rather than giving up and saying that we cannot use it, it turns out we ''can'' if we are willing to allow one or more of these ratios to use an alternative approximation, especially considering that we probably do not mind using the second-best approximation in more complex intervals if we can guarantee that the arithmetic never fails us. A val will show us how to do that.		Unfortunately, it is not possible to fix inconsistency, except by avoiding this particular harmony in this particular edo, but rather than giving up and saying that we cannot use it, it turns out we ''can'' if we are willing to allow one or more of these ratios to use an alternative approximation, especially considering that we probably do not mind using the second-best approximation in more complex intervals if we can guarantee that the arithmetic never fails us. A val will show us how to do that.

	== Definition ==		== Definition ==

FloraC: This is just unnecessary obfuscation

2026-05-07T18:48:44Z

This is just unnecessary obfuscation

← Older revision		Revision as of 18:48, 7 May 2026
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	{{Beginner\|Vals and tuning space}}		{{Beginner\|Vals and tuning space}}

	A [[val]] – short for ''valuation'' – is like an algorithm or procedure for finding out how to represent intervals ~~(aka. [[frequency ratio]]s)~~ of [[just intonation\|just intonation (JI)]] with the pitches of an [[equal tuning]] such as an [[edo]]. They are typically written using the notation {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>4</sub> ''a''<sub>5</sub> ''a''<sub>6</sub> … }}, where ''a''<sub>''i''</sub> are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, are represented in edosteps, up to some [[harmonic limit\|prime limit]].		A [[val]] – short for ''valuation'' – is like an algorithm or procedure for finding out how to represent intervals of [[just intonation\|just intonation (JI)]] with the pitches of an [[equal tuning]] such as an [[edo]]. They are typically written using the notation {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>4</sub> ''a''<sub>5</sub> ''a''<sub>6</sub> … }}, where ''a''<sub>''i''</sub> are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, are represented in edosteps, up to some [[harmonic limit\|prime limit]].

	The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.		The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.

	== Motivation ==		== Motivation ==
	One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2, though practically this is usually true for tunings of interest, so the realistic examples appear for more complex cases like (the direct approximations of) [[~]][[9/8]] and [[~]][[5/4]] combined not being equal to the direct approximation of [[45/32]] (= 9/8 * 5/4), as discussed in the 26edo example below. More generally, combining the approximations in an edo does not necessarily give you the same result as multiplying their ratios first and then using the direct approximation of that in the edo. When this happens, we say that the arithmetic is ''inconsistent''. Therefore when this does not happen, we say that the result is [[consistent]].		One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2. More generally, combining the approximations in an edo does not necessarily give you the same result as multiplying their ratios first and then using the direct approximation of that in the edo. When this happens, we say that the arithmetic is ''inconsistent''. Therefore when this does not happen, we say that the result is [[consistent]].

	Unfortunately, it is not possible to fix inconsistency, except by avoiding this particular harmony in this particular edo, but rather than giving up and saying that we cannot use it, it turns out we ''can'' if we are willing to allow one or more of these ratios to use an alternative approximation, especially considering that we probably do not mind using the second-best approximation in more complex intervals if we can guarantee that the arithmetic never fails us. A val will show us how to do that.		Unfortunately, it is not possible to fix inconsistency, except by avoiding this particular harmony in this particular edo, but rather than giving up and saying that we cannot use it, it turns out we ''can'' if we are willing to allow one or more of these ratios to use an alternative approximation, especially considering that we probably do not mind using the second-best approximation in more complex intervals if we can guarantee that the arithmetic never fails us. A val will show us how to do that.

	== Definition ==		== Definition ==

Godtone: /* Motivation */ elaborate

2026-05-07T17:39:11Z

Motivation: elaborate

← Older revision		Revision as of 17:39, 7 May 2026
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	== Motivation ==		== Motivation ==
	One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it ~~creates~~ contradictions. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2. More generally, combining the approximations in an edo does not necessarily give you the same result as multiplying their ratios first and then using the direct approximation of that in the edo. When this happens, we say that the arithmetic is ''inconsistent''. Therefore when this does not happen, we say that the result is [[consistent]].		One obvious way to find an approximation to a just interval is to use [[direct approximation]], that is, rounding the interval to the nearest edostep. While this may seem simple, it can create contradictions in arithmetic. For example, a [[just major triad]] consists of a [[5/4]] major third and a [[6/5]] minor third combining to a [[3/2]] perfect fifth, but the sum of direct approximations of 5/4 and 6/5 might not be the direct approximation of 3/2, though practically this is usually true for tunings of interest, so the realistic examples appear for more complex cases like (the direct approximations of) [[~]][[9/8]] and [[~]][[5/4]] combined not being equal to the direct approximation of [[45/32]] (= 9/8 * 5/4), as discussed in the 26edo example below. More generally, combining the approximations in an edo does not necessarily give you the same result as multiplying their ratios first and then using the direct approximation of that in the edo. When this happens, we say that the arithmetic is ''inconsistent''. Therefore when this does not happen, we say that the result is [[consistent]].

	Unfortunately, it is not possible to fix inconsistency, except by avoiding this particular harmony in this particular edo, but rather than giving up and saying that we cannot use it, it turns out we ''can'' if we are willing to allow one or more of these ratios to use an alternative approximation, especially considering that we probably do not mind using the second-best approximation in more complex intervals if we can guarantee that the arithmetic never fails us. A val will show us how to do that.		Unfortunately, it is not possible to fix inconsistency, except by avoiding this particular harmony in this particular edo, but rather than giving up and saying that we cannot use it, it turns out we ''can'' if we are willing to allow one or more of these ratios to use an alternative approximation, especially considering that we probably do not mind using the second-best approximation in more complex intervals if we can guarantee that the arithmetic never fails us. A val will show us how to do that.

	== Definition ==		== Definition ==

Godtone: i sincerely believe this wording is more beginner-friendly. i am doing a larger review of the page

2026-05-03T15:21:37Z

i sincerely believe this wording is more beginner-friendly. i am doing a larger review of the page

← Older revision		Revision as of 15:21, 3 May 2026
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	{{Beginner\|Vals and tuning space}}		{{Beginner\|Vals and tuning space}}

	A [[val]] – short for ''valuation'' – is like an algorithm or procedure for finding out how to represent intervals of [[just intonation\|just intonation (JI)]] with the pitches of an [[equal tuning]] such as an [[edo]]. They are typically written using the notation {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>4</sub> ''a''<sub>5</sub> ''a''<sub>6</sub> … }}, where ''a''<sub>''i''</sub> are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, are represented in edosteps, up to some [[harmonic limit\|prime limit]].		A [[val]] – short for ''valuation'' – is like an algorithm or procedure for finding out how to represent intervals (aka. [[frequency ratio]]s) of [[just intonation\|just intonation (JI)]] with the pitches of an [[equal tuning]] such as an [[edo]]. They are typically written using the notation {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>4</sub> ''a''<sub>5</sub> ''a''<sub>6</sub> … }}, where ''a''<sub>''i''</sub> are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, are represented in edosteps, up to some [[harmonic limit\|prime limit]].

	The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.		The basic principle of using a val is to assign [[prime harmonic]]s to edosteps, and then deduce the number of edosteps of an arbitrary interval based on its [[prime factorization]]. This therefore assumes either that you want to use an equal tuning to approximate specific harmonies or that you have some other more indirect use in mind.

Overthink: main article

2026-03-29T00:53:10Z

main article

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	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Tuning]]		[[Category:Tuning]]
	[[Category:Val]]		[[Category:Val\| ]] <!-- main article -->
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Notation]]		[[Category:Notation]]
	{{Todo\| cleanup }}		{{Todo\| cleanup }}