Technical data guide for regular temperaments - Revision history

FloraC: Tip for editors

2026-09-21T16:41:39Z

Tip for editors

← Older revision		Revision as of 16:41, 21 September 2026
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	Not all temperament tables provide the same information, nor do they all provide it in exactly the same way, but the following properties should cover most needs.		Not all temperament tables provide the same information, nor do they all provide it in exactly the same way, but the following properties should cover most needs.

			{{Tip\| See [[XW:TEMPCAT]] if you wish to document a temperament. }}

	== Structure properties ==		== Structure properties ==

FloraC: /* Optimal tuning(s) */ update to reflect current progress

2026-09-21T09:41:18Z

Optimal tuning(s): update to reflect current progress

← Older revision		Revision as of 09:41, 21 September 2026
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	{{Main\| Optimization }}		{{Main\| Optimization }}

	Tuning ''optimization'' is, essentially, the task of finding a tuning for a given regular temperament that has the lowest error in some way. While many approaches exist to going about this, the most widely used are algorithms based on the ''TE metric'', which weight all intervals in the infinite set available to the temperament by a measure of their complexity, and tune in order to minimize deviation from just across all of them.		Tuning ''optimization'' is, essentially, the task of finding a tuning for a given regular temperament that has the lowest error in some way. While many approaches exist to going about this, the most widely used are algorithms based on the ''Tenney–Euclidean'' (''TE'') ''metric'', which weight all intervals in the infinite set available to the temperament by a measure of their complexity, and tune in order to minimize deviation from just across all of them.

	It is common to optimize under the constraint that the octave (or equave) is tuned pure, and therefore that the generator known as the ''period'' is either an exact equave or a fraction thereof, whereas the other ''generators'' are tuned to inexact values. The simplest rational interpretation of these intervals are given in cents. ~~Multiple optimization algorithms~~ are offered, for both tempered-octave (WE, in progress) and pure-octave tuning (~~most commonly CTE and POTE~~, ~~but we're unifying at CWE as of late~~); ~~different~~ algorithms have subtle differences and ~~one or the other~~ may be chosen for a specific use. ~~However~~, optimal tunings are used ~~more often~~ as guidelines for where good tunings of a temperament are than as exact ways to tune, and for that purpose the algorithms usually agree sufficiently (aside from extreme exotemperaments and other special cases).		It is common to optimize under the constraint that the octave (or equave) is tuned pure, and therefore that the generator known as the ''period'' is either an exact equave or a fraction thereof, whereas the other ''generators'' are tuned to inexact values. The simplest rational interpretation of these intervals are given in cents. Optima are offered for both tempered-octave (WE, in progress) and pure-octave tuning (CWE, in progress); other algorithms have subtle differences and each may be chosen for a specific use. Besides, optimal tunings are often used as guidelines for where good tunings of a temperament are than as exact ways to tune, and for that purpose the algorithms usually agree sufficiently (aside from extreme exotemperaments and other special cases).

	In the ~~future, temperaments may appear with optimal tunings of ''~~prime harmonics'~~' (and their deviation~~ from just~~) in a collapsible table~~, though ~~this will be~~ merely for the sake of convenience as all tunings of intervals can be derived from the generators and the mapping.		Optimal tunings come with [[error map]]s (in progress). These represent the prime harmonics' deviations from just, though it is merely for the sake of convenience as all tunings of intervals can be derived from the generators and the mapping.

	=== Optimal ET sequence ===		=== Optimal ET sequence ===

FloraC: /* Advanced properties */ explain gencom mapping

2026-06-29T04:01:55Z

Advanced properties: explain gencom mapping

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	=== Advanced properties ===		=== Advanced properties ===
	==== Gencom mapping ====		==== Gencom mapping ====
	{{Main\|Gencom}}		{{Main\|Gencom #Gencom mapping}}
	~~{{todo\|complete section\|inline=1}}~~
			The gencom mapping is similar to the subgroup-val mapping for subgroup temperaments, but can be fractional and can involve zero columns for elements not included in the subgroup. Whilst the subgroup-val mapping takes subgroup monzos and outputs its steps in the temperament, the gencom mapping takes ordinary monzos and outputs the same steps.

	==== Mapping to lattice ====		==== Mapping to lattice ====
	{{Main\|Mapping to lattice}}		{{Main\|Mapping to lattice}}
	{{See also\|~~Harmonic lattice diagram~~}}		{{See also\|Lattice}}
	{{todo\|complete section\|inline=1}}		{{todo\|complete section\|inline=1}}

FloraC: /* Structure properties */ domain, not domain basis

2026-06-28T17:02:25Z

Structure properties: domain, not domain basis

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	== Structure properties ==		== Structure properties ==
	=== Subgroup ~~(domain basis)~~ ===		=== Subgroup ===
	{{Main\|Just intonation subgroup}}		{{Main\|Just intonation subgroup}}
	{{See also\|Domain basis}}		{{See also\|Domain basis}}
	The ''subgroup'' (or ''domain ~~basis~~'') of a regular temperament is the set of all [[interval]]s which are considered to be approximated by the temperament. For example, it is common to consider that the [[frequency ratio]] of [[3/2]] is approximated by [[12edo\|12-tone equal temperament]], therefore 3/2 would be included in this set, but other intervals like [[11/8]] could be excluded. Most of the time, a subgroup exclusively contains [[just intonation]] (JI) intervals.
			The ''subgroup'' (or ''domain'') of a regular temperament is the set of all [[interval]]s which are considered to be approximated by the temperament. For example, it is common to consider that the [[frequency ratio]] of [[3/2]] is approximated by [[12edo\|12-tone equal temperament]], therefore 3/2 would be included in this set, but other intervals like [[11/8]] could be excluded. Most of the time, a subgroup exclusively contains [[just intonation]] (JI) intervals.

	In a subgroup, all intervals are reachable by stacking (up and down) copies of a few "generating intervals", called ''[[Periods and generators\|generator]]s''. Continuing the previous example, if [[3/2]] is taken as a generator of the subgroup, then [[9/4]] is also included in the subgroup {{nowrap\|(3/2 × 3/2 {{=}} 9/4)}}, and so on. If [[2/1]] is added to the list of subgroup generators, then intervals like [[4/3]] can be reached by combining a 3/2 down with a 2/1 up {{nowrap\|(2/3 × 2/1 {{=}} 4/3)}}.		In a subgroup, all intervals are reachable by stacking (up and down) copies of a few "generating intervals", called ''[[Periods and generators\|generator]]s''. Continuing the previous example, if [[3/2]] is taken as a generator of the subgroup, then [[9/4]] is also included in the subgroup {{nowrap\|(3/2 × 3/2 {{=}} 9/4)}}, and so on. If [[2/1]] is added to the list of subgroup generators, then intervals like [[4/3]] can be reached by combining a 3/2 down with a 2/1 up {{nowrap\|(2/3 × 2/1 {{=}} 4/3)}}.
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	=== Comma list ===		=== Comma list ===
	{{Main\|Comma basis}}		{{Main\|Comma basis}}

	An abstract regular temperament can be thought of as a family of ''valuations'' (''tunings'' of the temperament) of the primes in its subgroup that satisfy certain equations; if we bold the numbers to make it clear that we are speaking of them as abstract variables, an example of such an equation would be {{nowrap\|'''3'''<sup>4</sup> {{=}} '''2'''<sup>4</sup> × '''5'''}}. Each of these equations corresponds to setting a JI interval to be equal to the unison ([[1/1]]); the equation specified here sets [[81/80]] (with factorization {{nowrap\|2<sup>−4</sup> × 3<sup>4</sup> × 5<sup>−1</sup>}}) to the unison, in other words ''tempering out'' 81/80.		An abstract regular temperament can be thought of as a family of ''valuations'' (''tunings'' of the temperament) of the primes in its subgroup that satisfy certain equations; if we bold the numbers to make it clear that we are speaking of them as abstract variables, an example of such an equation would be {{nowrap\|'''3'''<sup>4</sup> {{=}} '''2'''<sup>4</sup> × '''5'''}}. Each of these equations corresponds to setting a JI interval to be equal to the unison ([[1/1]]); the equation specified here sets [[81/80]] (with factorization {{nowrap\|2<sup>−4</sup> × 3<sup>4</sup> × 5<sup>−1</sup>}}) to the unison, in other words ''tempering out'' 81/80.

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	{{Main\| Mapping }}		{{Main\| Mapping }}
	{{See also\| Subgroup monzos and vals }}		{{See also\| Subgroup monzos and vals }}

	A regular temperament has a structure defined by a set of ''generators'', whose number is equivalent to the ''rank'' of the temperament. Like JI itself, the set of all distinct intervals available to the regular temperament can be created by stacking these generators. Unlike JI, the determination of which intervals are generators is often highly nontrivial given the comma basis or other information.		A regular temperament has a structure defined by a set of ''generators'', whose number is equivalent to the ''rank'' of the temperament. Like JI itself, the set of all distinct intervals available to the regular temperament can be created by stacking these generators. Unlike JI, the determination of which intervals are generators is often highly nontrivial given the comma basis or other information.

FloraC: /* Mapping and sval mapping */ update on terminology

2025-08-30T13:44:06Z

Mapping and sval mapping: update on terminology

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	As a last note, factorizations are generally abbreviated in the form of a (subgroup) [[monzo]], which is simply a list of the exponents in a factorization that are attached to each (formal) prime in the subgroup, so that for instance 225/224 would be {{monzo\|-5 2 2 -1}} (in this case the subgroup is 2.3.5.7; it should be specified if there is any ambiguity, but if not it can be assumed to be the temperament's subgroup).		As a last note, factorizations are generally abbreviated in the form of a (subgroup) [[monzo]], which is simply a list of the exponents in a factorization that are attached to each (formal) prime in the subgroup, so that for instance 225/224 would be {{monzo\|-5 2 2 -1}} (in this case the subgroup is 2.3.5.7; it should be specified if there is any ambiguity, but if not it can be assumed to be the temperament's subgroup).

	=== Mapping and ~~sval~~ mapping ===		=== Mapping and subgroup-val mapping ===
	{{Main\|Mapping}}		{{Main\| Mapping }}
	{{See also\|~~Smonzos~~ and ~~svals~~}}		{{See also\| Subgroup monzos and vals }}
	A regular temperament has a structure defined by a set of ''generators'', whose number is equivalent to the ''rank'' of the temperament. Like JI itself, the set of all distinct intervals available to the regular temperament can be created by stacking these generators. Unlike JI, the determination of which intervals are generators is often highly nontrivial given the comma basis or other information.		A regular temperament has a structure defined by a set of ''generators'', whose number is equivalent to the ''rank'' of the temperament. Like JI itself, the set of all distinct intervals available to the regular temperament can be created by stacking these generators. Unlike JI, the determination of which intervals are generators is often highly nontrivial given the comma basis or other information.

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	For an example, let us look at meanpop, an [[11-limit]] extension of meantone. Its mapping is given by {{mapping\| 1 0 -4 -13 24 \| 0 1 4 10 -13 }}, with the "mapping generators" being ~2 and ~3 (with the tilde used to indicate ''tunings'' of these intervals under the temperament), where each ''vector'' within the mapping indicates the number of each generator in the stack used to reach a prime harmonic. This particular mapping tells us that 2/1 and 3/1 are reached by one of the generators ~2 and ~3 (trivially) each; that 5/1 is reached by 4 times ~3 upward and 4 times ~2 downward; that 7/1 is reached by 10 times ~3 upward and 13 times ~2 downward; and that 11/1 is reached by 24 times ~2 upward and 13 times ~3 downward. Therefore, the 11th harmonic in this temperament is quite complex (even if we regard ~2, the octave, as "free"), especially because it is reached the ''opposite'' way that 3, 5, and 7 are and so ratios of 11 with these other primes are even more complex. Thus intervals of 11 will not appear until quite a long way down the [[chain of fifths]], and only in rather large scales built out of tempered intervals.		For an example, let us look at meanpop, an [[11-limit]] extension of meantone. Its mapping is given by {{mapping\| 1 0 -4 -13 24 \| 0 1 4 10 -13 }}, with the "mapping generators" being ~2 and ~3 (with the tilde used to indicate ''tunings'' of these intervals under the temperament), where each ''vector'' within the mapping indicates the number of each generator in the stack used to reach a prime harmonic. This particular mapping tells us that 2/1 and 3/1 are reached by one of the generators ~2 and ~3 (trivially) each; that 5/1 is reached by 4 times ~3 upward and 4 times ~2 downward; that 7/1 is reached by 10 times ~3 upward and 13 times ~2 downward; and that 11/1 is reached by 24 times ~2 upward and 13 times ~3 downward. Therefore, the 11th harmonic in this temperament is quite complex (even if we regard ~2, the octave, as "free"), especially because it is reached the ''opposite'' way that 3, 5, and 7 are and so ratios of 11 with these other primes are even more complex. Thus intervals of 11 will not appear until quite a long way down the [[chain of fifths]], and only in rather large scales built out of tempered intervals.

	In subgroups other than full prime-limits, mappings are sometimes called ~~"sval~~ mappings"; the only distinction here is that the columns of the mapping do not indicate all consecutive primes but only the basis elements of the subgroup. These are distinct from "gencom mappings" with zero entries for primes not included in the subgroup.		In subgroups other than full prime-limits, mappings are sometimes called ''subgroup-val mappings''; the only distinction here is that the columns of the mapping do not indicate all consecutive primes but only the basis elements of the subgroup. These are distinct from ''gencom mappings'' with zero entries for primes not included in the subgroup.

	One last note is that mappings may use a slightly different (if equivalent) set of generators from elsewhere in the temperament data: for meanpop, for instance, the ~~"canonical"~~ generator, for which optimal tunings are specified, is in fact ~3/2, rather than ~3. In these cases, the mapping should (but does not always) specify the set of generators used for the ''mapping''.		One last note is that mappings may use a slightly different (if equivalent) set of generators from elsewhere in the temperament data: for meanpop, for instance, the conventional generator, for which optimal tunings are specified, is in fact ~3/2, rather than ~3. In these cases, the mapping should (but does not always) specify the set of generators used for the ''mapping''.

	=== Extensions and restrictions ===		=== Extensions and restrictions ===

FloraC: /* Tuning properties */ update to reflect our progress

2025-07-17T06:05:41Z

Tuning properties: update to reflect our progress

← Older revision		Revision as of 06:05, 17 July 2025
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	== Tuning properties ==		== Tuning properties ==
	While an abstract regular temperament does not specify the values to which its intervals (meaning, its generators or valuations of primes) are tuned, it is clear that there are ways to set these values that are reasonable, and those that are absurd. For instance, meantone could be tuned with its "[[3/2]]" set to ~~675c~~, and therefore its {{nowrap\|"[[5/4]]" ~ "81/64"}} set to {{nowrap\|~~4×675¢ − 2400¢~~ {{=}} ~~300¢~~}}. However, this is an absurd tuning for meantone since ~~300¢~~ has a far better interpretation as [[6/5]] than 5/4, and the temperament providing that interpretation is instead [[mavila]].		While an abstract regular temperament does not specify the values to which its intervals (meaning, its generators or valuations of primes) are tuned, it is clear that there are ways to set these values that are reasonable, and those that are absurd. For instance, meantone could be tuned with its "[[3/2]]" set to 675{{c}}, and therefore its {{nowrap\| "[[5/4]]" ~ "81/64" }} set to {{nowrap\| 4×675{{c}} − 2400{{c}} {{=}} 300{{c}} }}. However, this is an absurd tuning for meantone since 300{{c}} has a far better interpretation as [[6/5]] than 5/4, and the temperament providing that interpretation is instead [[mavila]].

	Therefore, one can speak of temperaments as having finite "tuning ranges" for their generator, which is useful in the picture of building [[~~MOS~~ scale]]s as finite subsets of the intervals available in the temperament.		Therefore, one can speak of temperaments as having finite tuning ranges for their generator, which is useful in the picture of building [[mos scale]]s as finite subsets of the intervals available in the temperament.

	=== Optimal tuning(s) ===		=== Optimal tuning(s) ===
	{{Main\|Optimization}}		{{Main\| Optimization }}

	Tuning ''optimization'' is, essentially, the task of finding a tuning for a given regular temperament that has the lowest error in some way. While many approaches exist to going about this, the most widely used are algorithms based on the ''TE metric'', which weight all intervals in the infinite set available to the temperament by a measure of their complexity, and tune in order to minimize deviation from just across all of them.		Tuning ''optimization'' is, essentially, the task of finding a tuning for a given regular temperament that has the lowest error in some way. While many approaches exist to going about this, the most widely used are algorithms based on the ''TE metric'', which weight all intervals in the infinite set available to the temperament by a measure of their complexity, and tune in order to minimize deviation from just across all of them.

	It is ~~conventional on the wiki~~ to optimize under the constraint that the octave (or equave) is tuned pure, and therefore that the generator known as the ''period'' is either an exact equave or a fraction thereof~~. The rational interpretation of the period is depicted as equated to this fraction. However~~, the other ''generators'' are tuned to inexact values ~~expressed~~ in cents~~, and appear as rational interpretations equated to these values~~. Multiple optimization algorithms (most commonly CTE and POTE) ~~may appear~~; different algorithms have subtle differences and one or the other may be chosen for a specific use. However, optimal tunings are used more often as guidelines for where "good tunings" of a temperament are than as exact ways to tune, and for that purpose the algorithms usually agree sufficiently (aside from extreme exotemperaments and other special cases).		It is common to optimize under the constraint that the octave (or equave) is tuned pure, and therefore that the generator known as the ''period'' is either an exact equave or a fraction thereof, whereas the other ''generators'' are tuned to inexact values. The simplest rational interpretation of these intervals are given in cents. Multiple optimization algorithms are offered, for both tempered-octave (WE, in progress) and pure-octave tuning (most commonly CTE and POTE, but we're unifying at CWE as of late); different algorithms have subtle differences and one or the other may be chosen for a specific use. However, optimal tunings are used more often as guidelines for where good tunings of a temperament are than as exact ways to tune, and for that purpose the algorithms usually agree sufficiently (aside from extreme exotemperaments and other special cases).

	In the future, temperaments may appear with optimal tunings of ''prime harmonics'' (and their deviation from just) in a collapsible table, though this will be merely for the sake of convenience as all tunings of intervals can be derived from the generators and the mapping.		In the future, temperaments may appear with optimal tunings of ''prime harmonics'' (and their deviation from just) in a collapsible table, though this will be merely for the sake of convenience as all tunings of intervals can be derived from the generators and the mapping.

VectorGraphics: Operation Loosen Underpants

2025-06-12T19:12:09Z

Operation Loosen Underpants

← Older revision		Revision as of 19:12, 12 June 2025
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	{{Main\|Mapping to lattice}}		{{Main\|Mapping to lattice}}
	{{See also\|Harmonic lattice diagram}}		{{See also\|Harmonic lattice diagram}}
	~~{{todo\|complete section\|inline=1}}~~

	~~==== Wedgie ====~~
	~~{{Main\|Plücker coordinates}}~~
	{{todo\|complete section\|inline=1}}		{{todo\|complete section\|inline=1}}

Lériendil: puntuazione

2025-04-13T03:43:45Z

puntuazione

← Older revision		Revision as of 03:43, 13 April 2025
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	The ''comma basis'' is a list of such intervals–''commas''–that are tempered out by the temperament, thereby restricting the set of possible tunings. If a tuning only tempers out one comma, then the only intervals within the subgroup that are set to the unison are that comma, and its positive and negative powers (in the case above, 80/81, {{nowrap\|6561/6400 {{=}} (81/80)<sup>2</sup>}}, etc.); therefore there is only one logical choice for "which comma" you claim is tempered, that being the simplest of these powers greater than the unison. Rank-2 temperaments in subgroups including 3 primes, such as the [[5-limit]], only temper out one comma, for instance.		The ''comma basis'' is a list of such intervals–''commas''–that are tempered out by the temperament, thereby restricting the set of possible tunings. If a tuning only tempers out one comma, then the only intervals within the subgroup that are set to the unison are that comma, and its positive and negative powers (in the case above, 80/81, {{nowrap\|6561/6400 {{=}} (81/80)<sup>2</sup>}}, etc.); therefore there is only one logical choice for "which comma" you claim is tempered, that being the simplest of these powers greater than the unison. Rank-2 temperaments in subgroups including 3 primes, such as the [[5-limit]], only temper out one comma, for instance.

	However, if a tuning tempers out multiple independent commas, the situation gets more complicated, for the set of intervals that are tempered out in fact forms a lattice ''generated'' by more than one generator (in other words, a nontrivial subgroup of JI), and the choice of which specific intervals to consider generators (which in this context are ''basis commas'') is not always obvious. For instance, septimal meantone tempers out the intervals {{nowrap\|[[126/125]] {{=}} 2 × 3<sup>2</sup> × 5<sup>−3</sup> × 7}}; {{nowrap\|[[225/224]] {{=}} 2<sup>−5</sup> × 3<sup>2</sup> × 5<sup>2</sup> × 7<sup>−1</sup>}}, and 81/80, but {{nowrap\|81/80 {{=}} (126/125) × (225/224)}}, and therefore these three commas are ''not all independent''–but all of them are useful, in that all three define prominent ''temperament families'' (collections of regular temperaments that share a tempered comma in common): 81/80 defines [[meantone]], 126/125 defines [[starling]], and 225/224 defines [[marvel]]. Various methods exist for choosing which commas are selected to be basis commas, which are associated with the technique of [[matrix echelon forms]]; in the case of septimal meantone, the basis commas are chosen to be 81/80 and 126/125 at the price of obscuring the fact that it also tempers out 225/224.		However, if a tuning tempers out multiple independent commas, the situation gets more complicated, for the set of intervals that are tempered out in fact forms a lattice ''generated'' by more than one generator (in other words, a nontrivial subgroup of JI), and the choice of which specific intervals to consider generators (which in this context are ''basis commas'') is not always obvious. For instance, septimal meantone tempers out the intervals {{nowrap\|[[126/125]] {{=}} 2 × 3<sup>2</sup> × 5<sup>−3</sup> × 7}}, {{nowrap\|[[225/224]] {{=}} 2<sup>−5</sup> × 3<sup>2</sup> × 5<sup>2</sup> × 7<sup>−1</sup>}}, and 81/80, but {{nowrap\|81/80 {{=}} (126/125) × (225/224)}}, and therefore these three commas are ''not all independent''–but all of them are useful, in that all three define prominent ''temperament families'' (collections of regular temperaments that share a tempered comma in common): 81/80 defines [[meantone]], 126/125 defines [[starling]], and 225/224 defines [[marvel]]. Various methods exist for choosing which commas are selected to be basis commas, which are associated with the technique of [[matrix echelon forms]]; in the case of septimal meantone, the basis commas are chosen to be 81/80 and 126/125 at the price of obscuring the fact that it also tempers out 225/224.

	As a last note, factorizations are generally abbreviated in the form of a (subgroup) [[monzo]], which is simply a list of the exponents in a factorization that are attached to each (formal) prime in the subgroup, so that for instance 225/224 would be {{monzo\|-5 2 2 -1}} (in this case the subgroup is 2.3.5.7; it should be specified if there is any ambiguity, but if not it can be assumed to be the temperament's subgroup).		As a last note, factorizations are generally abbreviated in the form of a (subgroup) [[monzo]], which is simply a list of the exponents in a factorization that are attached to each (formal) prime in the subgroup, so that for instance 225/224 would be {{monzo\|-5 2 2 -1}} (in this case the subgroup is 2.3.5.7; it should be specified if there is any ambiguity, but if not it can be assumed to be the temperament's subgroup).

Lériendil: added section

2025-03-26T18:21:26Z

added section

← Older revision		Revision as of 18:21, 26 March 2025
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	One last note is that mappings may use a slightly different (if equivalent) set of generators from elsewhere in the temperament data: for meanpop, for instance, the "canonical" generator, for which optimal tunings are specified, is in fact ~3/2, rather than ~3. In these cases, the mapping should (but does not always) specify the set of generators used for the ''mapping''.		One last note is that mappings may use a slightly different (if equivalent) set of generators from elsewhere in the temperament data: for meanpop, for instance, the "canonical" generator, for which optimal tunings are specified, is in fact ~3/2, rather than ~3. In these cases, the mapping should (but does not always) specify the set of generators used for the ''mapping''.

			=== Extensions and restrictions ===
			{{Main\|Extension and restriction}}
			{{todo\|complete section\|inline=1}}

	=== Advanced properties ===		=== Advanced properties ===

Lériendil at 14:30, 26 March 2025

2025-03-26T14:30:05Z

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	In a subgroup, all intervals are reachable by stacking (up and down) copies of a few "generating intervals", called ''[[Periods and generators\|generator]]s''. Continuing the previous example, if [[3/2]] is taken as a generator of the subgroup, then [[9/4]] is also included in the subgroup {{nowrap\|(3/2 × 3/2 {{=}} 9/4)}}, and so on. If [[2/1]] is added to the list of subgroup generators, then intervals like [[4/3]] can be reached by combining a 3/2 down with a 2/1 up {{nowrap\|(2/3 × 2/1 {{=}} 4/3)}}.		In a subgroup, all intervals are reachable by stacking (up and down) copies of a few "generating intervals", called ''[[Periods and generators\|generator]]s''. Continuing the previous example, if [[3/2]] is taken as a generator of the subgroup, then [[9/4]] is also included in the subgroup {{nowrap\|(3/2 × 3/2 {{=}} 9/4)}}, and so on. If [[2/1]] is added to the list of subgroup generators, then intervals like [[4/3]] can be reached by combining a 3/2 down with a 2/1 up {{nowrap\|(2/3 × 2/1 {{=}} 4/3)}}.

	The entirety of JI can be generated by the infinite set of [[prime number]]s {{nowrap\|(2, 3, 5, 7, …)}}. In practice, most subgroups are generated by a few primes only (hence the term ''subgroup'', where JI is the larger ''group''). A common kind of subgroups are [[prime limit]]s, which are generated by all prime harmonics up to a certain limit. For example, the [[5-limit]] is generated by all primes up to 5 (i.e. 2, 3 and 5).		The entirety of JI can be generated by the infinite set of [[prime number]]s {{nowrap\|(2, 3, 5, 7, …)}}. In practice, most subgroups are generated by a few primes only (hence the term ''subgroup'', where JI as a whole is the larger ''group''). A common kind of subgroups are [[prime limit]]s, which are generated by all prime harmonics up to a certain limit. For example, the [[5-limit]] is generated by all primes up to 5 (i.e. 2, 3 and 5).

	A subgroup is generally expressed as a list of its generators separated by dots. For example, "2.3.5" denotes the aforementioned 5-limit. Primes are not required to be consecutive; [[2.3.7 subgroup\|2.3.7]] is an equally valid subgroup. A shorthand exists where full ''p''-limits within an extended subgroup are denoted by L''p'', e.g. the 2.3.5.7.11.17.29.31 subgroup can be written as "L11.17.29.31"; however this notation is not common and therefore remains discouraged for clarity.		A subgroup is generally expressed as a list of its generators separated by dots. For example, "2.3.5" denotes the aforementioned 5-limit. Primes are not required to be consecutive; [[2.3.7 subgroup\|2.3.7]] is an equally valid subgroup. A shorthand exists where full ''p''-limits within an extended subgroup are denoted by L''p'', e.g. the 2.3.5.7.11.17.29.31 subgroup can be written as "L11.17.29.31"; however this notation is not common and therefore remains discouraged for clarity.

← Older revision		Revision as of 19:12, 12 June 2025
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	{{Main\|Mapping to lattice}}		{{Main\|Mapping to lattice}}
	{{See also\|Harmonic lattice diagram}}		{{See also\|Harmonic lattice diagram}}
	~~{{todo\|complete section\|inline=1}}~~

	~~==== Wedgie ====~~
	~~{{Main\|Plücker coordinates}}~~
	{{todo\|complete section\|inline=1}}		{{todo\|complete section\|inline=1}}

← Older revision		Revision as of 03:43, 13 April 2025
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	The ''comma basis'' is a list of such intervals–''commas''–that are tempered out by the temperament, thereby restricting the set of possible tunings. If a tuning only tempers out one comma, then the only intervals within the subgroup that are set to the unison are that comma, and its positive and negative powers (in the case above, 80/81, {{nowrap\|6561/6400 {{=}} (81/80)<sup>2</sup>}}, etc.); therefore there is only one logical choice for "which comma" you claim is tempered, that being the simplest of these powers greater than the unison. Rank-2 temperaments in subgroups including 3 primes, such as the [[5-limit]], only temper out one comma, for instance.		The ''comma basis'' is a list of such intervals–''commas''–that are tempered out by the temperament, thereby restricting the set of possible tunings. If a tuning only tempers out one comma, then the only intervals within the subgroup that are set to the unison are that comma, and its positive and negative powers (in the case above, 80/81, {{nowrap\|6561/6400 {{=}} (81/80)<sup>2</sup>}}, etc.); therefore there is only one logical choice for "which comma" you claim is tempered, that being the simplest of these powers greater than the unison. Rank-2 temperaments in subgroups including 3 primes, such as the [[5-limit]], only temper out one comma, for instance.

	However, if a tuning tempers out multiple independent commas, the situation gets more complicated, for the set of intervals that are tempered out in fact forms a lattice ''generated'' by more than one generator (in other words, a nontrivial subgroup of JI), and the choice of which specific intervals to consider generators (which in this context are ''basis commas'') is not always obvious. For instance, septimal meantone tempers out the intervals {{nowrap\|[[126/125]] {{=}} 2 × 3<sup>2</sup> × 5<sup>−3</sup> × 7}}; {{nowrap\|[[225/224]] {{=}} 2<sup>−5</sup> × 3<sup>2</sup> × 5<sup>2</sup> × 7<sup>−1</sup>}}, and 81/80, but {{nowrap\|81/80 {{=}} (126/125) × (225/224)}}, and therefore these three commas are ''not all independent''–but all of them are useful, in that all three define prominent ''temperament families'' (collections of regular temperaments that share a tempered comma in common): 81/80 defines [[meantone]], 126/125 defines [[starling]], and 225/224 defines [[marvel]]. Various methods exist for choosing which commas are selected to be basis commas, which are associated with the technique of [[matrix echelon forms]]; in the case of septimal meantone, the basis commas are chosen to be 81/80 and 126/125 at the price of obscuring the fact that it also tempers out 225/224.		However, if a tuning tempers out multiple independent commas, the situation gets more complicated, for the set of intervals that are tempered out in fact forms a lattice ''generated'' by more than one generator (in other words, a nontrivial subgroup of JI), and the choice of which specific intervals to consider generators (which in this context are ''basis commas'') is not always obvious. For instance, septimal meantone tempers out the intervals {{nowrap\|[[126/125]] {{=}} 2 × 3<sup>2</sup> × 5<sup>−3</sup> × 7}}, {{nowrap\|[[225/224]] {{=}} 2<sup>−5</sup> × 3<sup>2</sup> × 5<sup>2</sup> × 7<sup>−1</sup>}}, and 81/80, but {{nowrap\|81/80 {{=}} (126/125) × (225/224)}}, and therefore these three commas are ''not all independent''–but all of them are useful, in that all three define prominent ''temperament families'' (collections of regular temperaments that share a tempered comma in common): 81/80 defines [[meantone]], 126/125 defines [[starling]], and 225/224 defines [[marvel]]. Various methods exist for choosing which commas are selected to be basis commas, which are associated with the technique of [[matrix echelon forms]]; in the case of septimal meantone, the basis commas are chosen to be 81/80 and 126/125 at the price of obscuring the fact that it also tempers out 225/224.

	As a last note, factorizations are generally abbreviated in the form of a (subgroup) [[monzo]], which is simply a list of the exponents in a factorization that are attached to each (formal) prime in the subgroup, so that for instance 225/224 would be {{monzo\|-5 2 2 -1}} (in this case the subgroup is 2.3.5.7; it should be specified if there is any ambiguity, but if not it can be assumed to be the temperament's subgroup).		As a last note, factorizations are generally abbreviated in the form of a (subgroup) [[monzo]], which is simply a list of the exponents in a factorization that are attached to each (formal) prime in the subgroup, so that for instance 225/224 would be {{monzo\|-5 2 2 -1}} (in this case the subgroup is 2.3.5.7; it should be specified if there is any ambiguity, but if not it can be assumed to be the temperament's subgroup).

← Older revision		Revision as of 14:30, 26 March 2025
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	In a subgroup, all intervals are reachable by stacking (up and down) copies of a few "generating intervals", called ''[[Periods and generators\|generator]]s''. Continuing the previous example, if [[3/2]] is taken as a generator of the subgroup, then [[9/4]] is also included in the subgroup {{nowrap\|(3/2 × 3/2 {{=}} 9/4)}}, and so on. If [[2/1]] is added to the list of subgroup generators, then intervals like [[4/3]] can be reached by combining a 3/2 down with a 2/1 up {{nowrap\|(2/3 × 2/1 {{=}} 4/3)}}.		In a subgroup, all intervals are reachable by stacking (up and down) copies of a few "generating intervals", called ''[[Periods and generators\|generator]]s''. Continuing the previous example, if [[3/2]] is taken as a generator of the subgroup, then [[9/4]] is also included in the subgroup {{nowrap\|(3/2 × 3/2 {{=}} 9/4)}}, and so on. If [[2/1]] is added to the list of subgroup generators, then intervals like [[4/3]] can be reached by combining a 3/2 down with a 2/1 up {{nowrap\|(2/3 × 2/1 {{=}} 4/3)}}.

	The entirety of JI can be generated by the infinite set of [[prime number]]s {{nowrap\|(2, 3, 5, 7, …)}}. In practice, most subgroups are generated by a few primes only (hence the term ''subgroup'', where JI is the larger ''group''). A common kind of subgroups are [[prime limit]]s, which are generated by all prime harmonics up to a certain limit. For example, the [[5-limit]] is generated by all primes up to 5 (i.e. 2, 3 and 5).		The entirety of JI can be generated by the infinite set of [[prime number]]s {{nowrap\|(2, 3, 5, 7, …)}}. In practice, most subgroups are generated by a few primes only (hence the term ''subgroup'', where JI as a whole is the larger ''group''). A common kind of subgroups are [[prime limit]]s, which are generated by all prime harmonics up to a certain limit. For example, the [[5-limit]] is generated by all primes up to 5 (i.e. 2, 3 and 5).

	A subgroup is generally expressed as a list of its generators separated by dots. For example, "2.3.5" denotes the aforementioned 5-limit. Primes are not required to be consecutive; [[2.3.7 subgroup\|2.3.7]] is an equally valid subgroup. A shorthand exists where full ''p''-limits within an extended subgroup are denoted by L''p'', e.g. the 2.3.5.7.11.17.29.31 subgroup can be written as "L11.17.29.31"; however this notation is not common and therefore remains discouraged for clarity.		A subgroup is generally expressed as a list of its generators separated by dots. For example, "2.3.5" denotes the aforementioned 5-limit. Primes are not required to be consecutive; [[2.3.7 subgroup\|2.3.7]] is an equally valid subgroup. A shorthand exists where full ''p''-limits within an extended subgroup are denoted by L''p'', e.g. the 2.3.5.7.11.17.29.31 subgroup can be written as "L11.17.29.31"; however this notation is not common and therefore remains discouraged for clarity.