Technical data guide for regular temperaments - Revision history

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

2025-08-30T13:44:06Z

Mapping and sval mapping: update on terminology

← Older revision		Revision as of 13:44, 30 August 2025
Line 28:		Line 28:
	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.

Line 37:		Line 37:
	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
Line 64:		Line 64:

	== 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
Line 53:		Line 53:
	{{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
Line 24:		Line 24:
	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
Line 40:		Line 40:

	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

← Older revision		Revision as of 14:30, 26 March 2025
Line 12:		Line 12:
	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.

Lériendil: added note about svals

2025-03-26T14:29:10Z

added note about svals

← Older revision		Revision as of 14:29, 26 March 2025
Line 36:		Line 36:

	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.

	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''.

Lériendil at 14:20, 26 March 2025

2025-03-26T14:20:50Z

← Older revision		Revision as of 14:20, 26 March 2025
Line 35:		Line 35:
	Therefore, it is useful to specify a way to "translate" between JI intervals and stacks of temperament generators—for example, to know how a given JI interval is retuned under the temperament (and a specific tuning of the generators). The ''mapping'' provides this way, by specifying how each of the ''prime harmonics'' (subgroup generators) are equated to a stack of generators of the temperament.		Therefore, it is useful to specify a way to "translate" between JI intervals and stacks of temperament generators—for example, to know how a given JI interval is retuned under the temperament (and a specific tuning of the generators). The ''mapping'' provides this way, by specifying how each of the ''prime harmonics'' (subgroup generators) are equated to a stack of generators of the temperament.

	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 [[~~circle~~ 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.

	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''.

Lériendil at 14:19, 26 March 2025

2025-03-26T14:19:04Z

← Older revision		Revision as of 14:19, 26 March 2025
Line 16:		Line 16:
	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.

	However, it may be reasonable in some cases to include composite numbers in a subgroup: the subgroup 2.9.15.7.11 (note that these are, by convention, sorted in order of prime limit rather than numerical order) includes ''some'' intervals that contain 3 and 5 in their factorization, such as 9/7, 15/8, or 5/3—the last being interpreted as 15/9, but not others: it would not contain an interval like 3/2 or 5/4, since these can't be reached from multiplying and dividing 9 and 15 with primes. Fractions may be included as well, like the subgroup 2.3.11.13/5.17 (note that this is interpreted as 2.3.11.(13/5).17), which includes intervals of 13 and intervals of 5, but only when a power of 13 is matched by an equal power of 5 on the other side of the fraction; or the subgroup 2.5/3.7/3.11/3, which additionally includes intervals like 7/~~5 and~~ 11/5 but not intervals like 7/4 or 11/8. Composites or fractions treated as primes in this context are often called "formal primes" or "basis elements."		However, it may be reasonable in some cases to include composite numbers in a subgroup: the subgroup 2.9.15.7.11 (note that these are, by convention, sorted in order of prime limit rather than numerical order) includes ''some'' intervals that contain 3 and 5 in their factorization, such as 9/7, 15/8, or 5/3—the last being interpreted as 15/9, but not others: it would not contain an interval like 3/2 or 5/4, since these can't be reached from multiplying and dividing 9 and 15 with primes. Fractions may be included as well, like the subgroup 2.3.11.13/5.17 (note that this is interpreted as 2.3.11.(13/5).17), which includes intervals of 13 and intervals of 5, but only when a power of 13 is matched by an equal power of 5 on the other side of the fraction; or the subgroup 2.3.7/5.11/5.13/5, which additionally includes intervals like 11/7 and 13/11 but not intervals like 5/4 or 11/8. Composites or fractions treated as primes in this context are often called "formal primes" or "basis elements."

	=== Comma list ===		=== Comma list ===

Lériendil at 14:18, 26 March 2025

2025-03-26T14:18:25Z

← Older revision		Revision as of 14:18, 26 March 2025
Line 16:		Line 16:
	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.

	However, it may be reasonable in some cases to include composite numbers in a subgroup: the subgroup 2.9.15.7.11 (note that these are, by convention, sorted in order of prime limit rather than numerical order) includes ''some'' intervals that contain 3 and 5 in their factorization, such as 9/7, 15/8, or 5/3—the last being interpreted as 15/9, but not others: it would not contain an interval like 3/2 or 5/4, since these can't be reached from multiplying and dividing 9 and 15 with primes. Fractions may be included as well, like the subgroup 2.3.11.13/5.17 (note that this is interpreted as 2.3.11.(13/5).17), which includes intervals of 13 and intervals of 5, but only when a power of 13 is matched by an equal power of 5 on the other side of the fraction, or the subgroup 2.5/3.7/3.11/3; which additionally includes intervals like 7/5 and 11/5 but not intervals like 7/4 or 11/8. Composites or fractions treated as primes in this context are often called "formal primes" or "basis elements."		However, it may be reasonable in some cases to include composite numbers in a subgroup: the subgroup 2.9.15.7.11 (note that these are, by convention, sorted in order of prime limit rather than numerical order) includes ''some'' intervals that contain 3 and 5 in their factorization, such as 9/7, 15/8, or 5/3—the last being interpreted as 15/9, but not others: it would not contain an interval like 3/2 or 5/4, since these can't be reached from multiplying and dividing 9 and 15 with primes. Fractions may be included as well, like the subgroup 2.3.11.13/5.17 (note that this is interpreted as 2.3.11.(13/5).17), which includes intervals of 13 and intervals of 5, but only when a power of 13 is matched by an equal power of 5 on the other side of the fraction; or the subgroup 2.5/3.7/3.11/3, which additionally includes intervals like 7/5 and 11/5 but not intervals like 7/4 or 11/8. Composites or fractions treated as primes in this context are often called "formal primes" or "basis elements."

	=== Comma list ===		=== Comma list ===

← Older revision		Revision as of 19:12, 12 June 2025
Line 53:		Line 53:
	{{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
Line 24:		Line 24:
	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
Line 12:		Line 12:
	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.