Just intonation subgroup - Revision history

FloraC: Misc. cleanup

2026-08-22T05:11:32Z

Misc. cleanup

← Older revision		Revision as of 05:11, 22 August 2026
Line 23:		Line 23:

	== Normalization ==		== Normalization ==
	A canonical ~~naming~~ system for just intonation subgroups is to give a [[normal forms #Normal forms for commas\|normal form]] for the generators of the group, which will also show the ~~[[Wikipedia: Rank~~ of an abelian group\|rank]] of the group by the number of generators in the list ~~(the~~ [[Hermite normal form]] should be used here, not the [[canonical form]], because in the case of subgroups, [[enfactoring]] is ~~sometimes desirable~~, such as in the subgroup 2.9.7 which should not be reduced to 2.3.7 by subgroup canonicalization).		A canonical notation system for just intonation subgroups is to give a [[normal forms #Normal forms for commas\|normal form]] for the generators of the group, which will also show the {{w\|rank of an abelian group\|rank}} of the group by the number of generators in the list. The [[Hermite normal form]] should be used here, not the [[canonical form]], because in the case of subgroups, [[enfactoring]] is usually desired, such as in the subgroup 2.9.7 which should not be reduced to 2.3.7 by subgroup canonicalization.

	== Index ==		== Index ==
	{{~~See also~~\| ~~Wikipedia:~~ Index of a subgroup }}		{{Wikipedia\|Index of a subgroup}}

	Intuitively speaking, the '''index''' measures the relative size of the subgroup within another subgroup, which is usually the ~~''p''-~~limit.		Intuitively speaking, the '''index''' measures the relative size of the subgroup within another subgroup, which is usually the minimal prime subgroup or the minimal prime limit.

	Subgroups in the strict sense come in two flavors: finite index and infinite index. For example, the subgroups generated by 4 and 3, by 2 and 9, and by 4 and 6 all have index 2 in the full [[3-limit]] (Pythagorean) group. Half of the 3-limit intervals will belong to any one of them, and half will not, and all three groups are distinct. On the other hand, the group generated by 2, 3, and 7 is of infinite index in the full [[7-limit]] group, which is generated by 2, 3, 5 and 7. The index can be computed by taking the determinant of the [[subgroup basis matrix]], whose columns are the [[monzo]]s of the generators.		Subgroups in the strict sense come in two flavors: finite index and infinite index. For example, the subgroups generated by 4 and 3, by 2 and 9, and by 4 and 6 all have index 2 in the full [[3-limit]] (Pythagorean) group. Half of the 3-limit intervals will belong to any one of them, and half will not, and all three groups are distinct. On the other hand, the group generated by 2, 3, and 7 is of infinite index in the full [[7-limit]] group, which is generated by 2, 3, 5 and 7. The index can be computed by taking the {{w\|determinant}} of the [[subgroup basis matrix]], whose columns are the [[monzo]]s of the generators.

	== Generalization ==		== Generalization ==
	Non-JI intervals can also be used as basis elements, when the subgroup in question contains non-JI intervals. For example, 2.sqrt(3~~/2) (sometimes written 2.2ed3~~/2) is the group generated by [[2/1]] and [[sqrt(3/2)]] (a neutral third which is exactly one half of 3/2, 350.978 [[cent]]s). This is closely related to the [[3L 4s]] mos tuning with neutral third generator sqrt(3/2).		Non-JI intervals can also be used as basis elements, when the subgroup in question contains non-JI intervals. For example, 2.sqrt(3/2) is the group generated by [[2/1]] and [[sqrt(3/2)]] (a neutral third which is exactly one half of 3/2, 350.978 [[cent]]s). This is closely related to the [[3L 4s]] mos tuning with neutral third generator sqrt(3/2).

	== List of selected subgroups ==		== List of selected subgroups ==

FloraC: Rework intro for accessibility (2/2). Not sure why the indices in the formula didn't line up; I think they should.

2026-08-22T04:53:20Z

Rework intro for accessibility (2/2). Not sure why the indices in the formula didn't line up; I think they should.

← Older revision		Revision as of 04:53, 22 August 2026
Line 9:		Line 9:
	Just intonation subgroups can be described by listing their [[generator]]s in [[frequency ratio]]s with full stops between them; we use said convention below. For example, the [[2.3.7 subgroup]] is a subgroup consisting of intervals that are combinations of [[2/1\|2]], [[3/1\|3]], and [[7/1\|7]].		Just intonation subgroups can be described by listing their [[generator]]s in [[frequency ratio]]s with full stops between them; we use said convention below. For example, the [[2.3.7 subgroup]] is a subgroup consisting of intervals that are combinations of [[2/1\|2]], [[3/1\|3]], and [[7/1\|7]].

	In standard mathematical notation, let ''c''<sub>1</sub>, …, ''c''<sub>''r''</sub> be positive rationals, and suppose ''v''<sub>''k''</sub> is the musical interval of log<sub>2</sub>(''c''<sub>''k''</sub>) octaves. Then		In standard mathematical notation, let ''r''<sub>1</sub>, …, ''r''<sub>''n''</sub> be positive rationals, and suppose ''s''<sub>''i''</sub> is the musical interval of log<sub>2</sub>(''r''<sub>''i''</sub>) octaves. Then

	$$ ~~c_1~~.~~c_2~~.\cdots.~~c_r~~ := \operatorname{span}_\mathbb{Z} \{v_1, \cdots, ~~v_k~~\}. $$		$$ r_1.r_2.\cdots.r_n := \operatorname{span}_\mathbb{Z} \{v_1, \cdots, v_n\}. $$

	~~There~~ are ~~three categories~~ of ~~subgroups~~:		If any redundant generators are eliminated, the set of generators is a [[basis]]. In general, given a subgroup written as generated by such a set: ''r''<sub>1</sub>.''r''<sub>2</sub>.''r''<sub>3</sub>.[…].''r''<sub>''n''</sub>, each member of this set is called a '''basis element''', '''structural prime''', or "'''formal prime'''".<ref group="note">The meaning of "formal" this term is using is "of external form or structure, rather than nature or content", which is to say that a formal prime is not necessarily ''actually'' a prime, but we treat them as if they were. The original coiner of this term, [[Inthar]], has recommended its disuse, in favor of the mathematically accurate and generic ''basis element'', or possibly something else which indicates the co-uniqueness of the elements.</ref>
	* ''~~Prime subgroups~~'' ~~(e.g~~. 2.3.~~7) contain only primes~~
	* ''~~Composite subgroups~~'' (e.g. ~~2.9.5) contain composite numbers~~ and ~~perhaps prime numbers too~~
	* ''~~Fractional subgroups~~'' ~~(e.g~~. ~~2.3.7/5) contain fractional numbers and perhaps prime and~~/~~or composite numbers too~~

	~~For composite and fractional subgroups, not all combinations of numbers are mathematically valid~~ [[~~basis~~\|~~bases~~]] ~~for~~ subgroups. ~~For example,~~ 2.3.~~9 has a redundant generator 9, and both~~ 2.3.15 and 2.3.~~5/3 can be simplified to~~ 2.3.5.		Subgroups have been categorized as follows (after [[#Normalization\|normalization]]):
			* ''Prime subgroups'' (e.g. 2.3.7) contain only prime basis elements;
			* ''Composite subgroups'' (e.g. 2.9.5) contain composite and perhaps prime basis elements too;
			* ''Fractional subgroups'' (e.g. 2.3.7/5) contain fractional numbers and perhaps prime and/or composite numbers too.

	A prime subgroup that does not omit any primes < ''p'' (e.g. 2.3.5, 2.3.5.7, 2.3.5.7.11, etc. but not 2.3.7 or 3.5.7) is simply called [[harmonic limit\|''p''-limit JI]]. It is customary of just intonation subgroups to refer only to prime subgroups that do omit such primes, as well as the other two categories.		A prime subgroup that does not omit any primes less than ''p'' (e.g. 2.3.5, 2.3.5.7, 2.3.5.7.11, etc. but not 2.3.7 or 3.5.7) is simply called [[harmonic limit\|''p''-limit JI]]. It is customary of just intonation subgroups to refer only to prime subgroups that do omit such primes, as well as the other two categories.

	The following terminology has been proposed for streamlining pedagogy: Given a subgroup written as generated by a fixed (non-redundant) set: ''a''.''b''.''c''.[…].''d'', call any member of this set a '''basis element''', '''structural prime''', or "'''formal prime'''".<ref group="note">The meaning of "formal" this term is using is "of external form or structure, rather than nature or content", which is to say that a formal prime is not necessarily ''actually'' a prime, but we treat them as if they were. The original coiner of this term, [[Inthar]], has recommended its disuse, in favor of the mathematically accurate and generic "basis element", or possibly something else which indicates the co-uniqueness of the elements.</ref> For example, if the group is written 2.5/3.7/3, the basis elements are 2, 5/3 and 7/3.

	== Normalization ==		== Normalization ==
	A canonical naming system for just intonation subgroups is to give a [[normal forms #Normal forms for commas\|normal form]] for the generators of the group, which will also show the [[Wikipedia: Rank of an abelian group\|rank]] of the group by the number of generators in the list (the [[Hermite normal form]] should be used here, not the [[canonical form]], because in the case of subgroups, [[enfactoring]] is sometimes desirable, such as in the subgroup 2.9.7 which should not be reduced to 2.3.7 by subgroup canonicalization)~~. Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale~~.		A canonical naming system for just intonation subgroups is to give a [[normal forms #Normal forms for commas\|normal form]] for the generators of the group, which will also show the [[Wikipedia: Rank of an abelian group\|rank]] of the group by the number of generators in the list (the [[Hermite normal form]] should be used here, not the [[canonical form]], because in the case of subgroups, [[enfactoring]] is sometimes desirable, such as in the subgroup 2.9.7 which should not be reduced to 2.3.7 by subgroup canonicalization).

	== Index ==		== Index ==
Line 38:		Line 36:

	== List of selected subgroups ==		== List of selected subgroups ==
			Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale.

	=== 7-limit subgroups ===		=== 7-limit subgroups ===
	* [[2.3.7 subgroup]]		* [[2.3.7 subgroup]]

FloraC: Rework intro for accessibility (1/2); also correction (subgroups aren't necessarily finite)

2026-08-22T04:08:47Z

Rework intro for accessibility (1/2); also correction (subgroups aren't necessarily finite)

← Older revision		Revision as of 04:08, 22 August 2026
Line 1:		Line 1:
	{{~~interwiki~~		{{Interwiki
			\| en = Just intonation subgroup
	\| de = Untergruppe der reinen Stimmung		\| de = Untergruppe der reinen Stimmung
	~~\| en = Just intonation subgroup~~
	\| es =		\| es =
	\| ja = 純正律部分群		\| ja = 純正律部分群
	}}		}}
	A '''just intonation subgroup''' is a {{w\|~~free abelian~~ group\|group}} ~~generated by~~ a ~~finite set of positive rational numbers via arbitrary multiplications and divisions~~. ~~Using~~ subgroups ~~implies~~ a ~~way to organize~~ [[~~just intonation~~]] ~~intervals such that they form a lattice. Therefore it~~ is closely related to [[regular temperament theory]].		A '''just intonation subgroup''' consists of all [[just intonation]] intervals formed by arbitrarily [[stacking]] a set of intervals and their inverses finitely many times. The term ''{{w\|subgroup}}'' refers to its mathematical structure with respect to JI – a subset of a {{w\|group (mathematics)\|group}} that is also a group. Just intonation subgroups organize intervals consistently, with each subgroup corresponding to a [[lattice]]; the use of these as an approach to JI is closely related to [[regular temperament theory]].

			Just intonation subgroups can be described by listing their [[generator]]s in [[frequency ratio]]s with full stops between them; we use said convention below. For example, the [[2.3.7 subgroup]] is a subgroup consisting of intervals that are combinations of [[2/1\|2]], [[3/1\|3]], and [[7/1\|7]].

	~~Just intonation subgroups can be described by listing their [[generator]]s with full stops between them; we use said convention below.~~ In standard mathematical notation, let ''c''<sub>1</sub>, …, ''c''<sub>''r''</sub> be positive ~~reals~~, and suppose ''v''<sub>''k''</sub> is the musical interval of log<sub>2</sub>(''c''<sub>''k''</sub>) octaves. Then		In standard mathematical notation, let ''c''<sub>1</sub>, …, ''c''<sub>''r''</sub> be positive rationals, and suppose ''v''<sub>''k''</sub> is the musical interval of log<sub>2</sub>(''c''<sub>''k''</sub>) octaves. Then

	~~<math>~~c_1.c_2.\cdots.c_r := \operatorname{span}_\mathbb{Z} \{v_1, \cdots, v_k\}.~~</math>~~		$$ c_1.c_2.\cdots.c_r := \operatorname{span}_\mathbb{Z} \{v_1, \cdots, v_k\}. $$

	There are three categories of subgroups:		There are three categories of subgroups:
Line 20:		Line 22:
	A prime subgroup that does not omit any primes < ''p'' (e.g. 2.3.5, 2.3.5.7, 2.3.5.7.11, etc. but not 2.3.7 or 3.5.7) is simply called [[harmonic limit\|''p''-limit JI]]. It is customary of just intonation subgroups to refer only to prime subgroups that do omit such primes, as well as the other two categories.		A prime subgroup that does not omit any primes < ''p'' (e.g. 2.3.5, 2.3.5.7, 2.3.5.7.11, etc. but not 2.3.7 or 3.5.7) is simply called [[harmonic limit\|''p''-limit JI]]. It is customary of just intonation subgroups to refer only to prime subgroups that do omit such primes, as well as the other two categories.

	The following terminology has been proposed for streamlining pedagogy: Given a subgroup written as generated by a fixed (non-redundant) set: ''a''.''b''.''c''.[…].''d'', call any member of this set a '''basis element''', '''structural prime''', or "'''formal prime'''".<ref>The meaning of "formal" this term is using is "of external form or structure, rather than nature or content", which is to say that a formal prime is not necessarily ''actually'' a prime, but we treat them as if they were. The original coiner of this term, [[Inthar]], has recommended its disuse, in favor of the mathematically accurate and generic "basis element", or possibly something else which indicates the co-uniqueness of the elements.</ref> For example, if the group is written 2.5/3.7/3, the basis elements are 2, 5/3 and 7/3.		The following terminology has been proposed for streamlining pedagogy: Given a subgroup written as generated by a fixed (non-redundant) set: ''a''.''b''.''c''.[…].''d'', call any member of this set a '''basis element''', '''structural prime''', or "'''formal prime'''".<ref group="note">The meaning of "formal" this term is using is "of external form or structure, rather than nature or content", which is to say that a formal prime is not necessarily ''actually'' a prime, but we treat them as if they were. The original coiner of this term, [[Inthar]], has recommended its disuse, in favor of the mathematically accurate and generic "basis element", or possibly something else which indicates the co-uniqueness of the elements.</ref> For example, if the group is written 2.5/3.7/3, the basis elements are 2, 5/3 and 7/3.

	== Normalization ==		== Normalization ==
Line 122:		Line 124:

	== Notes ==		== Notes ==
			<references group="note"/>

	[[Category:Subgroup\| ]] <!-- main article -->		[[Category:Subgroup\| ]] <!-- main article -->
	[[Category:Just intonation]]		[[Category:Just intonation]]

FloraC: /* List of selected subgroups */

2026-08-09T08:03:29Z

List of selected subgroups

← Older revision		Revision as of 08:03, 9 August 2026
Line 99:		Line 99:
	* 2.3.13/5		* 2.3.13/5
	** {{EDOs\|legend=1\| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}		** {{EDOs\|legend=1\| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
	** The [[~~Barbados~~]] temperament subgroup.		** The [[barbados]] temperament subgroup.
	* 2.3.11/5.13/5		* 2.3.11/5.13/5
	** {{EDOs\|legend=1\| 5, 9, 14, 19, 24, 29 }}		** {{EDOs\|legend=1\| 5, 9, 14, 19, 24, 29 }}

FloraC: /* List of selected subgroups */ cleanup

2026-08-09T08:00:45Z

List of selected subgroups: cleanup

Show changes

Furcht968 at 13:33, 10 June 2026

2026-06-10T13:33:58Z

← Older revision		Revision as of 13:33, 10 June 2026
Line 3:		Line 3:
	\| en = Just intonation subgroup		\| en = Just intonation subgroup
	\| es =		\| es =
	\| ja = ~~純正律サブグループ~~		\| ja = 純正律部分群
	}}		}}
	A '''just intonation subgroup''' is a {{w\|free abelian group\|group}} generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Using subgroups implies a way to organize [[just intonation]] intervals such that they form a lattice. Therefore it is closely related to [[regular temperament theory]].		A '''just intonation subgroup''' is a {{w\|free abelian group\|group}} generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Using subgroups implies a way to organize [[just intonation]] intervals such that they form a lattice. Therefore it is closely related to [[regular temperament theory]].

Overthink: /* Higher-limit subgroups */ subgroup ordering

2026-05-11T23:01:59Z

Higher-limit subgroups: subgroup ordering

← Older revision		Revision as of 23:01, 11 May 2026
Line 143:		Line 143:
	* [[2.17/13.19/13 subgroup]]		* [[2.17/13.19/13 subgroup]]

	; 5.7~~.8.9~~.11.13.17.23:		; 8.9.5.7.11.13.17.23:
	* [[143ed11]]		* [[143ed11]]

Inthar at 01:45, 13 April 2026

2026-04-13T01:45:28Z

← Older revision		Revision as of 01:45, 13 April 2026
Line 1:		Line 1:
	{{interwiki		{{interwiki
	\| de =		\| de = Untergruppe der reinen Stimmung
	\| en = Just intonation subgroup		\| en = Just intonation subgroup
	\| es =		\| es =

FloraC: /* Generalization */ improve wording and + links

2026-02-11T10:31:18Z

Generalization: improve wording and + links

← Older revision		Revision as of 10:31, 11 February 2026
Line 33:		Line 33:

	== Generalization ==		== Generalization ==
	Non-JI intervals can also be used as basis elements, when the subgroup in question contains non-JI intervals. For example, 2.sqrt(3/2) (sometimes written 2.2ed3/2) is the group generated by 2/1 and ~~350.978 cents, the square root of~~ 3/2 (a neutral third which is exactly one half of 3/2). This is closely related to the [[3L 4s]] mos tuning with neutral third generator sqrt(3/2).		Non-JI intervals can also be used as basis elements, when the subgroup in question contains non-JI intervals. For example, 2.sqrt(3/2) (sometimes written 2.2ed3/2) is the group generated by [[2/1]] and [[sqrt(3/2)]] (a neutral third which is exactly one half of 3/2, 350.978 [[cent]]s). This is closely related to the [[3L 4s]] mos tuning with neutral third generator sqrt(3/2).

	== List of selected subgroups ==		== List of selected subgroups ==

FloraC: Cleanup

2026-02-06T08:47:39Z

Cleanup

← Older revision		Revision as of 08:47, 6 February 2026
Line 5:		Line 5:
	\| ja = 純正律サブグループ		\| ja = 純正律サブグループ
	}}		}}
	A '''just intonation subgroup''' is a {{w\|~~Free~~ abelian group\|group}} generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Using subgroups implies a way to organize [[just intonation]] intervals such that they form a lattice. Therefore it is closely related to [[regular temperament theory]].		A '''just intonation subgroup''' is a {{w\|free abelian group\|group}} generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Using subgroups implies a way to organize [[just intonation]] intervals such that they form a lattice. Therefore it is closely related to [[regular temperament theory]].

	Just intonation subgroups can be described by listing their [[generator]]s with full stops between them; we use said convention below. In standard mathematical notation, let ''c''<sub>1</sub>, ~~...~~, ''c''<sub>''r''</sub> be positive reals, and suppose ''v''<sub>''k''</sub> is the musical interval of log<sub>2</sub>(''c''<sub>''k''</sub>) octaves. Then		Just intonation subgroups can be described by listing their [[generator]]s with full stops between them; we use said convention below. In standard mathematical notation, let ''c''<sub>1</sub>, …, ''c''<sub>''r''</sub> be positive reals, and suppose ''v''<sub>''k''</sub> is the musical interval of log<sub>2</sub>(''c''<sub>''k''</sub>) octaves. Then

	<math>c_1.c_2.\cdots.c_r := \operatorname{span}_\mathbb{Z} \{v_1, ~~...~~, v_k\}.</math>		<math>c_1.c_2.\cdots.c_r := \operatorname{span}_\mathbb{Z} \{v_1, \cdots, v_k\}.</math>

	There are three categories of subgroups:		There are three categories of subgroups:
			* ''Prime subgroups'' (e.g. 2.3.7) contain only primes
	* '''Prime subgroups''' (e.g. 2.3.7) contain only primes		* ''Composite subgroups'' (e.g. 2.9.5) contain composite numbers and perhaps prime numbers too
	* '''Composite subgroups''' (e.g. 2.9.5) contain composite numbers and perhaps prime numbers too		* ''Fractional subgroups'' (e.g. 2.3.7/5) contain fractional numbers and perhaps prime and/or composite numbers too
	* '''Fractional subgroups''' (e.g. 2.3.7/5) contain fractional numbers and perhaps prime and/or composite numbers too

	For composite and fractional subgroups, not all combinations of numbers are mathematically valid [[basis\|bases]] for subgroups. For example, 2.3.9 has a redundant generator 9, and both 2.3.15 and 2.3.5/3 can be simplified to 2.3.5.		For composite and fractional subgroups, not all combinations of numbers are mathematically valid [[basis\|bases]] for subgroups. For example, 2.3.9 has a redundant generator 9, and both 2.3.15 and 2.3.5/3 can be simplified to 2.3.5.

	A prime subgroup that does not omit any primes < ''p'' (e.g. 2.3.5, 2.3.5.7, 2.3.5.7.11, etc. but not 2.3.7 or 3.5.7) is simply called [[~~Harmonic~~ limit\|''p''-limit JI]]. It is customary of just intonation subgroups to refer only to prime subgroups that do omit such primes, as well as the other two categories.		A prime subgroup that does not omit any primes < ''p'' (e.g. 2.3.5, 2.3.5.7, 2.3.5.7.11, etc. but not 2.3.7 or 3.5.7) is simply called [[harmonic limit\|''p''-limit JI]]. It is customary of just intonation subgroups to refer only to prime subgroups that do omit such primes, as well as the other two categories.

	The following terminology has been proposed for streamlining pedagogy: Given a subgroup written as generated by a fixed (non-redundant) set: ''a''.''b''.''c''.[…].''d'', call any member of this set a '''basis element''', '''structural prime''', or "'''formal prime'''".<ref>The meaning of "formal" this term is using is "of external form or structure, rather than nature or content", which is to say that a formal prime is not necessarily ''actually'' a prime, but we treat them as if they were. The original coiner of this term, [[Inthar]], has recommended its disuse, in favor of the mathematically accurate and generic "basis element", or possibly something else which indicates the co-uniqueness of the elements.</ref> For example, if the group is written 2.5/3.7/3, the basis elements are 2, 5/3 and 7/3.		The following terminology has been proposed for streamlining pedagogy: Given a subgroup written as generated by a fixed (non-redundant) set: ''a''.''b''.''c''.[…].''d'', call any member of this set a '''basis element''', '''structural prime''', or "'''formal prime'''".<ref>The meaning of "formal" this term is using is "of external form or structure, rather than nature or content", which is to say that a formal prime is not necessarily ''actually'' a prime, but we treat them as if they were. The original coiner of this term, [[Inthar]], has recommended its disuse, in favor of the mathematically accurate and generic "basis element", or possibly something else which indicates the co-uniqueness of the elements.</ref> For example, if the group is written 2.5/3.7/3, the basis elements are 2, 5/3 and 7/3.

	== Normalization ==		== Normalization ==
	A canonical naming system for just intonation subgroups is to give a [[~~Normal lists~~ #Normal ~~interval lists~~\|normal ~~interval list~~]] for the generators of the group, which will also show the [[Wikipedia: Rank of an abelian group\|rank]] of the group by the number of generators in the list (the [[Hermite normal form]] should be used here, not the [[canonical form]], because in the case of subgroups, [[enfactoring]] is sometimes desirable, such as in the subgroup 2.9.7 which should not be reduced to 2.3.7 by subgroup canonicalization). Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale.		A canonical naming system for just intonation subgroups is to give a [[normal forms #Normal forms for commas\|normal form]] for the generators of the group, which will also show the [[Wikipedia: Rank of an abelian group\|rank]] of the group by the number of generators in the list (the [[Hermite normal form]] should be used here, not the [[canonical form]], because in the case of subgroups, [[enfactoring]] is sometimes desirable, such as in the subgroup 2.9.7 which should not be reduced to 2.3.7 by subgroup canonicalization). Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale.

	== Index ==		== Index ==
Line 152:		Line 151:

	== See also ==		== See also ==
	* [[Subgroup basis ~~matrices~~]] – a formal discussion on matrix representations of subgroup bases		* [[Subgroup basis matrix]] – a formal discussion on matrix representations of subgroup bases

	== Notes ==		== Notes ==