Just intonation subgroup - Revision history

Furcht968 at 13:33, 10 June 2026

2026-06-10T13:33:58Z

← Older revision		Revision as of 13:33, 10 June 2026
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	\| 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
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	* [[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
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	{{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
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	== 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
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	\| 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 ==
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	== 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 ==

Xenllium at 11:14, 5 February 2026

2026-02-05T11:14:42Z

← Older revision		Revision as of 11:14, 5 February 2026
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	=== Irrational subgroups ===		=== Irrational subgroups ===
	* [[Hemipyth]] (√2.√3 subgroup)		* [[Hemipyth]] (√2.√3 subgroup)
			* [[Hemipent]] (√2.√3.√5 subgroup)

	== See also ==		== See also ==

Xenllium at 03:44, 5 February 2026

2026-02-05T03:44:03Z

← Older revision		Revision as of 03:44, 5 February 2026
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	; 5.7.8.9.11.13.17.23:		; 5.7.8.9.11.13.17.23:
	* [[143ed11]]		* [[143ed11]]

			=== Irrational subgroups ===
			* [[Hemipyth]] (√2.√3 subgroup)

	== See also ==		== See also ==

Contribution: /* List of selected subgroups */

2025-10-27T16:14:52Z

List of selected subgroups

← Older revision		Revision as of 16:14, 27 October 2025
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	* [[2.11.13.17.19 subgroup]]		* [[2.11.13.17.19 subgroup]]
	* [[2.17/13.19/13 subgroup]]		* [[2.17/13.19/13 subgroup]]

			; 5.7.8.9.11.13.17.23:
			* [[143ed11]]

	== See also ==		== See also ==

VectorGraphics: i so want to remove "formal prime" here but it's actually used on the discord :sob:

2025-04-07T23:28:11Z

i so want to remove "formal prime" here but it's actually used on the discord :sob:

← Older revision		Revision as of 23:28, 7 April 2025
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	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 ==

BudjarnLambeth: Bold prime, composite and fractional subgroups since sometimes those phrases link here from other pages

2024-12-30T06:09:28Z

Bold prime, composite and fractional subgroups since sometimes those phrases link here from other pages

← Older revision		Revision as of 06:09, 30 December 2024
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	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.