Just intonation subgroup: Difference between revisions

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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''1, ~~...~~, ''c''''r'' be positive reals, and suppose ''v''''k'' is the musical interval of log2(''c''''k'') 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''1, …, ''c''''r'' be positive reals, and suppose ''v''''k'' is the musical interval of log2(''c''''k'') 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:

* ''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.

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.

== 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 ==

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== 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 ==

@@ Line 5: / Line 5: @@
 | 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:
+* ''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.
-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.
 == 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 ==
@@ Line 152: / Line 151: @@
 == 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 ==