Just intonation subgroup: Difference between revisions

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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''1, …, ''c''''r'' be positive rationals, and suppose ''v''''k'' is the musical interval of log2(''c''''k'') octaves. Then

In standard mathematical notation, let ''r''1, …, ''r''''n'' be positive rationals, and suppose ''s''''i'' is the musical interval of log2(''r''''i'') 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''1.''r''2.''r''3.[…].''r''''n'', 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 ==

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 ==

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== 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 ===

* [[2.3.7 subgroup]]

@@ 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]].
-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 ==
-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 ==
@@ Line 38: / Line 36: @@
 == 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 ===
 * [[2.3.7 subgroup]]