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]]. | 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 '' | 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 | ||
$$ | $$ r_1.r_2.\cdots.r_n := \operatorname{span}_\mathbb{Z} \{v_1, \cdots, v_n\}. $$ | ||
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> | |||
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 | 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. | ||
== Normalization == | == Normalization == | ||
A canonical | 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 == | ||
{{ | {{Wikipedia|Index of a subgroup}} | ||
Intuitively speaking, the '''index''' measures the relative size of the subgroup within another subgroup, which is usually the | 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 | 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 == | ||
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]] | ||