Just intonation subgroup

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 subgroup refers to its mathematical structure with respect to JI – a subset of a 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 generators in frequency ratios 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, 3, and 7.

In standard mathematical notation, let c₁, …, c_r be positive rationals, and suppose v_k is the musical interval of log₂(c_k) octaves. Then

$$ c_1.c_2.\cdots.c_r := \operatorname{span}_\mathbb{Z} \{v_1, \cdots, v_k\}. $$

There are three categories of subgroups:

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 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 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".^{[note 1]} 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 form for the generators of the group, which will also show the 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

See also: 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.

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 monzos of the generators.

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 cents). This is closely related to the 3L 4s mos tuning with neutral third generator sqrt(3/2).

List of selected subgroups

7-limit subgroups

Others:

2.3.7/5
- EDOs: 10, 29, 31, 41, 70, 171, 241, 412
2.5/3.7
- EDOs: 12, 15, 42, 57, 270, 327
2.5.7/3
- EDOs: 9, 31, 40, 50, 81, 90, 171, 261
2.5/3.7/3
- EDOs: 27, 68, 72, 99, 171, 517
2.27/25.7/3
- EDOs: 9
- In effect, equivalent to 9edo, which has a 7-limit version given by [27/25, 7/6, 63/50, 49/36, 72/49, 100/63, 12/7, 50/27, 2]
2.9/5.9/7
- EDOs: 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558
- The terrain temperament subgroup

11-limit subgroups

Others:

2.5.11
- EDOs: 6, 7, 9, 13, 15, 22, 37, 87, 320
2.7.11
- EDOs: 6, 9, 11, 20, 26, 135, 161, 296
2.5.7.11
- EDOs: 6, 15, 31, 35, 37, 109, 618, 960
2.5/3.7/3.11/3
- EDOs: 33, 41, 49, 57, 106, 204, 253
- The indium temperament subgroup.

13-limit subgroups

Others:

2.3.13
- EDOs: 7, 10, 17, 60, 70, 130, 147, 277, 424
- Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]
2.3.5.13
- EDOs: 15, 19, 34, 53, 87, 130, 140, 246, 270
- The cata, trinidad and parizekmic temperaments subgroup
2.3.7.13
- EDOs: 10, 26, 27, 36, 77, 94, 104, 130, 234
- Buzurg [14/13, 16/13, 4/3, 56/39, 3/2]
- Safi al-Din tuning [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1]
- Ibn Sina tuning [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]
2.3.5.11.13
2.5.7.13
- EDOs: 7, 10, 17, 27, 37, 84, 121, 400
- The huntington temperament subgroup
2.5.7.11.13
- EDOs: 6, 7, 13, 19, 25, 31, 37
- The roulette temperament subgroup
2.3.13/5
- EDOs: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362
- The barbados temperament subgroup.
2.3.11/5.13/5
- EDOs: 5, 9, 14, 19, 24, 29
- The bridgetown temperament subgroup
2.3.11/7.13/7
- EDOs: 5, 7, 12, 17, 29, 46, 75, 196, 271
- The pepperoni temperament subgroup.
2.7/5.11/5.13/5
- EDOs: 5, 8, 21, 29, 37, 66, 169, 235
- The tridec temperament subgroup.

Higher-limit subgroups

Irrational subgroups

Hemipyth (√2.√3 subgroup)
Hemipent (√2.√3.√5 subgroup)

Notes

↑ 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.

[1] 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.

[note 1]