Just intonation subgroup

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Todo: add introduction
introduction needed that helps musicians/composers understand that this is relevant to them

A just intonation subgroup is a group generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Any such group will be contained in a p-limit group for some minimal choice of prime p, which is the prime limit of the subgroup.

There are three categories of subgroups:

  • Prime subgroups (e.g. 2.3.7) contain only primes
  • Composite subgroups (e.g. 2.5.9) contain composite numbers and perhaps prime numbers too
  • Rational subgroups (e.g. 2.3.7/5) contain rational numbers and perhaps prime and/or composite numbers too

For composite and rational 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 doesn't 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 5-limit JI, 7-limit JI, etc. Thus a just intonation subgroup in the strict sense refers only to prime subgroups that do omit such primes, as well as the other two categories.

Inthar proposes the following terminology 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 formal prime. (Mathematically, this is a synonym for an element of a fixed basis.) For example, if the group is written 2.5/3.7/3, the formal primes are 2, 5/3 and 7/3. Formal primes may not necessarily be actual primes, but they behave similarly to primes in the p-limit.

Subgroups in the strict sense come in two flavors: finite index and infinite index, where intuitively speaking the index measures the relative size of the subgroup within the entire p-limit group. 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 matrix whose rows are the monzos of the generators.

A canonical naming system for just intonation subgroups is to give a normal interval list for the generators of the group, which will also show the rank of the group by the number of generators in the list. 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. Just intonation subgroups can be described by listing their generators with dots between them; the purpose of using dots is to flag the fact that it is a subgroup which is being referred to. This naming convention is employed below.

Non-JI intervals can also be used as formal primes, 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).


7-limit subgroups

2.3.7
  • EDOs5, 17, 31, 36, 135, 571
  • Archytas Diatonic [8/7, 32/27, 4/3, 3/2, 12/7, 16/9, 2/1]
  • Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1]
2.5.7
2.3.7/5
2.5/3.7
2.5.7/3
2.5/3.7/3
2.27/25.7/3
  • EDOs9
  • 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

11-limit subgroups

2.3.11
2.5.11
2.7.11
2.3.5.11
2.3.7.11
2.5.7.11
2.5/3.7/3.11/3

13-limit subgroups

2.3.13
2.3.5.13
2.3.7.13
  • EDOs10, 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]
2.5.7.13
2.5.7.11.13
2.3.13/5
2.3.11/5.13/5
2.3.11/7.13/7
2.7/5.11/5.13/5