# Just intonation subgroups

# Definition

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.

It is only when the group in question is not the entire p-limit group that we have a just intonation subgroup in the strict sense. Such subgroups 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.

# 7-limit subgroups

2.3.7

Ets: 5, 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

Ets: 6, 25, 31, 35, 47, 171, 239, 379, 410, 789

2.3.7/5

Ets: 10, 29, 31, 41, 70, 171, 241, 412

2.5/3.7

Ets: 12, 15, 42, 57, 270, 327

2.5.7/3

Ets: 9, 31, 40, 50, 81, 90, 171, 261

2.5/3.7/3

Ets: 27, 68, 72, 99, 171, 517

2.27/25.7/3

Ets: 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

Ets: 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

2.3.11

Ets: 7, 15, 17, 24, 159, 494, 518, 653

Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]

2.5.11

Ets: 6, 7, 9, 13, 15, 22, 37, 87, 320

2.7.11

Ets: 6, 9, 11, 20, 26, 135, 161, 296

2.3.5.11

Ets: 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494

2.3.7.11

Ets: 9, 17, 26, 31, 41, 46, 63, 72, 135

The Radon temperament subgroup, generated by the Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1]

See: Gallery of 2.3.7.11 Subgroup Scales

2.5.7.11

Ets: 6, 15, 31, 35, 37, 109, 618, 960

2.5/3.7/3.11/3

Ets: 33, 41, 49, 57, 106, 204, 253

The Indium temperament subgroup.

# 13-limit subgroups

2.3.13

Ets: 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

Ets: 15, 19, 34, 53, 87, 130, 140, 246, 270

The Cata, Trinidad and Parizekmic temperaments subgroup.

2.3.7.13

Ets: 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]

2.5.7.13

Ets: 7, 10, 17, 27, 37, 84, 121, 400

The Huntington temperament subgroup.

2.5.7.11.13

Ets: 6, 7, 13, 19, 25, 31, 37

The Roulette temperament subgroup

2.3.13/5

Ets: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362

The Barbados temperament subgroup.

2.3.11/5.13/5

5, 9, 14, 19, 24, 29

The Bridgetown temperament subgroup.

2.3.11/7.13/7

Ets: 5, 7, 12, 17, 29, 46, 75, 196, 271

The Pepperoni temperament subgroup.

2.7/5.11/5.13/5

Ets: 5, 8, 21, 29, 37, 66, 169, 235

The Tridec temperament subgroup.