# 2.3.7.11 subgroup

The 2.3.7.11 subgroup (laza in color notation) is a just intonation subgroup consisting of rational intervals where 2, 3, 7, and 11 are the only allowable prime factors, so that every such interval may be written as a ratio of integers which are products of 2, 3, 7, and 11. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the octave include 3/2, 7/4, 9/7, 21/16, 11/9, 22/21, and so on.

The 2.3.7.11 subgroup is a retraction of the 11-limit, obtained by removing prime 5. Its simplest expansion is the 2.3.7.11.13 subgroup, which adds prime 13. It can also be retracted to the 2.3.7 subgroup by removing prime 11.

A notable subset of the 2.3.7.11 subgroup is the 1.3.7.9.11 tonality diamond, comprised of all intervals in which 1, 3, 7, 9, and 11 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. The complete list of intervals in the 1.3.7.9.11 tonality diamond within the octave is 1/1, 12/11, 9/8, 8/7, 7/6, 11/9, 14/11, 9/7, 4/3, 11/8, 16/11, 3/2, 14/9, 11/7, 18/11, 12/7, 7/4, 16/9, 11/6, and 2/1.

When octave equivalence is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3, 7, and 11, which can be represented in a 3-dimensional lattice diagram, each prime represented by a different dimension, such that each point on the lattice represents a different interval class.

## Scales

• Genus(3*7*11) 12/11, 14/11, 21/16, 16/11, 3/2, 7/4, 21/11, 2
• Genus(3*7^2*11) 49/48, 49/44, 7/6, 14/11, 4/3, 16/11, 49/33, 49/32, 56/33, 7/4, 64/33, 2
• Ptolemy's Intense Chromatic at 1/1 22/21 8/7 4/3 3/2 11/7 12/7 2/1 (disjunct form)
• A conjunct Rast: 1/1 9/8 27/22 4/3 3/2 11/8 16/9 2/1 (note: actually 2.3.11 subgroup)
• Qutb al-Din al-Shirazi's version of Hijaz: 1/1-12/11-14/11-4/3 (12:11-7:6-22:21), itself a permutation of Ptolemy's Intense Chromatic
• Margo Schulter notes: "A modern form of Maqam Hijaz based on this tuning might be 1/1-12/11-14/11-4/3-3/2-18/11-16/9-2/1 ascending, and 1/1-12/11-14/11-4/3-3/2-128/81-16/9-2/1 descending (with the minor sixth maybe a bit smaller, say 11/7 or the like)."
• A septimal flavor of Sazkar, which has a minor third above the 1/1 ascending but a tone descending: thus ascending 1/1-7/6-11/9-4/3-3/2-27/16-11/6-2/1, and descending 1/1-9/8-11/9-4/3-3/2-27/16-11/6-2/1
• Margo Schulter adds: "I should caution that an Arab Rast, of which Sazkar is an offshoot, might usually have a Zalzalian third more like 27/22 or 16/13 rather than 11/9, so this may be more of a new tuning than a traditional Arab Sazkar."
• The 1-3-7-9-11 2(5 Dekany within a 12-tone constant structure (see link): 1*3, 9*11, 3*9, 1*7, 3*7*11, 7*9, 3*11, 1*9, 3*9*11, 7*11, 3*7, 1*11.
• (with prime 13 added) A 12f, {352/351, 364/363} 2.3.7.11.13 elf transversal: 28/27-9/8-13/11-9/7-4/3-11/8-3/2-14/9-22/13-16/9-27/14-2