2.3.7.11 subgroup

The 2.3.7.11 subgroup (zala 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, comprising 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 this 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

Regular temperaments

Rank-1 temperaments (edos)

The 2.3.7.11 subgroup is relatively well approximated by the following edos (decreasing TE error, bold ones do particularly well in this subgroup): 5, 9, 10, 12, 14, 17, 31, 41, 58, 63, 72, 94, 118, 130, 135, 342, …

Rank-2 temperaments

Supra, which extends archy, provides a simple yet high-damage approximation to the subgroup. It is generated by a perfect fifth, tuned a little sharp so that two make 8/7~9/8 and six make 16/11, tempering out 64/63 and 99/98. Alternatively, suhajira can be considered an extension of archy that adds neutral intervals, with the perfect fifth split into two neutral third generators, each representing 11/9~27/22, thus tempering out 64/63 and 243/242.

Skwares takes the 11/9~27/22 neutral third, adds an octave to it and splits it in halves for ~11/7, tempering out 99/98 and 243/242.

Radon, which adds prime 11 to slendric by tempering out 896/891, provides a more complex entry, well represented by 41edo and 46edo. A similar temperament at this level is hemif, which tempers out 243/242 and 896/891 and can be tuned to 58edo. In both cases, the diatonic major third represents 14/11.

On the high-accuracy side, gary is an important temperament that finds 7 and 11 far into the chain of fifths.

Rank-3 temperaments

Parapyth equates 28/27 with 33/32 and uses this interval as a spacer added to the chain of fifths to finds intervals of 7 and 11.

Symbiotian, which makes 33/32 and 28/27 sum to the Pythagorean apotome, is an efficient high-accuracy counterpart of parapyth.

Olympian equates 33/32 with a stack of two 64/63's. It is very accurate yet still easy to notate on the staff, using the septimal comma as spacer added to the chain of fifths to finds intervals of 7 and 11.