2.3.7 subgroup

The 2.3.7-limit or 2.3.7-prime-limit is a Just intonation subgroup consisting of rational intervals where 2, 3, and 7 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, and 7. 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, 7/6, 9/7, 9/8, 21/16, and so on.

The 1.3.7-odd-limit refers to a constraint on the selection of just intervals for a scale or composition such that 3 and 7 are the only allowable odd numbers, either for the intervals of the scale, or the ratios between successive or simultaneously sounding notes of the composition. The complete list of 1.3.7-odd-limit intervals within the octave is 1/1, 8/7, 7/6, 4/3, 3/2, 12/7, 7/4, and 2/1, which is known as the 1.3.7-limit tonality diamond.

The phrase "2.3.7-limit just intonation" usually refers to the 2.3.7-prime-limit and includes primes 2, 3, and 7. When octave equivalence is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3 and 7, which can be represented in 2-dimensional lattice diagrams, each prime represented by a different dimension.