2.5.7 subgroup

The 2.5.7 subgroup, or the no-threes 7-limit (yaza nowa in color notation) is a just intonation subgroup consisting of rational intervals where 2, 5, 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 2.5.7 subgroup is a retraction of the 7-limit, obtained by removing prime 3. Its simplest expansion is the 2.5.7.11 subgroup, which adds prime 11.

A notable subset of the 2.5.7 subgroup is the 1.5.7 tonality diamond, comprised of all intervals in which 1, 5 and 7 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 1.5.7 tonality diamond within the octave is 1/1, 8/7, 5/4, 7/5, 10/7, 8/5, 7/4, and 2/1.

Another such subset is the 1.5.7.25.35 tonality diamond, which adds the following intervals to the previous list: 25/16, 25/14, 35/32, 64/35, 28/25, and 32/25.

When octave equivalence is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 5 and 7, which can be represented in a 2-dimensional lattice diagram, each prime represented by a different dimension, such that each point on the lattice represents a different interval class.

Properties

This subgroup is modestly well-represented by 6edo for its size, enough so that many of its simple intervals tend to cluster around the notes of 6edo. Therefore, one way to approach the 2.5.7 subgroup is to think of a hexatonic framework for composition as natural to it, rather than the diatonic framework associated with the 5-limit.

Regular temperaments

Commas and rank-2 temperaments

Main article: Tour of regular temperaments § Clans defined by a 2.5.7 (yaza nowa) comma

In the below tables, the generator of the temperament is highlighted in bold. Intervals in the tables reflect the 1.5.7.25.35 tonality diamond. It is structurally notable that they come in five clusters, each centered around one note of 6edo - 35/32 ~ 28/25 ~ 8/7 close to a wholetone, 5/4 and 32/25 close to a ditone, 7/5 and 10/7 close to a tritone, and so on - with intervals within each cluster separated by the commas 50/49 and 128/125.

Jubilic

Jubilic temperament tempers out the comma 50/49 in the 2.5.7 subgroup, which makes 7/5 ~ 10/7 a single half-octave tritone which serves as the period. Similarly to dicot, jubilic can be regarded as an exotemperament that elides fundamental distinctions within the subgroup, though the comma involved is half the size of dicot's 25/24.

The DKW (2.5.7) optimum tuning states ~5/4 is tuned to 385.002c and therefore that ~8/7 is tuned to 214.998c; a chart of mistunings of simple intervals is below.

Jubilic (50/49)

Interval	Just tuning	Tunings*
		Optimal tuning	Deviation
35/32	155.140	170.005	+14.865
28/25	196.198	214.998	+18.799
8/7	231.174	214.998	-16.176
5/4	386.314	385.002	-1.311
32/25	427.373	429.995	+2.623
7/5	582.512	600.000	+17.488
10/7	617.488	600.000	-17.488
25/16	772.627	770.005	-2.623
8/5	813.686	814.998	+1.311
7/4	968.826	985.002	+16.176
25/14	1003.802	985.002	-18.799
64/35	1044.860	1029.995	-14.865

Augmented

Augmentsept temperament tempers out the comma 128/125 = S4/S5. Strictly speaking, augmented is a 2.3.5 temperament, with which the 2.5.7 temperament shares only the same comma, hence this version should be known under a different name. However, both temperaments make the interval 5/4 an exact third of the octave, which serves as the period. The generator in this case is then naturally ~8/7.

The DKW (2.5.7) optimum tuning states ~8/7 is tuned to 218.297c; a chart of mistunings of simple intervals is below.

Augmented (128/125)

Interval	Just tuning	Tunings*
		Optimal tuning	Deviation
35/32	155.140	181.703	-26.563
28/25	196.198	181.703	-14.495
8/7	231.174	218.297	-12.877
5/4	386.314	400.000	+13.686
32/25	427.373	400.000	-27.373
7/5	582.512	581.703	-0.809
10/7	617.488	618.297	+0.809
25/16	772.627	800.000	+27.373
8/5	813.686	800.000	-13.686
7/4	968.826	981.703	+12.877
25/14	1003.802	1018.297	+14.495
64/35	1044.860	1018.297	-26.563

Didacus

Didacus temperament, the 2.5.7 restriction of hemimean, tempers out the comma 3136/3125 in the 2.5.7 subgroup, which splits the major third (5/4) into two intervals of 28/25. It is one of the most accurate temperaments of its simplicity. While augmented and jubilic merge the middle interval of the triplets surrounding 1\6 and 5\6 with one of the two nearby intervals, didacus makes it an exact mean between the two.

The DKW (2.5.7) optimum tuning states ~5/4 is tuned to 388.122c, and therefore ~28/25 to 194.061c; a chart of mistunings of simple intervals is below.

Didacus (3136/3125)

Interval	Just tuning	Tunings*
		Optimal tuning	Deviation
35/32	155.140	158.427	+3.288
28/25	196.198	194.061	-2.137
8/7	231.174	229.695	-1.479
5/4	386.314	388.122	+1.808
32/25	427.373	423.756	-3.617
7/5	582.512	582.183	-0.329
10/7	617.488	617.817	+0.329
25/16	772.627	776.244	+3.617
8/5	813.686	811.878	-1.808
7/4	968.826	970.305	+1.479
25/14	1003.802	1005.939	+2.137
64/35	1044.860	1041.573	-3.288