2.3.7 subgroup

The 2.3.7 subgroup (za in color notation) 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 2.3.7 subgroup is a retraction of the 7-limit, obtained by removing prime 5. Its simplest expansion is the 2.3.7.11 subgroup, which adds prime 11.

A notable subset of the 2.3.7 subgroup is the 1.3.7 tonality diamond, comprised of all intervals in which 1, 3 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 1.3.7 tonality diamond within the octave is 1/1, 8/7, 7/6, 4/3, 3/2, 12/7, 7/4, and 2/1.

Another such subset is the 1.3.7.9 tonality diamond, which adds the following intervals to the previous list: 9/8, 9/7, 14/9, and 16/9.

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 a 2-dimensional lattice diagram, each prime represented by a different dimension, such that each point on the lattice represents a different interval class.

Scales

• Zo minor pentatonic: 1/1 7/6 4/3 3/2 7/4 2/1
• Ru pentatonic: 1/1 9/8 9/7 3/2 12/7 2/1
• Zo in: 1/1 9/8 7/6 3/2 14/9 2/1
• Zo minor: 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1
• Za harmonic minor: 1/1 9/8 7/6 4/3 3/2 14/9 27/14 2/1 (zo minor with a ru 7th)
• Za diasem/Tas[9] (left-handed): 1/1 9/8 7/6 21/16 4/3 3/2 14/9 7/4 16/9 2/1

Regular temperaments

Rank-1 temperaments (edos)

A list of edos with progressively better tunings for the 2.3.7 subgroup: 5, 12, 14, 17, 22, 31, 36, 77, 94, 130, 135, 171, 265, 306, 400, 571, 706, 1277 and so on.

Another list of edos which provides relatively good tunings for the 2.3.7 subgroup (relative error < 2.5%): 36, 41, 77, 94, 99, 130, 135, 171, 207, 229, 265, 301, 306, 364, 400, 436, 441, 477, 494, 535, 571, 576, 607, 648, 665, 670, 701, 706, 742, 747, 783, 836, 841, 877, 913, 935, 971, 976, 1007, 1012, 1048, 1106, 1147, 1178, 1183, 1236, 1241, 1277 and so on.

Commas and rank-2 temperaments

Main article: Tour of regular temperaments § Clans defined by a 2.3.7 (za) comma

Semaphore

Semaphore temperament tempers out the comma 49/48 = S7 in the 2.3.7 subgroup, which equates 8/7 with 7/6, creating a single neutral semifourth. Similarly to dicot, semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup (from the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does), though the comma involved is half the size of dicot's 25/24.

The DKW (2.3.7) optimum tuning states ~3/2 is tuned to 696.230c (though most other optimizations tune this a few cents flatter); a chart of mistunings of simple intervals is below.

Semaphore (49/48)

Interval	Just tuning	Tunings
		Optimal tuning	Deviation
9/8	203.910	192.460	-11.450
8/7	231.174	251.885	+20.711
7/6	266.871	251.885	-14.986
9/7	435.084	444.345	+9.261
21/16	470.781	444.345	-26.436
4/3	498.045	503.770	+5.725
3/2	701.955	696.230	-5.725
32/21	729.219	755.655	+26.436
14/9	764.916	755.655	-9.261
12/7	933.129	948.115	+14.986
7/4	968.826	948.115	-20.711
16/9	996.090	1007.540	+11.450

Archy

Archy temperament tempers out the comma 64/63 = S8 in the 2.3.7 subgroup, which equates 9/8 with 8/7, and 4/3 with 21/16. It serves as a septimal analogue of meantone, favoring fifths sharp of just rather than flat.

The DKW (2.3.7) optimum tuning states ~3/2 is tuned to 712.585c (though most other optimizations tune this a few cents flatter); a chart of mistunings of simple intervals is below.

Archy (64/63)

Interval	Just tuning	Tunings
		Optimal tuning	Deviation
9/8	203.910	225.171	+21.261
8/7	231.174	225.171	-6.003
7/6	266.871	262.244	-4.627
9/7	435.084	450.341	+15.257
21/16	470.781	487.415	+16.634
4/3	498.045	487.415	-10.630
3/2	701.955	712.585	+10.630
32/21	729.219	712.585	-16.634
14/9	764.916	749.659	-15.257
12/7	933.129	937.756	+4.627
7/4	968.826	974.829	+6.003
16/9	996.090	974.829	-21.261

Gamelic

Gamelic temperament, better known as slendric, tempers out the comma 1029/1024 = S7/S8 in the 2.3.7 subgroup, which splits the perfect fifth into three intervals of 8/7. It is one of the most accurate temperaments of its simplicity.

The DKW (2.3.7) optimum tuning states ~3/2 is tuned to 699.126c, and therefore ~8/7 to 233.042c; a chart of mistunings of simple intervals is below.

Gamelic (1029/1024)

Interval	Just tuning	Tunings
		Optimal tuning	Deviation
9/8	203.910	198.253	-5.657
8/7	231.174	233.042	+1.868
7/6	266.871	267.831	+0.960
9/7	435.084	431.295	-3.789
21/16	470.781	466.084	-4.697
4/3	498.045	500.874	+2.829
3/2	701.955	699.126	-2.829
32/21	729.219	733.916	+4.697
14/9	764.916	768.705	+3.789
12/7	933.129	932.169	-0.960
7/4	968.826	966.958	-1.868
16/9	996.090	1001.747	+5.657

Music

Michael Harrison

• From Ancient Worlds (for harmonic piano), 1992
• Revelation: Music in Pure Intonation, 2007