Intergan

Intergan is a set of temperaments related to 24edo semiquartal. The term is short for "interseptimal Ganassi," after 12edo's 2.3.17.19 Ganassi temperament. It seeks to cover all of the notable concordances generated by a meantone-range semifourth.

• The most basic version is 2.3.11.13/5.17.19 5&24. 4/3 is split into two semifourths most simply represented by 15/13, and 22/17 is equated to 13/10 and 22/19 is equated to 15/13. It features the remarkably simple 10:13:15 ultramajor triad and its retroversion found in Island temperaments, along with the 16:19:22 +1+1 sub-diminished triad and 16:19:24:27 +3+5+3 minor add6 tetrad. 11:12:17:19 is present, although this adding 13, 14, or 15 to the subgroup will significantly improve its usefulness.
• Tempering out 81/80 is somewhat obvious, resulting in 2.3.5.11.13.17.19 5&24.
• Primes 31 and 37 are also obvious to add, equating 37/32 to 22/19 and 37/31 to 19/16, resulting in 2.3.5.11.13.17.19.31.37 5&24. 67edo is roughly optimal due to its very mild meantone fourth being split in half. The 2.31.37 restriction is highly accurate in 43edo.
• Primes 7, 23, and 29 are all more difficult to map due to their complexity, although their difficulty is in ascending order. They are more easily reached by splitting the semifourth generator in half in a manner similar to Negri.

Like other important temperaments by User:Ground, it may be understood as a relatively low-complexity rank-3 for greater flexibility. 81/80 is an obvious comma to temper out as seen in the standard rank-2 version, but leaving it in creates a significant amount of generator complexity. With this in mind, rank-3 Intergan is constructed as 2.3.5.11.13.17.19.31.37 5&9&24 with the following comma list:

• 4/3 / (15/13)^2 = 676/675
• 22/19 / (15/13) = 286/285
• 13/10 / (22/17) = 221/220
• 16/15 / (17/16) = 256/255
• 37/32 / (15/13) = 481/480
• 37/31 / (19/16) = 592/589

It seems obvious to add (22/17)^2/(5/3) = 1452/1445, but that's equivalent to adding 81/80, as is 5/4/(19/17)^2 = 1445/1444.

Rank-3 Intergan, 2.3.5.11.13.17.19.31.37 5&9&24 CE tuning; defining primes highlighted, along with the two most likely mappings of 7, 23, and 29

	-5	-4	-3	-2	-1	0	1	2	3	4	5
-5	482	877	71	466	860	55	449	844	38	433	827
-4	231	626	1020	215	609	1004	198	593	987	182	576
-3	1180	375	769	1164	358	753	1147	342	736	1131	325
-2	929	124	518	913	107	502	896	91	485	880	75
-1	678	1073	267	662	1056	251	645	1040	235	629	1024
0	427	822	16	411	805	0	395	789	1184	378	773
1	176	571	965	160	555	949	144	538	933	127	522
2	1125	320	715	1109	304	698	1093	287	682	1076	271
3	875	69	464	858	53	447	842	36	431	825	20
4	624	1018	213	607	1002	196	591	985	180	574	969
5	373	767	1162	356	751	1145	340	734	1129	323	718

Being so close to 24edo means that 1/2, 1/3, 1/4, 1/6, and 1/8-octave temperaments are all available. 12edo's remarkable accuracy in 2.3.17.19 provides many ways to temper the subgroup and its extensions, but this requires more effort to produce a temperament that isn't virtually 12edo; Intergan's interseptimals are one option. The rank-3 temperament may be attached to one of numerous often multi-period temperaments in this subgroup to reduce it to rank-2.

• Third-octave: 121172^-11 = 2057/2048 and (27/17)^3/4 = 19683/19652. The former is called the Blume comma, and it is obscure despite being both accurate and low-complexity. The third-octave version of Intergan is especially useful when adding the elusive prime 29 because 29/23 is 401¢.
• Quarter-octave: add (19/16)^4/2=130321/131072. This is one of very few useful commas dividing the octave into four equal parts.

The different versions of Intergan may be defined by the sheer number of intervals it maps to the ~19/16 interval: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37, and the otonal semifourth stack 26:31:37:44.