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. 22/17 is equated to 13/10 and 22/19 is equated to 15/13. It contains the 16:19:22 sub-diminished triad.
- 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. 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.
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.
| -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 accuracy provides many ways to temper 2.3.17.19, but this requires more effort to produce a temperament that isn't virtually 12edo. Intergan's interseptimals are one option. The rank-3 temperament should be attached to one of numerous often multi-period temperaments in this subgroup.
- Third-octave: 121172^-11 = 2057/2048 and (27/17)^3/4 = 19683/19652.
- Quarter-octave: add (19/16)^4/2=130321/131072
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.
