Gaster
Gaster is a regular temperament that tempers out a plethora of unnoticeable commas including [-70 72 -19⟩ (gaster comma, quadbila-negu). The generator for this temperament is a gassormic tritone, 55/39, and from this it derives its name. Aside from the psychoacoustic advantage of tempering out such small commas, a secondary concept of this temperament is that complex intervals in lower prime limits are identified with simple intervals in higher prime limits (for example, 55/39 = 38/27 = 31/22 = 24/17, 26/19 = 41/30 = 56/41, etc.) Since it tempers out the mirkwai comma, it qualifies as a mirkwai temperament.
Temperament data
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 533/532, 540/539, 575/574, 667/666
Mapping: [⟨1 11 38 37 -1 26 14 32 7 -11 0 -27 45], ⟨0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65 -80]]
- 7-limit: ~800/567 = 594.64130
- 11-limit: ~512/363 = 594.63909
- 13-limit: ~55/39 = 594.63899
- 17-limit: ~24/17 = 594.63553
- 19-limit: ~24/17 = 594.63620
- 23-limit: ~24/17 = 594.64147
- 29-limit: ~24/17 = 594.64556
- 31-limit: ~24/17 = 594.64392
- 37-limit: ~24/17 = 594.64383
- 41-limit: ~24/17 = 594.64299
- 7-limit: ~2 = 1199.99210, ~800/567 = 594.63738
- 11-limit: ~2 = 1199.93879, ~512/363 = 594.60876
- 13-limit: ~2 = 1199.91552, ~55/39 = 594.59713
- 17-limit: ~2 = 1199.80789, ~24/17 = 594.54033
- 19-limit: ~2 = 1199.75466, ~24/17 = 594.51463
- 23-limit: ~2 = 1199.87402, ~24/17 = 594.57904
- 29-limit: ~2 = 1199.94492, ~24/17 = 594.61827
- 31-limit: ~2 = 1199.91080, ~24/17 = 594.59972
- 37-limit: ~2 = 1199.90948, ~24/17 = 594.59897
- 41-limit: ~2 = 1199.91875, ~24/17 = 594.60272
- 7, 19 and 23-limit: 111, 224
- 11-limit: 111, 224, 783d, 1007d, 1231dd
- 13-limit: 111, 224, 783df, 1007df, 1231ddf
- 17-limit: 111, 224, 559dgg
- 29, 31, 37 and 41-limit: 111, 113, 224
- 7-limit: 0.154521
- 11-limit: 0.054060
- 13-limit: 0.024882
- 17-limit: 0.021436
- 19-limit: 0.018370
- 23-limit: 0.017619
- 29-limit: 0.016815
- 31-limit: 0.014790
- 37-limit: 0.014377
- 41-limit: 0.012858
Intervals
| Number of LGT(*1) |
Cents value(*2) |
Approximate Ratios |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 2 | 10.714 | 144/143, 145/144, 153/152, 154/153, 155/154, 156/155, 161/160, 162/161, 165/164, 169/168, 170/169, 171/170, 185/184, 186/185, 187/186, 190/189, 196/195, 204/203 |
| 4 | 21.428 | 77/76, 78/77, 81/80, 82/81, 85/84, 88/87, 91/90, 93/92 |
| 6 | 32.142 | 50/49, 52/51, 55/54, 56/55, 57/56 |
| 8 | 42.856 | 39/38, 40/39, 41/40, 42/41 |
| 10 | 53.570 | 32/31, 33/32, 34/33 |
| 12 | 64.284 | 27/26, 28/27 |
| 14 | 74.998 | 23/22, 24/23 |
| 16 | 85.712 | 20/19, 21/20, 41/39 |
| 18 | 96.426 | 18/17, 19/18 |
| 20 | 107.140 | 17/16, 33/31 |
| 22 | 117.854 | 15/14, 31/29 |
| 24 | 128.568 | 14/13, 41/38 |
| 26 | 139.282 | 13/12 |
| 28 | 149.996 | 12/11 |
| 30 | 160.710 | 34/31 |
| 32 | 171.424 | 21/19, 32/29 |
| 34 | 182.139 | 10/9 |
| 36 | 192.853 | 19/17 |
| 38 | 203.567 | 9/8 |
| 40 | 214.281 | 26/23 |
| 42 | 224.995 | 33/29, 41/36 |
| 44 | 235.709 | 39/34 |
| 46 | 246.423 | 15/13, 38/33 |
| 48 | 257.137 | 36/31 |
| 50 | 267.851 | 7/6 |
| 52 | 278.565 | 20/17, 27/23, 34/29 |
| 54 | 289.279 | 13/11 |
| 56 | 299.993 | 19/16, 25/21, 44/37 |
| 58 | 310.707 | |
| 60 | 321.421 | 41/34 |
| 62 | 332.135 | 40/33 |
| 64 | 342.849 | 28/23, 39/32, 50/41 |
| 66 | 353.563 | 27/22, 38/31 |
| 68 | 364.277 | 21/17 |
| 70 | 374.991 | 36/29, 41/33, 46/37 |
| 72 | 385.705 | 5/4 |
| 74 | 396.419 | 39/31 |
| 76 | 407.133 | 81/64 |
| 78 | 417.847 | 14/11 |
| 80 | 428.561 | 41/32, 50/39 |
| 82 | 439.275 | 40/31 |
*1: LGT stands for larger gassormic tritone.
*2: in 41-limit POTE tuning
| Number of SGT(*3) |
Cents value(*4) |
Approximate Ratios |
|---|---|---|
| 1 | 594.643 | 24/17, 31/22, 38/27, 55/39 |
| 3 | 583.929 | 7/5 |
| 5 | 573.215 | 32/23, 39/28, 46/33 |
| 7 | 562.501 | 18/13 |
| 9 | 551.787 | 11/8 |
| 11 | 541.073 | 26/19, 41/30, 56/41 |
| 13 | 530.359 | 19/14 |
| 15 | 519.645 | 23/17, 27/20, 31/23 |
| 17 | 508.931 | |
| 19 | 498.217 | 4/3 |
| 21 | 487.503 | |
| 23 | 476.789 | 29/22, 54/41 |
| 25 | 466.075 | 17/13 |
| 27 | 455.361 | 13/10 |
| 29 | 444.647 | 22/17, 31/24 |
| 31 | 433.933 | 9/7 |
| 33 | 423.219 | 23/18, 37/29 |
| 35 | 412.504 | 19/15, 33/26, 52/41 |
| 37 | 401.790 | 24/19, 29/23, 34/27 |
| 39 | 391.076 | |
| 41 | 380.362 | |
| 43 | 369.648 | 26/21 |
| 45 | 358.934 | 16/13 |
| 47 | 348.220 | 11/9 |
| 49 | 337.506 | 17/14 |
| 51 | 326.792 | 23/19, 29/24 |
| 53 | 316.078 | 6/5 |
| 55 | 305.364 | 31/26, 37/31 |
| 57 | 294.650 | 32/27 |
| 59 | 283.936 | 33/28, 46/39 |
| 61 | 273.222 | 41/35, 48/41 |
| 63 | 262.508 | |
| 65 | 251.794 | 22/19, 37/32 |
| 67 | 241.080 | 23/20, 31/27 |
| 69 | 230.366 | 8/7 |
| 71 | 219.652 | 17/15 |
| 73 | 208.938 | 44/39 |
| 75 | 198.224 | 28/25, 37/33, 46/41 |
| 77 | 187.510 | 29/26, 39/35 |
| 79 | 176.796 | 31/28 |
| 81 | 166.082 | |
| 83 | 155.368 |
*3: SGT stands for smaller gassormic tritone.
*4: in 41-limit POTE tuning