# Gaster

(Redirected from Gaster temperament)

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

Gaster temperament (111&113)

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-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