View source for Gaster

'''Gaster''' is a [[regular temperament]] that tempers out a plethora of [[unnoticeable comma]]s including {{monzo|-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. Coincidentally, one of its commas involves stacking seventeen 9/8s, and 17 is a number associated with the character Gaster from Undertale. 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 clan|mirkwai temperament]].

== Temperament data ==
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px; overflow:auto;">
<div style="line-height:1.6;">'''Gaster temperament (111&amp;113)'''</div>
<div class="mw-collapsible-content">
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]]: [{{val| 1 11 38 37 -1 26 14 32 7 -11 0 -27 45 }}, {{val| 0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65 -80 }}]

[[POTE generator]]: 
* 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

[[TOP tuning|TOP generator]]s: 
* 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

[[Optimal ET sequence]]s:
* 7, 19 and 23-limit: {{Optimal ET sequence| 111, 224 }}
* 11-limit: {{Optimal ET sequence| 111, 224, 783d, 1007d, 1231dd }}
* 13-limit: {{Optimal ET sequence| 111, 224, 783df, 1007df, 1231ddf }}
* 17-limit: {{Optimal ET sequence| 111, 224, 559dgg }}
* 29, 31, 37 and 41-limit: {{Optimal ET sequence| 111, 113, 224 }}

[[Badness]]: 
* 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
</div></div>

== Intervals ==
{| class="wikitable center-1 right-2"
! Number of <br>LGT(*1)
! Cents <br>value(*2)
! Approximate Ratios
|-
| 0
| 0.000
| 1/1
|-
| 2
| 10.714
| 144/143, 145/144, <br>153/152, 154/153, <br>155/154, 156/155, <br>161/160, 162/161, <br>165/164, 169/168, <br>170/169, 171/170, <br>185/184, 186/185, <br>187/186, 190/189, <br>196/195, 204/203
|-
| 4
| 21.428
| 77/76, 78/77, 81/80, <br>82/81, 85/84, 88/87, <br>91/90, 93/92
|-
| 6
| 32.142
| 50/49, 52/51, 55/54, <br>56/55, 57/56
|-
| 8
| 42.856
| 39/38, 40/39, 41/40, <br>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
|}
<nowiki>*</nowiki>1: LGT stands for larger gassormic tritone.<br><nowiki>*</nowiki>2: in 41-limit POTE tuning

{| class="wikitable center-1 right-2"
! Number of <br>SGT(*3)
! Cents <br>value(*4)
! Approximate Ratios
|-
| 1
| 594.643
| 24/17, 31/22, 38/27, <br>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
| 
|}
<nowiki>*</nowiki>3: SGT stands for smaller gassormic tritone.<br><nowiki>*</nowiki>4: in 41-limit POTE tuning

== See also ==
* [[Mirkwai clan]]
* [[Very high accuracy temperaments]]
* [[Unnoticeable comma]]

[[Category:Gaster| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Mirkwai clan]]