Gaster: Difference between revisions
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'''Gaster''' is a [[regular temperament]] that tempers out a plethora of [[unnoticeable comma]]s including {{monzo|-70 72 -19}} (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 = 31/22 = 24/17, 26/19 = 41/30, etc.) Since it tempers out the mirkwai comma, it qualifies as a [[Mirkwai clan|mirkwai temperament]]. | '''Gaster''' is a [[regular temperament]] that tempers out a plethora of [[unnoticeable comma]]s including {{monzo|-70 72 -19}} (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 clan|mirkwai temperament]]. | ||
== Temperament data == | == Temperament data == | ||
| Line 57: | Line 57: | ||
== Intervals == | == Intervals == | ||
{| class="wikitable center-1 right-2" | {| class="wikitable center-1 right-2" | ||
! Number of <br>LGT (*1) | ! Number of <br>LGT(*1) | ||
! Cents <br>value (*2) | ! Cents <br>value(*2) | ||
! Approximate Ratios | ! Approximate Ratios | ||
|- | |- | ||
| Line 67: | Line 67: | ||
| 2 | | 2 | ||
| 10.714 | | 10.714 | ||
| 144/143, 145/144, 153/152, | | 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 | | 4 | ||
| 21.428 | | 21.428 | ||
| 77/76, 78/77, 81/80, 82/81, | | 77/76, 78/77, 81/80, <br>82/81, 85/84, 88/87, <br>91/90, 93/92 | ||
|- | |- | ||
| 6 | | 6 | ||
| 32.142 | | 32.142 | ||
| 52/51, 55/54, 56/55, 57/56 | | 50/49, 52/51, 55/54, <br>56/55, 57/56 | ||
|- | |- | ||
| 8 | | 8 | ||
| 42.856 | | 42.856 | ||
| 39/38, 40/39, 41/40, 42/41 | | 39/38, 40/39, 41/40, <br>42/41 | ||
|- | |- | ||
| 10 | | 10 | ||
| Line 232: | Line 232: | ||
{| class="wikitable center-1 right-2" | {| class="wikitable center-1 right-2" | ||
! Number of <br>SGT (*3) | ! Number of <br>SGT(*3) | ||
! Cents <br>value (*4) | ! Cents <br>value(*4) | ||
! Approximate Ratios | ! Approximate Ratios | ||
|- | |- | ||
| 1 | | 1 | ||
| 594.643 | | 594.643 | ||
| 24/17, 31/22, 38/27, 55/39 | | 24/17, 31/22, 38/27, <br>55/39 | ||
|- | |- | ||
| 3 | | 3 | ||
| Line 405: | Line 405: | ||
|} | |} | ||
<nowiki>*</nowiki>3: SGT stands for smaller gassormic tritone.<br><nowiki>*</nowiki>4: in 41-limit POTE tuning | <nowiki>*</nowiki>3: SGT stands for smaller gassormic tritone.<br><nowiki>*</nowiki>4: in 41-limit POTE tuning | ||
== Gaster comma == | |||
'''Gaster comma''' is a 5-limit [[unnoticeable comma]] measuring about 0.7995 cents. It is the difference between a stack of seventeen [[9/8]]s and a stack of nineteen [[10/9]]s, (9/8)<sup>17</sup>/(10/9)<sup>19</sup> = {{monzo|-70 72 -19}} (quadbila-negu). Equal temperaments which temper out this comma include {{Vals| 111, 113, 224, 335, 559, 783, 1342, 1901, 3243 }} and others. | |||
=== 5-limit gaster temperament === | |||
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"> | |||
<div style="line-height:1.6;">'''5-limit gaster temperament (559&1342)'''</div> | |||
<div class="mw-collapsible-content"> | |||
Subgroup: 2.3.5 | |||
[[Comma]]: {{monzo|-70 72 -19}} | |||
[[Mapping]]: [{{val|1 11 38}}, {{val|0 -19 -72}}] | |||
[[POTE generator]]: ~1638400000/1162261467 = 594.63457 | |||
[[TOP tuning|TOP generator]]s: ~2 = 1200.00282, ~1638400000/1162261467 = 594.63596 | |||
{{Optimal GPV sequence|legend=1| 111, 224, 335, 559, 1342, 1901, 3243 }} | |||
[[Badness]]: 0.088530 | |||
</div></div> | |||
== See also == | == See also == | ||
* [[Mirkwai clan]] | * [[Mirkwai clan]] | ||
* [[Unnoticeable comma]] | |||
[[Category: Mirkwai clan]] | [[Category: Mirkwai clan]] | ||
[[Category: Index of temperaments]] | |||
[[Category: Unnoticeable comma]] | |||