31st-octave temperaments: Difference between revisions

Line 130:

== Gallium ==

The name of gallium temperament comes from the 31st element. Gallium preserves the 11-limit mapping of ~~31edo~~, while ~~keeping~~ 13, 17, 19 as independent ~~generators. In what way is~~ this ~~useful is unexplained~~.

The name of gallium temperament comes from the 31st element. Gallium preserves the 11-limit mapping of 31et, while adding 13, 17, and 19 on an independent generator chain, and this considerably improves the qualities of 13-limit and beyond.

Subgroup: 2.3.5.7.11.13

[[Subgroup]]: 2.3.5.7.11.13

[[Comma list]]: 81/80, 99/98, 121/120, 126/125

~~[[Mapping]]: [~~{{~~val~~| 31 49 72 87 107 115 ~~}}, {{val~~| 0 0 0 0 0 -1 }}]

[[~~POTE generator~~]]: ~~~16807~~/~~16640~~ = 15.~~541~~

[[Optimal tuning]] ([[CTE]]): ~45/44 = 1\31, ~13/8 = 840.5276 (~144/143 = 11.0853)

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Comma list: 81/80, 99/98, 121/120, 126/125, 273/272

Mapping: [{{~~val~~| 31 49 72 87 107 115 127 ~~}}, {{val~~| 0 0 0 0 0 -1 -1 }}]

Mapping: {{mapping| 31 49 72 87 107 115 127 | 0 0 0 0 0 -1 -1 }}

~~POTE generator~~: ~~~121~~/~~119~~ = 15.~~785~~

Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.4879 (~144/143 = 11.1250)

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Comma list: 81/80, 99/98, 121/120, 126/125, 153/152, 273/272

Mapping: [{{~~val~~| 31 49 72 87 107 115 127 132 ~~}}, {{val~~| 0 0 0 0 0 -1 -1 -1 }}]

Mapping: {{mapping| 31 49 72 87 107 115 127 132 | 0 0 0 0 0 -1 -1 -1 }}

~~POTE generator~~: ~77/76 = 16.~~206~~

Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.1820 (~144/143 = 11.4309)

Badness: 0.019963

@@ Line 130: / Line 130: @@
 == Gallium ==
-The name of gallium temperament comes from the 31st element. Gallium preserves the 11-limit mapping of 31edo, while keeping 13, 17, 19 as independent generators. In what way is this useful is unexplained.
+The name of gallium temperament comes from the 31st element. Gallium preserves the 11-limit mapping of 31et, while adding 13, 17, and 19 on an independent generator chain, and this considerably improves the qualities of 13-limit and beyond.
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 81/80, 99/98, 121/120, 126/125
-[[Mapping]]: [{{val| 31 49 72 87 107 115 }}, {{val| 0 0 0 0 0 -1 }}]
+{{Mapping|legend=1| 31 49 72 87 107 115 | 0 0 0 0 0 -1 }}
-[[POTE generator]]: ~16807/16640 = 15.541
+[[Optimal tuning]] ([[CTE]]): ~45/44 = 1\31, ~13/8 = 840.5276 (~144/143 = 11.0853)
 {{Optimal ET sequence|legend=1| 31, 62, 93e, 155bef }}
@@ Line 149: / Line 149: @@
 Comma list: 81/80, 99/98, 121/120, 126/125, 273/272
-Mapping: [{{val| 31 49 72 87 107 115 127 }}, {{val| 0 0 0 0 0 -1 -1 }}]
+Mapping: {{mapping| 31 49 72 87 107 115 127 | 0 0 0 0 0 -1 -1 }}
-POTE generator: ~121/119 = 15.785
+Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.4879 (~144/143 = 11.1250)
 {{Optimal ET sequence|legend=1| 31, 62, 93e, 155befg }}
@@ Line 162: / Line 162: @@
 Comma list: 81/80, 99/98, 121/120, 126/125, 153/152, 273/272
-Mapping: [{{val| 31 49 72 87 107 115 127 132 }}, {{val| 0 0 0 0 0 -1 -1 -1 }}]
+Mapping: {{mapping| 31 49 72 87 107 115 127 132 | 0 0 0 0 0 -1 -1 -1 }}
-POTE generator: ~77/76 = 16.206
+Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.1820 (~144/143 = 11.4309)
 {{Optimal ET sequence|legend=1| 31, 62, 155befg }}
 Badness: 0.019963