Canou family: Difference between revisions

Line 66:

== Canta ==

By adding [[896/891]], the pentacircle comma, [[33/32]] is equated with 28/27, so the scale is filled with this 33/32~28/27 mixture. This may be described as 75e & 80 & 99e, and 80edo makes the optimal.

By adding [[896/891]], the pentacircle comma, [[33/32]] is equated with 28/27, so the scale is filled with this 33/32~28/27 mixture. This may be described as 75e & 80 & 99e, and 80edo makes the optimal. It has a natural extension to the 13-limit since 896/891 = (352/351)(364/363), named ''gentcanta'' in earlier materials.

Subgroup: 2.3.5.7

Subgroup: 2.3.5.7.11

Comma list: 896/891, 472392/471625

[[Comma list]]: 896/891, 472392/471625

Mapping: [{{val| 1 0 0 -1 6 }}, {{val| 0 1 2 2 -2 }}, {{val| 0 0 4 -3 -3 }}]

[[Mapping]]: [{{val| 1 0 0 -1 6 }}, {{val| 0 1 2 2 -2 }}, {{val| 0 0 4 -3 -3 }}]

POTE ~~generators~~: ~3/2 = 703.7418, ~64/55 = 254.6133

[[POTE generator]]s: ~3/2 = 703.7418, ~64/55 = 254.6133

~~Badness: 4.523 × 10-3~~

[[Badness]]: 4.523 × 10-3

~~=== Cantawolf ===~~

~~This extension was named ''canta'' in the earlier materials. It adds~~ [[~~351/350~~]]~~, the ratwolfsma, to the comma list. Since 351/350 = (81/70)/(15/13). The 81/70-generator simultaneously represents 15/13, adding a lot of fun to the scale. Again 80edo makes the optimal.~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list~~: ~~351/350, 832/825, 13013/12960~~

~~Mapping: [<1 0 0 -1 6 0|, <0 1 2 2 -2 3|, <0 0~~ 4 ~~-3 -3 5|]~~

~~POTE generators: ~3/2 = 703~~.~~8423, ~15/13 = 254.3605~~

~~Optimal GPV sequence: {{val list| 75ef, 80, 99e, 104c, 179e, 184c, 203ce }}~~

~~Badness: 3.470~~ × 10-3

~~=== Cantamint ===~~

This extension was named ''gentcanta'' in the earlier materials. It adds [[352/351]], the minthma, as well as [[364/363]], the gentle comma, to the comma list. It is a natural extension of canta, as 896/891 factors neatly into (352/351)×(364/363). Again 80edo makes the optimal.

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

@@ Line 66: / Line 66: @@
 == Canta ==
-By adding [[896/891]], the pentacircle comma, [[33/32]] is equated with 28/27, so the scale is filled with this 33/32~28/27 mixture. This may be described as 75e & 80 & 99e, and 80edo makes the optimal.
+By adding [[896/891]], the pentacircle comma, [[33/32]] is equated with 28/27, so the scale is filled with this 33/32~28/27 mixture. This may be described as 75e & 80 & 99e, and 80edo makes the optimal. It has a natural extension to the 13-limit since 896/891 = (352/351)(364/363), named ''gentcanta'' in earlier materials.
-Subgroup: 2.3.5.7
+Subgroup: 2.3.5.7.11
-Comma list: 896/891, 472392/471625
+[[Comma list]]: 896/891, 472392/471625
-Mapping: [{{val| 1 0 0 -1 6 }}, {{val| 0 1 2 2 -2 }}, {{val| 0 0 4 -3 -3 }}]
+[[Mapping]]: [{{val| 1 0 0 -1 6 }}, {{val| 0 1 2 2 -2 }}, {{val| 0 0 4 -3 -3 }}]
-POTE generators: ~3/2 = 703.7418, ~64/55 = 254.6133
+[[POTE generator]]s: ~3/2 = 703.7418, ~64/55 = 254.6133
 {{Val list|legend=1| 75e, 80, 99e, 179e }}
-Badness: 4.523 × 10<sup>-3</sup>
+[[Badness]]: 4.523 × 10<sup>-3</sup>
-=== Cantawolf ===
-This extension was named ''canta'' in the earlier materials. It adds [[351/350]], the ratwolfsma, to the comma list. Since 351/350 = (81/70)/(15/13). The 81/70-generator simultaneously represents 15/13, adding a lot of fun to the scale. Again 80edo makes the optimal.
-Subgroup: 2.3.5.7.11
-Comma list: 351/350, 832/825, 13013/12960
-Mapping: [<1 0 0 -1 6 0|, <0 1 2 2 -2 3|, <0 0 4 -3 -3 5|]
-POTE generators: ~3/2 = 703.8423, ~15/13 = 254.3605
-Optimal GPV sequence: {{val list| 75ef, 80, 99e, 104c, 179e, 184c, 203ce }}
-Badness: 3.470 × 10<sup>-3</sup>
-=== Cantamint ===
-This extension was named ''gentcanta'' in the earlier materials. It adds [[352/351]], the minthma, as well as [[364/363]], the gentle comma, to the comma list. It is a natural extension of canta, as 896/891 factors neatly into (352/351)×(364/363). Again 80edo makes the optimal.
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13