Canou family: Difference between revisions

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-The '''canou family''' of rank-3 temperaments tempers out the [[canousma]], 4802000/4782969 = {{monzo|4 -14 3 4}}, a 7-limit comma measuring about 6.9 cents.
+The '''canou family''' of rank-3 temperaments tempers out the [[canousma]], 4802000/4782969 = {{monzo| 4 -14 3 4 }}, a 7-limit comma measuring about 6.9 cents.
 == Canou ==
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 [[Mapping]]: [{{val| 1 0 0 -1 }}, {{val| 0 1 2 2 }}, {{val| 0 0 -4 3 }}]
-{{Multival|legend=1|rank=3| 4 -3 -14 -4 }}
+Lattice basis:
+: 3/2 length = 0.8110, 81/70 length = 0.5135
+: Angle (3/2, 81/70) = 73.88 deg
 [[POTE generator]]s: ~3/2 = 702.3728, ~81/70 = 254.6253
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 * [[9-odd-limit]]: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just
 : [[Eigenmonzo]]s: 2, 7/5
-Lattice basis:
-: 3/2 length = 0.8110, 81/70 length = 0.5135
-: Angle (3/2, 81/70) = 73.88 deg
 {{Val list|legend=1| 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b }}
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 [[Complexity spectrum]]: 4/3, 9/7, 9/8, 7/6, 6/5, 10/9, 5/4, 8/7, 7/5
-=== Extensions ===
+=== Overview to extensions ===
 Canou has a neat extension to the 2.3.5.7.17.19 subgroup with virtually no additional errors. The [[comma basis]] is {1216/1215, 1225/1224, 1445/1444}. Otherwise, 11- and 13-limit extensions are somewhat less ideal.
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 Badness: 2.555 × 10<sup>-3</sup>
+== 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.
+Subgroup: 2.3.5.7
+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 }}]
+POTE generators: ~3/2 = 703.7418, ~64/55 = 254.6133
+{{Val list|legend=1| 75e, 80, 99e, 179e }}
+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
+Vals: {{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.
+Subgroup: 2.3.5.7.11.13
+Comma list: 352/351, 364/363, 472392/471625
+Mapping: [{{val| 1 0 0 -1 6 11 }}, {{val| 0 1 2 2 -2 -5 }}, {{val| 0 0 4 -3 -3 -3 }}]
+POTE generators: ~3/2 = 703.8695, ~64/55 = 254.6321
+Vals: {{Val list| 75e, 80, 99ef, 179ef }}
+Badness: 4.781 × 10<sup>-3</sup>
 == Semicanou ==
 Semicanou adds 9801/9800, the kalisma, to the comma list, and may be described as 80 & 94 & 118. It splits the octave into two equal parts, each representing ~99/70. Note that 99/70 = (81/70)(11/9), this extension is more than natural.
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 Badness: 3.511 × 10<sup>-3</sup>
-== 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.
-Subgroup: 2.3.5.7
-Comma list: 896/891, 472392/471625
-POTE generators: ~3/2 = 703.7418, ~64/55 = 254.6133
-Mapping: [{{val| 1 0 0 -1 6 }}, {{val| 0 1 2 2 -2 }}, {{val| 0 0 4 -3 -3 }}]
-{{Val list|legend=1| 75e, 80, 99e, 179e }}
-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
-POTE generators: ~3/2 = 703.8423, ~15/13 = 254.3605
-Mapping: [<1 0 0 -1 6 0|, <0 1 2 2 -2 3|, <0 0 4 -3 -3 5|]
-Vals: {{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.
-Subgroup: 2.3.5.7.11.13
-Comma list: 352/351, 364/363, 472392/471625
-POTE generators: ~3/2 = 703.8695, ~64/55 = 254.6321
-Mapping: [{{val| 1 0 0 -1 6 11 }}, {{val| 0 1 2 2 -2 -5 }}, {{val| 0 0 4 -3 -3 -3 }}]
-Vals: {{Val list| 75e, 80, 99ef, 179ef }}
-Badness: 4.781 × 10<sup>-3</sup>
 [[Category:Temperament families]]
 [[Category:Canou family| ]] <!-- main article -->
 [[Category:Rank 3]]