Canou family: Difference between revisions

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== 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.

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.

The other comma necessary to define it is 14641/14580, the [[semicanousma]], which is the difference between [[121/120]] and [[243/242]]. By flattening the 11th harmonic by one cent, it identifies [[20/11]] by three [[11/9]]'s stacked, so an octave can be divided into 11/9-11/9-11/9-11/10.

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Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 14641/14580

[[Comma list]]: 9801/9800, 14641/14580

POTE ~~generators~~: ~3/2 = 702.3850, ~81/70 = 254.6168 or ~11/9 = 345.3832

[[POTE generator]]s: ~3/2 = 702.3850, ~81/70 = 254.6168 or ~11/9 = 345.3832

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

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

Badness: 2.197 × 10-3

[[Badness]]: 2.197 × 10-3

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 14641/14580

Mapping: [{{val| 2 0 0 -2 1 -11 }}, {{val| 0 1 2 2 2 5 }}, {{val| 0 0 -4 3 -1 6 }}]

POTE generators: ~3/2 = 702.5046, ~81/70 = 254.6501 or ~11/9 = 345.3499

Vals: {{Val list| 80f, 94, 118f, 198, 410 }}

Badness: 2.974 × 10-3

=== Semicanoumint ===

This extension was named ''semicanou'' in the earlier materials. It adds [[352/351]], the minthma, to the comma list~~. It is a natural extension to~~ the ~~13-limit~~.

This extension was named ''semicanou'' in the earlier materials. It adds [[352/351]], the minthma, to the comma list, so that the flat ~11/9 simultaneously represents ~39/32.

Subgroup: 2.3.5.7.11.13

@@ Line 69: / Line 69: @@
 == 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.
+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.
 The other comma necessary to define it is 14641/14580, the [[semicanousma]], which is the difference between [[121/120]] and [[243/242]]. By flattening the 11th harmonic by one cent, it identifies [[20/11]] by three [[11/9]]'s stacked, so an octave can be divided into 11/9-11/9-11/9-11/10.
@@ Line 77: / Line 77: @@
 Subgroup: 2.3.5.7.11
-Comma list: 9801/9800, 14641/14580
+[[Comma list]]: 9801/9800, 14641/14580
-POTE generators: ~3/2 = 702.3850, ~81/70 = 254.6168 or ~11/9 = 345.3832
+[[POTE generator]]s: ~3/2 = 702.3850, ~81/70 = 254.6168 or ~11/9 = 345.3832
-Mapping: [{{val| 2 0 0 -2 1 }}, {{val| 0 1 2 2 2 }}, {{val| 0 0 4 -3 1 }}]
+[[Mapping]]: [{{val| 2 0 0 -2 1 }}, {{val| 0 1 2 2 2 }}, {{val| 0 0 4 -3 1 }}]
 {{Val list|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}
-Badness: 2.197 × 10<sup>-3</sup>
+[[Badness]]: 2.197 × 10<sup>-3</sup>
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 1716/1715, 2080/2079, 14641/14580
+Mapping: [{{val| 2 0 0 -2 1 -11 }}, {{val| 0 1 2 2 2 5 }}, {{val| 0 0 -4 3 -1 6 }}]
+POTE generators: ~3/2 = 702.5046, ~81/70 = 254.6501 or ~11/9 = 345.3499
+Vals: {{Val list| 80f, 94, 118f, 198, 410 }}
+Badness: 2.974 × 10<sup>-3</sup>
 === Semicanoumint ===
-This extension was named ''semicanou'' in the earlier materials. It adds [[352/351]], the minthma, to the comma list. It is a natural extension to the 13-limit.
+This extension was named ''semicanou'' in the earlier materials. It adds [[352/351]], the minthma, to the comma list, so that the flat ~11/9 simultaneously represents ~39/32.
 Subgroup: 2.3.5.7.11.13