Canousmic temperaments: Difference between revisions

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These are rank-2 temperaments that temper out the [[canousma]], 4802000/4782969 = {{monzo|4 -14 3 4}}. For the rank-3 temperament, see [[Canou family]].  
These are rank-2 temperaments that temper out the [[canousma]], 4802000/4782969 = {{monzo| 4 -14 3 4 }}. For the rank-3 temperament, see [[Canou family]].  


Note that 4802000/4782969 = 2×([[10/9]])<sup>3</sup>/([[9/7]])<sup>4</sup>, these intervals tend to have lower complexity.  
Note that 4802000/4782969 = 2×([[10/9]])<sup>3</sup>/([[9/7]])<sup>4</sup>, these intervals tend to have lower complexity.  
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= Satin =
= Satin =
== 5-limit ==
== 5-limit ==
Subgroup: 2.3.5
[[Comma list]]: {{monzo|104 -70 3}}
[[Comma list]]: {{monzo|104 -70 3}}
[[Mapping]]: [{{val| 1 2 12 }}, {{val| 0 -3 -70 }}]


[[POTE generator]]: ~{{monzo|-34 23 -1}} = 165.907
[[POTE generator]]: ~{{monzo|-34 23 -1}} = 165.907
[[Mapping]]: [{{val| 1 2 12 }}, {{val| 0 -3 -70 }}]


{{Val list|legend=1| 94, 217, 528, 745, 1273 }}
{{Val list|legend=1| 94, 217, 528, 745, 1273 }}
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== 7-limit ==
== 7-limit ==
Comma list: 2100875/2097152, 4802000/4782969


POTE generator: ~8575/7776 = 165.913
Subgroup: 2.3.5.7


Mapping: [{{val| 1 2 12 -3 }}, {{val| 0 -3 -70 42 }}]
[[Comma list]]: 2100875/2097152, 4802000/4782969
 
[[Mapping]]: [{{val| 1 2 12 -3 }}, {{val| 0 -3 -70 42 }}]
 
[[POTE generator]]: ~8575/7776 = 165.913


{{Val list|legend=1| 94, 217, 311, 839, 1150 }}
{{Val list|legend=1| 94, 217, 311, 839, 1150 }}


Badness: 0.1972
[[Badness]]: 0.1972


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 4000/3993, 19712/19683, 41503/41472
Comma list: 4000/3993, 19712/19683, 41503/41472
Mapping: [{{val| 1 2 12 -3 13 }}, {{val| 0 -3 -70 42 -69 }}]


POTE generator: ~11/10 = 165.915
POTE generator: ~11/10 = 165.915
Mapping: [{{val| 1 2 12 -3 13 }}, {{val| 0 -3 -70 42 -69 }}]


{{Val list|legend=1| 94, 217, 311 }}
{{Val list|legend=1| 94, 217, 311 }}
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== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689
Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689
Mapping: [{{val| 1 2 12 -3 13 -1 }}, {{val| 0 -3 -70 42 -69 34 }}]


POTE generator: ~11/10 = 165.914
POTE generator: ~11/10 = 165.914
Mapping: [{{val| 1 2 12 -3 13 -1 }}, {{val| 0 -3 -70 42 -69 34 }}]


{{Val list|legend=1| 94, 217, 311, 839e, 1150e }}
{{Val list|legend=1| 94, 217, 311, 839e, 1150e }}
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== 17-limit ==
== 17-limit ==
Subgroup: 2.3.5.7.11.13.17
Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095
Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095
Mapping: [{{val| 1 2 12 -3 13 -1 11 }}, {{val| 0 -3 -70 42 -69 34 -50 }}]


POTE generator: ~11/10 = 165.913
POTE generator: ~11/10 = 165.913
Mapping: [{{val| 1 2 12 -3 13 -1 11 }}, {{val| 0 -3 -70 42 -69 34 -50 }}]


{{Val list|legend=1| 94, 217, 311, 839e, 1150eg }}
{{Val list|legend=1| 94, 217, 311, 839e, 1150eg }}
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== 19-limit ==
== 19-limit ==
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573
Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573
Mapping: [{{val| 1 2 12 -3 13 -1 11 16 }}, {{val| 0 -3 -70 42 -69 34 -50 -85 }}]


POTE generator: ~11/10 = 165.913
POTE generator: ~11/10 = 165.913
Mapping: [{{val| 1 2 12 -3 13 -1 11 16 }}, {{val| 0 -3 -70 42 -69 34 -50 -85 }}]


{{Val list|legend=1| 94, 217, 311, 839e, 1150eg }}
{{Val list|legend=1| 94, 217, 311, 839e, 1150eg }}
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== 23-limit ==
== 23-limit ==
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155
Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155
Mapping: [{{val| 1 2 12 -3 13 -1 11 16 16 }}, {{val| 0 -3 -70 42 -69 34 -50 -85 -83 }}]


POTE generator: ~11/10 = 165.914
POTE generator: ~11/10 = 165.914
Mapping: [{{val| 1 2 12 -3 13 -1 11 16 16 }}, {{val| 0 -3 -70 42 -69 34 -50 -85 -83 }}]


{{Val list|legend=1| 94, 217, 311, 839ei, 1150egi }}
{{Val list|legend=1| 94, 217, 311, 839ei, 1150egi }}
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= Semiluna =
= Semiluna =
{{see also|Luna family #Semiluna}}
{{see also|Luna family #Semiluna}}
Subgroup: 2.3.5.7


[[Comma list]]: 4802000/4782969, 95703125/95551488
[[Comma list]]: 4802000/4782969, 95703125/95551488
[[Mapping]]: [{{val| 2 8 4 23 }}, {{val| 0 -15 2 -54 }}]


[[POTE generator]]: ~2187/1960 = 193.1725
[[POTE generator]]: ~2187/1960 = 193.1725
[[Mapping]]: [{{val| 2 8 4 23 }}, {{val| 0 -15 2 -54 }}]


{{Val list|legend=1| 56d, 118, 292, 410 }}
{{Val list|legend=1| 56d, 118, 292, 410 }}
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== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 5632/5625, 9801/9800, 14641/14580
Comma list: 5632/5625, 9801/9800, 14641/14580
Mapping: [{{val| 2 8 4 23 14 }}, {{val| 0 -15 2 -54 -22 }}]


POTE generator: ~121/108 = 193.1732
POTE generator: ~121/108 = 193.1732
Mapping: [{{val| 2 8 4 23 14 }}, {{val| 0 -15 2 -54 -22 }}]


{{Val list|legend=1| 56d, 118, 292, 410 }}
{{Val list|legend=1| 56d, 118, 292, 410 }}

Revision as of 05:04, 6 April 2021

These are rank-2 temperaments that temper out the canousma, 4802000/4782969 = [4 -14 3 4⟩. For the rank-3 temperament, see Canou family.

Note that 4802000/4782969 = 2×(10/9)3/(9/7)4, these intervals tend to have lower complexity.

Temperaments not dicussed here include godzilla (a trivial case) and these:

Discussed below are satin and semiluna.

Satin

5-limit

Subgroup: 2.3.5

Comma list: [104 -70 3⟩

Mapping: [⟨1 2 12], ⟨0 -3 -70]]

POTE generator: ~[-34 23 -1⟩ = 165.907

Template:Val list

Badness: 2.8530

7-limit

Subgroup: 2.3.5.7

Comma list: 2100875/2097152, 4802000/4782969

Mapping: [⟨1 2 12 -3], ⟨0 -3 -70 42]]

POTE generator: ~8575/7776 = 165.913

Template:Val list

Badness: 0.1972

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4000/3993, 19712/19683, 41503/41472

Mapping: [⟨1 2 12 -3 13], ⟨0 -3 -70 42 -69]]

POTE generator: ~11/10 = 165.915

Template:Val list

Badness: 0.0580

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689

Mapping: [⟨1 2 12 -3 13 -1], ⟨0 -3 -70 42 -69 34]]

POTE generator: ~11/10 = 165.914

Template:Val list

Badness: 0.0303

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095

Mapping: [⟨1 2 12 -3 13 -1 11], ⟨0 -3 -70 42 -69 34 -50]]

POTE generator: ~11/10 = 165.913

Template:Val list

Badness: 0.0200

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573

Mapping: [⟨1 2 12 -3 13 -1 11 16], ⟨0 -3 -70 42 -69 34 -50 -85]]

POTE generator: ~11/10 = 165.913

Template:Val list

Badness: 0.0145

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155

Mapping: [⟨1 2 12 -3 13 -1 11 16 16], ⟨0 -3 -70 42 -69 34 -50 -85 -83]]

POTE generator: ~11/10 = 165.914

Template:Val list

Badness: 0.0122

Semiluna

See also: Luna family § Semiluna

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 95703125/95551488

Mapping: [⟨2 8 4 23], ⟨0 -15 2 -54]]

POTE generator: ~2187/1960 = 193.1725

Template:Val list

Badness: 0.1922

11-limit

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 9801/9800, 14641/14580

Mapping: [⟨2 8 4 23 14], ⟨0 -15 2 -54 -22]]

POTE generator: ~121/108 = 193.1732

Template:Val list

Badness: 0.0678

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