Canousmic temperaments: Difference between revisions

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= Satin =
= Satin =
Comma: {{monzo|104 -70 3}}
== 5-limit ==
[[Comma list]]: {{monzo|104 -70 3}}


[[POTE_tuning|POTE generator]]: ~{{monzo|-34 23 -1}} = 165.907
[[POTE generator]]: ~{{monzo|-34 23 -1}} = 165.907


Map: [<1 2 12|, <0 -3 -70|]
[[Mapping]]: [{{val| 1 2 12 }}, {{val| 0 -3 -70 }}]


EDOs: {{EDOs|94, 217, 528, 745, 1273}}
{{Val list|legend=1| 94, 217, 528, 745, 1273 }}


Badness: 2.8530
[[Badness]]: 2.8530


==7-limit==
== 7-limit ==
Commas: 2100875/2097152, 4802000/4782969
Comma list: 2100875/2097152, 4802000/4782969


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


Map: [<1 2 12 -3|, <0 -3 -70 42|]
Mapping: [{{val| 1 2 12 -3 }}, {{val| 0 -3 -70 42 }}]


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


Badness: 0.1972
Badness: 0.1972


==11-limit==
== 11-limit ==
Commas: 4000/3993, 19712/19683, 41503/41472
Comma list: 4000/3993, 19712/19683, 41503/41472


POTE generator: ~11/10 = 165.915
POTE generator: ~11/10 = 165.915


Map: [<1 2 12 -3 13|, <0 -3 -70 42 -69|]
Mapping: [{{val| 1 2 12 -3 13 }}, {{val| 0 -3 -70 42 -69 }}]


EDOs: {{EDOs|94, 217, 311}}
{{Val list|legend=1| 94, 217, 311 }}


Badness: 0.0580
Badness: 0.0580


==13-limit==
== 13-limit ==
Commas: 1575/1573, 2080/2079, 4096/4095, 13720/13689
Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689


POTE generator: ~11/10 = 165.914
POTE generator: ~11/10 = 165.914


Map: [<1 2 12 -3 13 -1|, <0 -3 -70 42 -69 34|]
Mapping: [{{val| 1 2 12 -3 13 -1 }}, {{val| 0 -3 -70 42 -69 34 }}]


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


Badness: 0.0303
Badness: 0.0303


==17-limit==
== 17-limit ==
Commas: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095
Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095


POTE generator: ~11/10 = 165.913
POTE generator: ~11/10 = 165.913


Map: [<1 2 12 -3 13 -1 11|, <0 -3 -70 42 -69 34 -50|]
Mapping: [{{val| 1 2 12 -3 13 -1 11 }}, {{val| 0 -3 -70 42 -69 34 -50 }}]


EDOs: {{EDOs|94, 217, 311, 839e, 1150eg}}
{{Val list|legend=1| 94, 217, 311, 839e, 1150eg }}


Badness: 0.0200
Badness: 0.0200


==19-limit==
== 19-limit ==
Commas: 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


POTE generator: ~11/10 = 165.913
POTE generator: ~11/10 = 165.913


Map: [<1 2 12 -3 13 -1 11 16|, <0 -3 -70 42 -69 34 -50 -85|]
Mapping: [{{val| 1 2 12 -3 13 -1 11 16 }}, {{val| 0 -3 -70 42 -69 34 -50 -85 }}]


EDOs: {{EDOs|94, 217, 311, 839e, 1150eg}}
{{Val list|legend=1| 94, 217, 311, 839e, 1150eg }}


Badness: 0.0145
Badness: 0.0145


==23-limit==
== 23-limit ==
Commas: 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


POTE generator: ~11/10 = 165.914
POTE generator: ~11/10 = 165.914


Map: [<1 2 12 -3 13 -1 11 16 16|, <0 -3 -70 42 -69 34 -50 -85 -83|]
Mapping: [{{val| 1 2 12 -3 13 -1 11 16 16 }}, {{val| 0 -3 -70 42 -69 34 -50 -85 -83 }}]


EDOs: {{EDOs|94, 217, 311, 839ei, 1150egi}}
{{Val list|legend=1| 94, 217, 311, 839ei, 1150egi }}


Badness: 0.0122
Badness: 0.0122
Line 92: Line 93:
{{see also|Luna family #Semiluna}}
{{see also|Luna family #Semiluna}}


Commas: 4802000/4782969, 95703125/95551488
[[Comma list]]: 4802000/4782969, 95703125/95551488


POTE generator: ~2187/1960 = 193.1725
[[POTE generator]]: ~2187/1960 = 193.1725


Map: [<2 8 4 23|, <0 -15 2 -54|]
[[Mapping]]: [{{val| 2 8 4 23 }}, {{val| 0 -15 2 -54 }}]


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


Badness: 0.1922
[[Badness]]: 0.1922


== 11-limit ==
== 11-limit ==
Commas: 5632/5625, 9801/9800, 14641/14580
Comma list: 5632/5625, 9801/9800, 14641/14580


POTE generator: ~121/108 = 193.1732
POTE generator: ~121/108 = 193.1732


Map: [<2 8 4 23 14|, <0 -15 2 -54 -22|]
Mapping: [{{val| 2 8 4 23 14 }}, {{val| 0 -15 2 -54 -22 }}]


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


Badness: 0.0678
Badness: 0.0678


== 13-limit ==
== 13-limit ==
Commas: 352/351, 625/624, 9801/9800, 14641/14580
Comma list: 352/351, 625/624, 9801/9800, 14641/14580


POTE generator: ~121/108 = 193.1550
POTE generator: ~121/108 = 193.1550


Map: [<2 8 4 23 14 0|, <0 -15 2 -54 -22 23|]
Mapping: [{{val| 2 8 4 23 14 0 }}, {{val| 0 -15 2 -54 -22 23 }}]


EDOs: {{EDOs|56d, 118, 174d, 292}}
{{Val list|legend=1| 56d, 118, 174d, 292 }}


Badness: 0.0620
Badness: 0.0620


[[Category:Theory]]
[[Category:Temperament]]
[[Category:Temperament]]
[[Category:Canou]]
[[Category:Canou]]
[[Category:Rank 2]]

Revision as of 09:38, 20 March 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

Comma list: [104 -70 3⟩

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

Mapping: [⟨1 2 12], ⟨0 -3 -70]]

Template:Val list

Badness: 2.8530

7-limit

Comma list: 2100875/2097152, 4802000/4782969

POTE generator: ~8575/7776 = 165.913

Mapping: [⟨1 2 12 -3], ⟨0 -3 -70 42]]

Template:Val list

Badness: 0.1972

11-limit

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

POTE generator: ~11/10 = 165.915

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

Template:Val list

Badness: 0.0580

13-limit

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

POTE generator: ~11/10 = 165.914

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

Template:Val list

Badness: 0.0303

17-limit

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

POTE generator: ~11/10 = 165.913

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

Template:Val list

Badness: 0.0200

19-limit

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

POTE generator: ~11/10 = 165.913

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

Template:Val list

Badness: 0.0145

23-limit

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

POTE generator: ~11/10 = 165.914

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

Template:Val list

Badness: 0.0122

Semiluna

See also: Luna family § Semiluna

Comma list: 4802000/4782969, 95703125/95551488

POTE generator: ~2187/1960 = 193.1725

Mapping: [⟨2 8 4 23], ⟨0 -15 2 -54]]

Template:Val list

Badness: 0.1922

11-limit

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

POTE generator: ~121/108 = 193.1732

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

Template:Val list

Badness: 0.0678

13-limit

Comma list: 352/351, 625/624, 9801/9800, 14641/14580

POTE generator: ~121/108 = 193.1550

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

Template:Val list

Badness: 0.0620

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