Canousmic temperaments: Difference between revisions

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+proposed temperament: superlimmal (80 & 311)
m Precision
Line 54: Line 54:
POTE generator: ~11/10 = 165.915
POTE generator: ~11/10 = 165.915


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


Badness: 0.0580
Badness: 0.0580
Line 68: Line 68:
POTE generator: ~11/10 = 165.914
POTE generator: ~11/10 = 165.914


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


Badness: 0.0303
Badness: 0.0303
Line 82: Line 82:
POTE generator: ~11/10 = 165.913
POTE generator: ~11/10 = 165.913


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


Badness: 0.0200
Badness: 0.0200
Line 96: Line 96:
POTE generator: ~11/10 = 165.913
POTE generator: ~11/10 = 165.913


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


Badness: 0.0145
Badness: 0.0145
Line 110: Line 110:
POTE generator: ~11/10 = 165.914
POTE generator: ~11/10 = 165.914


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


Badness: 0.0122
Badness: 0.0122
Line 142: Line 142:
Vals: {{val list| 80, 231, 311, 1013e, 1324be }}
Vals: {{val list| 80, 231, 311, 1013e, 1324be }}


Badness: 0.06067
Badness: 0.0607


=== 13-limit ===
=== 13-limit ===
Line 155: Line 155:
Vals: {{val list| 80, 231, 311, 702, 1013e }}
Vals: {{val list| 80, 231, 311, 702, 1013e }}


Badness: 0.03902
Badness: 0.0390


=== 17-limit ===
=== 17-limit ===
Line 168: Line 168:
Vals: {{val list| 80, 231, 311 }}
Vals: {{val list| 80, 231, 311 }}


Badness: 0.03008
Badness: 0.0301


=== 19-limit ===
=== 19-limit ===
Line 181: Line 181:
Vals: {{val list| 80, 231, 311 }}
Vals: {{val list| 80, 231, 311 }}


Badness: 0.02046
Badness: 0.0205


=== 23-limit ===
=== 23-limit ===
Line 194: Line 194:
Vals: {{val list| 80, 231, 311 }}
Vals: {{val list| 80, 231, 311 }}


Badness: 0.01615
Badness: 0.0161


=== 29-limit ===
=== 29-limit ===
Line 207: Line 207:
Vals: {{val list| 80, 231, 311 }}
Vals: {{val list| 80, 231, 311 }}


Badness: 0.01305
Badness: 0.0131


== Semiluna ==
== Semiluna ==
Line 234: Line 234:
POTE generator: ~121/108 = 193.1732
POTE generator: ~121/108 = 193.1732


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


Badness: 0.0678
Badness: 0.0678

Revision as of 06:51, 12 June 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:

Discussed below are satin, superlimmal 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: Template:Val list

Badness: 0.0122

Superlimmal

The superlimmal temperament uses an ever slightly sharpened large limma as the generator, nine exceed the octave by 126/125. It can be described as 80 & 311. It gets all the primes up to 29 reasonably covered, but still acceptible just as a 13-limit microtemperament, judging from its comma basis. The 80-tone MOS scale is presumably the place to start, and a 151-tone MOS is possible.

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 52734375/52706752

Mapping: [⟨1 8 12 18], ⟨0 -57 -86 -135]]

Wedgie: ⟨⟨ 57 86 135 3 53 72 ]]

POTE generator: ~27/25 = 135.0464

Template:Val list

Badness: 0.2524

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 1479016/1476225

Mapping: [⟨1 8 12 18 11], ⟨0 -57 -86 -135 -67]]

POTE generator: ~27/25 = 135.0455

Vals: Template:Val list

Badness: 0.0607

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455

Mapping: [⟨1 8 12 18 11 1], ⟨0 -57 -86 -135 -67 24]]

POTE generator: ~27/25 = 135.0446

Vals: Template:Val list

Badness: 0.0390

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224

Mapping: [⟨1 8 12 18 11 1 6], ⟨0 -57 -86 -135 -67 24 -17]]

POTE generator: ~27/25 = 135.0462

Vals: Template:Val list

Badness: 0.0301

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499

Mapping: [⟨1 8 12 18 11 1 6 11], ⟨0 -57 -86 -135 -67 24 -17 -60]]

POTE generator: ~27/25 = 135.0464

Vals: Template:Val list

Badness: 0.0205

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495

Mapping: [⟨1 8 12 18 11 1 6 11 7], ⟨0 -57 -86 -135 -67 24 -17 -60 -22]]

POTE generator: ~27/25 = 135.0458

Vals: Template:Val list

Badness: 0.0161

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495

Mapping: [⟨1 8 12 18 11 1 6 11 7 16], ⟨0 -57 -86 -135 -67 24 -17 -60 -22 -99]]

POTE generator: ~27/25 = 135.0460

Vals: Template:Val list

Badness: 0.0131

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

Vals: Template:Val list

Badness: 0.0678

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