Canousmic temperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
m Wording; correction; improve categories
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,025 edits
mNo edit summary
Line 14: Line 14:


== Satin ==
== Satin ==
The satin temperament (94&217) uses 11/10 as a generator, three of them gives 4/3, and tempers out both the [[rainy comma]] and the canousma.
=== 5-limit ===
=== 5-limit ===
Subgroup: 2.3.5
Subgroup: 2.3.5


Line 26: Line 27:
{{Val list|legend=1| 94, 217, 528, 745, 1273 }}
{{Val list|legend=1| 94, 217, 528, 745, 1273 }}


[[Badness]]: 2.8530
[[Badness]]: 2.853049


=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 40: Line 40:
{{Val list|legend=1| 94, 217, 311, 839, 1150 }}
{{Val list|legend=1| 94, 217, 311, 839, 1150 }}


[[Badness]]: 0.1972
[[Badness]]: 0.197207


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 54: Line 53:
Vals: {{Val list| 94, 217, 311 }}
Vals: {{Val list| 94, 217, 311 }}


Badness: 0.0580
Badness: 0.057972


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 68: Line 66:
Vals: {{Val list| 94, 217, 311, 839e, 1150e }}
Vals: {{Val list| 94, 217, 311, 839e, 1150e }}


Badness: 0.0303
Badness: 0.030316


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 82: Line 79:
Vals: {{Val list| 94, 217, 311, 839e, 1150eg }}
Vals: {{Val list| 94, 217, 311, 839e, 1150eg }}


Badness: 0.0200
Badness: 0.020007


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 96: Line 92:
Vals: {{Val list| 94, 217, 311, 839e, 1150eg }}
Vals: {{Val list| 94, 217, 311, 839e, 1150eg }}


Badness: 0.0145
Badness: 0.014479


=== 23-limit ===
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19.23


Line 110: Line 105:
Vals: {{Val list| 94, 217, 311, 839ei, 1150egi }}
Vals: {{Val list| 94, 217, 311, 839ei, 1150egi }}


Badness: 0.0122
Badness: 0.012158


== Superlimmal ==
== Superlimmal ==
The superlimmal temperament uses an ever slightly sharpened [[27/25|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.  
The superlimmal temperament uses an ever slightly sharpened [[27/25|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
Subgroup: 2.3.5.7
Line 127: Line 122:
{{Val list|legend=1| 80, 231, 311, 1324b, 1635b }}
{{Val list|legend=1| 80, 231, 311, 1324b, 1635b }}


[[Badness]]: 0.2524
[[Badness]]: 0.252387


=== 11-limit ===
=== 11-limit ===
Line 140: Line 135:
Vals: {{val list| 80, 231, 311, 1013e, 1324be }}
Vals: {{val list| 80, 231, 311, 1013e, 1324be }}


Badness: 0.0607
Badness: 0.060667


=== 13-limit ===
=== 13-limit ===
Line 153: Line 148:
Vals: {{val list| 80, 231, 311, 702, 1013e }}
Vals: {{val list| 80, 231, 311, 702, 1013e }}


Badness: 0.0390
Badness: 0.039017


=== 17-limit ===
=== 17-limit ===
Line 166: Line 161:
Vals: {{val list| 80, 231, 311 }}
Vals: {{val list| 80, 231, 311 }}


Badness: 0.0301
Badness: 0.030077


=== 19-limit ===
=== 19-limit ===
Line 179: Line 174:
Vals: {{val list| 80, 231, 311 }}
Vals: {{val list| 80, 231, 311 }}


Badness: 0.0205
Badness: 0.020460


=== 23-limit ===
=== 23-limit ===
Line 192: Line 187:
Vals: {{val list| 80, 231, 311 }}
Vals: {{val list| 80, 231, 311 }}


Badness: 0.0161
Badness: 0.016146


=== 29-limit ===
=== 29-limit ===
Line 205: Line 200:
Vals: {{val list| 80, 231, 311 }}
Vals: {{val list| 80, 231, 311 }}


Badness: 0.0131
Badness: 0.013054


== Semiluna ==
== Semiluna ==
Line 220: Line 215:
{{Val list|legend=1| 56d, 118, 292, 410 }}
{{Val list|legend=1| 56d, 118, 292, 410 }}


[[Badness]]: 0.1922
[[Badness]]: 0.192167


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 234: Line 228:
Vals: {{Val list| 56d, 118, 292, 410 }}
Vals: {{Val list| 56d, 118, 292, 410 }}


Badness: 0.0678
Badness: 0.067783


[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]

Revision as of 12:59, 26 June 2021

This is a collection of rank-2 temperaments that temper out the canousma, 4802000/4782969 = [4 -14 3 4⟩. For the rank-3 temperament, see Canou family.

Temperaments discussed elsewhere are:

Considered below are satin, superlimmal and semiluna.

Satin

The satin temperament (94&217) uses 11/10 as a generator, three of them gives 4/3, and tempers out both the rainy comma and the canousma.

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: 2.853049

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: 0.197207

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: 0.057972

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: 0.030316

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: 0.020007

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: 0.014479

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: 0.012158

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: 0.252387

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: 0.060667

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: 0.039017

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: 0.030077

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: 0.020460

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: 0.016146

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: 0.013054

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: 0.192167

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: 0.067783

Retrieved from "https://en.xen.wiki/index.php?title=Canousmic_temperaments&oldid=73208"