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]].  
{{Technical data page}}
This is a collection of 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.
Temperaments discussed elsewhere are:
* [[Godzilla]] (+49/48 or 81/80) → [[Slendro clan #Godzilla|Slendro clan]]
* ''[[Betic]]'' (+225/224) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Pentorwell]]'' (+1728/1715) → [[Orwellismic temperaments #Pentorwell|Orwellismic temperaments]]
* ''[[Amicable]]'' (+2401/2400) → [[Breedsmic temperaments #Amicable|Breedsmic temperaments]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Septiquarter]]'' (+5120/5103) → [[Hemifamity temperaments #Septiquarter|Hemifamity temperaments]]
* ''[[Marthirds]]'' (+15625/15552) → [[Kleismic family #Marthirds|Kleismic family]]
* ''[[Kleischismic]]'' (+32805/32768) → [[Schismatic family #Kleischismic|Schismatic family]]
* ''[[Kaboom]]'' (+65625/65536) → [[Vavoom family #Kaboom|Vavoom family]]
* ''[[Quartiquart]]'' (+390625/388962) → [[Quartonic family #Quartiquart|Quartonic family]]
* ''[[Turkey (temperament)|Turkey]]'' (+5250987/5242880) → [[Vulture family #Turkey|Vulture family]]
* ''[[Hemiquindromeda]]'' (+67108864/66976875) → [[Quindromeda family #Hemiquindromeda|Quindromeda family]]
* ''[[Semiluna]]'' (+95703125/95551488) → [[Luna family #Semiluna|Luna family]]


Temperaments not dicussed here include [[Meantone family #Godzilla|godzilla]] (a trivial case) and these:
Considered below are satin and superlimmal.
* [[Sycamore family #Betic|Betic]]
* [[Amicable]]
* [[Ragismic microtemperaments #Parakleismic|Parakleismic]]
* [[Hemifamity temperaments #Septiquarter|Septiquarter]]
* [[Schismatic family #Kleischismic|Kleischismic]]


Discussed below are satin and semiluna.  
== Satin ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Satin]].''


= Satin =
The satin temperament (94 &amp; 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 ==
[[Comma list]]: {{monzo|104 -70 3}}


[[POTE generator]]: ~{{monzo|-34 23 -1}} = 165.907
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 2 12 }}, {{val| 0 -3 -70 }}]
[[Comma list]]: 2100875/2097152, 4802000/4782969


{{Val list|legend=1| 94, 217, 528, 745, 1273 }}
{{Mapping|legend=1| 1 2 12 -3 | 0 -3 -70 42 }}


[[Badness]]: 2.8530
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8575/7776 = 165.913


== 7-limit ==
{{Optimal ET sequence|legend=1| 94, 217, 311, 839, 1150 }}
Comma list: 2100875/2097152, 4802000/4782969


POTE generator: ~8575/7776 = 165.913
[[Badness]]: 0.197207


Mapping: [{{val| 1 2 12 -3 }}, {{val| 0 -3 -70 42 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 94, 217, 311, 839, 1150 }}
Comma list: 4000/3993, 19712/19683, 41503/41472


Badness: 0.1972
Mapping: {{mapping| 1 2 12 -3 13 | 0 -3 -70 42 -69 }}


== 11-limit ==
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.915
Comma list: 4000/3993, 19712/19683, 41503/41472


POTE generator: ~11/10 = 165.915
{{Optimal ET sequence|legend=1| 94, 217, 311 }}


Mapping: [{{val| 1 2 12 -3 13 }}, {{val| 0 -3 -70 42 -69 }}]
Badness: 0.057972


{{Val list|legend=1| 94, 217, 311 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0580
Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689


== 13-limit ==
Mapping: {{mapping| 1 2 12 -3 13 -1 | 0 -3 -70 42 -69 34 }}
Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689


POTE generator: ~11/10 = 165.914
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.914


Mapping: [{{val| 1 2 12 -3 13 -1 }}, {{val| 0 -3 -70 42 -69 34 }}]
{{Optimal ET sequence|legend=1| 94, 217, 311, 839e, 1150e }}


{{Val list|legend=1| 94, 217, 311, 839e, 1150e }}
Badness: 0.030316


Badness: 0.0303
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== 17-limit ==
Comma list: 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
Mapping: {{mapping| 1 2 12 -3 13 -1 11 | 0 -3 -70 42 -69 34 -50 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.913


Mapping: [{{val| 1 2 12 -3 13 -1 11 }}, {{val| 0 -3 -70 42 -69 34 -50 }}]
{{Optimal ET sequence|legend=1| 94, 217, 311, 839e, 1150eg }}


{{Val list|legend=1| 94, 217, 311, 839e, 1150eg }}
Badness: 0.020007


Badness: 0.0200
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


== 19-limit ==
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


POTE generator: ~11/10 = 165.913
Mapping: {{mapping| 1 2 12 -3 13 -1 11 16 | 0 -3 -70 42 -69 34 -50 -85 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.913


Mapping: [{{val| 1 2 12 -3 13 -1 11 16 }}, {{val| 0 -3 -70 42 -69 34 -50 -85 }}]
{{Optimal ET sequence|legend=1| 94, 217, 311, 839e, 1150eg }}


{{Val list|legend=1| 94, 217, 311, 839e, 1150eg }}
Badness: 0.014479


Badness: 0.0145
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


== 23-limit ==
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


POTE generator: ~11/10 = 165.914
Mapping: {{mapping| 1 2 12 -3 13 -1 11 16 16 | 0 -3 -70 42 -69 34 -50 -85 -83 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.914
 
{{Optimal ET sequence|legend=1| 94, 217, 311, 839ei, 1150egi }}
 
Badness: 0.012158
 
== Superlimmal ==
The superlimmal temperament (80 &amp; 311) uses an ever slightly sharpened [[27/25|large limma]] as the generator, nine exceed the octave by [[126/125]]. It gets all the primes up to 29 reasonably covered, but still acceptable just as a 13-limit microtemperament, judging from its [[comma basis]]. While the [[mos scale]] may not be the most effective approach, the 80-tone mos is presumably the place to start if it is used. It can also be extended to prime 37 by tempering out ([[27/25]])/([[40/37]]) = [[1000/999]], where 40/37 is notably the mediant of [[27/25]] and [[13/12]], which could be interpreted as an explanation of the sharpened limma.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4802000/4782969, 52734375/52706752
 
{{Mapping|legend=1| 1 8 12 18 | 0 -57 -86 -135 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 135.0464
 
{{Optimal ET sequence|legend=1| 80, 231, 311, 1324b, 1635b }}
 
[[Badness]]: 0.252387
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4000/3993, 1479016/1476225
 
Mapping: {{mapping| 1 8 12 18 11 | 0 -57 -86 -135 -67 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0455
 
{{Optimal ET sequence|legend=1| 80, 231, 311, 1013e, 1324be }}
 
Badness: 0.060667
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455
 
Mapping: {{mapping| 1 8 12 18 11 1 | 0 -57 -86 -135 -67 24 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0446
 
{{Optimal ET sequence|legend=1| 80, 231, 311, 702, 1013e }}
 
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: {{mapping| 1 8 12 18 11 1 6 | 0 -57 -86 -135 -67 24 -17 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0462
 
{{Optimal ET sequence|legend=1| 80, 231, 311 }}
 
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: {{mapping| 1 8 12 18 11 1 6 11 | 0 -57 -86 -135 -67 24 -17 -60 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0464
 
{{Optimal ET sequence|legend=1| 80, 231, 311 }}


Mapping: [{{val| 1 2 12 -3 13 -1 11 16 16 }}, {{val| 0 -3 -70 42 -69 34 -50 -85 -83 }}]
Badness: 0.020460


{{Val list|legend=1| 94, 217, 311, 839ei, 1150egi }}
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


Badness: 0.0122
Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495


= Semiluna =
Mapping: {{mapping| 1 8 12 18 11 1 6 11 7 | 0 -57 -86 -135 -67 24 -17 -60 -22 }}
{{see also|Luna family #Semiluna}}


[[Comma list]]: 4802000/4782969, 95703125/95551488
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0458


[[POTE generator]]: ~2187/1960 = 193.1725
{{Optimal ET sequence|legend=1| 80, 231, 311 }}


[[Mapping]]: [{{val| 2 8 4 23 }}, {{val| 0 -15 2 -54 }}]
Badness: 0.016146


{{Val list|legend=1| 56d, 118, 292, 410 }}
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29


[[Badness]]: 0.1922
Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495


== 11-limit ==
Mapping: {{mapping| 1 8 12 18 11 1 6 11 7 16 | 0 -57 -86 -135 -67 24 -17 -60 -22 -99 }}
Comma list: 5632/5625, 9801/9800, 14641/14580


POTE generator: ~121/108 = 193.1732
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0460


Mapping: [{{val| 2 8 4 23 14 }}, {{val| 0 -15 2 -54 -22 }}]
{{Optimal ET sequence|legend=1| 80, 231, 311 }}


{{Val list|legend=1| 56d, 118, 292, 410 }}
Badness: 0.013054


Badness: 0.0678
=== No-31's 37-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.37


== 13-limit ==
Comma list: 595/594, 760/759, 784/783, 925/924, 969/968, 1000/999, 1045/1044, 1105/1104, 1275/1274
Comma list: 352/351, 625/624, 9801/9800, 14641/14580


POTE generator: ~121/108 = 193.1550
Mapping: {{mapping| 1 8 12 18 11 1 6 11 7 16 15 | 0 -57 -86 -135 -67 24 -17 -60 -22 -99 -87 }}


Mapping: [{{val| 2 8 4 23 14 0 }}, {{val| 0 -15 2 -54 -22 23 }}]
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0460


{{Val list|legend=1| 56d, 118, 174d, 292 }}
{{Optimal ET sequence|legend=1| 80, 231, 311 }}


Badness: 0.0620
Badness: 0.010901


[[Category:Theory]]
[[Category:Temperament collections]]
[[Category:Temperament]]
[[Category:Pages with mostly numerical content]]
[[Category:Canou]]
[[Category:Canousmic temperaments| ]] <!-- main article -->
[[Category:Canou| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:32, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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 and superlimmal.

Satin

For the 5-limit version of this temperament, see High badness temperaments #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.

Subgroup: 2.3.5.7

Comma list: 2100875/2097152, 4802000/4782969

Mapping: [⟨1 2 12 -3], ⟨0 -3 -70 42]]

Optimal tuning (POTE): ~2 = 1\1, ~8575/7776 = 165.913

Optimal ET sequence: 94, 217, 311, 839, 1150

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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.915

Optimal ET sequence: 94, 217, 311

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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.914

Optimal ET sequence: 94, 217, 311, 839e, 1150e

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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.913

Optimal ET sequence: 94, 217, 311, 839e, 1150eg

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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.913

Optimal ET sequence: 94, 217, 311, 839e, 1150eg

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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.914

Optimal ET sequence: 94, 217, 311, 839ei, 1150egi

Badness: 0.012158

Superlimmal

The superlimmal temperament (80 & 311) uses an ever slightly sharpened large limma as the generator, nine exceed the octave by 126/125. It gets all the primes up to 29 reasonably covered, but still acceptable just as a 13-limit microtemperament, judging from its comma basis. While the mos scale may not be the most effective approach, the 80-tone mos is presumably the place to start if it is used. It can also be extended to prime 37 by tempering out (27/25)/(40/37) = 1000/999, where 40/37 is notably the mediant of 27/25 and 13/12, which could be interpreted as an explanation of the sharpened limma.

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 52734375/52706752

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

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0464

Optimal ET sequence: 80, 231, 311, 1324b, 1635b

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]]

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0455

Optimal ET sequence: 80, 231, 311, 1013e, 1324be

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]]

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0446

Optimal ET sequence: 80, 231, 311, 702, 1013e

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]]

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0462

Optimal ET sequence: 80, 231, 311

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]]

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0464

Optimal ET sequence: 80, 231, 311

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]]

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0458

Optimal ET sequence: 80, 231, 311

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]]

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0460

Optimal ET sequence: 80, 231, 311

Badness: 0.013054

No-31's 37-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.37

Comma list: 595/594, 760/759, 784/783, 925/924, 969/968, 1000/999, 1045/1044, 1105/1104, 1275/1274

Mapping: [⟨1 8 12 18 11 1 6 11 7 16 15], ⟨0 -57 -86 -135 -67 24 -17 -60 -22 -99 -87]]

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0460

Optimal ET sequence: 80, 231, 311

Badness: 0.010901

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