Canousmic temperaments: Difference between revisions

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-5-limit satin (addressed in high badness temperaments)
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{{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]].  
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]].  


Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* [[Godzilla]], {49/48, 81/80} → [[Meantone family #Godzilla|Meantone family]]
* [[Godzilla]] (+49/48 or 81/80) → [[Slendro clan #Godzilla|Slendro clan]]
* ''[[Betic]]'', {225/224, 1071875/1062882} → [[Sycamore family #Betic|Sycamore family]]
* ''[[Betic]]'' (+225/224) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Pentorwell]]'', {1728/1715, 179200/177147} → [[Orwellismic temperaments #Pentorwell|Orwellismic temperaments]]
* ''[[Pentorwell]]'' (+1728/1715) → [[Orwellismic temperaments #Pentorwell|Orwellismic temperaments]]
* ''[[Amicable]]'', {2401/2400, 1600000/1594323} → [[Breedsmic temperaments #Amicable|Breedsmic temperaments]]
* ''[[Amicable]]'' (+2401/2400) → [[Breedsmic temperaments #Amicable|Breedsmic temperaments]]
* [[Parakleismic]], {3136/3125, 4375/4374} → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Septiquarter]]'', {5120/5103, 420175/419904} → [[Hemifamity temperaments #Septiquarter|Hemifamity temperaments]]
* ''[[Septiquarter]]'' (+5120/5103) → [[Hemifamity temperaments #Septiquarter|Hemifamity temperaments]]
* ''[[Marthirds]]'', {15625/15552, 2460375/2458624} → [[Kleismic family #Marthirds|Kleismic family]]
* ''[[Marthirds]]'' (+15625/15552) → [[Kleismic family #Marthirds|Kleismic family]]
* ''[[Kleischismic]]'', {32805/32768, 1500625/1492992} → [[Schismatic family #Kleischismic|Schismatic family]]
* ''[[Kleischismic]]'' (+32805/32768) → [[Schismatic family #Kleischismic|Schismatic family]]
* ''[[Turkey (temperament)|Turkey]]'', {4802000/4782969, 5250987/5242880} → [[Vulture family #Turkey|Vulture family]]
* ''[[Kaboom]]'' (+65625/65536) → [[Vavoom family #Kaboom|Vavoom family]]
* ''[[Semiluna]]'', {4802000/4782969, 95703125/95551488} → [[Luna family #Semiluna|Luna 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]]


Considered below are satin and superlimmal.  
Considered below are satin and superlimmal.


== 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.
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Satin]].''


Subgroup: 2.3.5.7
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
[[Comma list]]: 2100875/2097152, 4802000/4782969


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


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


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


[[Badness]]: 0.197207
[[Badness]]: 0.197207
Line 35: Line 41:
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 }}]
Mapping: {{mapping| 1 2 12 -3 13 | 0 -3 -70 42 -69 }}


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


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


Badness: 0.057972
Badness: 0.057972
Line 48: Line 54:
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 }}]
Mapping: {{mapping| 1 2 12 -3 13 -1 | 0 -3 -70 42 -69 34 }}


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


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


Badness: 0.030316
Badness: 0.030316
Line 61: Line 67:
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 }}]
Mapping: {{mapping| 1 2 12 -3 13 -1 11 | 0 -3 -70 42 -69 34 -50 }}


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


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


Badness: 0.020007
Badness: 0.020007
Line 74: Line 80:
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 }}]
Mapping: {{mapping| 1 2 12 -3 13 -1 11 16 | 0 -3 -70 42 -69 34 -50 -85 }}


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


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


Badness: 0.014479
Badness: 0.014479
Line 87: Line 93:
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 }}]
Mapping: {{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
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.914


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


Badness: 0.012158
Badness: 0.012158


== Superlimmal ==
== Superlimmal ==
The superlimmal temperament (80 & 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 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 (80 & 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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4802000/4782969, 52734375/52706752
[[Comma list]]: 4802000/4782969, 52734375/52706752


[[Mapping]]: [{{val| 1 8 12 18 }}, {{val| 0 -57 -86 -135 }}]
{{Mapping|legend=1| 1 8 12 18 | 0 -57 -86 -135 }}
 
{{Multival|legend=1| 57 86 135 3 53 72 }}


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


{{Val list|legend=1| 80, 231, 311, 1324b, 1635b }}
{{Optimal ET sequence|legend=1| 80, 231, 311, 1324b, 1635b }}


[[Badness]]: 0.252387
[[Badness]]: 0.252387
Line 117: Line 121:
Comma list: 3025/3024, 4000/3993, 1479016/1476225
Comma list: 3025/3024, 4000/3993, 1479016/1476225


Mapping: [{{val| 1 8 12 18 11 }}, {{val| 0 -57 -86 -135 -67 }}]
Mapping: {{mapping| 1 8 12 18 11 | 0 -57 -86 -135 -67 }}


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


Vals: {{val list| 80, 231, 311, 1013e, 1324be }}
{{Optimal ET sequence|legend=1| 80, 231, 311, 1013e, 1324be }}


Badness: 0.060667
Badness: 0.060667
Line 130: Line 134:
Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455
Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455


Mapping: [{{val| 1 8 12 18 11 1 }}, {{val| 0 -57 -86 -135 -67 24 }}]
Mapping: {{mapping| 1 8 12 18 11 1 | 0 -57 -86 -135 -67 24 }}


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


Vals: {{val list| 80, 231, 311, 702, 1013e }}
{{Optimal ET sequence|legend=1| 80, 231, 311, 702, 1013e }}


Badness: 0.039017
Badness: 0.039017
Line 143: Line 147:
Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224
Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224


Mapping: [{{val| 1 8 12 18 11 1 6 }}, {{val| 0 -57 -86 -135 -67 24 -17 }}]
Mapping: {{mapping| 1 8 12 18 11 1 6 | 0 -57 -86 -135 -67 24 -17 }}


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


Vals: {{val list| 80, 231, 311 }}
{{Optimal ET sequence|legend=1| 80, 231, 311 }}


Badness: 0.030077
Badness: 0.030077
Line 156: Line 160:
Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499
Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499


Mapping: [{{val| 1 8 12 18 11 1 6 11 }}, {{val| 0 -57 -86 -135 -67 24 -17 -60 }}]
Mapping: {{mapping| 1 8 12 18 11 1 6 11 | 0 -57 -86 -135 -67 24 -17 -60 }}


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


Vals: {{val list| 80, 231, 311 }}
{{Optimal ET sequence|legend=1| 80, 231, 311 }}


Badness: 0.020460
Badness: 0.020460
Line 169: Line 173:
Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495
Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495


Mapping: [{{val| 1 8 12 18 11 1 6 11 7 }}, {{val| 0 -57 -86 -135 -67 24 -17 -60 -22 }}]
Mapping: {{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
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0458


Vals: {{val list| 80, 231, 311 }}
{{Optimal ET sequence|legend=1| 80, 231, 311 }}


Badness: 0.016146
Badness: 0.016146
Line 182: Line 186:
Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495
Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495


Mapping: [{{val| 1 8 12 18 11 1 6 11 7 16 }}, {{val| 0 -57 -86 -135 -67 24 -17 -60 -22 -99 }}]
Mapping: {{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
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0460


Vals: {{val list| 80, 231, 311 }}
{{Optimal ET sequence|legend=1| 80, 231, 311 }}


Badness: 0.013054
Badness: 0.013054


[[Category:Regular temperament theory]]
=== No-31's 37-limit ===
[[Category:Temperament collection]]
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: {{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|legend=1| 80, 231, 311 }}
 
Badness: 0.010901
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Canousmic temperaments| ]] <!-- main article -->
[[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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