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]] ({{monzo|legend=1| 4 -14 3 4 }}, [[ratio]]: 4802000/4782969). 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) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
* ''[[Satin]]'' (+2100875/2097152) → [[Garischismic clan #Satin|Garischismic clan]]
* ''[[Kleischismic]]'' (+32805/32768) → [[Schismatic family #Kleischismic|Schismatic family]]
* ''[[Pentaorwell]]'' (+1728/1715) → [[Orwellismic temperaments #Pentaorwell|Orwellismic temperaments]]
* ''[[Septiquarter]]'' (+5120/5103) → [[Hemifamity temperaments #Septiquarter|Hemifamity temperaments]]
* ''[[Hemiquindromeda]]'' (+67108864/66976875) → [[Quindromeda family #Hemiquindromeda|Quindromeda family]]
* ''[[Betic]]'' (+225/224) → [[Sycamore family #Betic|Sycamore family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Turkey (temperament)|Turkey]]'' (+5250987/5242880) → [[Vulture family #Turkey|Vulture family]]
* ''[[Kaboom]]'' (+65625/65536) → [[Vavoom family #Kaboom|Vavoom family]]
* ''[[Amicable]]'' (+2401/2400) → [[Amity family #Amicable|Amity family]]
* ''[[Marthirds]]'' (+15625/15552) → [[Kleismic family #Marthirds|Kleismic family]]
* ''[[Semiluna]]'' (+95703125/95551488) → [[Luna family #Semiluna|Luna family]]
* ''[[Quartiquart]]'' (+390625/388962) → [[Quartonic family #Quartiquart|Quartonic family]]


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


Discussed below are satin and semiluna.  
== Superlimmal ==
Superlimmal is essentially an 80-form, and may be described as the {{nowrap| 80 & 311 }} temperament. It uses an ever slightly sharpened [[27/25|large limma]] as the generator, nine exceed the octave by [[126/125]]. Note that in the data that follow, the generator is its [[octave complement]], [[~]][[50/27]], so that 57 of them [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]].  


= Satin =
Superlimmal gets all the primes up to [[29/1|29]] reasonably covered, but is acceptable just as a 13-limit microtemperament, given a relatively simple [[comma basis]]. It can also be extended to include prime [[37/1|37]] by mapping it to 87 generator steps, tempering out ([[27/25]])/([[40/37]]) = [[1000/999]]. Since 40/37 is the mediant of [[27/25]] and [[13/12]], this extension further consolidates the sharpened limma.
Comma: {{monzo|104 -70 3}}


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


Map: [&lt;1 2 12|, &lt;0 -3 -70|]
[[Comma list]]: 4802000/4782969, 52734375/52706752


EDOs: {{EDOs|94, 217, 528, 745, 1273}}
{{Mapping|legend=1| 1 -49 -74 -117 | 0 57 86 135 }}
: mapping generators: ~2, ~50/27


Badness: 2.8530
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9770{{c}}, ~50/27 = 1064.9332{{c}}
: [[error map]]: {{val| -0.023 +0.365 -0.356 -0.152 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9533{{c}}
: error map: {{val| 0.000 +0.386 -0.326 -0.124 }}


==7-limit==
{{Optimal ET sequence|legend=1| 80, 231, 311, 1324b, 1635b }}
Commas: 2100875/2097152, 4802000/4782969


POTE generator: ~8575/7776 = 165.913
[[Badness]] (Sintel): 6.39


Map: [&lt;1 2 12 -3|, &lt;0 -3 -70 42|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: {{EDOs|94, 217, 311, 839, 1150}}
Comma list: 3025/3024, 4000/3993, 1479016/1476225


Badness: 0.1972
Mapping: {{mapping| 1 -49 -74 -117 -56 | 0 57 86 135 67 }}


==11-limit==
Optimal tuning:
Commas: 4000/3993, 19712/19683, 41503/41472
* WE: ~2 = 1199.9235{{c}}, ~50/27 = 1064.8866{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9536{{c}}


POTE generator: ~11/10 = 165.915
{{Optimal ET sequence|legend=0| 80, 231, 311, 1013e, 1324be }}


Map: [&lt;1 2 12 -3 13|, &lt;0 -3 -70 42 -69|]
Badness (Sintel): 2.01


EDOs: {{EDOs|94, 217, 311}}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0580
Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455


==13-limit==
Mapping: {{mapping| 1 -49 -74 -117 -56 25 | 0 57 86 135 67 -24 }}
Commas: 1575/1573, 2080/2079, 4096/4095, 13720/13689


POTE generator: ~11/10 = 165.914
Optimal tuning:  
* WE: ~2 = 1199.8904{{c}}, ~50/27 = 1064.8582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9547{{c}}


Map: [&lt;1 2 12 -3 13 -1|, &lt;0 -3 -70 42 -69 34|]
{{Optimal ET sequence|legend=0| 80, 231, 311, 702, 1013e }}


EDOs: {{EDOs|94, 217, 311, 839e, 1150e}}
Badness (Sintel): 1.61


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


==17-limit==
Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224
Commas: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095


POTE generator: ~11/10 = 165.913
Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 | 0 57 86 135 67 -24 17 }}


Map: [&lt;1 2 12 -3 13 -1 11|, &lt;0 -3 -70 42 -69 34 -50|]
Optimal tuning:  
* WE: ~2 = 1199.9634{{c}}, ~50/27 = 1064.9213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9536{{c}}


EDOs: {{EDOs|94, 217, 311, 839e, 1150eg}}
{{Optimal ET sequence|legend=0| 80, 231, 311 }}


Badness: 0.0200
Badness (Sintel): 1.53


==19-limit==
=== 19-limit ===
Commas: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~11/10 = 165.913
Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499


Map: [&lt;1 2 12 -3 13 -1 11 16|, &lt;0 -3 -70 42 -69 34 -50 -85|]
Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 -49 | 0 57 86 135 67 -24 17 60 }}


EDOs: {{EDOs|94, 217, 311, 839e, 1150eg}}
Optimal tuning:  
* WE: ~2 = 1199.9800{{c}}, ~50/27 = 1064.9358{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9535{{c}}


Badness: 0.0145
{{Optimal ET sequence|legend=0| 80, 231, 311 }}


==23-limit==
Badness (Sintel): 1.24
Commas: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155


POTE generator: ~11/10 = 165.914
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


Map: [&lt;1 2 12 -3 13 -1 11 16 16|, &lt;0 -3 -70 42 -69 34 -50 -85 -83|]
Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495


EDOs: {{EDOs|94, 217, 311, 839ei, 1150egi}}
Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 -49 -15 | 0 57 86 135 67 -24 17 60 22 }}


Badness: 0.0122
Optimal tuning:  
* WE: ~2 = 1199.9546{{c}}, ~50/27 = 1064.9138{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9539{{c}}


= Semiluna =
{{Optimal ET sequence|legend=0| 80, 231, 311 }}
{{see also|Luna family #Semiluna}}


Commas: 4802000/4782969, 95703125/95551488
Badness (Sintel): 1.16


POTE generator: ~2187/1960 = 193.1725
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29


Map: [<2 8 4 23|, <0 -15 2 -54|]
Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495


EDOs: {{EDOs|56d, 118, 292, 410}}
Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 -49 -15 -83 | 0 57 86 135 67 -24 17 60 22 99 }}


Badness: 0.1922
Optimal tuning:  
* WE: ~2 = 1199.9430{{c}}, ~50/27 = 1064.9035{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9538{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 80, 231, 311 }}
Commas: 5632/5625, 9801/9800, 14641/14580


POTE generator: ~121/108 = 193.1732
Badness (Sintel): 1.09


Map: [<2 8 4 23 14|, <0 -15 2 -54 -22|]
[[Category:Temperament collections]]
 
[[Category:Canousmic temperaments| ]] <!-- main article -->
EDOs: {{EDOs|56d, 118, 292, 410}}
[[Category:Rank 2]]
 
Badness: 0.0678
 
== 13-limit ==
Commas: 352/351, 625/624, 9801/9800, 14641/14580
 
POTE generator: ~121/108 = 193.1550
 
Map: [<2 8 4 23 14 0|, <0 -15 2 -54 -22 23|]
 
EDOs: {{EDOs|56d, 118, 174d, 292}}
 
Badness: 0.0620
 
[[Category:Temperament]]
[[Category:Canou]]

Latest revision as of 15:54, 17 May 2026

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 (monzo: [4 -14 3 4⟩, ratio: 4802000/4782969). For the rank-3 temperament, see Canou family.

Temperaments discussed elsewhere are:

Considered below is superlimmal.

Superlimmal

Superlimmal is essentially an 80-form, and may be described as the 80 & 311 temperament. It uses an ever slightly sharpened large limma as the generator, nine exceed the octave by 126/125. Note that in the data that follow, the generator is its octave complement, ~50/27, so that 57 of them octave reduced make the perfect fifth.

Superlimmal gets all the primes up to 29 reasonably covered, but is acceptable just as a 13-limit microtemperament, given a relatively simple comma basis. It can also be extended to include prime 37 by mapping it to 87 generator steps, tempering out (27/25)/(40/37) = 1000/999. Since 40/37 is the mediant of 27/25 and 13/12, this extension further consolidates the sharpened limma.

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 52734375/52706752

Mapping: [⟨1 -49 -74 -117], ⟨0 57 86 135]]

mapping generators: ~2, ~50/27

Optimal tunings:

  • WE: ~2 = 1199.9770 ¢, ~50/27 = 1064.9332 ¢
error map: ⟨-0.023 +0.365 -0.356 -0.152]
  • CWE: ~2 = 1200.0000 ¢, ~50/27 = 1064.9533 ¢
error map: ⟨0.000 +0.386 -0.326 -0.124]

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

Badness (Sintel): 6.39

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -49 -74 -117 -56], ⟨0 57 86 135 67]]

Optimal tuning:

  • WE: ~2 = 1199.9235 ¢, ~50/27 = 1064.8866 ¢
  • CWE: ~2 = 1200.0000 ¢, ~50/27 = 1064.9536 ¢

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

Badness (Sintel): 2.01

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -49 -74 -117 -56 25], ⟨0 57 86 135 67 -24]]

Optimal tuning:

  • WE: ~2 = 1199.8904 ¢, ~50/27 = 1064.8582 ¢
  • CWE: ~2 = 1200.0000 ¢, ~50/27 = 1064.9547 ¢

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

Badness (Sintel): 1.61

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -49 -74 -117 -56 25 -11], ⟨0 57 86 135 67 -24 17]]

Optimal tuning:

  • WE: ~2 = 1199.9634 ¢, ~50/27 = 1064.9213 ¢
  • CWE: ~2 = 1200.0000 ¢, ~50/27 = 1064.9536 ¢

Optimal ET sequence: 80, 231, 311

Badness (Sintel): 1.53

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 -49 -74 -117 -56 25 -11 -49], ⟨0 57 86 135 67 -24 17 60]]

Optimal tuning:

  • WE: ~2 = 1199.9800 ¢, ~50/27 = 1064.9358 ¢
  • CWE: ~2 = 1200.0000 ¢, ~50/27 = 1064.9535 ¢

Optimal ET sequence: 80, 231, 311

Badness (Sintel): 1.24

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 -49 -74 -117 -56 25 -11 -49 -15], ⟨0 57 86 135 67 -24 17 60 22]]

Optimal tuning:

  • WE: ~2 = 1199.9546 ¢, ~50/27 = 1064.9138 ¢
  • CWE: ~2 = 1200.0000 ¢, ~50/27 = 1064.9539 ¢

Optimal ET sequence: 80, 231, 311

Badness (Sintel): 1.16

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 -49 -74 -117 -56 25 -11 -49 -15 -83], ⟨0 57 86 135 67 -24 17 60 22 99]]

Optimal tuning:

  • WE: ~2 = 1199.9430 ¢, ~50/27 = 1064.9035 ¢
  • CWE: ~2 = 1200.0000 ¢, ~50/27 = 1064.9538 ¢

Optimal ET sequence: 80, 231, 311

Badness (Sintel): 1.09

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