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:
+Considered below is superlimmal.
-* [[godzilla]], {49/48, 81/80} → [[Meantone family #Godzilla]]
-* ''[[betic]]'', {225/224, 1071875/1062882} → [[Sycamore family #Betic]]
-* ''[[pentorwell]]'', {1728/1715, 179200/177147} → [[Orwellismic temperaments #Pentorwell]]
-* ''[[amicable]]'', {2401/2400, 1600000/1594323} → [[Breedsmic temperaments #Amicable]]
-* [[parakleismic]], {3136/3125, 4375/4374} → [[Ragismic microtemperaments #Parakleismic]]
-* ''[[septiquarter]]'', {5120/5103, 420175/419904} → [[Hemifamity temperaments #Septiquarter]]
-* ''[[marthirds]]'', {15625/15552, 2460375/2458624} → [[Kleismic family #Marthirds]]
-* ''[[kleischismic]]'', {32805/32768, 1562500/1492992} → [[Schismatic family #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.
-== 5-limit ==
-Subgroup: 2.3.5
+[[Subgroup]]: 2.3.5.7
-[[Comma list]]: {{monzo|104 -70 3}}
+[[Comma list]]: 4802000/4782969, 52734375/52706752
-[[Mapping]]: [{{val| 1 2 12 }}, {{val| 0 -3 -70 }}]
+{{Mapping|legend=1| 1 -49 -74 -117 | 0 57 86 135 }}
+: mapping generators: ~2, ~50/27
-[[POTE generator]]: ~{{monzo|-34 23 -1}} = 165.907
+[[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 }}
-{{Val list|legend=1| 94, 217, 528, 745, 1273 }}
+{{Optimal ET sequence|legend=1| 80, 231, 311, 1324b, 1635b }}
-[[Badness]]: 2.8530
+[[Badness]] (Sintel): 6.39
-== 7-limit ==
-Subgroup: 2.3.5.7
-[[Comma list]]: 2100875/2097152, 4802000/4782969
-[[Mapping]]: [{{val| 1 2 12 -3 }}, {{val| 0 -3 -70 42 }}]
-[[POTE generator]]: ~8575/7776 = 165.913
-{{Val list|legend=1| 94, 217, 311, 839, 1150 }}
-[[Badness]]: 0.1972
-== 11-limit ==
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 4000/3993, 19712/19683, 41503/41472
+Comma list: 3025/3024, 4000/3993, 1479016/1476225
-Mapping: [{{val| 1 2 12 -3 13 }}, {{val| 0 -3 -70 42 -69 }}]
+Mapping: {{mapping| 1 -49 -74 -117 -56 | 0 57 86 135 67 }}
-POTE generator: ~11/10 = 165.915
+Optimal tuning:
+* WE: ~2 = 1199.9235{{c}}, ~50/27 = 1064.8866{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9536{{c}}
-{{Val list|legend=1| 94, 217, 311 }}
+{{Optimal ET sequence|legend=0| 80, 231, 311, 1013e, 1324be }}
-Badness: 0.0580
+Badness (Sintel): 2.01
-== 13-limit ==
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689
+Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455
-Mapping: [{{val| 1 2 12 -3 13 -1 }}, {{val| 0 -3 -70 42 -69 34 }}]
+Mapping: {{mapping| 1 -49 -74 -117 -56 25 | 0 57 86 135 67 -24 }}
-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}}
-{{Val list|legend=1| 94, 217, 311, 839e, 1150e }}
+{{Optimal ET sequence|legend=0| 80, 231, 311, 702, 1013e }}
-Badness: 0.0303
+Badness (Sintel): 1.61
-== 17-limit ==
+=== 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095
+Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224
-Mapping: [{{val| 1 2 12 -3 13 -1 11 }}, {{val| 0 -3 -70 42 -69 34 -50 }}]
+Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 | 0 57 86 135 67 -24 17 }}
-POTE generator: ~11/10 = 165.913
+Optimal tuning:
+* WE: ~2 = 1199.9634{{c}}, ~50/27 = 1064.9213{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9536{{c}}
-{{Val list|legend=1| 94, 217, 311, 839e, 1150eg }}
+{{Optimal ET sequence|legend=0| 80, 231, 311 }}
-Badness: 0.0200
+Badness (Sintel): 1.53
-== 19-limit ==
+=== 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
+Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499
-Mapping: [{{val| 1 2 12 -3 13 -1 11 16 }}, {{val| 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 }}
-POTE generator: ~11/10 = 165.913
+Optimal tuning:
+* WE: ~2 = 1199.9800{{c}}, ~50/27 = 1064.9358{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9535{{c}}
-{{Val list|legend=1| 94, 217, 311, 839e, 1150eg }}
+{{Optimal ET sequence|legend=0| 80, 231, 311 }}
-Badness: 0.0145
+Badness (Sintel): 1.24
-== 23-limit ==
+=== 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
+Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495
-Mapping: [{{val| 1 2 12 -3 13 -1 11 16 16 }}, {{val| 0 -3 -70 42 -69 34 -50 -85 -83 }}]
-POTE generator: ~11/10 = 165.914
-{{Val list|legend=1| 94, 217, 311, 839ei, 1150egi }}
-Badness: 0.0122
-= Semiluna =
-{{see also|Luna family #Semiluna}}
-Subgroup: 2.3.5.7
+Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 -49 -15 | 0 57 86 135 67 -24 17 60 22 }}
-[[Comma list]]: 4802000/4782969, 95703125/95551488
+Optimal tuning:
+* WE: ~2 = 1199.9546{{c}}, ~50/27 = 1064.9138{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9539{{c}}
-[[Mapping]]: [{{val| 2 8 4 23 }}, {{val| 0 -15 2 -54 }}]
+{{Optimal ET sequence|legend=0| 80, 231, 311 }}
-[[POTE generator]]: ~2187/1960 = 193.1725
+Badness (Sintel): 1.16
-{{Val list|legend=1| 56d, 118, 292, 410 }}
+=== 29-limit ===
+Subgroup: 2.3.5.7.11.13.17.19.23.29
-[[Badness]]: 0.1922
-== 11-limit ==
-Subgroup: 2.3.5.7.11
-Comma list: 5632/5625, 9801/9800, 14641/14580
+Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495
-Mapping: [{{val| 2 8 4 23 14 }}, {{val| 0 -15 2 -54 -22 }}]
+Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 -49 -15 -83 | 0 57 86 135 67 -24 17 60 22 99 }}
-POTE generator: ~121/108 = 193.1732
+Optimal tuning:
+* WE: ~2 = 1199.9430{{c}}, ~50/27 = 1064.9035{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9538{{c}}
-{{Val list|legend=1| 56d, 118, 292, 410 }}
+{{Optimal ET sequence|legend=0| 80, 231, 311 }}
-Badness: 0.0678
+Badness (Sintel): 1.09
-[[Category:Regular temperament theory]]
+[[Category:Temperament collections]]
-[[Category:Temperament collection]]
+[[Category:Canousmic temperaments| ]] <!-- main article -->
-[[Category:Canou]]
 [[Category:Rank 2]]