Canousmic temperaments: Difference between revisions

Revision as of 08:43, 14 September 2023

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:

Godzilla (+49/48 or 81/80) → Meantone family
Betic (+225/224) → Sycamore family
Pentorwell (+1728/1715) → Orwellismic temperaments
Amicable (+2401/2400) → Breedsmic temperaments
Parakleismic (+3136/3125 or 4375/4374) → Ragismic microtemperaments
Septiquarter (+5120/5103) → Hemifamity temperaments
Marthirds (+15625/15552) → Kleismic family
Kleischismic (+32805/32768) → Schismatic family
Turkey (+5250987/5242880) → Vulture family
Hemiquindromeda (+67108864/66976875) → Quindromeda family
Semiluna (+95703125/95551488) → Luna family

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]]

Wedgie: ⟨⟨ 57 86 135 3 53 72 ]]

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

@@ Line 2: / Line 2: @@
 Temperaments discussed elsewhere are:
-* [[Godzilla]], {49/48, 81/80} → [[Meantone family #Godzilla|Meantone family]]
+* [[Godzilla]] (+49/48 or 81/80) → [[Meantone family #Godzilla|Meantone family]]
-* ''[[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]]
+* ''[[Turkey (temperament)|Turkey]]'' (+5250987/5242880) → [[Vulture family #Turkey|Vulture family]]
-* ''[[Hemiquindromeda]]'', {4802000/4782969, 67108864/66976875} → [[Quindromeda family #Hemiquindromeda|Quindromeda family]]
+* ''[[Hemiquindromeda]]'' (+67108864/66976875) → [[Quindromeda family #Hemiquindromeda|Quindromeda family]]
-* ''[[Semiluna]]'', {4802000/4782969, 95703125/95551488} → [[Luna family #Semiluna|Luna family]]
+* ''[[Semiluna]]'' (+95703125/95551488) → [[Luna family #Semiluna|Luna family]]
 Considered below are satin and superlimmal.
@@ Line 19: / Line 19: @@
 : ''For the 5-limit version of this temperament, see [[High badness temperaments #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.
+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.
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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
 {{Optimal ET sequence|legend=1| 94, 217, 311, 839, 1150 }}
@@ Line 38: / Line 38: @@
 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
 {{Optimal ET sequence|legend=1| 94, 217, 311 }}
@@ Line 51: / Line 51: @@
 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
 {{Optimal ET sequence|legend=1| 94, 217, 311, 839e, 1150e }}
@@ Line 64: / Line 64: @@
 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
 {{Optimal ET sequence|legend=1| 94, 217, 311, 839e, 1150eg }}
@@ Line 77: / Line 77: @@
 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
 {{Optimal ET sequence|legend=1| 94, 217, 311, 839e, 1150eg }}
@@ Line 90: / Line 90: @@
 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
 {{Optimal ET sequence|legend=1| 94, 217, 311, 839ei, 1150egi }}
@@ Line 99: / Line 99: @@
 == 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 acceptible 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.
+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
+[[Subgroup]]: 2.3.5.7
 [[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
 {{Optimal ET sequence|legend=1| 80, 231, 311, 1324b, 1635b }}
@@ Line 120: / Line 120: @@
 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
 {{Optimal ET sequence|legend=1| 80, 231, 311, 1013e, 1324be }}
@@ Line 133: / Line 133: @@
 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
 {{Optimal ET sequence|legend=1| 80, 231, 311, 702, 1013e }}
@@ Line 146: / Line 146: @@
 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
 {{Optimal ET sequence|legend=1| 80, 231, 311 }}
@@ Line 159: / Line 159: @@
 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
 {{Optimal ET sequence|legend=1| 80, 231, 311 }}
@@ Line 172: / Line 172: @@
 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
 {{Optimal ET sequence|legend=1| 80, 231, 311 }}
@@ Line 185: / Line 185: @@
 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
 {{Optimal ET sequence|legend=1| 80, 231, 311 }}
@@ Line 198: / Line 198: @@
 Comma list: 595/594, 760/759, 784/783, 925/924, 969/968, 1000/999, 1045/1044, 1105/1104, 1275/1274
-Mapping: [{{val| 1 8 12 18 11 1 6 11 7 16 15 }}, {{val| 0 -57 -86 -135 -67 24 -17 -60 -22 -99 -87 }}]
+Mapping: {{mapping| 1 8 12 18 11 1 6 11 7 16 15 | 0 -57 -86 -135 -67 24 -17 -60 -22 -99 -87 }}
-POTE generator: ~27/25 = 135.0460
+Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0460
 {{Optimal ET sequence|legend=1| 80, 231, 311 }}
@@ Line 208: / Line 208: @@
 [[Category:Temperament collections]]
 [[Category:Canousmic temperaments| ]] <!-- main article -->
+[[Category:Canou| ]] <!-- key article -->
 [[Category:Rank 2]]

Canousmic temperaments: Difference between revisions

Revision as of 08:43, 14 September 2023

Contents

Satin

11-limit

13-limit

17-limit

19-limit

23-limit

Superlimmal

11-limit

13-limit

17-limit

19-limit

23-limit

29-limit

No-31's 37-limit

Navigation menu

Canousmic temperaments: Difference between revisions

Revision as of 08:43, 14 September 2023

Satin

11-limit

13-limit

17-limit

19-limit

23-limit

Superlimmal

11-limit

13-limit

17-limit

19-limit

23-limit

29-limit

No-31's 37-limit

Navigation menu

Search