Xenharmonic Wiki

Meantone family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''meantone family''' is the family of rank-2 temperaments that temper out the syntonic comma, [[81/80]], and thus can all be seen as extensions of [[meantone]]. The [[period]] is an [[octave]], the [[generator]] is a [[3/2|fifth]], and four fifths go to make up a [[5/1]] interval.
The '''meantone family''' is the family of [[rank-2 temperament]]s that [[tempering out|temper out]] the syntonic comma, [[81/80]], and thus can all be seen as [[extension]]s of [[meantone]].  


== Meantone ==
== Meantone ==
{{Main| Meantone }}
{{Main| Meantone }}
Meantone is characterized by an [[2/1|octave]] [[period]], a [[3/2|fifth]] [[generator]], and the relationship that four fifths go to make up a [[5/1|5th harmonic]].


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 20: Line 22:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 697.214
* [[WE]]: ~2 = 1201.3906{{c}}, ~3/2 = 697.0455{{c}}
: [[error map]]: {{val| 0 -4.741 +2.544 }}
: [[error map]]: {{val| +1.391 -3.519 +1.868 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 696.239
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6512{{c}}
: [[error map]]: {{val| 0 -5.716 -1.359 }}
: error map: {{val| 0.000 -5.304 +0.291 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 35: Line 37:
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }}


[[Badness]]: 0.007381
[[Badness]] (Sintel): 0.173


=== Extensions ===
=== Overview to extensions ===
The second comma of the normal comma list defines which [[7-limit]] family member we are looking at.
The second comma of the normal comma list defines which [[7-limit]] family member we are looking at.
* Flattertone adds {{monzo| -24 17 0 -1 }}, finding the ~7/4 at the double-augmented sixth. It is well-tuned flat of 26edo.
* Flattertone adds {{monzo| -24 17 0 -1 }}, finding the [[~]][[7/4]] at the double-augmented sixth, for a tuning between 33edo and 26edo.
* Flattone adds {{monzo| -17 9 0 1 }}, finding the ~7/4 at the diminished seventh, It is well-tuned between 19edo and 26edo.
* Flattone adds {{monzo| -17 9 0 1 }}, finding the ~7/4 at the diminished seventh, for a tuning between 26edo and 19edo.
* Septimal meantone adds [[Harrison's comma|{{monzo| -13 10 0 -1 }}]], finding the ~[[7/4]] at the augmented sixth (diesis-flat minor seventh). It is well-tuned between 12edo and 19edo.
* Septimal meantone adds [[Harrison's comma|{{monzo| -13 10 0 -1 }}]], finding the ~7/4 at the augmented sixth, for a tuning between 19edo and 12edo.
* Dominant adds [[64/63|{{monzo| 6 -2 0 -1 }}]], finding the ~7/4 at the minor seventh. It is well-tuned between 5edo and 12edo.
* Dominant adds [[64/63|{{monzo| 6 -2 0 -1 }}]], finding the ~7/4 at the minor seventh, for a tuning between 12edo and 5edo.
* Sharptone adds [[28/27|{{monzo| 2 -3 0 1 }}]], finding the ~7/4 at the major sixth. It is "well-tuned" sharp of 5edo, however it horribly mistunes 5/4, so it can be considered an exotemperament.
* Sharptone adds [[28/27|{{monzo| 2 -3 0 1 }}]], finding the ~7/4 at the major sixth, for an [[exotemperament]] never exactly well-tuned, and where 5edo is the only [[diamond monotone]] tuning, with a terrible 5-limit part.  
Those all have a fifth as generator.
Those all have a fifth as generator.
* Injera adds {{monzo| -7 8 0 -2 }} with a half-octave period.
* Injera adds {{monzo| -7 8 0 -2 }} with a half-octave period.
Line 74: Line 76:
* ''[[Plutus]]'' (+15/14) → [[Very low accuracy temperaments #Plutus|Very low accuracy temperaments]]
* ''[[Plutus]]'' (+15/14) → [[Very low accuracy temperaments #Plutus|Very low accuracy temperaments]]
* [[Godzilla]] (+49/48) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
* [[Godzilla]] (+49/48) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
* ''[[Mothra]]'' (+1029/1024) → [[Gamelismic clan #Mothra|Gamelismic clan]]
* [[Mothra]] (+1029/1024) → [[Gamelismic clan #Mothra|Gamelismic clan]]
* ''[[Mohaha]]'' (+121/120) → [[Rastmic clan #Mohaha|Rastmic clan]]
* ''[[Mohaha]]'' (+121/120) → [[Rastmic clan #Mohaha|Rastmic clan]]


Line 93: Line 95:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 696.952
* [[WE]]: ~2 = 1201.2358{{c}}, ~3/2 = 697.2122{{c}}
: [[error map]]: {{val| 0 -5.003 +1.495 +0.695 }}
: [[error map]]: {{val| +1.236 -3.507 +2.535 -0.412 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 696.495
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6562{{c}}
: [[error map]]: {{val| 0 -5.460 -0.334 -3.877 }}
: error map: {{val| 0.000 -5.299 +0.311 -2.264 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 112: Line 114:
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}


[[Badness]]: 0.013707
[[Badness]] (Sintel): 0.347


=== Undecimal meantone (huygens) ===
=== Undecimal meantone (huygens) ===
Line 127: Line 129:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 697.168
* WE: ~2 = 1200.7636{{c}}, ~3/2 = 697.4122{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.967
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0315{{c}}


Minimax tuning:  
Minimax tuning:  
Line 143: Line 145:
{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }}
{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }}


Badness: 0.017027
Badness (Sintel): 0.563


; Music
; Music
Line 156: Line 158:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.855
* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.642
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}


Minimax tuning:  
Minimax tuning:  
Line 165: Line 167:
{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}
{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}


Badness: 0.018048
Badness (Sintel): 0.746


===== Meantonic =====
===== Meantonic =====
Line 177: Line 179:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.649
* WE: ~2 = 1201.2376{{c}}, ~3/2 = 697.0954{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.377
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4563{{c}}


{{Optimal ET sequence|legend=0| 12fg, 19eg, 31, 50e }}
{{Optimal ET sequence|legend=0| 12fg, 19eg, 31, 50e }}


Badness: 0.019037
Badness (Sintel): 0.970


====== 19-limit ======
====== 19-limit ======
Line 192: Line 194:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.555
* WE: ~2 = 1201.4134{{c}}, ~3/2 = 697.0933{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.273
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.3526{{c}}


{{Optimal ET sequence|legend=0| 12fghh, 19egh, 31, 50e }}
{{Optimal ET sequence|legend=0| 12fghh, 19egh, 31, 50e }}


Badness: 0.017846
Badness (Sintel): 1.09
 
===== Meantoid =====
Dubbed ''meantoid'' here, this extension maps 17/16~19/18 to the augmented unison (C–C♯) and 19/16 to the augmented second (C–D♯). For any tuning flatter than 12edo, the sizes of 17/16 (augmented unison) and 18/17 (minor second) are inverted, so genuine septendecimal and undevicesimal harmony cannot be expected.
 
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 51/50, 66/65, 81/80, 85/84, 99/98
 
Mapping: {{mapping| 1 0 -4 -13 -25 -20 -7 | 0 1 4 10 18 15 7 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 697.036
* POTE: ~2 = 1200.000, ~3/2 = 696.448
 
{{Optimal ET sequence|legend=0| 12f, 19eg, 31g }}
 
Badness: 0.019433
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 51/50, 57/56, 66/65, 81/80, 85/84, 99/98
 
Mapping: {{mapping| 1 0 -4 -13 -25 -20 -7 -10 | 0 1 4 10 18 15 7 9 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 697.216
* POTE: ~2 = 1200.000, ~3/2 = 696.394
 
{{Optimal ET sequence|legend=0| 12f, 19egh, 31gh }}
 
Badness: 0.017437


===== Huygens =====
===== Huygens =====
Line 241: Line 211:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.908
* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 697.003
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}}


{{Optimal ET sequence|legend=0| 12f, 31 }}
{{Optimal ET sequence|legend=0| 12f, 31 }}


Badness: 0.019982
Badness (Sintel): 1.02


====== 19-limit ======
====== 19-limit ======
Line 256: Line 226:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.931
* WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 697.140
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}}


{{Optimal ET sequence|legend=0| 12f, 31 }}
{{Optimal ET sequence|legend=0| 12f, 31 }}


Badness: 0.018047
Badness (Sintel): 1.10


==== Grosstone ====
==== Grosstone ====
Line 273: Line 243:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 697.258
* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 697.264
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}


Minimax tuning:  
Minimax tuning:  
Line 286: Line 256:
{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}
{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}


Badness: 0.025899
Badness (Sintel): 1.07


===== 17-limit =====
===== 17-limit =====
Line 296: Line 266:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 697.300
* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 697.335
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}


{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}
{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}


Badness: 0.020889
Badness (Sintel): 1.06


===== 19-limit =====
===== 19-limit =====
Line 311: Line 281:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 697.327
* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 697.380
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}


{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}
{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}


Badness: 0.017611
Badness (Sintel): 1.07


==== Meridetone ====
==== Meridetone ====
Line 328: Line 298:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 697.516
* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 697.529
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}}


Minimax tuning:  
Minimax tuning:  
Line 337: Line 307:
{{Optimal ET sequence|legend=0| 12f, 31f, 43 }}
{{Optimal ET sequence|legend=0| 12f, 31f, 43 }}


Badness: 0.026421
Badness (Sintel): 1.09


===== Meridetonic =====
===== Meridetonic =====
Line 347: Line 317:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 697.508
* WE: ~2 = 1199.9428{{c}}, ~3/2 = 697.4804{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 697.514
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5113{{c}}


{{Optimal ET sequence|legend=0| 12fg, 31fg, 43 }}
{{Optimal ET sequence|legend=0| 12fg, 31fg, 43 }}


Badness: 0.027706
Badness (Sintel): 1.41


====== 19-limit ======
====== 19-limit ======
Line 362: Line 332:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 697.485
* WE: ~2 = 1200.0089{{c}}, ~3/2 = 697.4864{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 697.481
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.4815{{c}}


{{Optimal ET sequence|legend=0| 12fghh, 31fgh, 43 }}
{{Optimal ET sequence|legend=0| 12fghh, 31fgh, 43 }}


Badness: 0.025315
Badness (Sintel): 1.54


===== Meridetoid =====
===== Sauveuric =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 51/50, 78/77, 81/80, 85/84, 99/98
Comma list: 78/77, 81/80, 99/98, 120/119, 126/125


Mapping: {{mapping| 1 0 -4 -13 -25 -39 -7 | 0 1 4 10 18 27 7 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 697.610
* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 697.376
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}}


{{Optimal ET sequence|legend=0| 12f, 31fg, 43g }}
{{Optimal ET sequence|legend=0| 12f, 43 }}


Badness: 0.027518
Badness (Sintel): 1.22


====== 19-limit ======
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 51/50, 57/56, 78/77, 81/80, 85/84, 99/98
Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125


Mapping: {{mapping| 1 0 -4 -13 -25 -39 -7 -10 | 0 1 4 10 18 27 7 9 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 697.701
* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 697.316
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}


{{Optimal ET sequence|legend=0| 12f, 19effgh, 31fgh, 43gh }}
{{Optimal ET sequence|legend=0| 12f, 43 }}


Badness: 0.023613
Badness (Sintel): 1.25


===== Sauveuric =====
==== Hemimeantone ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13
 
Comma list: 78/77, 81/80, 99/98, 120/119, 126/125
 
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 697.538
* POTE: ~2 = 1200.000, ~3/2 = 697.644
 
{{Optimal ET sequence|legend=0| 12f, 43 }}
 
Badness: 0.023881
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125
 
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 697.555
* POTE: ~2 = 1200.000, ~3/2 = 697.715
 
{{Optimal ET sequence|legend=0| 12f, 43 }}
 
Badness: 0.020540
 
==== Hemimeantone ====
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 99/98, 126/125, 169/168
Comma list: 81/80, 99/98, 126/125, 169/168
Line 439: Line 379:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~26/15 = 948.611
* WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}}
* POTE: ~2 = 1200.000, ~26/15 = 948.465
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5065{{c}}


{{Optimal ET sequence|legend=0| 19e, 43, 62 }}
{{Optimal ET sequence|legend=0| 19e, 43, 62 }}


Badness: 0.031433
Badness (Sintel): 1.30


===== 17-limit =====
===== 17-limit =====
Line 454: Line 394:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~26/15 = 948.617
* WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}}
* POTE: ~2 = 1200.000, ~26/15 = 948.477
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}}


{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}
{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}


Badness: 0.023380
Badness (Sintel): 1.19


===== 19-limit =====
===== 19-limit =====
Line 469: Line 409:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~19/11 = 948.609
* WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}}
* POTE: ~2 = 1200.000, ~19/11 = 948.473
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}}


{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}
{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}


Badness: 0.018952
Badness (Sintel): 1.15


==== Semimeantone ====
==== Semimeantone ====
Line 486: Line 426:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 600.000, ~3/2 = 697.168
* WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}}
* POTE: ~55/39 = 600.000, ~3/2 = 697.005
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}}


{{Optimal ET sequence|legend=0| 12f, 38deefff, 50eff, 62, 136b }}
{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }}


Badness: 0.040668
Badness (Sintel): 1.68


===== 17-limit =====
===== 17-limit =====
Line 501: Line 441:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~3/2 = 697.174
* WE: ~17/12 = 600.5426{{c}}, ~3/2 = 697.5571{{c}}
* POTE: ~17/12 = 600.000, ~3/2 = 696.927
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9858{{c}}


{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}
{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}


Badness: 0.031491
Badness (Sintel): 1.60


===== 19-limit =====
===== 19-limit =====
Line 516: Line 456:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~3/2 = 697.187
* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}}
* POTE: ~17/12 = 600.000, ~3/2 = 696.906
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}}


{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}
{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}


Badness: 0.024206
Badness (Sintel): 1.47


=== Meanpop ===
=== Meanpop ===
Line 537: Line 477:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.531
* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.434
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}}


Minimax tuning:  
Minimax tuning:  
Line 553: Line 493:
{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}
{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}


Badness: 0.021543
Badness (Sintel): 0.712


; Music
; Music
Line 567: Line 507:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.356
* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.211
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}


Minimax tuning:  
Minimax tuning:  
Line 580: Line 520:
{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }}
{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }}


Badness: 0.020883
Badness (Sintel): 0.863


===== Meanpoppic =====
===== Meanpoppic =====
Line 590: Line 530:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.351
* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.194
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}}


{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}
{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}


Badness: 0.019953
Badness (Sintel): 1.02


====== 19-limit ======
====== 19-limit ======
Line 605: Line 545:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.347
* WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.188
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}}


{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}
{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}


Badness: 0.017791
Badness (Sintel): 1.08


===== Meanpoid =====
===== Meanpoid =====
Line 620: Line 560:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.439
* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.408
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}}


{{Optimal ET sequence|legend=0| 19, 31 }}
{{Optimal ET sequence|legend=0| 19, 31 }}


Badness: 0.022870
Badness (Sintel): 1.17


====== 19-limit ======
====== 19-limit ======
Line 635: Line 575:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.484
* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.499
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}}


{{Optimal ET sequence|legend=0| 12ef, 19, 31 }}
{{Optimal ET sequence|legend=0| 19, 31 }}


Badness: 0.020488
Badness (Sintel): 1.25


==== Meanplop ====
==== Meanplop ====
Line 650: Line 590:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.283
* WE: ~2 = 1202.3237{{c}}, ~3/2 = 697.5502{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.202
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2135{{c}}


Minimax tuning:  
Minimax tuning:  
Line 659: Line 599:
{{Optimal ET sequence|legend=0| 12e, 19, 31f }}
{{Optimal ET sequence|legend=0| 12e, 19, 31f }}


Badness: 0.027666
Badness (Sintel): 1.14


===== 17-limit =====
===== 17-limit =====
Line 669: Line 609:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.407
* WE: ~2 = 1201.4737{{c}}, ~3/2 = 697.2690{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.414
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4129{{c}}


{{Optimal ET sequence|legend=0| 12e, 19 }}
{{Optimal ET sequence|legend=0| 12e, 19 }}


Badness: 0.026836
Badness (Sintel): 1.37


====== 19-limit ======
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 684: Line 624:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.473
* WE: ~2 = 1200.8839{{c}}, ~3/2 = 697.0104{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.497
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4949{{c}}


{{Optimal ET sequence|legend=0| 12e, 19 }}
{{Optimal ET sequence|legend=0| 12e, 19 }}


Badness: 0.023540
Badness (Sintel): 1.43


===== Meanploid =====
=== Meanenneadecal ===
Subgroup: 2.3.5.7.11.13.17
Meanenneadecal maps the 11/8 to the augmented fourth (C–F♯), and tridecimal meanenneadecal still maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a major second; 12/11~14/13, minor second; and 13/12, double-augmented unison.  
 
Comma list: 51/50, 65/64, 78/77, 81/80, 85/84
 
Mapping: {{mapping| 1 0 -4 -13 24 10 -7 | 0 1 4 10 -13 -4 7 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 696.661
* POTE: ~2 = 1200.000, ~3/2 = 696.415
 
{{Optimal ET sequence|legend=0| 12e, 19g, 31fg }}
 
Badness: 0.026094
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 51/50, 57/56, 65/64, 76/75, 78/77, 81/80
 
Mapping: {{mapping| 1 0 -4 -13 24 10 -7 -10 | 0 1 4 10 -13 -4 7 9 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 697.016
* POTE: ~2 = 1200.000, ~3/2 = 696.583
 
{{Optimal ET sequence|legend=0| 12e, 19gh, 31fgh }}
 
Badness: 0.023104
 
=== Meanenneadecal ===
Meanenneadecal maps the 11/8 to the augmented fourth (C–F♯), and tridecimal meanenneadecal still maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a major second; 12/11~14/13, minor second; and 13/12, double-augmented unison.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 731: Line 641:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.153
* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.250
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}


Tuning ranges:  
Tuning ranges:  
Line 740: Line 650:
{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}
{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}


Badness: 0.021423
Badness (Sintel): 0.708


==== 13-limit ====
==== 13-limit ====
Line 750: Line 660:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.098
* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.146
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}


{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}
{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}


Badness: 0.021182
Badness (Sintel): 0.875


===== 17-limit =====
===== 17-limit =====
Line 765: Line 675:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.216
* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.575
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}


{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}


Badness: 0.022980
Badness (Sintel): 1.17


====== 19-limit ======
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 780: Line 690:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.277
* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.706
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}


{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}


Badness: 0.020293
Badness (Sintel): 1.23


===== Meanenneadecoid =====
==== Vincenzo ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 34/33, 45/44, 51/50, 56/55, 78/77
Comma list: 45/44, 56/55, 65/64, 81/80


Mapping: {{mapping| 1 0 -4 -13 -6 -20 -7 | 0 1 4 10 6 15 7 }}
Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.450
* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.025
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}


{{Optimal ET sequence|legend=0| 7dfg, 12f, 19g }}
{{Optimal ET sequence|legend=0| 7d, 12, 19 }}


Badness: 0.020171
Badness (Sintel): 1.02


====== 19-limit ======
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 34/33, 45/44, 51/50, 56/55, 57/55, 78/77
 
Mapping: {{mapping| 1 0 -4 -13 -6 -20 -7 -10 | 0 1 4 10 6 15 7 9 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 696.793
* POTE: ~2 = 1200.000, ~3/2 = 696.121
 
{{Optimal ET sequence|legend=0| 7dfgh, 12f, 19gh }}
 
Badness: 0.018045
 
==== Vincenzo ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 45/44, 56/55, 65/64, 81/80
 
Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 695.790
* POTE: ~2 = 1200.000, ~3/2 = 695.060
 
{{Optimal ET sequence|legend=0| 7d, 12, 19 }}
 
Badness: 0.024763
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Comma list: 45/44, 52/51, 56/55, 65/64, 81/80
Comma list: 45/44, 52/51, 56/55, 65/64, 81/80
Line 840: Line 720:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.011
* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 695.858
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}


{{Optimal ET sequence|legend=0| 12, 19 }}
{{Optimal ET sequence|legend=0| 12, 19 }}


Badness: 0.025535
Badness (Sintel): 1.30


====== 19-limit ======
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 855: Line 735:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.120
* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.131
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}


{{Optimal ET sequence|legend=0| 12, 19 }}
{{Optimal ET sequence|legend=0| 12, 19 }}


Badness: 0.022302
Badness (Sintel): 1.36


===== Vincenzoid =====
==== Meanundec ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 34/33, 45/44, 51/50, 56/55, 65/64
Comma list: 27/26, 40/39, 45/44, 56/55


Mapping: {{mapping| 1 0 -4 -13 -6 10 -7 | 0 1 4 10 6 -4 7 }}
Mapping: {{mapping| 1 0 -4 -13 -6 -1 | 0 1 4 10 6 3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.412
* WE: ~2 = 1196.0359{{c}}, ~3/2 = 694.9504{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 695.358
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7474{{c}}


{{Optimal ET sequence|legend=0| 7dg, 12, 19g }}
{{Optimal ET sequence|legend=0| 7d, 12f, 19f }}


Badness: 0.022099
Badness (Sintel): 1.00


====== 19-limit ======
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 34/33, 45/44, 51/50, 56/55, 57/55, 65/64
 
Mapping: {{mapping| 1 0 -4 -13 -6 10 -7 -10 | 0 1 4 10 6 -4 7 9 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 696.950
* POTE: ~2 = 1200.000, ~3/2 = 695.725
 
{{Optimal ET sequence|legend=0| 7dgh, 12, 19gh }}
 
Badness: 0.019904
 
==== Meanundec ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 40/39, 45/44, 56/55
 
Mapping: {{mapping| 1 0 -4 -13 -6 -1 | 0 1 4 10 6 3 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 695.620
* POTE: ~2 = 1200.000, ~3/2 = 697.254
 
{{Optimal ET sequence|legend=0| 7d, 12f, 19f }}
 
Badness: 0.024243
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 915: Line 765:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.279
* WE: ~2 = 1196.8604{{c}}, ~3/2 = 695.7613{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 697.586
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1744{{c}}


{{Optimal ET sequence|legend=0| 7dg, 12f }}
{{Optimal ET sequence|legend=0| 7dg, 12f }}


Badness: 0.021400
Badness (Sintel): 1.09


===== 19-limit =====
===== 19-limit =====
Line 930: Line 780:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.849
* WE: ~2 = 1196.9296{{c}}, ~3/2 = 696.3321{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 698.118
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.7122{{c}}


{{Optimal ET sequence|legend=0| 7dgh, 12f }}
{{Optimal ET sequence|legend=0| 7dgh, 12f }}


Badness: 0.018996
Badness (Sintel): 1.16


=== Meanundeci ===
=== Meanundeci ===
Line 947: Line 797:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.702
* WE: ~2 = 1205.7146{{c}}, ~3/2 = 697.9977{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 694.689
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.1805{{c}}


{{Optimal ET sequence|legend=0| 7d, 12e, 19e }}
{{Optimal ET sequence|legend=0| 7d, 12e, 19e }}


Badness: 0.031539
Badness (Sintel): 1.04


==== 13-limit ====
==== 13-limit ====
Line 962: Line 812:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.241
* WE: ~2 = 1205.5631{{c}}, ~3/2 = 697.9847{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 694.764
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.0144{{c}}


{{Optimal ET sequence|legend=0| 7d, 12e, 19e }}
{{Optimal ET sequence|legend=0| 7d, 12e, 19e }}


Badness: 0.026288
Badness (Sintel): 1.09


=== Bimeantone ===
=== Bimeantone ===
Line 981: Line 831:


Optimal tunings:  
Optimal tunings:  
* CTE: ~63/44 = 600.000, ~3/2 = 696.520
* WE: ~63/44 = 600.7492{{c}}, ~3/2 = 696.8853{{c}}
* POTE: ~63/44 = 600.000, ~3/2 = 696.016
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 696.1908{{c}}


{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}
{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}


Badness: 0.038122
Badness (Sintel): 1.26


==== 13-limit ====
==== 13-limit ====
Line 996: Line 846:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 600.000, ~3/2 = 696.341
* WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}}
* POTE: ~55/39 = 600.000, ~3/2 = 695.836
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}}


{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}


Badness: 0.028817
Badness (Sintel): 1.19


==== 17-limit ====
==== 17-limit ====
Line 1,011: Line 861:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~3/2 = 696.353
* WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}}
* POTE: ~17/12 = 600.000, ~3/2 = 695.783
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}}


{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}
{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}


Badness: 0.022666
Badness (Sintel): 1.15


==== 19-limit ====
==== 19-limit ====
Line 1,026: Line 876:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~3/2 = 696.384
* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
* POTE: ~17/12 = 600.000, ~3/2 = 695.752
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}


{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}


Badness: 0.017785
Badness (Sintel): 1.08


=== Trimean ===
=== Trimean ===
{{see also|No-sevens subgroup temperaments #Superpine}}
{{See also| No-sevens subgroup temperaments #Superpine }}


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,045: Line 895:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 167.707
* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 167.805
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}


{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}
{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}


Badness: 0.050729
Badness (Sintel): 1.68


==== 13-limit ====
==== 13-limit ====
Line 1,060: Line 910:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 167.712
* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 167.790
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}


{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}
{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}


Badness: 0.035445
Badness (Sintel): 1.46


==== 17-limit ====
==== 17-limit ====
Line 1,075: Line 925:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 167.705
* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 167.786
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}


{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}
{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}


Badness: 0.025221
Badness (Sintel): 1.28


== Flattone ==
== Flattone ==
Line 1,094: Line 944:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 693.552
* [[WE]]: ~2 = 1203.6308{{c}}, ~3/2 = 695.8782{{c}}
: [[error map]]: {{val| 0.000 -8.403 -12.106 -10.794 }}
: [[error map]]: {{val| +3.631 -2.446 -2.801 -2.684 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 693.779
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.7334{{c}}
: error map: {{val| 0.000 -8.176 -11.197 -12.838 }}
: error map: {{val| 0.000 -8.222 -11.380 -12.426 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 1,116: Line 966:
{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}
{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}


[[Badness]]: 0.038553
[[Badness]] (Sintel): 0.976


=== 11-limit ===
=== 11-limit ===
Line 1,128: Line 978:


Optimal tuning:  
Optimal tuning:  
* CTE: ~2 = 1200.000, ~3/2 = 693.251
* WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 693.126
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}}


Tuning ranges:  
Tuning ranges:  
Line 1,137: Line 987:
{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}
{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}


Badness: 0.033839
Badness (Sintel): 1.12


=== 13-limit ===
=== 13-limit ===
Line 1,147: Line 997:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 693.029
* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 693.058
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}


Tuning ranges:  
Tuning ranges:  
Line 1,156: Line 1,006:
{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}
{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}


Badness: 0.022260
Badness (Sintel): 0.920


== Flattertone ==
== Flattertone ==
Line 1,170: Line 1,020:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 692.698
* [[WE]]: ~2 = 1204.4511{{c}}, ~3/2 = 694.3258{{c}}
: [[error map]]: {{val| 0.000 -9.257 -15.520 +7.047 }}
: [[error map]]: {{val| +4.451 -3.178 -9.011 +3.554 }}
* [[CWE]]: ~2 = 1200.000, ~3/2 = 692.048
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 692.0479{{c}}
: error map: {{val| 0.000 -9.907 -18.122 -4.012 }}
: error map: {{val| 0.000 -9.907 -18.122 -4.012 }}


{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}
{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}


[[Badness]]: 0.0961
[[Badness]] (Sintel): 2.43


==== 11-limit ====
==== 11-limit ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 45/44, 81/80, 1375/1344
Comma list: 45/44, 81/80, 1375/1344


{{Mapping|legend=1| 1 0 -4 -24 0| 0 1 4 17 6 }}
Mapping: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }}


: mapping generators: ~2, ~3
Optimal tunings:  
* WE: ~2 = 1203.4653{{c}}, ~3/2 = 693.8144{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 692.0422{{c}}


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=0| 7d, 19d, 26 }}
* [[CTE]]: ~2 = 1200.000, ~3/2 = 692.642
* [[CWE]]: ~2 = 1200.000, ~3/2 = 692.042


{{Optimal ET sequence|legend=0| 7d, 19d, 26, 59bcd, 85bccd }}
Badness (Sintel): 1.53


; Music
; Music
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] - [[Budjarn Lambeth]] (2024)
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024)


== Dominant ==
== Dominant ==
Line 1,212: Line 1,062:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 699.622
* [[WE]]: ~2 = 1195.3384{{c}}, ~3/2 = 698.8478{{c}}
: [[error map]]: {{val| 0.000 -2.333 +12.173 +31.931 }}
: [[error map]]: {{val| -4.662 -7.769 +9.077 +14.832 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 701.573
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.1125{{c}}
: error map: {{val| 0.000 -0.382 +19.979 +28.028 }}
: error map: {{val| 0.000 -0.842 +18.136 +28.949 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 1,224: Line 1,074:
{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}


[[Badness]]: 0.020690
[[Badness]] (Sintel): 0.524


=== 11-limit ===
=== 11-limit ===
Line 1,238: Line 1,088:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 703.334
* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 703.254
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}}


{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}
{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}


Badness: 0.024180
Badness (Sintel): 0.799


==== 13-limit ====
==== 13-limit ====
Line 1,253: Line 1,103:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 704.847
* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 703.636
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}}


Tuning ranges:  
Tuning ranges:  
Line 1,260: Line 1,110:
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]


{{Optimal ET sequence|legend=0|12f, 17c, 29cdef }}
{{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }}


Badness: 0.024108
Badness (Sintel): 0.996


==== Dominion ====
==== Dominion ====
Line 1,272: Line 1,122:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 704.034
* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 704.905
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}


{{Optimal ET sequence|legend=0| 5, 12, 17c, 46cde }}
{{Optimal ET sequence|legend=0| 5, 12, 17c }}


Badness: 0.027295
Badness (Sintel): 1.13


=== Domineering ===
=== Domineering ===
Line 1,287: Line 1,137:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.240
* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 698.776
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}}


{{Optimal ET sequence|legend=0| 5e, 7, 12, 19d, 43de }}
{{Optimal ET sequence|legend=0| 5e, 7, 12 }}


Badness: 0.021978
Badness (Sintel): 0.727


==== 13-limit ====
==== 13-limit ====
Line 1,302: Line 1,152:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 695.315
* WE: ~2 = 1198.1958{{c}}, ~3/2 = 694.7159{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 695.762
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.6809{{c}}


{{Optimal ET sequence|legend=0| 5ef, 7, 12, 19d, 31def }}
{{Optimal ET sequence|legend=0| 7, 12 }}


Badness: 0.027039
Badness (Sintel): 1.12


===== 17-limit =====
===== 17-limit =====
Line 1,317: Line 1,167:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 695.894
* WE: ~2 = 1197.7959{{c}}, ~3/2 = 694.8362{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.115
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.0834{{c}}


{{Optimal ET sequence|legend=0| 5ef, 7, 12, 19d, 31def }}
{{Optimal ET sequence|legend=0| 7, 12 }}


Badness: 0.024539
Badness (Sintel): 1.25


===== 19-limit =====
===== 19-limit =====
Line 1,332: Line 1,182:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.139
* WE: ~2 = 1197.6198{{c}}, ~3/2 = 694.8362{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.217
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2075{{c}}


{{Optimal ET sequence|legend=0| 5ef, 7, 12, 19d, 31def }}
{{Optimal ET sequence|legend=0| 5ef, 7, 12, 19d, 31def }}


Badness: 0.020398
Badness (Sintel): 1.24


==== Dominatrix ====
==== Dominatrix ====
Line 1,347: Line 1,197:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 694.840
* WE: ~2 = 1193.1574{{c}}, ~3/2 = 694.5610{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 698.544
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.7268{{c}}


{{Optimal ET sequence|legend=0| 5e, 7, 12f, 19df }}
{{Optimal ET sequence|legend=0| 5e, 7, 12f }}


Badness: 0.018289
Badness (Sintel): 0.756


=== Domination ===
=== Domination ===
Line 1,362: Line 1,212:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 703.268
* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 705.004
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}


{{Optimal ET sequence|legend=0| 5e, 12e, 17c, 46cd }}
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}


Badness: 0.036562
Badness (Sintel): 1.21


==== 13-limit ====
==== 13-limit ====
Line 1,377: Line 1,227:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 703.719
* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 705.496
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}


{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}


Badness: 0.027435
Badness (Sintel): 1.13


=== Arnold ===
=== Arnold ===
Line 1,392: Line 1,242:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 698.546
* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 698.491
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}}


{{Optimal ET sequence|legend=0| 5, 7, 12e }}
{{Optimal ET sequence|legend=0| 5, 7, 12e }}


Badness: 0.026141
Badness (Sintel): 0.864


==== 13-limit ====
==== 13-limit ====
Line 1,407: Line 1,257:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 695.929
* WE: ~2 = 1197.8123{{c}}, ~3/2 = 695.4727{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.743
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.5713{{c}}


{{Optimal ET sequence|legend=0| 5, 7, 12ef, 19def }}
{{Optimal ET sequence|legend=0| 5, 7 }}


Badness: 0.023300
Badness (Sintel): 0.963


==== 17-limit ====
==== 17-limit ====
Line 1,422: Line 1,272:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.683
* WE: ~2 = 1197.6327{{c}}, ~3/2 = 695.6030{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.978
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9316{{c}}


{{Optimal ET sequence|legend=0| 5, 7, 12ef, 19def }}
{{Optimal ET sequence|legend=0| 5, 7 }}


Badness: 0.024535
Badness (Sintel): 1.25


==== 19-limit ====
==== 19-limit ====
Line 1,437: Line 1,287:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 696.996
* WE: ~2 = 1197.5295{{c}}, ~3/2 = 695.6325{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 697.068
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0579{{c}}


{{Optimal ET sequence|legend=0| 5, 7, 12ef, 19def }}
{{Optimal ET sequence|legend=0| 5, 7, 12ef, 19def }}


Badness: 0.021098
Badness (Sintel): 1.28


== Sharptone ==
== Sharptone ==
Sharptone is a low-accuracy temperament tempering out [[21/20]] and [[28/27]]. In sharptone, 7/4 is a major sixth, 7/6 a whole tone, and 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.
Sharptone is a low-accuracy temperament tempering out [[21/20]] and [[28/27]]. In sharptone, 7/4 is a major sixth, 7/6 a whole tone, and 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.
However, while 12edo ends up near-optimal, the only valid [[diamond monotone]] tuning for sharptone is [[5edo]]. Anything flat of it has ~12/7 and ~7/4 in the wrong order (and so should be dominant) and anything sharp of it has ~5/4 and ~4/3 in the wrong order (and so should not be meantone).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,454: Line 1,306:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 703.732
* [[WE]]: ~2 = 1204.2961{{c}}, ~3/2 = 702.6463{{c}}
: [[error map]]: {{val| 0.000 +1.777 +28.614 -57.630 }}
: [[error map]]: {{val| +4.296 +4.987 +24.271 -56.591 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 700.140
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4928{{c}}
: error map: {{val| 0.000 -1.815 +14.245 -68.407 }}
: error map: {{val| 0.000 -0.462 +19.657 -64.347 }}


{{Optimal ET sequence|legend=1| 5, 7d, 12d }}
{{Optimal ET sequence|legend=1| 5, 7d, 12d }}


[[Badness]]: 0.024848
[[Badness]] (Sintel): 0.629


=== Meanertone ===
=== Meanertone ===
Line 1,471: Line 1,323:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 702.730
* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 696.615
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}}


{{Optimal ET sequence|legend=0| 5, 7d, 12de }}
{{Optimal ET sequence|legend=0| 5, 7d, 12de }}


Badness: 0.025167
Badness (Sintel): 0.832


== Supermean ==
== Supermean ==
Line 1,486: Line 1,338:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 703.811
* [[WE]]: ~2 = 1195.4372{{c}}, ~3/2 = 702.2086{{c}}
: [[error map]]: {{val| 0.000 +1.856 +28.929 -11.665 }}
: [[error map]]: {{val| -4.563 -4.309 +22.521 -8.319 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 704.889
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5375{{c}}
: error map: {{val| 0.000 +2.934 +33.242 +4.507 }}
: error map: {{val| 0.000 +2.583 +31.836 -0.763 }}


{{Optimal ET sequence|legend=1| 5d, 12d, 17c, 29c }}
{{Optimal ET sequence|legend=1| 5d, 12d, 17c }}


[[Badness]]: 0.134204
[[Badness]] (Sintel): 3.40


=== 11-limit ===
=== 11-limit ===
Line 1,503: Line 1,355:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 704.016
* WE: ~2 = 1195.7270{{c}}, ~3/2 = 702.5848{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 705.096
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7471{{c}}


{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }}
{{Optimal ET sequence|legend=0| 5de, 12de, 17c }}


Badness: 0.063262
Badness (Sintel): 2.09


=== 13-limit ===
=== 13-limit ===
Line 1,518: Line 1,370:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 704.121
* WE: ~2 = 1196.3958{{c}}, ~3/2 = 702.9766{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 705.094
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7940{{c}}


{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }}
{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }}


Badness: 0.040324
Badness (Sintel): 1.67


== Mohajira ==
== Mohajira ==
Line 1,539: Line 1,391:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~128/105 = 348.437
* [[WE]]: ~2 = 1200.8160{{c}}, ~128/105 = 348.6518{{c}}
: [[error map]]: {{val| 0.000 -5.080 +1.186 -1.637 }}
: [[error map]]: {{val| +0.816 -3.835 +2.901 +0.900 }}
* [[POTE]]: ~2 = 1200.000, ~128/105 = 348.415
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 348.4194{{c}}
: error map: {{val| 0.000 -5.125 +1.005 -1.390 }}
: error map: {{val| 0.000 -5.116 +1.041 -1.439 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 1,558: Line 1,410:
{{Optimal ET sequence|legend=1| 7, 24, 31 }}
{{Optimal ET sequence|legend=1| 7, 24, 31 }}


[[Badness]]: 0.055714
[[Badness]] (Sintel): 1.41


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 1,570: Line 1,422:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/9 = 348.561
* WE: ~2 = 1201.1562{{c}}, ~11/9 = 348.8124{{c}}
* POTE: ~2 = 1200.000, ~11/9 = 348.477
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.4910{{c}}


Minimax tuning:  
Minimax tuning:  
Line 1,584: Line 1,436:
{{Optimal ET sequence|legend=0| 7, 24, 31 }}
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


Badness: 0.026064
Badness (Sintel): 0.862


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 1,596: Line 1,448:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/9 = 348.587
* WE: ~2 = 1200.4256{{c}}, ~11/9 = 348.6819{{c}}
* POTE: ~2 = 1200.000, ~11/9 = 348.558
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5622{{c}}


{{Optimal ET sequence|legend=0| 7, 24, 31 }}
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


Badness: 0.023388
Badness (Sintel): 0.966


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 1,613: Line 1,465:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/9 = 348.735
* WE: ~2 = 1200.0382{{c}}, ~11/9 = 348.7471{{c}}
* POTE: ~2 = 1200.000, ~11/9 = 348.736
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.7360{{c}}


{{Optimal ET sequence|legend=0| 7, 24, 31 }}
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


Badness: 0.020576
Badness (Sintel): 1.05


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 1,630: Line 1,482:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/9 = 348.821
* WE: ~2 = 1199.7469{{c}}, ~11/9 = 348.7367{{c}}
* POTE: ~2 = 1200.000, ~11/9 = 348.810
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.8117{{c}}


{{Optimal ET sequence|legend=0| 7, 24, 31, 55 }}
{{Optimal ET sequence|legend=0| 7, 24, 31, 55 }}


Badness: 0.017302
Badness (Sintel): 1.05


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 1,649: Line 1,501:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~25/21 = 350.352
* [[WE]]: ~2 = 1199.0661{{c}}, ~25/21 = 350.3127{{c}}
: [[error map]]: {{val| 0.000 -1.250 +16.505 -14.245 }}
: [[error map]]: {{val| -0.934 -2.264 +16.188 -13.827 }}
* [[POTE]]: ~2 = 1200.000, ~25/21 = 350.586
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 350.4856{{c}}
: error map: {{val| 0.000 -0.784 +18.370 -11.214 }}
: error map: {{val| 0.000 -0.984 +17.571 -12.513 }}


{{Optimal ET sequence|legend=1| 7d, 17c, 24 }}
{{Optimal ET sequence|legend=1| 7d, 17c, 24 }}


[[Badness]]: 0.077734
[[Badness]] (Sintel): 1.97


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 1,668: Line 1,520:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/9 = 350.347
* WE: ~2 = 1199.1924{{c}}, ~11/9 = 350.3286{{c}}
* POTE: ~2 = 1200.000, ~11/9 = 350.565
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4821{{c}}


{{Optimal ET sequence|legend=0| 7d, 17c, 24 }}
{{Optimal ET sequence|legend=0| 7d, 17c, 24 }}


Badness: 0.036207
Badness (Sintel): 1.20


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 1,685: Line 1,537:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/9 = 350.365
* WE: ~2 = 1198.5986{{c}}, ~11/9 = 350.3353{{c}}
* POTE: ~2 = 1200.000, ~11/9 = 350.745
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.6459{{c}}


{{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }}
{{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }}


Badness: 0.028738
Badness (Sintel): 1.19


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 1,708: Line 1,560:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~10/7 = 632.783
* [[WE]]: ~2 = 1201.5548{{c}}, ~10/7 = 633.2251{{c}}
: [[error map]]: {{val| 0.000 -3.606 +7.084 -8.212 }}
: [[error map]]: {{val| +1.555 -2.280 +6.168 -8.015 }}
* [[POTE]]: ~2 = 1200.000, ~10/7 = 632.406
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 632.5640{{c}}
: error map: {{val| 0.000 -4.738 +2.554 -12.363 }}
: error map: {{val| 0.000 -4.263 +4.454 -10.622 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 1,722: Line 1,574:
{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}
{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}


[[Badness]]: 0.046706
[[Badness]] (Sintel): 1.18


=== Liesel ===
=== Liesel ===
Line 1,732: Line 1,584:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 632.812
* WE: ~2 = 1198.8507{{c}}, ~10/7 = 632.4668{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 633.073
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 632.9963{{c}}


{{Optimal ET sequence|legend=0| 17c, 19, 36 }}
{{Optimal ET sequence|legend=0| 17c, 19, 36 }}


Badness: 0.040721
Badness (Sintel): 1.35


==== 13-limit ====
==== 13-limit ====
Line 1,747: Line 1,599:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 632.925
* WE: ~2 = 1199.4968{{c}}, ~10/7 = 632.7766{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 633.042
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.0082{{c}}


{{Optimal ET sequence|legend=0| 17c, 19, 36, 91ceef }}
{{Optimal ET sequence|legend=0| 17c, 19, 36 }}


Badness: 0.027304
Badness (Sintel): 1.13


=== Elisa ===
=== Elisa ===
Line 1,762: Line 1,614:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 633.317
* WE: ~2 = 1201.0489{{c}}, ~10/7 = 633.6147{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 633.061
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1644{{c}}


{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}
{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}


Badness: 0.041592
Badness (Sintel): 1.37


==== 13-limit ====
==== 13-limit ====
Line 1,777: Line 1,629:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 633.370
* WE: ~2 = 1201.4815{{c}}, ~10/7 = 633.7720{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 632.991
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1281{{c}}


{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}
{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}


Badness: 0.026922
Badness (Sintel): 1.11


=== Lisa ===
=== Lisa ===
Line 1,792: Line 1,644:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 631.981
* WE: ~2 = 1202.6773{{c}}, ~10/7 = 632.7783{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 631.370
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.6175{{c}}


{{Optimal ET sequence|legend=0| 17cee, 19 }}
{{Optimal ET sequence|legend=0| 17cee, 19 }}


Badness: 0.054829
Badness (Sintel): 1.81


==== 13-limit ====
==== 13-limit ====
Line 1,807: Line 1,659:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 632.093
* WE: ~2 = 1203.6086{{c}}, ~10/7 = 633.1193{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 631.221
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.5346{{c}}


{{Optimal ET sequence|legend=0| 17cee, 19 }}
{{Optimal ET sequence|legend=0| 17cee, 19 }}


Badness: 0.036144
Badness (Sintel): 1.49


== Superpine ==
== Superpine ==
{{see also|No-sevens subgroup temperaments #Superpine}}
{{See also| No-sevens subgroup temperaments #Superpine }}


The superpine temperament is generated by 1/3 of a fourth, represented by [[~]][[35/32]], which resembles [[porcupine]], but it favors flat fifths instead of sharp ones. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent [[6/5]]–harmonics other than 3 all require the 15-tone mos to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as [[11/10]] as in porcupine, which makes [[11/8]] high-[[complexity]] like the other harmonics, but in the 13-limit 5 generators up closely approximates [[13/8]]. [[43edo]] is a good tuning especially for the higher-limit extensions.
The superpine temperament is generated by 1/3 of a fourth, represented by [[~]][[35/32]], which resembles [[porcupine]], but it favors flat fifths instead of sharp ones. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent [[6/5]]–harmonics other than 3 all require the 15-tone mos to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as [[11/10]] as in porcupine, which makes [[11/8]] high-[[complexity]] like the other harmonics, but in the 13-limit 5 generators up closely approximates [[13/8]]. [[43edo]] is a good tuning especially for the higher-limit extensions.
Line 1,826: Line 1,678:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~35/32 = 167.279
* [[WE]]: ~2 = 1199.3652{{c}}, ~35/32 = 167.1615{{c}}
: [[error map]]: {{val| 0.000 -3.793 +6.336 +5.804 }}
: [[error map]]: {{val| -0.635 -4.709 +5.209 +3.639 }}
* [[CWE]]: ~2 = 1200.000, ~35/32 = 167.256
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/32 = 167.2561{{c}}
: error map: {{val| 0.000 -3.723 +6.613 +5.503 }}
: error map: {{val| 0.000 -3.723 +6.613 +5.503 }}


{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }}
{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }}


[[Badness]]: 0.137
[[Badness]] (Sintel): 3.46


=== 11-limit ===
=== 11-limit ===
Line 1,843: Line 1,695:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 167.407
* WE: ~2 = 1199.0522{{c}}, ~11/10 = 167.1904{{c}}
* CWE: ~2 = 1200.000, ~11/10 = 167.338
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3382{{c}}


{{Optimal ET sequence|legend=0| 7, 36, 43 }}
{{Optimal ET sequence|legend=0| 7, 36, 43 }}


Badness: 0.0576
Badness (Sintel): 1.90


=== 13-limit ===
=== 13-limit ===
Line 1,858: Line 1,710:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 167.427
* WE: ~2 = 1199.4286{{c}}, ~11/10 = 167.3105{{c}}
* CWE: ~2 = 1200.000, ~11/10 = 167.396
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3958{{c}}


{{Optimal ET sequence|legend=0| 7, 36, 43 }}
{{Optimal ET sequence|legend=0| 7, 36, 43 }}


Badness: 0.0368
Badness (Sintel): 1.52


== Lithium ==
== Lithium ==
Line 1,877: Line 1,729:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~56/45 = 400.000, ~3/2 = 695.827 (~15/14 = 104.173)
* [[WE]]: ~56/45 = 400.6744{{c}}, ~3/2 = 695.8474{{c}} {~15/14 = 105.5015{{c}})
: [[error map]]: {{val| 0.000 -6.128 -3.007 +6.135 }}
: [[error map]]: {{val| +2.023 -4.084 -2.924 +4.910 }}
* [[CWE]]: ~56/45 = 400.000, ~3/2 = 695.141 {~15/14 = 104.859)
* [[CWE]]: ~56/45 = 400.0000{{c}}, ~3/2 = 695.1413{{c}} {~15/14 = 104.8587{{c}})
: error map: {{val| 0.000 -6.814 -5.748 +2.022 }}
: error map: {{val| 0.000 -6.814 -5.748 +2.022 }}


{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}
{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}


[[Badness]]: 0.0692
[[Badness]] (Sintel): 1.75


== Squares ==
== Squares ==
Line 1,895: Line 1,747:
[[Comma list]]: 81/80, 2401/2400
[[Comma list]]: 81/80, 2401/2400


{{Mapping|legend=1| 1 3 8 6 | 0 -4 -16 -9 }}
{{Mapping|legend=1| 1 -1 -8 -3 | 0 4 16 9 }}


: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~14/9


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~9/7 = 425.695
* [[WE]]: ~2 = 1201.2488{{c}}, ~14/9 = 774.8640{{c}}
: [[error map]]: {{val| 0.000 -4.734 +2.570 -0.079 }}
: [[error map]]: {{val| +1.249 -3.748 +1.520 +1.204 }}
* [[POTE]]: ~2 = 1200.000, ~9/7 = 425.942
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 774.1560{{c}}
: error map: {{val| 0.000 -5.721 -1.378 -2.300 }}
: error map: {{val| 0.000 -5.331 +0.183 -1.422 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 1,914: Line 1,766:
{{Optimal ET sequence|legend=1| 14c, 17c, 31, 169b, 200b }}
{{Optimal ET sequence|legend=1| 14c, 17c, 31, 169b, 200b }}


[[Badness]]: 0.045993
[[Badness]] (Sintel): 1.16


Scales: [[skwares8]], [[skwares11]], [[skwares14]]
Scales: [[skwares8]], [[skwares11]], [[skwares14]]
Line 1,923: Line 1,775:
Comma list: 81/80, 99/98, 121/120
Comma list: 81/80, 99/98, 121/120


Mapping: {{mapping| 1 3 8 6 7 | 0 -4 -16 -9 -10 }}
Mapping: {{mapping| 1 -1 -8 -3 -3 | 0 4 16 9 10 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 425.599
* WE: ~2 = 1201.6657{{c}}, ~11/7 = 775.1171{{c}}
* POTE: ~2 = 1200.000, ~9/7 = 425.957
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.1754{{c}}


{{Optimal ET sequence|legend=0| 14c, 17c, 31, 130bee, 169beee }}
{{Optimal ET sequence|legend=0| 14c, 17c, 31, 130bee, 169beee }}


Badness: 0.021636
Badness (Sintel): 0.715


==== 13-limit ====
==== 13-limit ====
Line 1,938: Line 1,790:
Comma list: 66/65, 81/80, 99/98, 121/120
Comma list: 66/65, 81/80, 99/98, 121/120


Mapping: {{mapping| 1 3 8 6 7 3 | 0 -4 -16 -9 -10 2 }}
Mapping: {{mapping| 1 -1 -8 -3 -3 5 | 0 4 16 9 10 -2 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 425.578
* WE: ~2 = 1199.8419{{c}}, ~11/7 = 774.3484{{c}}
* POTE: ~2 = 1200.000, ~9/7 = 425.550
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4422{{c}}


{{Optimal ET sequence|legend=0| 14c, 17c, 31, 79cf }}
{{Optimal ET sequence|legend=0| 14c, 17c, 31, 79cf }}


Badness: 0.025514
Badness (Sintel): 1.05


==== Squad ====
==== Squad ====
Line 1,953: Line 1,805:
Comma list: 78/77, 81/80, 91/90, 99/98
Comma list: 78/77, 81/80, 91/90, 99/98


Mapping: {{mapping| 1 3 8 6 7 9 | 0 -4 -16 -9 -10 -15 }}
Mapping: {{mapping| 1 -1 -8 -3 -3 -6 | 0 4 16 9 10 15 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 425.297
* WE: ~2 = 1202.0312{{c}}, ~11/7 = 775.5589{{c}}
* POTE: ~2 = 1200.000, ~9/7 = 425.752
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4140{{c}}


{{Optimal ET sequence|legend=0| 14cf, 17c, 31f }}
{{Optimal ET sequence|legend=0| 14cf, 17c, 31f }}


Badness: 0.026877
Badness (Sintel): 1.11


==== Agora ====
==== Agora ====
Line 1,968: Line 1,820:
Comma list: 81/80, 99/98, 105/104, 121/120
Comma list: 81/80, 99/98, 105/104, 121/120


Mapping: {{mapping| 1 3 8 6 7 14 | 0 -4 -16 -9 -10 -29 }}
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 | 0 4 16 9 10 29 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 425.870
* WE: ~2 = 1202.3228{{c}}, ~11/7 = 775.2214{{c}}
* POTE: ~2 = 1200.000, ~9/7 = 426.276
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8617{{c}}


{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 76e }}
{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 76e }}


Badness: 0.024522
Badness (Sintel): 1.01


===== 17-limit =====
===== 17-limit =====
Line 1,983: Line 1,835:
Comma list: 81/80, 99/98, 105/104, 120/119, 121/119
Comma list: 81/80, 99/98, 105/104, 120/119, 121/119


Mapping: {{mapping| 1 3 8 6 7 14 8 | 0 -4 -16 -9 -10 -29 -11 }}
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 | 0 4 16 9 10 29 11 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 425.919
* WE: ~2 = 1201.4340{{c}}, ~11/7 = 774.7375{{c}}
* POTE: ~2 = 1200.000, ~9/7 = 426.187
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8955{{c}}


{{Optimal ET sequence|legend=0| 14cf, 31 }}
{{Optimal ET sequence|legend=0| 14cf, 31 }}


Badness: 0.022573
Badness (Sintel): 1.15


===== 19-limit =====
===== 19-limit =====
Line 1,998: Line 1,850:
Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119
Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119


Mapping: {{mapping| 1 3 8 6 7 14 8 11 | 0 -4 -16 -9 -10 -29 -11 -19 }}
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 -8 | 0 4 16 9 10 29 11 19 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 425.984
* WE: ~2 = 1201.2461{{c}}, ~11/7 = 774.5783{{c}}
* POTE: ~2 = 1200.000, ~9/7 = 426.225
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8479{{c}}


{{Optimal ET sequence|legend=0| 14cf, 31 }}
{{Optimal ET sequence|legend=0| 14cf, 31 }}


Badness: 0.018839
Badness (Sintel): 1.15


=== Cuboctahedra ===
=== Cuboctahedra ===
Line 2,013: Line 1,865:
Comma list: 81/80, 385/384, 1375/1372
Comma list: 81/80, 385/384, 1375/1372


Mapping: {{mapping| 1 3 8 6 -4 | 0 -4 -16 -9 21 }}
Mapping: {{mapping| 1 -1 -8 -3 17 | 0 4 16 9 -21 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 425.899
* WE: ~2 = 1201.4436{{c}}, ~14/9 = 774.9386{{c}}
* POTE: ~2 = 1200.000, ~9/7 = 425.993
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.0243{{c}}


{{Optimal ET sequence|legend=0| 31, 107b, 138b, 169be, 200be }}
{{Optimal ET sequence|legend=0| 31, 107b, 138b, 169be, 200be }}


Badness: 0.056826
Badness (Sintel): 1.88


== Jerome ==
== Jerome ==
Line 2,035: Line 1,887:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~54/49 = 139.371
* [[WE]]: ~2 = 1200.1640{{c}}, ~54/49 = 139.3624{{c}}
: [[error map]]: {{val| 0.000 -5.098 +1.114 +6.774 }}
: [[error map]]: {{val| +0.164 -4.979 +0.934 +7.039 }}
* [[POTE]]: ~2 = 1200.000, ~54/49 = 139.343
* [[CWE]]: ~2 = 1200.0000{{c}}, ~54/49 = 139.3528{{c}}
: error map: {{val| 0.000 -5.238 +0.553 +6.577 }}
: error map: {{val| 0.000 -5.191 +0.741 +6.643 }}


{{Optimal ET sequence|legend=1| 17c, 26, 43 }}
{{Optimal ET sequence|legend=1| 17c, 26, 43 }}


[[Badness]]: 0.108656
[[Badness]] (Sintel): 2.75


=== 11-limit ===
=== 11-limit ===
Line 2,052: Line 1,904:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~12/11 = 139.349
* WE: ~2 = 1201.4436{{c}}, ~12/11 = 139.3714{{c}}
* POTE: ~2 = 1200.000, ~12/11 = 139.428
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.4038{{c}}


{{Optimal ET sequence|legend=0| 17c, 26, 43 }}
{{Optimal ET sequence|legend=0| 17c, 26, 43 }}


Badness: 0.047914
Badness (Sintel): 1.58


=== 13-limit ===
=== 13-limit ===
Line 2,067: Line 1,919:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/12 = 139.369
* WE: ~2 = 1199.8860{{c}}, ~13/12 = 139.3737{{c}}
* POTE: ~2 = 1200.000, ~13/12 = 139.387
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3817{{c}}


{{Optimal ET sequence|legend=0| 17c, 26, 43 }}
{{Optimal ET sequence|legend=0| 17c, 26, 43 }}


Badness: 0.029285
Badness (Sintel): 1.21


=== 17-limit ===
=== 17-limit ===
Line 2,082: Line 1,934:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/12 = 139.334
* WE: ~2 = 1199.8346{{c}}, ~13/12 = 139.3431{{c}}
* POTE: ~2 = 1200.000, ~13/12 = 139.362
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3544{{c}}


{{Optimal ET sequence|legend=0| 17cg, 26, 43 }}
{{Optimal ET sequence|legend=0| 17cg, 26, 43 }}


Badness: 0.020878
Badness (Sintel): 1.06


=== 19-limit ===
=== 19-limit ===
Line 2,097: Line 1,949:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/12 = 139.296
* WE: ~2 = 1199.8891{{c}}, ~13/12 = 139.3001{{c}}
* POTE: ~2 = 1200.000, ~13/12 = 139.313
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3080{{c}}


{{Optimal ET sequence|legend=0| 17cgh, 26, 43, 69 }}
{{Optimal ET sequence|legend=0| 17cgh, 26, 43, 69 }}


Badness: 0.018229
Badness (Sintel): 1.11


== Meantritone ==
== Meantritone ==
Line 2,111: Line 1,963:
[[Comma list]]: 81/80, 16875/16807
[[Comma list]]: 81/80, 16875/16807


{{Mapping|legend=1| 1 4 12 12 | 0 -5 -20 -19 }}
{{Mapping|legend=1| 1 -1 -8 -7 | 0 5 20 19 }}
 
: mapping generators: ~2, ~10/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~7/5 = 580.568
* [[WE]]: ~2 = 1201.3832{{c}}, ~10/7 = 619.9478{{c}}
: [[error map]]: {{val| 0.000 -4.795 +2.325 +0.381 }}
: [[error map]]: {{val| +1.383 -3.599 +1.576 +0.499 }}
* [[POTE]]: ~2 = 1200.000, ~7/5 = 580.766
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.3176{{c}}
: error map: {{val| 0.000 -5.785 -1.634 -3.380 }}
: error map: {{val| 0.000 -5.367 +0.038 -1.791 }}


{{Optimal ET sequence|legend=1| 29cd, 31, 188bcd, 219bbcd }}
{{Optimal ET sequence|legend=1| 29cd, 31, 188bcd, 219bbcd }}


[[Badness]]: 0.082239
[[Badness]] (Sintel): 2.08


=== 11-limit ===
=== 11-limit ===
Line 2,128: Line 1,982:
Comma list: 81/80, 99/98, 2541/2500
Comma list: 81/80, 99/98, 2541/2500


Mapping: {{mapping| 1 4 12 12 17 | 0 -5 -20 -19 -28 }}
Mapping: {{mapping| 1 -1 -8 -7 -11 | 0 5 20 19 28 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/5 = 580.482
* WE: ~2 = 1201.2054{{c}}, ~10/7 = 619.9752{{c}}
* POTE: ~2 = 1200.000, ~7/5 = 580.647
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.4223{{c}}


{{Optimal ET sequence|legend=0| 29cde, 31 }}
{{Optimal ET sequence|legend=0| 29cde, 31 }}


Badness: 0.042869
Badness (Sintel): 1.42


== Injera ==
== Injera ==
Line 2,152: Line 2,006:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000, ~3/2 = 695.330 (~21/20 = 95.330)
* [[WE]]: ~7/5 = 600.6662{{c}}, ~3/2 = 695.1463{{c}} (~21/20 = 94.4801{{c}})
: [[error map]]: {{val| 0.000 -6.625 -4.993 +12.495 }}
: [[error map]]: {{val| +1.332 -5.476 -5.729 +12.425 }}
* [[POTE]]: ~7/5 = 600.000, ~3/2 = 694.375 (~21/20 = 94.375)
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 694.7712{{c}} (~21/20 = 94.7712{{c}})
: error map: {{val| 0.000 -7.580 -8.813 +8.675 }}
: error map: {{val| 0.000 -7.184 -7.229 +10.259 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 2,164: Line 2,018:
{{Optimal ET sequence|legend=1| 12, 26, 38 }}
{{Optimal ET sequence|legend=1| 12, 26, 38 }}


[[Badness]]: 0.031130
[[Badness]] (Sintel): 0.788


; Music
; Music
Line 2,177: Line 2,031:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~3/2 = 694.098 (~21/20 = 94.098)
* WE: ~7/5 = 600.9350{{c}}, ~3/2 = 693.9198{{c}} (~21/20 = 92.9848{{c}})
* POTE: ~7/5 = 600.000, ~3/2 = 692.840 (~21/20 = 92.840)
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.3539{{c}} (~21/20 = 93.3539{{c}})


Tuning ranges:  
Tuning ranges:  
Line 2,186: Line 2,040:
{{Optimal ET sequence|legend=0| 12, 26 }}
{{Optimal ET sequence|legend=0| 12, 26 }}


Badness: 0.023124
Badness (Sintel): 0.764


==== 13-limit ====
==== 13-limit ====
Line 2,196: Line 2,050:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~3/2 = 693.806 (~21/20 = 93.806)
* WE: ~7/5 = 600.9982{{c}}, ~3/2 = 693.8249{{c}} (~21/20 = 92.8267{{c}})
* POTE: ~7/5 = 600.000, ~3/2 = 692.673 (~21/20 = 92.673)
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.0992{{c}} (~21/20 = 93.0992{{c}})


Tuning ranges:  
Tuning ranges:  
Line 2,205: Line 2,059:
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}


Badness: 0.021565
Badness (Sintel): 0.891


===== 17-limit =====
===== 17-limit =====
Line 2,215: Line 2,069:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~3/2 = 693.852 (~21/20 = 93.852)
* WE: ~7/5 = 601.1757{{c}}, ~3/2 = 693.8441{{c}} (~21/20 = 92.6684{{c}})
* POTE: ~7/5 = 600.000, ~3/2 = 692.487 (~21/20 = 92.487)
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 692.8879{{c}} (~21/20 = 92.8879{{c}})


{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}


Badness: 0.018358
Badness (Sintel): 0.935


===== 19-limit =====
===== 19-limit =====
Line 2,230: Line 2,084:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~3/2 = 694.031 (~21/20 = 94.031)
* WE: ~7/5 = 601.4245{{c}}, ~3/2 = 693.9426{{c}} (~21/20 = 92.5181{{c}})
* POTE: ~7/5 = 600.000, ~3/2 = 692.299 (~21/20 = 92.299)
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 692.7606{{c}} (~21/20 = 92.7606{{c}})


{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}


Badness: 0.015118
Badness (Sintel): 0.920


==== Enjera ====
==== Enjera ====
Line 2,245: Line 2,099:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~3/2 = 693.088 (~21/20 = 93.088)
* WE: ~7/5 = 599.1863{{c}}, ~3/2 = 693.1791{{c}} (~21/20 = 93.9929{{c}})
* POTE: ~7/5 = 600.000, ~3/2 = 694.121 (~21/20 = 94.121)
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.6809{{c}} (~21/20 = 93.6809{{c}})


{{Optimal ET sequence|legend=0| 10cdeef, 12f }}
{{Optimal ET sequence|legend=0| 10cdeef, 12f }}


Badness: 0.026542
Badness (Sintel): 1.10


=== Injerous ===
=== Injerous ===
Line 2,260: Line 2,114:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~3/2 = 694.619 (~21/20 = 94.619)
* WE: ~7/5 = 603.1682{{c}}, ~3/2 = 694.1945{{c}} (~21/20 = 91.0264{{c}})
* POTE: ~7/5 = 600.000, ~3/2 = 690.548 (~21/20 = 90.548)
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 691.6107{{c}} (~21/20 = 91.6107{{c}})


{{Optimal ET sequence|legend=0| 12e, 14c, 26e, 40cee }}
{{Optimal ET sequence|legend=0| 12e, 14c, 26e, 40cee }}


Badness: 0.038577
Badness (Sintel): 1.28


=== Lahoh ===
=== Lahoh ===
Line 2,275: Line 2,129:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~3/2 = 695.330 (~21/20 = 95.330)
* WE: ~7/5 = 597.3179{{c}}, ~3/2 = 695.8759{{c}} (~21/20 = 98.5581{{c}})
* POTE: ~7/5 = 600.000, ~3/2 = 699.001 (~21/20 = 99.001)
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 697.8757{{c}} (~21/20 = 97.8757{{c}})


{{Optimal ET sequence|legend=0| 10cd, 12 }}
{{Optimal ET sequence|legend=0| 10cd, 12 }}


Badness: 0.043062
Badness (Sintel): 1.42


=== Teff ===
=== Teff ===
Line 2,296: Line 2,150:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~11/8 = 552.331 (~33/32 = 47.669)
* WE: ~7/5 = 600.2802{{c}}, ~16/11 = 647.7720{{c}} (~33/32 = 47.4918{{c}})
* POTE: ~7/5 = 600.000, ~11/8 = 552.530 (~33/32 = 47.470)
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.5224{{c}} (~33/32 = 47.5224{{c}})


{{Optimal ET sequence|legend=0| 24d, 26, 50d }}
{{Optimal ET sequence|legend=0| 24d, 26, 50d }}


Badness: 0.070689
Badness (Sintel): 2.34


==== 13-limit ====
==== 13-limit ====
Line 2,311: Line 2,165:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~11/8 = 552.297 (~33/32 = 47.703)
* WE: ~7/5 = 600.3037{{c}}, ~16/11 = 647.7954{{c}} (~33/32 = 47.4917{{c}})
* POTE: ~7/5 = 600.000, ~11/8 = 552.532 (~33/32 = 47.468)
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.5256{{c}} (~33/32 = 47.5256{{c}})


{{Optimal ET sequence|legend=0| 24d, 26, 50d }}
{{Optimal ET sequence|legend=0| 24d, 26, 50d }}


Badness: 0.040047
Badness (Sintel): 1.65


==== 17-limit ====
==== 17-limit ====
Line 2,326: Line 2,180:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~11/8 = 552.249 (~34/33 = 47.751)
* WE: ~7/5 = 600.5123{{c}}, ~16/11 = 647.8970{{c}} (~34/33 = 47.3846{{c}})
* POTE: ~7/5 = 600.000, ~11/8 = 552.656 (~34/33 = 47.344)
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.4314{{c}} (~34/33 = 47.4314{{c}})


{{Optimal ET sequence|legend=0| 24d, 26 }}
{{Optimal ET sequence|legend=0| 24d, 26 }}


Badness: 0.029499
Badness (Sintel): 1.50


==== 19-limit ====
==== 19-limit ====
Line 2,341: Line 2,195:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~11/8 = 552.107 (~34/33 = 47.893)
* WE: ~7/5 = 600.6308{{c}}, ~16/11 = 648.0424{{c}} (~34/33 = 47.4116{{c}})
* POTE: ~7/5 = 600.000, ~11/8 = 552.638 (~34/33 = 47.362)
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.4715{{c}} (~34/33 = 47.4715{{c}})


{{Optimal ET sequence|legend=0| 24d, 26 }}
{{Optimal ET sequence|legend=0| 24d, 26 }}


Badness: 0.023133
Badness (Sintel): 1.41


== Pombe ==
== Pombe ==
Line 2,360: Line 2,214:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~735/512 = 600.000, ~48/35 = 551.845 (~36/35 = 48.155)
* [[WE]]: ~735/512 = 601.0652{{c}}, ~35/24 = 648.9295{{c}} (~36/35 = 47.8642{{c}})
: [[error map]]: {{val| 0.000 -5.645 -1.075 -9.600 }}
: [[error map]]: {{val| +2.130 -3.031 +0.861 -1.756 }}
* [[POTE]]: ~735/512 = 600.000, ~48/35 = 552.221 (~36/35 = 48.779)
* [[CWE]]: ~735/512 = 600.0000{{c}}, ~35/24 = 647.8628{{c}} (~36/35 = 47.8628{{c}})
: error map: {{val| 0.000 -6.229 -3.411 -8.140 }}
: error map: {{val| 0.000 -6.229 -3.411 -8.140 }}


{{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}
{{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}


[[Badness]]: 0.116104
[[Badness]] (Sintel): 2.94


=== 11-limit ===
=== 11-limit ===
Line 2,377: Line 2,231:


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000, ~11/8 = 551.843 (~36/35 = 48.157)
* WE: ~99/70 = 600.7890{{c}}, ~16/11 = 648.7592{{c}} (~36/35 = 47.9701{{c}})
* POTE: ~99/70 = 600.000, ~11/8 = 552.093 (~36/35 = 47.907)
* CWE: ~99/70 = 600.0000{{c}}, ~16/11 = 647.9516{{c}} (~36/35 = 47.9516{{c}})


{{Optimal ET sequence|legend=0| 24, 26, 50 }}
{{Optimal ET sequence|legend=0| 24, 26, 50 }}


Badness: 0.052099
Badness (Sintel): 1.72


=== 13-limit ===
=== 13-limit ===
Line 2,392: Line 2,246:


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000, ~11/8 = 551.848 (~36/35 = 48.152)
* WE: ~99/70 = 600.6971{{c}}, ~16/11 = 648.6029{{c}} (~36/35 = 47.9058{{c}})
* POTE: ~99/70 = 600.000, ~11/8 = 552.150 (~36/35 = 47.850)
* CWE: ~99/70 = 600.0000{{c}}, ~16/11 = 647.8990{{c}} (~36/35 = 47.8990{{c}})


{{Optimal ET sequence|legend=0| 24, 26, 50 }}
{{Optimal ET sequence|legend=0| 24, 26, 50 }}


Badness: 0.031039
Badness (Sintel): 1.28


=== 17-limit ===
=== 17-limit ===
Line 2,407: Line 2,261:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~11/8 = 551.793 (~36/35 = 48.207)
* WE: ~17/12 = 600.7610{{c}}, ~16/11 = 648.6638{{c}} (~36/35 = 47.9028{{c}})
* POTE: ~17/12 = 600.000, ~11/8 = 552.158 (~36/35 = 47.842)
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 647.8990{{c}} (~36/35 = 47.8990{{c}})


{{Optimal ET sequence|legend=0| 24, 26, 50 }}
{{Optimal ET sequence|legend=0| 24, 26, 50 }}


Badness: 0.021260
Badness (Sintel): 1.08


=== 19-limit ===
=== 19-limit ===
Line 2,422: Line 2,276:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~11/8 = 551.660 (~36/35 = 48.340)
* WE: ~17/12 = 600.8048{{c}}, ~16/11 = 648.7494{{c}} (~36/35 = 47.9446{{c}})
* POTE: ~17/12 = 600.000, ~11/8 = 552.120 (~36/35 = 47.880)
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 647.9425{{c}} (~36/35 = 47.9425{{c}})


{{Optimal ET sequence|legend=0| 24, 26, 50 }}
{{Optimal ET sequence|legend=0| 24, 26, 50 }}


Badness: 0.016548
Badness (Sintel): 1.01


== Orphic ==
== Orphic ==
Line 2,439: Line 2,293:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2401/1728 = 600.000, ~343/288 = 324.275 (~7/6 = 275.725)
* [[WE]]: ~2401/1728 = 600.1767{{c}}, ~343/288 = 324.3015{{c}} (~7/6 = 275.8751{{c}})
: [[error map]]: {{val| 0.000 -4.854 +2.091 +4.000 }}
: [[error map]]: {{val| +0.353 -4.572 +1.804 +4.785 }}
* [[POTE]]: ~2401/1728 = 600.000, ~343/288 = 324.206 (~7/6 = 275.794)
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~343/288 = 324.2285{{c}} (~7/6 = 275.7715{{c}})
: error map: {{val| 0.000 -5.131 +0.983 +3.792 }}
: error map: {{val| 0.000 -5.041 +1.342 +3.860 }}


{{Optimal ET sequence|legend=1| 26, 48c, 74 }}
{{Optimal ET sequence|legend=1| 26, 48c, 74 }}


[[Badness]]: 0.258825
[[Badness]] (Sintel): 6.55


=== 11-limit ===
=== 11-limit ===
Line 2,453: Line 2,307:
Comma list: 81/80, 99/98, 73728/73205
Comma list: 81/80, 99/98, 73728/73205


Mapping: {{mapping| 2 1 -4 4 8 | 0 -4 -16 -3 2 }}
Mapping: {{mapping| 2 1 -4 4 8 | 0 4 16 3 -2 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~363/256 = 600.000, ~77/64 = 324.276 (~7/6 = 275.724)
* WE: ~363/256 = 600.1011{{c}}, ~77/64 = 324.2923{{c}} (~7/6 = 275.8088{{c}})
* POTE: ~363/256 = 600.000, ~77/64 = 324.238 (~7/6 = 275.762)
* CWE: ~363/256 = 600.0000{{c}}, ~77/64 = 324.2463{{c}} (~7/6 = 275.7537{{c}})


{{Optimal ET sequence|legend=0| 26, 48c, 74 }}
{{Optimal ET sequence|legend=0| 26, 48c, 74 }}


Badness: 0.101499
Badness (Sintel): 3.36


=== 13-limit ===
=== 13-limit ===
Line 2,468: Line 2,322:
Comma list: 81/80, 99/98, 144/143, 2200/2197
Comma list: 81/80, 99/98, 144/143, 2200/2197


Mapping: {{mapping| 2 1 -4 4 8 2 | 0 -4 -16 -3 2 -10 }}
Mapping: {{mapping| 2 1 -4 4 8 2 | 0 4 16 3 -2 10 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 600.000, ~77/64 = 324.250 (~7/6 = 275.750)
* WE: ~55/39 = 600.0540{{c}}, ~77/64 = 324.2551{{c}} (~7/6 = 275.7989{{c}})
* POTE: ~55/39 = 600.000, ~77/64 = 324.226 (~7/6 = 275.774)
* CWE: ~55/39 = 600.0000{{c}}, ~77/64 = 324.2307{{c}} (~7/6 = 275.7693{{c}})


{{Optimal ET sequence|legend=0| 26, 48c, 74 }}
{{Optimal ET sequence|legend=0| 26, 48c, 74 }}


Badness: 0.053482
Badness (Sintel): 2.21


== Cloudtone ==
== Cloudtone ==
The cloudtone temperament (5 & 50) tempers out the [[cloudy comma]], 16807/16384 and the [[81/80|syntonic comma]], 81/80 in the 7-limit. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.
The cloudtone temperament ({{nowrap| 5 & 50 }}) tempers out the [[cloudy comma]], 16807/16384 and the [[81/80|syntonic comma]], 81/80 in the 7-limit. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 2,490: Line 2,344:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~8/7 = 240.000, ~3/2 = 697.214 (~49/48 = 22.786)
* [[WE]]: ~8/7 = 240.4267{{c}}, ~3/2 = 696.9566{{c}} (~49/48 = 24.3235{{c}})
: [[error map]]: {{val| 0.000 -4.741 +2.544 -8.826 }}
: [[error map]]: {{val| +2.133 -2.865 +1.513 -2.852 }}
* [[POTE]]: ~8/7 = 240.000, ~3/2 = 695.720 (~49/48 = 24.280)
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~3/2 = 696.1637{{c}} (~49/48 = 23.8373{{c}})
: error map: {{val| 0.000 -6.235 -3.435 -8.826 }}
: error map: {{val| 0.000 -5.791 -1.659 -8.826 }}


{{Optimal ET sequence|legend=1| 5, 40c, 45, 50 }}
{{Optimal ET sequence|legend=1| 5, 40c, 45, 50 }}


[[Badness]]: 0.102256
[[Badness]] (Sintel): 2.59


=== 11-limit ===
=== 11-limit ===
Line 2,507: Line 2,361:


Optimal tunings:  
Optimal tunings:  
* CTE: ~8/7 = 240.000, ~3/2 = 697.034 (~56/55 = 22.966)
* WE: ~8/7 = 240.2740{{c}}, ~3/2 = 697.3317{{c}} (~56/55 = 23.4904{{c}})
* POTE: ~8/7 = 240.000, ~3/2 = 696.536 (~56/55 = 23.464)
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 696.6269{{c}} (~56/55 = 23.3731{{c}})


{{Optimal ET sequence|legend=0| 5, 45, 50 }}
{{Optimal ET sequence|legend=0| 5, 45, 50 }}


Badness: 0.070378
Badness (Sintel): 2.33


=== 13-limit ===
=== 13-limit ===
Line 2,522: Line 2,376:


Optimal tunings:  
Optimal tunings:  
* CTE: ~8/7 = 240.000, ~3/2 = 696.749 (~91/90 = 23.251)
* WE: ~8/7 = 240.2435{{c}}, ~3/2 = 696.8686{{c}} (~91/90 = 23.8618{{c}})
* POTE: ~8/7 = 240.000, ~3/2 = 696.162 (~91/90 = 23.838)
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 696.2653{{c}} (~91/90 = 23.7347{{c}})


{{Optimal ET sequence|legend=0| 5, 45f, 50 }}
{{Optimal ET sequence|legend=0| 5, 45f, 50 }}


Badness: 0.048829
Badness (Sintel): 2.02


== Subgroup extensions ==
== Subgroup extensions ==
Line 2,536: Line 2,390:


{{Mapping|legend=2| 1 0 -4 9 | 0 1 4 -3 }}
{{Mapping|legend=2| 1 0 -4 9 | 0 1 4 -3 }}
: sval mapping generators: ~2, ~3


{{Mapping|legend=3| 1 0 -4 0 0 0 0 9 | 0 1 4 0 0 0 0 -3 }}
{{Mapping|legend=3| 1 0 -4 0 0 0 0 9 | 0 1 4 0 0 0 0 -3 }}


: [[gencom]]: [2 3; 81/80 96/95]
: mapping generators: ~2, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 697.681
* [[WE]]: ~2 = 1199.5513{{c}}, ~3/2 = 697.6058{{c}}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 697.867
: [[error map]]: {{val| -0.448 -4.798 +4.110 +6.977 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.8222{{c}}
: error map: {{val| 0.000 -4.133 +4.975 +9.020 }}


{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43 }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43, 98h }}


[[Tp tuning #T2 tuning|RMS error]]: 1.378 cents
[[Badness]] (Sintel): 0.324


=== Hypnotone ===
=== Hypnotone ===
Line 2,560: Line 2,414:
{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}
{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}


: sval mapping generators: ~2, ~3
{{Mapping|legend=3| 1 0 -4 0 -6 | 0 1 4 0 6 }}
 
: mapping generators: ~2, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 694.700
* [[WE]]: ~2 = 1202.0621{{c}}, ~3/2 = 694.5448{{c}}
* [[CWE]]: ~2 = 1200.000, ~3/2 = 693.908
: [[error map]]: {{val| +2.062 -5.348 -8.135 +15.951 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.9085{{c}}
: error map: {{val| 0.000 -8.047 -10.680 +12.133 }}


{{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }}
{{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }}


[[Badness]]: 0.0104
[[Badness]] (Sintel): 0.326


==== 2.3.5.11.13 subgroup ====
==== 2.3.5.11.13 subgroup ====
Line 2,575: Line 2,433:
Comma list: 45/44, 65/64, 81/80
Comma list: 45/44, 65/64, 81/80


Sval mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}
Subgroup-val mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}


: sval mapping generators: ~2, ~3
Gencom mapping: {{mapping| 1 0 -4 0 -6 10 | 0 1 4 0 6 -4 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 693.951
* WE: ~2 = 1202.6916{{c}}, ~3/2 = 694.4181{{c}}
* CTE: ~2 = 1200.000, ~3/2 = 693.087
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0870{{c}}


{{Optimal ET sequence|legend=0| 7, 12, 19, 26, 45f }}
{{Optimal ET sequence|legend=0| 7, 12, 19, 26, 45f }}


Badness: 0.0141
Badness (Sintel): 0.561


=== Dequarter ===
=== Dequarter ===
Line 2,594: Line 2,452:
{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}
{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}


: sval mapping generators: ~2, ~3
{{Mapping|legend=3| 1 0 -4 0 5 | 0 1 4 0 -1 }}
 
: mapping generators: ~2, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 696.039
* [[WE]]: ~2 = 1206.5832{{c}}, ~3/2 = 695.8763{{c}}
* [[CWE]]: ~2 = 1200.000, ~3/2 = 693.121
: [[error map]]: {{val| +6.583 +0.504 -2.809 -20.862 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 693.1206{{c}}
: error map: {{val| 0.000 -8.834 -13.831 -44.439 }}


{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }}
{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }}


[[Badness]]: 0.0145
[[Badness]] (Sintel): 0.451


==== Dreamtone ====
==== Dreamtone ====
Line 2,609: Line 2,471:
Comma list: 33/32, 55/54, 975/968
Comma list: 33/32, 55/54, 975/968


Sval mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}
Subgroup-val mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}


: sval mapping generators: ~2, ~3
Gencom mapping: {{mapping| 1 0 -4 0 5 21 | 0 1 4 0 -1 -11 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 689.699
* WE: ~2 = 1207.8248{{c}}, ~3/2 = 694.7806{{c}}
* CWE: ~2 = 1200.000, ~3/2 = 690.183
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 690.1826{{c}}


{{Optimal ET sequence|legend=0| 7, 19eff, 26eff, 33ceeff, 40ceeff }}
{{Optimal ET sequence|legend=0| 7, 19eff, 26eff, 33ceeff, 40ceeff }}


Badness: 0.0353
Badness (Sintel): 1.40


[[Category:Temperament families]]
[[Category:Temperament families]]
Retrieved from "https://en.xen.wiki/w/Meantone_family"