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Badness: 0.057931
Badness: 0.057931


== Casablanca ==
== Nusecond ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Casablanca]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Nusecond]].''
 
Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&70. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. [[31edo|31EDO]] can be used as a tuning, or [[132edo|132EDO]] with a val which is the sum of the [[patent val]]s for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. MOS of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note MOS might also be considered from the melodic point of view.


Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described as 31&73. 74\135 or 91\166 supply good tunings for the generator, and 20 and 31 note MOS are available.
Subgroup: 2.3.5.7


It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the 35/24 generator is particularly interesting; like 15/14 and 21/20, it represents an interval between one vertex of a [[hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads. For another, if we add 385/384 to the list of commas, 35/24 is identified with 16/11, and casablanca is revealed as an 11-limit temperament with a very low complexity for 11 and not too high a one for 7; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit meantone.
[[Comma list]]: 126/125, 2430/2401


Subgroup: 2.3.5.7
[[Mapping]]: [{{val| 1 3 4 5 }}, {{val| 0 -11 -13 -17 }}]


[[Comma list]]: 126/125, 589824/588245
Mapping generators: ~2, ~49/45


[[Mapping]]: [{{val| 1 12 10 5 }}, {{val| 0 -19 -14 -4 }}]
{{Multival|legend=1| 11 13 17 -5 -4 3 }}


{{Multival|legend=1| 19 14 4 -22 -47 -30 }}
[[POTE generator]]: ~49/45 = 154.579


[[POTE generator]]: ~35/24 = 657.818
[[Minimax tuning]]:
* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| -5/13 0 11/13 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| -3/13 0 17/13 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5
* [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 5/11 13/11 0 0 }}, {{monzo| 4/11 17/11 0 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 3


{{Val list|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}
{{Val list|legend=1| 8d, 23d, 31, 101, 132c, 163c }}


[[Badness]]: 0.101191
[[Badness]]: 0.050389


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 385/384, 2420/2401
Comma list: 99/98, 121/120, 126/125
 
Mapping: [{{val| 1 3 4 5 5 }}, {{val| 0 -11 -13 -17 -12 }}]
 
Mapping generators: ~2, ~11/10
 
POTE generator: ~11/10 = 154.645


Mapping: [{{val| 1 12 10 5 4 }}, {{val| 0 -19 -14 -4 -1 }}]
Minimax tuning:  
* [[11-odd-limit]]: ~11/10 = {{monzo| 1/10 -1/5 0 0 1/10 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]
: Eigenmonzos (unchanged intervals): 2, 11/9


POTE generator: ~16/11 = 657.923
Algebraic generator: positive root of 15''x''<sup>2</sup> - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.


Optimal GPV sequence: {{Val list| 11b, 20b, 31 }}
Optimal GPV sequence: {{Val list| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}


Badness: 0.067291
Badness: 0.025621


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 196/195, 385/384, 2420/2401
Comma list: 66/65, 99/98, 121/120, 126/125
 
Mapping: [{{val| 1 3 4 5 5 5 }}, {{val| 0 -11 -13 -17 -12 -10 }}]
 
POTE generator: ~11/10 = 154.478
 
Optimal GPV sequence: {{Val list| 8d, 23de, 31, 70f, 101ff }}
 
Badness: 0.023323
 
== Oolong ==
{{Main| Oolong }}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Oolong]].''
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 126/125, 117649/116640
 
[[Mapping]]: [{{val| 1 6 7 8 }}, {{val| 0 -17 -18 -20 }}]
 
{{Multival|legend=1| 17 18 20 -11 -16 -4 }}


Mapping: [{{val| 1 12 10 5 4 7 }}, {{val| 0 -19 -14 -4 -1 -6 }}]
[[POTE generator]]: ~6/5 = 311.679


POTE generator: ~16/11 = 657.854
{{Val list|legend=1| 27, 50, 77 }}


Optimal GPV sequence: {{Val list| 11b, 20b, 31 }}
[[Badness]]: 0.073509


=== Marrakesh ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 176/175, 14641/14580
Comma list: 126/125, 176/175, 26411/26244


Mapping: [{{val| 1 12 10 5 21 }}, {{val| 0 -19 -14 -4 -32 }}]
Mapping: [{{val| 1 6 7 8 18 }}, {{val| 0 -17 -18 -20 -56 }}]


POTE generator: ~22/15 = 657.791
POTE generator: ~6/5 = 311.587


Optimal GPV sequence: {{Val list| 31, 73, 104c, 135c }}
Optimal GPV sequence: {{Val list| 27e, 77, 104c, 181c }}


Badness: 0.040539
Badness: 0.056915


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 176/175, 196/195, 14641/14580
Comma list: 126/125, 176/175, 196/195, 13013/12960
 
Mapping: [{{val| 1 6 7 8 18 5 }}, {{val| 0 -17 -18 -20 -56 -5 }}]
 
POTE generator: ~6/5 = 311.591
 
Optimal GPV sequence: {{Val list| 27e, 77, 104c, 181c }}
 
Badness: 0.035582
 
== Vines ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Vines]].''
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 126/125, 84035/82944


Mapping: [{{val| 1 12 10 5 21 -10 }}, {{val| 0 -19 -14 -4 -32 25 }}]
[[Mapping]]: [{{val| 2 7 8 8 }}, {{val| 0 -8 -7 -5 }}]


POTE generator: ~22/15 = 657.756
[[POTE generator]]: ~6/5 = 312.602


Optimal GPV sequence: {{Val list| 31, 73, 104c, 135c, 239ccf }}
{{Val list|legend=1| 42, 46, 96d, 142d, 238dd }}


Badness: 0.040774
[[Badness]]: 0.078049


==== Murakuc ====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 126/125, 144/143, 176/175, 1540/1521
Comma list: 126/125, 385/384, 2401/2376


Mapping: [{{val| 1 12 10 5 21 7 }}, {{val| 0 -19 -14 -4 -32 -6 }}]
Mapping: [{{val| 2 7 8 8 5 }}, {{val| 0 -8 -7 -5 4 }}]


POTE generator: ~22/15 = 657.700
POTE generator: ~6/5 = 312.601


Optimal GPV sequence: {{Val list| 31, 104cff, 135cff }}
Optimal GPV sequence: {{Val list| 42, 46, 96d, 142d, 238dd }}


Badness: 0.041395
Badness: 0.044499


== Nusecond ==
=== 13-limit ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Nusecond]].''
Subgroup: 2.3.5.7.11.13


Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&amp;70. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. [[31edo|31EDO]] can be used as a tuning, or [[132edo|132EDO]] with a val which is the sum of the [[patent val]]s for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. MOS of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note MOS might also be considered from the melodic point of view.
Comma list: 126/125, 196/195, 364/363, 385/384


Subgroup: 2.3.5.7
Mapping: [{{val| 2 7 8 8 5 5 }}, {{val| 0 -8 -7 -5 4 5 }}]


[[Comma list]]: 126/125, 2430/2401
POTE generator: ~6/5 = 312.564


[[Mapping]]: [{{val| 1 3 4 5 }}, {{val| 0 -11 -13 -17 }}]
Optimal GPV sequence: {{Val list| 42, 46, 96d, 238ddf }}


Mapping generators: ~2, ~49/45
Badness: 0.029693


{{Multival|legend=1| 11 13 17 -5 -4 3 }}
== Kumonga ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Kumonga]].''


[[POTE generator]]: ~49/45 = 154.579
Subgroup: 2.3.5.7


[[Minimax tuning]]:  
[[Comma list]]: 126/125, 12288/12005
* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| -5/13 0 11/13 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| -3/13 0 17/13 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5
* [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 5/11 13/11 0 0 }}, {{monzo| 4/11 17/11 0 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 3


{{Val list|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
[[Mapping]]: [{{val| 1 4 4 3 }}, {{val| 0 -13 -9 -1 }}]


[[Badness]]: 0.050389
{{Multival|legend=1| 13 9 1 -16 -35 -23 }}


=== 11-limit ===
[[POTE generator]]: ~8/7 = 222.797
Subgroup: 2.3.5.7.11


Comma list: 99/98, 121/120, 126/125
{{Val list|legend=1| 16, 27, 43, 70, 167ccdd }}


Mapping: [{{val| 1 3 4 5 5 }}, {{val| 0 -11 -13 -17 -12 }}]
[[Badness]]: 0.087500


Mapping generators: ~2, ~11/10
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~11/10 = 154.645
Comma list: 126/125, 176/175, 864/847


Minimax tuning:  
Mapping: [{{val| 1 4 4 3 7 }}, {{val| 0 -13 -9 -1 -19 }}]
* [[11-odd-limit]]: ~11/10 = {{monzo| 1/10 -1/5 0 0 1/10 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]
: Eigenmonzos (unchanged intervals): 2, 11/9


Algebraic generator: positive root of 15''x''<sup>2</sup> - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.
POTE generator: ~8/7 = 222.898


Optimal GPV sequence: {{Val list| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}
Optimal GPV sequence: {{Val list| 16, 27e, 43, 70e }}


Badness: 0.025621
Badness: 0.043336


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 66/65, 99/98, 121/120, 126/125
Comma list: 78/77, 126/125, 144/143, 176/175


Mapping: [{{val| 1 3 4 5 5 5 }}, {{val| 0 -11 -13 -17 -12 -10 }}]
Mapping: [{{val| 1 4 4 3 7 5 }}, {{val| 0 -13 -9 -1 -19 -7 }}]


POTE generator: ~11/10 = 154.478
POTE generator: ~8/7 = 222.961


Optimal GPV sequence: {{Val list| 8d, 23de, 31, 70f, 101ff }}
Optimal GPV sequence: {{Val list| 16, 27e, 43, 70e, 113cdee }}


Badness: 0.023323
Badness: 0.028920


== Thuja ==
== Thuja ==
Line 643: Line 687:
Badness: 0.023877
Badness: 0.023877


== Vines ==
== Casablanca ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Vines]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Casablanca]].''
 
Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described as 31&amp;73. 74\135 or 91\166 supply good tunings for the generator, and 20 and 31 note MOS are available.
 
It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the 35/24 generator is particularly interesting; like 15/14 and 21/20, it represents an interval between one vertex of a [[hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads. For another, if we add 385/384 to the list of commas, 35/24 is identified with 16/11, and casablanca is revealed as an 11-limit temperament with a very low complexity for 11 and not too high a one for 7; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit meantone.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 126/125, 84035/82944
[[Comma list]]: 126/125, 589824/588245
 
[[Mapping]]: [{{val| 1 12 10 5 }}, {{val| 0 -19 -14 -4 }}]


[[Mapping]]: [{{val| 2 7 8 8 }}, {{val| 0 -8 -7 -5 }}]
{{Multival|legend=1| 19 14 4 -22 -47 -30 }}


[[POTE generator]]: ~6/5 = 312.602
[[POTE generator]]: ~35/24 = 657.818


{{Val list|legend=1| 42, 46, 96d, 142d, 238dd }}
{{Val list|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}


[[Badness]]: 0.078049
[[Badness]]: 0.101191


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 385/384, 2401/2376
Comma list: 126/125, 385/384, 2420/2401


Mapping: [{{val| 2 7 8 8 5 }}, {{val| 0 -8 -7 -5 4 }}]
Mapping: [{{val| 1 12 10 5 4 }}, {{val| 0 -19 -14 -4 -1 }}]


POTE generator: ~6/5 = 312.601
POTE generator: ~16/11 = 657.923


Optimal GPV sequence: {{Val list| 42, 46, 96d, 142d, 238dd }}
Optimal GPV sequence: {{Val list| 11b, 20b, 31 }}


Badness: 0.044499
Badness: 0.067291


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 196/195, 364/363, 385/384
Comma list: 126/125, 196/195, 385/384, 2420/2401


Mapping: [{{val| 2 7 8 8 5 5 }}, {{val| 0 -8 -7 -5 4 5 }}]
Mapping: [{{val| 1 12 10 5 4 7 }}, {{val| 0 -19 -14 -4 -1 -6 }}]


POTE generator: ~6/5 = 312.564
POTE generator: ~16/11 = 657.854


Optimal GPV sequence: {{Val list| 42, 46, 96d, 238ddf }}
Optimal GPV sequence: {{Val list| 11b, 20b, 31 }}


Badness: 0.029693
=== Marrakesh ===
Subgroup: 2.3.5.7.11


== Kumonga ==
Comma list: 126/125, 176/175, 14641/14580
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Kumonga]].''


Subgroup: 2.3.5.7
Mapping: [{{val| 1 12 10 5 21 }}, {{val| 0 -19 -14 -4 -32 }}]


[[Comma list]]: 126/125, 12288/12005
POTE generator: ~22/15 = 657.791


[[Mapping]]: [{{val| 1 4 4 3 }}, {{val| 0 -13 -9 -1 }}]
Optimal GPV sequence: {{Val list| 31, 73, 104c, 135c }}


{{Multival|legend=1| 13 9 1 -16 -35 -23 }}
Badness: 0.040539


[[POTE generator]]: ~8/7 = 222.797
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 16, 27, 43, 70, 167ccdd }}
Comma list: 126/125, 176/175, 196/195, 14641/14580


[[Badness]]: 0.087500
Mapping: [{{val| 1 12 10 5 21 -10 }}, {{val| 0 -19 -14 -4 -32 25 }}]


=== 11-limit ===
POTE generator: ~22/15 = 657.756
Subgroup: 2.3.5.7.11


Comma list: 126/125, 176/175, 864/847
Optimal GPV sequence: {{Val list| 31, 73, 104c, 135c, 239ccf }}


Mapping: [{{val| 1 4 4 3 7 }}, {{val| 0 -13 -9 -1 -19 }}]
Badness: 0.040774


POTE generator: ~8/7 = 222.898
==== Murakuc ====
 
Optimal GPV sequence: {{Val list| 16, 27e, 43, 70e }}
 
Badness: 0.043336
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 78/77, 126/125, 144/143, 176/175
Comma list: 126/125, 144/143, 176/175, 1540/1521


Mapping: [{{val| 1 4 4 3 7 5 }}, {{val| 0 -13 -9 -1 -19 -7 }}]
Mapping: [{{val| 1 12 10 5 21 7 }}, {{val| 0 -19 -14 -4 -32 -6 }}]


POTE generator: ~8/7 = 222.961
POTE generator: ~22/15 = 657.700


Optimal GPV sequence: {{Val list| 16, 27e, 43, 70e, 113cdee }}
Optimal GPV sequence: {{Val list| 31, 104cff, 135cff }}


Badness: 0.028920
Badness: 0.041395


== Amigo ==
== Amigo ==
Line 770: Line 814:
Badness: 0.030666
Badness: 0.030666


== Oolong ==
== Supersensi ==
{{Main| Oolong }}
Supersensi (8d &amp; 43) has supermajor third as a generator like [[sensi]], but the no-fives comma 17496/16807 rather than 245/243 tempered out.
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Oolong]].''


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 126/125, 117649/116640
[[Comma list]]: 126/125, 17496/16807


[[Mapping]]: [{{val| 1 6 7 8 }}, {{val| 0 -17 -18 -20 }}]
[[Mapping]]: [{{val| 1 -4 -4 -5 }}, {{val| 0 15 17 21 }}]


{{Multival|legend=1| 17 18 20 -11 -16 -4 }}
{{Multival|legend=1| 15 17 21 -8 -9 1 }}


[[POTE generator]]: ~6/5 = 311.679
[[POTE generator]]: ~343/270 = 446.568


{{Val list|legend=1| 27, 50, 77 }}
{{Val list|legend=1| 8d, 35, 43 }}


[[Badness]]: 0.073509
[[Badness]]: 0.148531


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 176/175, 26411/26244
Comma list: 99/98, 126/125, 864/847


Mapping: [{{val| 1 6 7 8 18 }}, {{val| 0 -17 -18 -20 -56 }}]
Mapping: [{{val| 1 -4 -4 -5 -1 }}, {{val| 0 15 17 21 12 }}]


POTE generator: ~6/5 = 311.587
POTE generator: ~72/55 = 446.616


Optimal GPV sequence: {{Val list| 27e, 77, 104c, 181c }}
Optimal GPV sequence: {{Val list| 8d, 35, 43 }}


Badness: 0.056915
Badness: 0.059449


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 176/175, 196/195, 13013/12960
Comma list: 78/77, 99/98, 126/125, 144/143
 
Mapping: [{{val| 1 -4 -4 -5 -1 -3 }}, {{val| 0 15 17 21 12 18 }}]
 
POTE generator: ~13/10 = 446.598
 
Optimal GPV sequence: {{Val list| 8d, 35f, 43 }}
 
Badness: 0.035258
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 78/77, 99/98, 120/119, 126/125, 144/143


Mapping: [{{val| 1 6 7 8 18 5 }}, {{val| 0 -17 -18 -20 -56 -5 }}]
Mapping: [{{val| 1 -4 -4 -5 -1 -3 0 }}, {{val| 0 15 17 21 12 18 11 }}]


POTE generator: ~6/5 = 311.591
POTE generator: ~13/10 = 446.631


Optimal GPV sequence: {{Val list| 27e, 77, 104c, 181c }}
Optimal GPV sequence: {{Val list| 8d, 35f, 43 }}


Badness: 0.035582
Badness: 0.025907


== Cobalt ==
== Cobalt ==
Retrieved from "https://en.xen.wiki/w/Starling_temperaments"