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{{Technical data page}}
{{Technical data page}}
This page discusses miscellaneous rank-2 temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the marvel comma or septimal kleisma.  
This page discusses miscellaneous [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] [[225/224]], the marvel comma or septimal kleisma.  


Temperaments considered in families and clans are:  
Temperaments considered in families and clans are:  
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Considered below are wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, alphorn, tertiosec, gwazy, and gracecordial.  
Considered below are wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, alphorn, tertiosec, gwazy, and gracecordial.  


Since {{nowrap|(5/4)<sup>2</sup> {{=}} 225/224 × 14/9}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.


The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.
The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.
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== Wizard ==
== Wizard ==
{{Main| Wizard }}
{{Main| Wizard }}
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Wizard]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''


Wizard has a semi-octave period and is generated by an interval that is best treated as ~17/15. The semi-octave complement of this interval is ~5/4. Wizard can be described as {{nowrap|22 &amp; 72}}. Its ploidacot is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.  
Wizard has a semi-octave period and is generated by an interval that is best treated as ~17/15. The semi-octave complement of this interval is ~5/4. Wizard can be described as {{nowrap| 22 & 72 }}. Its ploidacot is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216
: mapping generators: ~1225/864, ~245/216


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~1225/864 = 600.000, ~245/216 = 216.919
* [[CTE]]: ~1225/864 = 600.000{{c}}, ~245/216 = 216.919{{c}}
: [[error map]]: {{val| 0.000 -0.443 -3.232 +0.361 }}
: [[error map]]: {{val| 0.000 -0.443 -3.232 +0.361 }}
* [[POTE]]: ~1225/864 = 600.000, ~245/216 = 216.744
* [[POTE]]: ~1225/864 = 600.000{{c}}, ~245/216 = 216.744{{c}}
: error map: {{val| 0.000 -1.492 -3.057 -1.388 }}
: error map: {{val| 0.000 -1.492 -3.057 -1.388 }}


Line 73: Line 72:


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000, ~25/22 = 216.900
* CTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.900{{c}}
* POTE: ~99/70 = 600.000, ~25/22 = 216.768
* POTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.768{{c}}


{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
Line 88: Line 87:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 600.000, ~25/22 = 216.703
* CTE: ~55/39 = 600.000{{c}}, ~25/22 = 216.703{{c}}
* POTE: ~55/39 = 600.000, ~25/22 = 216.711
* POTE: ~55/39 = 600.000{{c}}, ~25/22 = 216.711{{c}}


{{Optimal ET sequence|legend=0| 22, 50, 72, 122, 194df }}
{{Optimal ET sequence|legend=0| 22, 50, 72, 122, 194df }}
Line 103: Line 102:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~17/15 = 216.748
* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.748{{c}}
* POTE: ~17/12 = 600.000, ~17/15 = 216.619
* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.619{{c}}


{{Optimal ET sequence|legend=0| 22, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 22, 50, 72, 122g, 194dfg }}
Line 118: Line 117:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~17/15 = 216.677
* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.677{{c}}
* POTE: ~17/12 = 600.000, ~17/15 = 216.523
* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.523{{c}}


{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
Line 133: Line 132:


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000, ~25/22 = 216.928
* CTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.928{{c}}
* POTE: ~99/70 = 600.000, ~25/22 = 216.830
* POTE: ~99/70 = 600.000{{c}}, ~25/22 = 216.830{{c}}


{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~17/15 = 216.943
* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.943{{c}}
* POTE: ~17/12 = 600.000, ~17/15 = 216.825
* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.825{{c}}


{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
Line 163: Line 162:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~17/15 = 216.918
* CTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.918{{c}}
* POTE: ~17/12 = 600.000, ~17/15 = 216.862
* POTE: ~17/12 = 600.000{{c}}, ~17/15 = 216.862{{c}}


{{Optimal ET sequence|legend=0| 72, 94, 166g }}
{{Optimal ET sequence|legend=0| 72, 94, 166g }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~77/54 = 600.000, ~55/48 = 217.247
* CTE: ~77/54 = 600.000{{c}}, ~55/48 = 217.247{{c}}
* POTE: ~77/54 = 600.000, ~55/48 = 216.876
* POTE: ~77/54 = 600.000{{c}}, ~55/48 = 216.876{{c}}


{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
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{{Mapping|legend=1| 1 4 -3 -3 | 0 -5 11 12 }}
{{Mapping|legend=1| 1 4 -3 -3 | 0 -5 11 12 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 580.286
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 580.286{{c}}


{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


[[Badness]]: 0.047578
[[Badness]] (Smith): 0.047578


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 4 -3 -3 2 | 0 -5 11 12 3 }}
Mapping: {{mapping| 1 4 -3 -3 2 | 0 -5 11 12 3 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.267
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.267{{c}}


{{Optimal ET sequence|legend=1| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


Badness: 0.023659
Badness (Smith): 0.023659


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 4 -3 -3 2 -5 | 0 -5 11 12 3 18 }}
Mapping: {{mapping| 1 4 -3 -3 2 -5 | 0 -5 11 12 3 18 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.108
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.108{{c}}


{{Optimal ET sequence|legend=1| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


Badness: 0.022993
Badness (Smith): 0.022993


==== 17-limit ====
==== 17-limit ====
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Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 | 0 -5 11 12 3 18 25 }}
Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 | 0 -5 11 12 3 18 25 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.055
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.055{{c}}


{{Optimal ET sequence|legend=1| 29g, 31, 60e }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness: 0.019093
Badness (Smith): 0.019093


==== 19-limit ====
==== 19-limit ====
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Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 -3 | 0 -5 11 12 3 18 25 15 }}
Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 -3 | 0 -5 11 12 3 18 25 15 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.026
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.026{{c}}


{{Optimal ET sequence|legend=1| 29g, 31, 60e }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness: 0.017010
Badness (Smith): 0.017010


==== 23-limit ====
==== 23-limit ====
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Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 -3 5 | 0 -5 11 12 3 18 25 15 -1 }}
Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 -3 5 | 0 -5 11 12 3 18 25 15 -1 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.009
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.009{{c}}


{{Optimal ET sequence|legend=1| 29g, 31, 60e }}
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Badness: 0.014471
Badness (Smith): 0.014471


=== Tritoni ===
=== Tritoni ===
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Mapping: {{mapping| 1 4 -3 -3 17 | 0 -5 11 12 -28 }}
Mapping: {{mapping| 1 4 -3 -3 17 | 0 -5 11 12 -28 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.389
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 580.389{{c}}


{{Optimal ET sequence|legend=1| 31, 91, 122, 153d }}
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Badness: 0.045456
Badness (Smith): 0.045456


== Septimin ==
== Septimin ==
Line 287: Line 286:
{{Mapping|legend=1| 1 4 1 5 | 0 -11 6 -10 }}
{{Mapping|legend=1| 1 4 1 5 | 0 -11 6 -10 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/6 = 263.632
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/6 = 263.632{{c}}


{{Optimal ET sequence|legend=1| 41, 91, 132d }}
{{Optimal ET sequence|legend=1| 41, 91, 132d }}


[[Badness]]: 0.054502
[[Badness]] (Smith): 0.054502


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 4 1 5 5 | 0 -11 6 -10 -7 }}
Mapping: {{mapping| 1 4 1 5 5 | 0 -11 6 -10 -7 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/6 = 263.634
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/6 = 263.634{{c}}


{{Optimal ET sequence|legend=1| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


Badness: 0.031309
Badness (Smith): 0.031309


=== 13-limit ===
=== 13-limit ===
Line 313: Line 312:
Mapping: {{mapping| 1 4 1 5 5 7 | 0 -11 6 -10 -7 -15 }}
Mapping: {{mapping| 1 4 1 5 5 7 | 0 -11 6 -10 -7 -15 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/6 = 263.700
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/6 = 263.700{{c}}


{{Optimal ET sequence|legend=1| 41, 91 }}
{{Optimal ET sequence|legend=0| 41, 91 }}


Badness: 0.023117
Badness (Smith): 0.023117


== Merman ==
== Merman ==
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{{Mapping|legend=1| 1 5 -5 -5 | 0 -7 15 16 }}
{{Mapping|legend=1| 1 5 -5 -5 | 0 -7 15 16 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 585.585
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 585.585{{c}}


{{Optimal ET sequence|legend=1| 41, 84, 125 }}
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


[[Badness]]: 0.055078
[[Badness]] (Smith): 0.055078


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 5 -5 -5 2 | 0 -7 15 16 3 }}
Mapping: {{mapping| 1 5 -5 -5 2 | 0 -7 15 16 3 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 585.606
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.606{{c}}


{{Optimal ET sequence|legend=1| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


Badness: 0.036383
Badness (Smith): 0.036383


=== 13-limit ===
=== 13-limit ===
Line 354: Line 353:
Mapping: {{mapping| 1 5 -5 -5 2 12 | 0 -7 15 16 3 -17 }}
Mapping: {{mapping| 1 5 -5 -5 2 12 | 0 -7 15 16 3 -17 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 585.657
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 585.657{{c}}


{{Optimal ET sequence|legend=1| 41, 84, 125e, 209ef, 293ef }}
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Badness: 0.027544
Badness (Smith): 0.027544


== Slender ==
== Slender ==
Slender ({{nowrap|31 &amp; 32}}) tempers out the [[Hewuermera temperaments|hewuermera comma]] in addition to the marvel comma. This temperament has a generator of [[49/48]], 3 of which equal marvel's 16/15~15/14, and 10 generators is 5/4.
Slender ({{nowrap| 31 & 32 }}) tempers out the [[hewuermera comma]] in addition to the marvel comma. This temperament has a generator of [[49/48]], 3 of which equal marvel's 16/15~15/14, and 10 generators is 5/4.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 369: Line 368:
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 38.413
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/48 = 38.413{{c}}


{{Optimal ET sequence|legend=1| 31, 94, 125 }}
{{Optimal ET sequence|legend=1| 31, 94, 125 }}


[[Badness]]: 0.056934
[[Badness]] (Smith): 0.056934


=== 11-limit ===
=== 11-limit ===
Line 382: Line 381:
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 38.387
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 38.387{{c}}


{{Optimal ET sequence|legend=1| 31, 63, 94, 125 }}
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Badness: 0.025342
Badness (Smith): 0.025342


=== 13-limit ===
=== 13-limit ===
Line 395: Line 394:
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 38.314
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 38.314{{c}}


{{Optimal ET sequence|legend=1| 31, 63, 94 }}
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Badness: 0.025913
Badness (Smith): 0.025913


== Triton ==
== Triton ==
Line 410: Line 409:
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 568.865
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 568.865{{c}}


{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


[[Badness]]: 0.059245
[[Badness]] (Smith): 0.059245


=== 11-limit ===
=== 11-limit ===
Line 423: Line 422:
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 569.144
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 569.144{{c}}


{{Optimal ET sequence|legend=1| 2, 17d, 19, 59bde, 78bde, 97bde }}
{{Optimal ET sequence|legend=0| 2, 17d, 19, 59bde, 78bde, 97bde }}


Badness: 0.045675
Badness (Smith): 0.045675


== Marvolo ==
== Marvolo ==
Line 436: Line 435:
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~21/20 = 83.348
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~21/20 = 83.348{{c}}


{{Optimal ET sequence|legend=1| 29, 43, 72, 619bcd, 691bcd }}
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bcd, 691bcd }}


[[Badness]]: 0.083338
[[Badness]] (Smith): 0.083338


=== 11-limit ===
=== 11-limit ===
Line 449: Line 448:
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.340
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.340{{c}}


{{Optimal ET sequence|legend=1| 29, 43, 72 }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.028965
Badness (Smith): 0.028965


=== 13-limit ===
=== 13-limit ===
Line 462: Line 461:
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.330
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.330{{c}}


{{Optimal ET sequence|legend=1| 29, 43, 72, 115f }}
{{Optimal ET sequence|legend=0| 29, 43, 72, 115f }}


Badness: 0.021470
Badness (Smith): 0.021470


=== 17-limit ===
=== 17-limit ===
Line 475: Line 474:
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.330
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.330{{c}}


{{Optimal ET sequence|legend=1| 29g, 43, 72 }}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Badness: 0.014924
Badness (Smith): 0.014924


=== 19-limit ===
=== 19-limit ===
Line 488: Line 487:
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.330
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 83.330{{c}}


{{Optimal ET sequence|legend=1| 29g, 43, 72 }}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Badness: 0.014722
Badness (Smith): 0.014722


== Enneaportent ==
== Enneaportent ==
Line 500: Line 499:


{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: Mapping generators: ~2592/2401, ~12005/6912
: Mapping generators: ~2592/2401, ~12005/6912


[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 133.3333, ~12005/6912 = 950.1680 (~1728/1715 = 16.8347)
[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.1680{{c}} (~1728/1715 = 16.8347{{c}})


{{Optimal ET sequence|legend=1| 9, 63, 72, 495bcd }}
{{Optimal ET sequence|legend=1| 9, 63, 72, 495bcd }}


[[Badness]]: 0.093679
[[Badness]] (Smith): 0.093679


=== 11-limit ===
=== 11-limit ===
Line 516: Line 514:
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


Optimal tuning (POTE): ~121/112 = 133.3333, ~210/121 = 950.1873 (~99/98 = 16.8540)
Optimal tuning (POTE): ~121/112 = 133.3333{{c}}, ~210/121 = 950.1873{{c}} (~99/98 = 16.8540{{c}})


{{Optimal ET sequence|legend=1| 9, 63, 72, 423cd, 495bcd }}
{{Optimal ET sequence|legend=0| 9, 63, 72, 423cd, 495bcd }}


Badness: 0.030426
Badness (Smith): 0.030426


=== 13-limit ===
=== 13-limit ===
Line 529: Line 527:
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


Optimal tuning (POTE): ~14/13 = 133.3333, ~26/15 = 950.2867 (~105/104 = 16.9534)
Optimal tuning (POTE): ~14/13 = 133.3333{{c}}, ~26/15 = 950.2867{{c}} (~105/104 = 16.9534{{c}})


{{Optimal ET sequence|legend=1| 9, 63, 72, 279cf }}
{{Optimal ET sequence|legend=0| 9, 63, 72, 279cf }}


Badness: 0.022322
Badness (Smith): 0.022322


== Gracecordial ==
== Gracecordial ==
Line 544: Line 542:
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3/2 = 700.824
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 700.824{{c}}


{{Optimal ET sequence|legend=1| 12, 113, 125, 238c, 363c }}
{{Optimal ET sequence|legend=1| 12, 113, 125, 238c, 363c }}


[[Badness]]: 0.096279
[[Badness]] (Smith): 0.096279


=== 11-limit ===
=== 11-limit ===
Line 557: Line 555:
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.834
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.834{{c}}


{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125, 238c }}
{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125, 238c }}


Badness: 0.089588
Badness (Smith): 0.089588


==== 13-limit ====
==== 13-limit ====
Line 570: Line 568:
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.841
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.841{{c}}


{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125f, 238cf }}


Badness: 0.052235
Badness (Smith): 0.052235


==== 17-limit ====
==== 17-limit ====
Line 583: Line 581:
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.841
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.841{{c}}


{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125f, 238cf }}


Badness: 0.038565
Badness (Smith): 0.038565


==== 19-limit ====
==== 19-limit ====
Line 596: Line 594:
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.842
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.842{{c}}


{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125f, 238cf }}


Badness: 0.028165
Badness (Smith): 0.028165


==== 23-limit ====
==== 23-limit ====
Line 609: Line 607:
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.843
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.843{{c}}


{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 238cfi }}
{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 238cfi }}


Badness: 0.021879
Badness (Smith): 0.021879


==== 29-limit ====
==== 29-limit ====
Line 622: Line 620:
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.842
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.842{{c}}


{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cfi }}
{{Optimal ET sequence|legend=0| 12e, 101cde, 113, 125f, 238cfi }}


Badness: 0.018011
Badness (Smith): 0.018011


==== 31-limit ====
==== 31-limit ====
Line 635: Line 633:
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.838
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.838{{c}}


{{Optimal ET sequence|legend=1| 12e, 101cdek, 113, 125f, 238cfi }}
{{Optimal ET sequence|legend=0| 12e, 101cdek, 113, 125f, 238cfi }}


Badness: 0.016007
Badness (Smith): 0.016007


=== Gracecord ===
=== Gracecord ===
Line 648: Line 646:
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.885
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.885{{c}}


{{Optimal ET sequence|legend=1| 12, 101cd, 113 }}
{{Optimal ET sequence|legend=0| 12, 101cd, 113 }}


Badness: 0.066964
Badness (Smith): 0.066964


==== 13-limit ====
==== 13-limit ====
Line 661: Line 659:
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.890
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.890{{c}}


{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Badness: 0.044196
Badness (Smith): 0.044196


==== 17-limit ====
==== 17-limit ====
Line 674: Line 672:
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.870
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.870{{c}}


{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Badness: 0.036637
Badness (Smith): 0.036637


==== 19-limit ====
==== 19-limit ====
Line 687: Line 685:
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.866
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 700.866{{c}}


{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Badness: 0.027559
Badness (Smith): 0.027559


== Alphorn ==
== Alphorn ==
Line 700: Line 698:
{{Mapping|legend=1| 1 9 0 13 | 0 -16 5 -22 }}
{{Mapping|legend=1| 1 9 0 13 | 0 -16 5 -22 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~48/35 = 556.221
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~48/35 = 556.221{{c}}


{{Optimal ET sequence|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
{{Optimal ET sequence|legend=1| 28d, 41, 151cd, 192cd, 233cd }}


[[Badness]]: 0.129258
[[Badness]] (Smith): 0.129258


=== 11-limit ===
=== 11-limit ===
Line 713: Line 711:
Mapping: {{mapping| 1 9 0 13 3 | 0 -16 5 -22 1 }}
Mapping: {{mapping| 1 9 0 13 3 | 0 -16 5 -22 1 }}


Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 556.144
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 556.144{{c}}


{{Optimal ET sequence|legend=1| 28d, 41, 315cde }}
{{Optimal ET sequence|legend=0| 28d, 41, 315cde }}


Badness: 0.073459
Badness (Smith): 0.073459


== Misneb ==
== Misneb ==
Line 728: Line 726:
{{Mapping|legend=1| 1 3 1 3 | 0 -15 14 -2 }}
{{Mapping|legend=1| 1 3 1 3 | 0 -15 14 -2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16/15 = 113.235
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~16/15 = 113.235{{c}}


{{Optimal ET sequence|legend=1| 21, 32, 53 }}
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


[[Badness]]: 0.140970
[[Badness]] (Smith): 0.140970


=== 11-limit ===
=== 11-limit ===
Line 741: Line 739:
Mapping: {{mapping| 1 3 1 3 1 | 0 -15 14 -2 26 }}
Mapping: {{mapping| 1 3 1 3 1 | 0 -15 14 -2 26 }}


Optimal tuning (POTE): ~2 = 1200.000, ~16/15 = 113.323
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 113.323{{c}}


{{Optimal ET sequence|legend=1| 21, 32e, 53, 127, 180de }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127, 180de }}


Badness: 0.085390
Badness (Smith): 0.085390


==== 13-limit ====
==== 13-limit ====
Line 754: Line 752:
Mapping: {{mapping| 1 3 1 3 1 2 | 0 -15 14 -2 26 18 }}
Mapping: {{mapping| 1 3 1 3 1 2 | 0 -15 14 -2 26 18 }}


Optimal tuning (POTE): ~2 = 1200.000, ~16/15 = 113.323
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 113.323{{c}}


{{Optimal ET sequence|legend=1| 21, 32e, 53, 127, 180de }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127, 180de }}


Badness: 0.045569
Badness (Smith): 0.045569


=== Musneb ===
=== Musneb ===
Line 767: Line 765:
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


Optimal tuning (POTE): ~2 = 1200.000, ~16/15 = 113.142
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 113.142{{c}}


{{Optimal ET sequence|legend=1| 32, 53, 191de, 244cddee, 297cddee }}
{{Optimal ET sequence|legend=0| 32, 53, 191de, 244cddee, 297cddee }}


Badness: 0.087333
Badness (Smith): 0.087333


== Untriton ==
== Untriton ==
Line 782: Line 780:
{{Mapping|legend=1| 1 6 -7 -7 | 0 -9 19 20 }}
{{Mapping|legend=1| 1 6 -7 -7 | 0 -9 19 20 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 588.641
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~7/5 = 588.641{{c}}


{{Optimal ET sequence|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}
{{Optimal ET sequence|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}


[[Badness]]: 0.143976
[[Badness]] (Smith): 0.143976


=== 11-limit ===
=== 11-limit ===
Line 795: Line 793:
Mapping: {{mapping| 1 6 -7 -7 1 | 0 -9 19 20 5 }}
Mapping: {{mapping| 1 6 -7 -7 1 | 0 -9 19 20 5 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 588.626
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 588.626{{c}}


{{Optimal ET sequence|legend=1| 51, 53 }}
{{Optimal ET sequence|legend=0| 51, 53 }}


Badness: 0.074295
Badness (Smith): 0.074295


=== 13-limit ===
=== 13-limit ===
Line 808: Line 806:
Mapping: {{mapping| 1 6 -7 -7 1 -12 | 0 -9 19 20 5 32 }}
Mapping: {{mapping| 1 6 -7 -7 1 -12 | 0 -9 19 20 5 32 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 588.654
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~7/5 = 588.654{{c}}


{{Optimal ET sequence|legend=1| 51f, 53 }}
{{Optimal ET sequence|legend=0| 51f, 53 }}


Badness: 0.047441
Badness (Smith): 0.047441


== Naiadical ==
== Naiadical ==
Line 822: Line 820:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1202.120, ~21/16 = 456.652
* [[WE]]: ~2 = 1202.120{{c}}, ~21/16 = 456.652{{c}}
* [[CWE]]: ~2 = 1200.000, ~21/16 = 455.868
* [[CWE]]: ~2 = 1200.000{{c}}, ~21/16 = 455.868{{c}}


{{Optimal ET sequence|legend=1| 21, 29, 50, 79d }}
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d }}
Line 837: Line 835:


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.901, ~21/16 = 456.514
* WE: ~2 = 1201.901, ~21/16{{c}} = 456.514{{c}}
* CWE: ~2 = 1200.000, ~21/16 = 455.822
* CWE: ~2 = 1200.000, ~21/16{{c}} = 455.822{{c}}


{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
Line 852: Line 850:


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.786, ~13/10 = 456.452
* WE: ~2 = 1201.786{{c}}, ~13/10 = 456.452{{c}}
* CWE: ~2 = 1200.000, ~13/10 = 455.807
* CWE: ~2 = 1200.000{{c}}, ~13/10 = 455.807{{c}}


{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
Line 867: Line 865:


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.921, ~13/10 = 456.523
* WE: ~2 = 1201.921{{c}}, ~13/10 = 456.523{{c}}
* CWE: ~2 = 1200.000, ~13/10 = 455.833
* CWE: ~2 = 1200.000{{c}}, ~13/10 = 455.833{{c}}


{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}
Line 881: Line 879:
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~10000/9261 = 139.838
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~10000/9261 = 139.838{{c}}


{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


[[Badness]]: 0.150565
[[Badness]] (Smith): 0.150565


=== 11-limit ===
=== 11-limit ===
Line 894: Line 892:
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


Optimal tuning (POTE): ~2 = 1200.000, ~320/297 = 139.827
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~320/297 = 139.827{{c}}


{{Optimal ET sequence|legend=1| 43, 60e, 103 }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 0.052590
Badness (Smith): 0.052590


=== 13-limit ===
=== 13-limit ===
Line 907: Line 905:
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


Optimal tuning (POTE): ~2 = 1200.000, ~13/12 = 139.812
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 139.812{{c}}


{{Optimal ET sequence|legend=1| 43, 60e, 103 }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 0.032730
Badness (Smith): 0.032730


=== 17-limit ===
=== 17-limit ===
Line 920: Line 918:
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


Optimal tuning (POTE): ~2 = 1200.000, ~13/12 = 139.815
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 139.815{{c}}


{{Optimal ET sequence|legend=1| 43, 60e, 103 }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 0.023038
Badness (Smith): 0.023038


== Gwazy ==
== Gwazy ==
Line 935: Line 933:
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}


[[Optimal tuning]] ([[POTE]]): ~2401/1728 = 600.000, ~35/32 = 162.658
[[Optimal tuning]] ([[POTE]]): ~2401/1728 = 600.000{{c}}, ~35/32 = 162.658{{c}}


{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


[[Badness]]: 0.178826
[[Badness]] (Smith): 0.178826


=== 11-limit ===
=== 11-limit ===
Line 948: Line 946:
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


Optimal tuning (POTE): ~363/256 = 600.000, ~11/10 = 162.592
Optimal tuning (POTE): ~363/256 = 600.000{{c}}, ~11/10 = 162.592{{c}}


{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}
{{Optimal ET sequence|legend=0| 22, 74, 96, 118d }}


Badness: 0.068410
Badness (Smith): 0.068410


== Tertiosec ==
== Tertiosec ==
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 963: Line 961:
{{Mapping|legend=1| 3 7 5 9 | 0 -8 7 -2 }}
{{Mapping|legend=1| 3 7 5 9 | 0 -8 7 -2 }}


[[Optimal tuning]] ([[POTE]]): ~3072/2401 = 400.000, ~15/14 = 112.283
[[Optimal tuning]] ([[POTE]]): ~3072/2401 = 400.000{{c}}, ~15/14 = 112.283{{c}}


{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


[[Badness]]: 0.431636
[[Badness]] (Smith): 0.431636


=== 11-limit ===
=== 11-limit ===
Line 976: Line 974:
Mapping: {{mapping| 3 7 5 9 9 | 0 -8 7 -2 5 }}
Mapping: {{mapping| 3 7 5 9 9 | 0 -8 7 -2 5 }}


Optimal tuning (POTE): ~44/35 = 400.000, ~15/14 = 112.171
Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~15/14 = 112.171{{c}}


{{Optimal ET sequence|legend=1| 21, 54, 75e }}
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


Badness: 0.173485
Badness (Smith): 0.173485


[[Category:Temperament collections]]
[[Category:Temperament collections]]
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"