Marvel temperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
4,516 edits
mNo edit summary
m →Gracecordial: linking
 
(28 intermediate revisions by 11 users not shown)
Line 1: Line 1:
{{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 temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the marvel comma or septimal kleisma.  


Temperaments considered in families and clans are:  
Temperaments considered in families and clans are:  
* ''[[Sharp]]'', {25/24, 28/27} → [[Dicot family #Sharp|Dicot family]]
* ''[[Pelogic]]'' → [[Mavila family #Pelogic|Mavila family]] (+21/20 or 135/128, generated by the fifth with 5/4 mapped to the m3)
* ''[[Pelogic]]'', {21/20, 135/128} → [[Pelogic family #Pelogic|Pelogic family]]
* [[Meantone]] → [[Meantone family #Septimal meantone|Meantone family]] (+81/80 or 126/125, generated by the fifth with 5/4 mapped to the M3)
* ''[[August]]'', {36/35, 128/125} → [[Augmented family #August|Augmented family]]
* [[Garibaldi]] → [[Schismatic family #Garibaldi|Schismatic family]] (+3125/3087, generated by the fifth with 5/4 mapped to the d4)
* [[Pajara]], {50/49, 64/63} → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Pajara]] → [[Diaschismic family #Pajara|Diaschismic family]] (+50/49 or 64/63, generated by the fifth with a semioctave period)
* [[Meantone]], {81/80, 126/125} → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Sharpie]]'' → [[Dicot family #Sharpie|Dicot family]] (+25/24 or 28/27, fifth sliced in two)
* [[Magic]], {225/224, 245/243} → [[Magic family #Magic|Magic family]]
* ''[[Immune]]'' → [[Immunity family #Immune|Immunity family]] (+781250/750141, twelfth sliced in two)
* ''[[Passive]]'', {225/224, 256/245} → [[Passion family #Passive|Passion family]]
* ''[[August]]'' → [[Augmented family #August|Augmented family]] (+36/35 or 128/125, generated by the fifth with a 1/3-octave period)
* [[Miracle]], {225/224, 1029/1024} → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Fog]]'' → [[Misty family #Fog|Misty family]] (+156250/151263, generated by the fifth with a 1/3-octave period)
* [[Orwell]], {225/224, 1728/1715} → [[Semicomma family #Orwell|Semicomma family]]
* [[Negri]] → [[Slendro clan #Negri|Slendro clan]] (+49/48, fourth sliced in four)
* [[Garibaldi temperament|Garibaldi]], {225/224, 3125/3087} → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Magic]] → [[Magic family #Magic|Magic family]] (+245/243, twelfth sliced in five)
* [[Catakleismic]], {225/224, 4375/4374} → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Passive]]'' → [[Passion family #Passive|Passion family]] (+256/245, fourth sliced in five)
* ''[[Snipes]]'', {225/224, 6125/5832} → [[Wesley family #Snipes|Wesley family]]
* ''[[Quintapole]]'' → [[Quintaleap family #Quintapole|Quintaleap family]] (+7812500/7411887, fourth sliced in five)
* ''[[Decic]]'', {225/224, 16807/16384} → [[Cloudy clan #Decic|Cloudy clan]]
* ''[[Houborizic]]'' → [[Amity family #Houborizic|Amity family]] (+1250000/1240029, eleventh sliced in five)
* ''[[Amavil]]'', {225/224, 17496/16807} → [[Mabila family #Amavil|Mabila family]]
* ''[[Qintosec]]'' → [[Quintosec family #Qintosec|Quintosec family]] (+2560000/2470629, generated by the classical minor second with a 1/5-octave period)
* ''[[Escapade]]'', {225/224, 65625/65536} → [[Escapade family #Escapade|Escapade family]]
* [[Miracle]] → [[Gamelismic clan #Miracle|Gamelismic clan]] (+1029/1024, fifth sliced in six)
* ''[[Fog]]'', {225/224, 156250/151263} → [[Misty family #Fog|Misty family]]
* [[Catakleismic]] → [[Kleismic family #Catakleismic|Kleismic family]] (+4375/4374, twelfth sliced in six)
* ''[[Compton]]'', {225/224, 250047/250000} → [[Compton family #Compton|Compton family]]
* ''[[Marvo]]'' → [[Gravity family #Marvo|Gravity family]] (+78125000/78121827, two octaves and a fifth sliced in six)
* ''[[Immune]]'', {225/224, 781250/750141} → [[Immunity family #Immune|Immunity family]]
* [[Orwell]] → [[Semicomma family #Orwell|Semicomma family]] (+1728/1715, twelfth sliced in seven)
* ''[[Betic]]'', {225/224, 1071875/1062882} → [[Sycamore family #Betic|Sycamore family]]
* ''[[Snipes]]'' → [[Wesley family #Snipes|Wesley family]] (+6125/5832, two octaves and a fourth sliced in seven)
* ''[[Houborizic]]'' {225/224, 1250000/1240029} → [[Amity family #Houborizic|Amity family]]
* ''[[Submajor]]'' → [[Buzzardsmic clan #Submajor|Buzzardsmic clan]] (+65536/64827, two octaves and a fourth sliced in eight)
* ''[[Qintosec]]'', {225/224, 2560000/2470629} → [[Qintosec family #Qintosec|Qintosec family]]
* ''[[Escapist]]'' → [[Escapade family #Escapist|Escapade family]] (+65625/65536, fourth sliced in nine)
* ''[[Quintapole]]'', {225/224, 7812500/7411887} → [[Quintaleap family #Quintapole|Quintaleap family]]
* ''[[Decic]]'' → [[Cloudy clan #Decic|Cloudy clan]] (+16807/16384, generated by the fifth with a 1/10-octave period)
* ''[[Maquila]]'', {225/224, 30233088/28824005} → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Amavil]]'' → [[Mabila family #Amavil|Mabila family]] (+17496/16807, four octaves and a fourth sliced in ten)
* ''[[Marvo]]'', {225/224, 78125000/78121827} → [[Gravity family #Marvo|Gravity family]]
* ''[[Betic]]'' → [[Sycamore family #Betic|Sycamore family]] (+1071875/1062882, fifth sliced in eleven)
* ''[[Hendeca]]'' → [[11th-octave temperaments #Hendeca|11th-octave temperaments]] (+122880/117649, generated by the fifth with a 1/11-octave period)
* ''[[Compton]]'' → [[Compton family #Compton|Compton family]] (+250047/250000, generated by the classical major third with a 1/12-octave period)
* ''[[Raccoon]]'' → [[Vavoom family #Raccoon|Vavoom family]] (+41943040/40353607, twelfth sliced in seventeen)
* ''[[Maquila]]'' → [[Maquila family #Septimal maquila|Maquila family]] (+30233088/28824005, seven octaves and a fifth sliced in seventeen)
* ''[[Gammy]]'' → [[Gammic family #Gammy|Gammic family]] (+94143178827/91913281250, fifth sliced in twenty)


Considered below are negri, wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, submajor, alphorn, tertiosec, gwazy, and gracecordial.  
Considered below are wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, alphorn, tertiosec, gwazy, and gracecordial.  


Since (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.


Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)
Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)


== Negri ==
== Wizard ==
{{main| Negri }}
{{Main| Wizard }}
 
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Wizard]].''
Negri tempers out the [[negri comma]] in the 5-limit, [[49/48]] and [[225/224]] in the 7-limit. It can be extended naturally to the 2.3.5.7.13 subgroup by adding [[91/90]] to the comma list; this will be discussed below under the title of negra.
 
Subgroup: 2.3.5
 
[[Comma list]]: 16875/16384
 
[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
 
{{Multival|legend=1| 4 -3 -14 }}
 
[[POTE generator]]: ~16/15 = 125.7549
 
{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
 
[[Badness]]: 0.086856
 
=== 7-limit ===
Subgroup: 2.3.5.7
 
[[Comma list]]: 49/48, 225/224
 
[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
 
{{Multival|legend=1| 4 -3 2 -14 -8 13 }}
 
[[POTE generator]]: ~15/14 = 125.608
 
{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
 
[[Badness]]: 0.026483
 
==== Negra ====
This is the 2.3.5.7.13 extension of negri.
 
Subgroup: 2.3.5.7.13
 
[[Comma list]]: 49/48, 65/64, 91/90
 
[[Sval]] [[mapping]]: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -3 }}]
 
[[Gencom]] [[mapping]]: [{{val| 1 2 2 3 0 4 }}, {{val| 0 -4 3 -2 0 -3 }}]
 
[[Gencom]]: [2 14/13; 49/48 65/64 91/90]
 
[[POTE generator]]: ~14/13 = 125.567
 
{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 49/48, 56/55
 
Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
 
POTE generator: ~15/14 = 126.474
 
Optimal GPV sequence: {{Val list| 9, 10, 19 }}
 
Badness: 0.026190
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 45/44, 49/48, 56/55, 78/77
 
Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
 
POTE generator: ~14/13 = 126.431
 
Optimal GPV sequence: {{Val list| 9, 10, 19 }}
 
Badness: 0.017833
 
=== Negril ===
Subgroup: 2.3.5.7.11
 
Comma list: 49/48, 100/99, 225/224
 
Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
 
POTE generator: ~15/14 = 124.767
 
Optimal GPV sequence: {{Val list| 19, 29, 48d, 77cdd }}
 
Badness: 0.038679
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 65/64, 91/90, 875/858
 
Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
 
POTE generator: ~14/13 = 124.716
 
Optimal GPV sequence: {{Val list| 19, 29, 48df, 77cddf }}
 
Badness: 0.024383
 
=== Negric ===
Subgroup: 2.3.5.7.11
 
Comma list: 33/32, 49/48, 77/75
 
Mapping: [{{val| 1 2 2 3 3 }}, {{val| 0 -4 3 -2 4 }}]
 
POTE generator: ~15/14 = 127.039
 
Optimal GPV sequence: {{Val list| 9, 19e }}
 
Badness: 0.030617
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 33/32, 49/48, 65/64, 91/90
 
Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
 
POTE generator: ~14/13 = 127.039
 
Optimal GPV sequence: {{Val list| 9, 19e }}
 
Badness: 0.020205
 
=== Negroni ===
Subgroup: 2.3.5.7.11
 
Comma list: 49/48, 55/54, 225/224
 
Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
 
POTE generator: ~15/14 = 124.539
 
Optimal GPV sequence: {{Val list| 10, 19e, 29, 77cddee }}
 
Badness: 0.035296
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 55/54, 65/64, 91/90
 
Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
 
POTE generator: ~14/13 = 124.545
 
Optimal GPV sequence: {{Val list| 10, 19e, 29, 77cddeef }}
 
Badness: 0.021559
 
=== Wilsec ===
Subgroup: 2.3.5.7.11
 
Comma list: 49/48, 121/120, 225/224
 
Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]
 
POTE generator: ~11/8 = 537.186
 
Optimal GPV sequence: {{Val list| 9, 20, 29, 38d, 67cdde }}
 
Badness: 0.041886
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 65/64, 91/90, 121/120
 
Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
 
POTE generator: ~11/8 = 537.208
 
Optimal GPV sequence: {{Val list| 9, 20, 29, 38df, 67cddef }}
 
Badness: 0.025192
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
 
Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
 
POTE generator: ~11/8 = 537.230
 
Optimal GPV sequence: {{Val list| 9, 20g, 29g, 38df, 67cddefg }}
 
Badness: 0.021778
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
 
Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
 
POTE generator: ~11/8 = 537.214
 
Optimal GPV sequence: {{Val list| 9, 20g, 29g, 38df, 67cddefgh }}
 
Badness: 0.016828


== 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.  
{{Main|Wizard}}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 118098/117649
[[Comma list]]: 225/224, 118098/117649


[[Mapping]]: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
 
Mapping generators: ~1225/864, ~245/216
 
[[POTE generator]]: ~5/4 = 383.256


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
: mapping generators: ~1225/864, ~245/216


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
[[Optimal tuning]]s:
* [[CTE]]: ~1225/864 = 600.000, ~245/216 = 216.919
: [[error map]]: {{val| 0.000 -0.443 -3.232 +0.361 }}
* [[POTE]]: ~1225/864 = 600.000, ~245/216 = 216.744
: error map: {{val| 0.000 -1.492 -3.057 -1.388 }}


[[Badness]]: 0.040846
{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}


Scales: [[wizard22]]
[[Badness]] (Smith): 0.040846


=== 11-limit ===
=== 11-limit ===
Line 269: Line 70:
Comma list: 225/224, 385/384, 4000/3993
Comma list: 225/224, 385/384, 4000/3993


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* CTE: ~99/70 = 600.000, ~25/22 = 216.900
* POTE: ~99/70 = 600.000, ~25/22 = 216.768


POTE generator: ~5/4 = 383.232
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


Optimal GPV sequence: {{Val list| 22, 50, 72, 166, 238c, 310c }}
Badness (Smith): 0.018539
 
Badness: 0.018539
 
Scales: [[wizard22]]


==== Lizard ====
==== Lizard ====
Line 286: Line 85:
Comma list: 225/224, 351/350, 364/363, 385/384
Comma list: 225/224, 351/350, 364/363, 385/384


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* CTE: ~55/39 = 600.000, ~25/22 = 216.703
* POTE: ~55/39 = 600.000, ~25/22 = 216.711


POTE generator: ~5/4 = 383.389
{{Optimal ET sequence|legend=0| 22, 50, 72, 122, 194df }}


Optimal GPV sequence: {{Val list| 22, 50, 72, 122, 194df }}
Badness (Smith): 0.021781
 
Badness: 0.021781


===== 17-limit =====
===== 17-limit =====
Line 301: Line 100:
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
 
Mapping generators: ~17/12, ~17/15


POTE generator: ~5/4 = 383.381
Optimal tunings:
* CTE: ~17/12 = 600.000, ~17/15 = 216.748
* POTE: ~17/12 = 600.000, ~17/15 = 216.619


Optimal GPV sequence: {{Val list| 22, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 22, 50, 72, 122g, 194dfg }}


Badness: 0.014536
Badness (Smith): 0.014536


===== 19-limit =====
===== 19-limit =====
Line 316: Line 115:
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
 
Mapping generators: ~17/12, ~17/15


POTE generator: ~5/4 = 383.477
Optimal tunings:
* CTE: ~17/12 = 600.000, ~17/15 = 216.677
* POTE: ~17/12 = 600.000, ~17/15 = 216.523


Optimal GPV sequence: {{Val list| 22h, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


Badness: 0.015702
Badness (Smith): 0.015702


==== Gizzard ====
==== Gizzard ====
Line 331: Line 130:
Comma list: 225/224, 325/324, 385/384, 1573/1568
Comma list: 225/224, 325/324, 385/384, 1573/1568


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
 
Mapping generators: ~99/70, ~25/22


POTE generator: ~5/4 = 383.170
Optimal tunings:
* CTE: ~99/70 = 600.000, ~25/22 = 216.928
* POTE: ~99/70 = 600.000, ~25/22 = 216.830


Optimal GPV sequence: {{Val list| 72, 166, 238cf }}
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}


Badness: 0.020252
Badness (Smith): 0.020252


===== 17-limit =====
===== 17-limit =====
Line 346: Line 145:
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
 
Mapping generators: ~17/12, ~17/15


POTE generator: ~5/4 = 383.175
Optimal tunings:
* CTE: ~17/12 = 600.000, ~17/15 = 216.943
* POTE: ~17/12 = 600.000, ~17/15 = 216.825


Optimal GPV sequence: {{Val list| 72, 166g, 238cfg }}
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}


Badness: 0.013624
Badness (Smith): 0.013624


===== 19-limit =====
===== 19-limit =====
Line 361: Line 160:
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}


Mapping generators: ~17/12, ~17/15
Optimal tunings:
* CTE: ~17/12 = 600.000, ~17/15 = 216.918
* POTE: ~17/12 = 600.000, ~17/15 = 216.862


POTE generator: ~5/4 = 383.138
{{Optimal ET sequence|legend=0| 72, 94, 166g }}


Optimal GPV sequence: {{Val list| 72, 94, 166g }}
Badness (Smith): 0.014810
 
Badness: 0.014810


=== Mage ===
=== Mage ===
Line 376: Line 175:
Comma list: 99/98, 176/175, 1331/1296
Comma list: 99/98, 176/175, 1331/1296


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}


Mapping generators: ~77/54, ~55/48
Optimal tunings:
* CTE: ~77/54 = 600.000, ~55/48 = 217.247
* POTE: ~77/54 = 600.000, ~55/48 = 216.876


POTE generator: ~5/4 = 383.124
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


Optimal GPV sequence: {{Val list| 22, 50e, 72ee, 94ee }}
Badness (Smith): 0.057799
 
Badness: 0.057799


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 50421/50000
[[Comma list]]: 225/224, 50421/50000


[[Mapping]]: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
{{Mapping|legend=1| 1 4 -3 -3 | 0 -5 11 12 }}


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 580.286


[[POTE generator]]: ~7/5 = 580.286
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
 
{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}


[[Badness]]: 0.047578
[[Badness]]: 0.047578
Line 408: Line 205:
Comma list: 121/120, 225/224, 441/440
Comma list: 121/120, 225/224, 441/440


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
Mapping: {{mapping| 1 4 -3 -3 2 | 0 -5 11 12 3 }}


POTE generator: ~7/5 = 580.267
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.267


Optimal GPV sequence: {{Val list| 29, 31, 60e }}
{{Optimal ET sequence|legend=1| 29, 31, 60e }}


Badness: 0.023659
Badness: 0.023659
Line 421: Line 218:
Comma list: 105/104, 121/120, 196/195, 275/273
Comma list: 105/104, 121/120, 196/195, 275/273


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
Mapping: {{mapping| 1 4 -3 -3 2 -5 | 0 -5 11 12 3 18 }}


POTE generator: ~7/5 = 580.108
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.108


Optimal GPV sequence: {{Val list| 29, 31, 60e, 151cde }}
{{Optimal ET sequence|legend=1| 29, 31, 60e }}


Badness: 0.022993
Badness: 0.022993
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
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 ET sequence|legend=1| 29g, 31, 60e }}
Badness: 0.019093
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
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 ET sequence|legend=1| 29g, 31, 60e }}
Badness: 0.017010
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
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 ET sequence|legend=1| 29g, 31, 60e }}
Badness: 0.014471


=== Tritoni ===
=== Tritoni ===
Line 434: Line 270:
Comma list: 225/224, 385/384, 27783/27500
Comma list: 225/224, 385/384, 27783/27500


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
Mapping: {{mapping| 1 4 -3 -3 17 | 0 -5 11 12 -28 }}


POTE generator: ~7/5 = 580.389
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.389


Optimal GPV sequence: {{Val list| 31, 91, 122, 153d }}
{{Optimal ET sequence|legend=1| 31, 91, 122, 153d }}


Badness: 0.045456
Badness: 0.045456


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 84035/82944
[[Comma list]]: 225/224, 84035/82944


[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
{{Mapping|legend=1| 1 4 1 5 | 0 -11 6 -10 }}
 
{{Multival|legend=1| 11 -6 10 -35 -15 40 }}


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


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


[[Badness]]: 0.054502
[[Badness]]: 0.054502
Line 464: Line 298:
Comma list: 225/224, 385/384, 2401/2376
Comma list: 225/224, 385/384, 2401/2376


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
Mapping: {{mapping| 1 4 1 5 5 | 0 -11 6 -10 -7 }}


POTE generator: ~7/6 = 263.634
Optimal tuning (POTE): ~2 = 1200.000, ~7/6 = 263.634


Optimal GPV sequence: {{Val list| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=1| 41, 91, 223cdef }}


Badness: 0.031309
Badness: 0.031309
Line 477: Line 311:
Comma list: 105/104, 144/143, 196/195, 245/242
Comma list: 105/104, 144/143, 196/195, 245/242


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Mapping: {{mapping| 1 4 1 5 5 7 | 0 -11 6 -10 -7 -15 }}


POTE generator: ~7/6 = 263.700
Optimal tuning (POTE): ~2 = 1200.000, ~7/6 = 263.700


Optimal GPV sequence: {{Val list| 41, 91 }}
{{Optimal ET sequence|legend=1| 41, 91 }}


Badness: 0.023117
Badness: 0.023117


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 2500000/2470629
[[Comma list]]: 225/224, 2500000/2470629


[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
{{Mapping|legend=1| 1 5 -5 -5 | 0 -7 15 16 }}


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 585.585


[[POTE generator]]: ~7/5 = 585.585
{{Optimal ET sequence|legend=1| 41, 84, 125 }}
 
{{Val list|legend=1| 41, 84, 125 }}


[[Badness]]: 0.055078
[[Badness]]: 0.055078
Line 507: Line 339:
Comma list: 225/224, 441/440, 1344/1331
Comma list: 225/224, 441/440, 1344/1331


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
Mapping: {{mapping| 1 5 -5 -5 2 | 0 -7 15 16 3 }}


POTE generator: ~7/5 = 585.606
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 585.606


Optimal GPV sequence: {{Val list| 41, 84, 125e }}
{{Optimal ET sequence|legend=1| 41, 84, 125e }}


Badness: 0.036383
Badness: 0.036383
Line 520: Line 352:
Comma list: 144/143, 225/224, 364/363, 441/440
Comma list: 144/143, 225/224, 364/363, 441/440


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Mapping: {{mapping| 1 5 -5 -5 2 12 | 0 -7 15 16 3 -17 }}


POTE generator: ~7/5 = 585.657
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 585.657


Optimal GPV sequence: {{Val list| 41, 84, 125e, 209ef, 293ef }}
{{Optimal ET sequence|legend=1| 41, 84, 125e, 209ef, 293ef }}


Badness: 0.027544
Badness: 0.027544


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 589824/588245
[[Comma list]]: 225/224, 589824/588245


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 38.413


[[POTE generator]]: ~49/48 = 38.413
{{Optimal ET sequence|legend=1| 31, 94, 125 }}
 
{{Val list|legend=1| 31, 94, 125 }}


[[Badness]]: 0.056934
[[Badness]]: 0.056934
Line 550: Line 380:
Comma list: 225/224, 385/384, 1331/1323
Comma list: 225/224, 385/384, 1331/1323


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


POTE generator: ~49/48 = 38.387
Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 38.387


Optimal GPV sequence: {{Val list| 31, 63, 94, 125 }}
{{Optimal ET sequence|legend=1| 31, 63, 94, 125 }}


Badness: 0.025342
Badness: 0.025342
Line 563: Line 393:
Comma list: 225/224, 275/273, 385/384, 1331/1323
Comma list: 225/224, 275/273, 385/384, 1331/1323


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


POTE generator: ~49/48 = 38.314
Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 38.314


Optimal GPV sequence: {{Val list| 31, 63, 94 }}
{{Optimal ET sequence|legend=1| 31, 63, 94 }}


Badness: 0.025913
Badness: 0.025913


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 1029/1000
[[Comma list]]: 225/224, 1029/1000


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 568.865


[[POTE generator]]: ~7/5 = 568.865
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
 
{{Val list|legend=1| 2, 17d, 19, 78bd, 97bd }}


[[Badness]]: 0.059245
[[Badness]]: 0.059245
Line 593: Line 421:
Comma list: 45/44, 56/55, 1029/1000
Comma list: 45/44, 56/55, 1029/1000


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


POTE generator: ~7/5 = 569.144
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 569.144


Optimal GPV sequence: {{Val list| 2, 17d, 19, 59bde, 78bde, 97bde }}
{{Optimal ET sequence|legend=1| 2, 17d, 19, 59bde, 78bde, 97bde }}


Badness: 0.045675
Badness: 0.045675


== Submajor ==
== Marvolo ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 69198046875/68719476736
[[Comma list]]: 225/224, 156250000/155649627


[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}


[[POTE generator]]: ~10125/8192 = 362.321
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~21/20 = 83.348


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bcd, 691bcd }}


[[Badness]]: 0.130236
[[Badness]]: 0.083338
 
=== 7-limit ===
Subgroup: 2.3.5.7
 
[[Comma list]]: 225/224, 51200/50421
 
[[Mapping]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
 
{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
 
[[POTE generator]]: ~49/40 = 362.255
 
{{Val list|legend=1| 10, 33, 43, 53 }}
 
[[Badness]]: 0.060533


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


Comma list: 225/224, 385/384, 6655/6561
Comma list: 225/224, 441/440, 4000/3993


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


POTE generator: ~27/22 = 362.101
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.340


Optimal GPV sequence: {{Val list| 10, 43e, 53, 116, 169de, 285cde }}
{{Optimal ET sequence|legend=1| 29, 43, 72 }}


Badness: 0.050582
Badness: 0.028965


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


Comma list: 169/168, 225/224, 275/273, 385/384
Comma list: 169/168, 225/224, 364/363, 441/440


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


POTE generator: ~16/13 = 362.105
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.330


Optimal GPV sequence: {{Val list| 10, 43e, 53, 116, 169de, 285cdef }}
{{Optimal ET sequence|legend=1| 29, 43, 72, 115f }}


Badness: 0.027689
Badness: 0.021470


=== Interpental ===
=== 17-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17


Comma list: 99/98, 176/175, 51200/50421
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


POTE generator: ~49/40 = 362.418
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.330


POTE generator: ~49/40 = 362.418
{{Optimal ET sequence|legend=1| 29g, 43, 72 }}


Optimal GPV sequence: {{Val list| 43, 53, 96, 149d }}
Badness: 0.014924


Badness: 0.051806
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 99/98, 169/168, 176/175, 640/637
 
POTE generator: ~16/13 = 362.402
 
Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
 
POTE generator: ~16/13 = 362.402
 
Optimal GPV sequence: {{Val list| 43, 53, 96, 149d }}
 
Badness: 0.029680
 
== Marvolo ==
Subgroup: 2.3.5.7
 
[[Comma list]]: 225/224, 156250000/155649627
 
[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
 
{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
 
[[POTE generator]]: ~21/20 = 83.348
 
{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
 
[[Badness]]: 0.083338
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 225/224, 441/440, 4000/3993
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


POTE generator: ~21/20 = 83.340
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.330


Optimal GPV sequence: {{Val list| 29, 43, 72 }}
{{Optimal ET sequence|legend=1| 29g, 43, 72 }}


Badness: 0.028965
Badness: 0.014722
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 364/363, 441/440
 
Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
 
POTE generator: ~21/20 = 83.330
 
Optimal GPV sequence: {{Val list| 29, 43, 72, 115f }}
 
Badness: 0.021470


== Enneaportent ==
== Enneaportent ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 40353607/40310784
[[Comma list]]: 225/224, 40353607/40310784


[[Mapping]]: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
: Mapping generators: ~2592/2401, ~12005/6912


[[POTE generator]]: ~5/4 = 383.165
[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 133.3333, ~12005/6912 = 950.1680 (~1728/1715 = 16.8347)


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


[[Badness]]: 0.093679
[[Badness]]: 0.093679
Line 746: Line 514:
Comma list: 225/224, 385/384, 12005/11979
Comma list: 225/224, 385/384, 12005/11979


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


POTE generator: ~5/4 = 383.146
Optimal tuning (POTE): ~121/112 = 133.3333, ~210/121 = 950.1873 (~99/98 = 16.8540)


Optimal GPV sequence: {{Val list| 9, 63, 72, 423cd, 495bcd }}
{{Optimal ET sequence|legend=1| 9, 63, 72, 423cd, 495bcd }}


Badness: 0.030426
Badness: 0.030426
Line 759: Line 527:
Comma list: 169/168, 225/224, 364/363, 1716/1715
Comma list: 169/168, 225/224, 364/363, 1716/1715


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


POTE generator: ~5/4 = 383.047
Optimal tuning (POTE): ~14/13 = 133.3333, ~26/15 = 950.2867 (~105/104 = 16.9534)


Optimal GPV sequence: {{Val list| 9, 63, 72, 279cf }}
{{Optimal ET sequence|legend=1| 9, 63, 72, 279cf }}


Badness: 0.022322
Badness: 0.022322


== Gracecordial ==
== Gracecordial ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Gracecordial]].''
: ''For the 5-limit version of this temperament, see [[Schismic–Pythagorean equivalence continuum #Gracecordial]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 781250000/771895089
[[Comma list]]: 225/224, 781250000/771895089


[[Mapping]]: [{{val| 1 0 34 63 }}, {{val| 0 1 -20 -38 }}]
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}


{{Multival|legend=1| 1 -20 -38 -34 -63 -32 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3/2 = 700.824


[[POTE generator]]: ~3/2 = 700.824
{{Optimal ET sequence|legend=1| 12, 113, 125, 238c, 363c }}
 
{{Val list|legend=1| 12, 113, 125, 238c, 363c }}


[[Badness]]: 0.096279
[[Badness]]: 0.096279
Line 789: Line 555:
Comma list: 225/224, 385/384, 236328125/234365481
Comma list: 225/224, 385/384, 236328125/234365481


Mapping: [{{val| 1 0 34 63 -90 }}, {{val| 0 1 -20 -38 59 }}]
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


POTE generator: ~3/2 = 700.834
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.834


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 125, 238c }}
{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125, 238c }}


Badness: 0.089588
Badness: 0.089588
Line 802: Line 568:
Comma list: 225/224, 325/324, 385/384, 831875/830466
Comma list: 225/224, 325/324, 385/384, 831875/830466


Mapping: [{{val| 1 0 34 63 -90 -66 }}, {{val| 0 1 -20 -38 59 44 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


POTE generator: ~3/2 = 700.841
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.841


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}


Badness: 0.052235
Badness: 0.052235
Line 815: Line 581:
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


Mapping: [{{val| 1 0 34 63 -90 -66 -7 }}, {{val| 0 1 -20 -38 59 44 7 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


POTE generator: ~3/2 = 700.841
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.841


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}


Badness: 0.038565
Badness: 0.038565
Line 828: Line 594:
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 }}, {{val| 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 }}


POTE generator: ~3/2 = 700.842
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.842


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}


Badness: 0.028165
Badness: 0.028165
Line 841: Line 607:
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 -43 }}, {{val| 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 }}


POTE generator: ~3/2 = 700.843
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.843


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 238cfi }}
{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 238cfi }}


Badness: 0.021879
Badness: 0.021879
Line 854: Line 620:
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 -43 -49 }}, {{val| 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 }}


POTE generator: ~3/2 = 700.842
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.842


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cfi }}
{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cfi }}


Badness: 0.018011
Badness: 0.018011
Line 867: Line 633:
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 }}, {{val| 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 }}


POTE generator: ~3/2 = 700.838
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.838


Optimal GPV sequence: {{Val list| 12e, 101cdek, 113, 125f, 238cfi }}
{{Optimal ET sequence|legend=1| 12e, 101cdek, 113, 125f, 238cfi }}


Badness: 0.016007
Badness: 0.016007
Line 880: Line 646:
Comma list: 225/224, 441/440, 109375/107811
Comma list: 225/224, 441/440, 109375/107811


Mapping: [{{val| 1 0 34 63 89 }}, {{val| 0 1 -20 -38 -54 }}]
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


POTE generator: ~3/2 = 700.885
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.885


Optimal GPV sequence: {{Val list| 12, 101cd, 113 }}
{{Optimal ET sequence|legend=1| 12, 101cd, 113 }}


Badness: 0.066964
Badness: 0.066964
Line 893: Line 659:
Comma list: 225/224, 364/363, 441/440, 6125/6084
Comma list: 225/224, 364/363, 441/440, 6125/6084


Mapping: [{{val| 1 0 34 63 89 113 }}, {{val| 0 1 -20 -38 -54 -69 }}]
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


POTE generator: ~3/2 = 700.890
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.890


Optimal GPV sequence: {{Val list| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}


Badness: 0.044196
Badness: 0.044196
Line 906: Line 672:
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Mapping: [{{val| 1 0 34 63 89 113 -7 }}, {{val| 0 1 -20 -38 -54 -69 7 }}]
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


POTE generator: ~3/2 = 700.870
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.870


Optimal GPV sequence: {{Val list| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}


Badness: 0.036637
Badness: 0.036637
Line 919: Line 685:
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


Mapping: [{{val| 1 0 34 63 89 113 -7 9 }}, {{val| 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 }}


POTE generator: ~3/2 = 700.866
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.866


Optimal GPV sequence: {{Val list| 12f, 101cdf, 113 }}
{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}


Badness: 0.027559
Badness: 0.027559


== Alphorn ==
== Alphorn ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 5764801/5668704
[[Comma list]]: 225/224, 5764801/5668704


[[Mapping]]: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
{{Mapping|legend=1| 1 9 0 13 | 0 -16 5 -22 }}


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~48/35 = 556.221


[[POTE generator]]: ~48/35 = 556.221
{{Optimal ET sequence|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
 
{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}


[[Badness]]: 0.129258
[[Badness]]: 0.129258
Line 947: Line 711:
Comma list: 225/224, 385/384, 12250/11979
Comma list: 225/224, 385/384, 12250/11979


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
Mapping: {{mapping| 1 9 0 13 3 | 0 -16 5 -22 1 }}


POTE generator: ~11/8 = 556.144
Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 556.144


Optimal GPV sequence: {{Val list| 28d, 41, 315cde }}
{{Optimal ET sequence|legend=1| 28d, 41, 315cde }}


Badness: 0.073459
Badness: 0.073459


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 4194304/4117715  
[[Comma list]]: 225/224, 4194304/4117715  


[[Mapping]]: [{{Val|1 3 1 3}}, {{Val|0 -15 14 -2}}]
{{Mapping|legend=1| 1 3 1 3 | 0 -15 14 -2 }}


{{Multival|legend=1|15 -14 2 -57 -39 44}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16/15 = 113.235


[[POTE generator]]: ~16/15 = 113.235
{{Optimal ET sequence|legend=1| 21, 32, 53 }}
 
{{Val list|legend=1| 21, 32, 53 }}


[[Badness]]: 0.140970
[[Badness]]: 0.140970
Line 977: Line 739:
Comma list: 99/98, 176/175, 1310720/1294139
Comma list: 99/98, 176/175, 1310720/1294139


Mapping: [{{Val|1 3 1 3 1}}, {{Val|0 -15 14 -2 26}}]
Mapping: {{mapping| 1 3 1 3 1 | 0 -15 14 -2 26 }}


POTE generator: ~16/15 = 113.323
Optimal tuning (POTE): ~2 = 1200.000, ~16/15 = 113.323


Optimal GPV sequence: {{Val list| 21, 32e, 53, 127, 180de }}
{{Optimal ET sequence|legend=1| 21, 32e, 53, 127, 180de }}


Badness: 0.085390
Badness: 0.085390
Line 990: Line 752:
Comma list: 99/98, 176/175, 640/637, 847/845
Comma list: 99/98, 176/175, 640/637, 847/845


Mapping: [{{Val|1 3 1 3 1 2}}, {{Val|0 -15 14 -2 26 18}}]
Mapping: {{mapping| 1 3 1 3 1 2 | 0 -15 14 -2 26 18 }}


POTE generator: ~16/15 = 113.323
Optimal tuning (POTE): ~2 = 1200.000, ~16/15 = 113.323


Optimal GPV sequence: {{Val list| 21, 32e, 53, 127, 180de }}
{{Optimal ET sequence|legend=1| 21, 32e, 53, 127, 180de }}


Badness: 0.045569
Badness: 0.045569
Line 1,003: Line 765:
Comma list: 225/224, 385/384, 66550/64827
Comma list: 225/224, 385/384, 66550/64827


Mapping: [{{Val|1 3 1 3 6}}, {{Val|0 -15 14 -2 -27}}]
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


POTE generator: ~16/15 = 113.142
Optimal tuning (POTE): ~2 = 1200.000, ~16/15 = 113.142


Optimal GPV sequence: {{Val list| 32, 53, 191de, 244cddee, 297cddee }}
{{Optimal ET sequence|legend=1| 32, 53, 191de, 244cddee, 297cddee }}


Badness: 0.087333
Badness: 0.087333


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 125000000/121060821
[[Comma list]]: 225/224, 125000000/121060821


[[Mapping]]: [{{Val|1 6 -7 -7}}, {{Val|0 -9 19 20}}]
{{Mapping|legend=1| 1 6 -7 -7 | 0 -9 19 20 }}
 
{{Multival|legend=1|9 -19 -20 -51 -57 7}}


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


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


[[Badness]]: 0.143976
[[Badness]]: 0.143976
Line 1,033: Line 793:
Comma list: 121/120, 225/224, 22000/21609
Comma list: 121/120, 225/224, 22000/21609


Mapping: [{{Val|1 6 -7 -7 1}}, {{Val|0 -9 19 20 5}}]
Mapping: {{mapping| 1 6 -7 -7 1 | 0 -9 19 20 5 }}


POTE generator: ~7/5 = 588.626
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 588.626


Optimal GPV sequence: {{Val list| 51, 53 }}
{{Optimal ET sequence|legend=1| 51, 53 }}


Badness: 0.074295
Badness: 0.074295
Line 1,046: Line 806:
Comma list: 121/120, 225/224, 275/273, 1040/1029
Comma list: 121/120, 225/224, 275/273, 1040/1029


Mapping: [{{Val|1 6 -7 -7 1 -12}}, {{Val|0 -9 19 20 5 32}}]
Mapping: {{mapping| 1 6 -7 -7 1 -12 | 0 -9 19 20 5 32 }}


POTE generator: ~7/5 = 588.654
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 588.654


Optimal GPV sequence: {{Val list| 51f, 53 }}
{{Optimal ET sequence|legend=1| 51f, 53 }}


Badness: 0.047441
Badness: 0.047441


== Quintannic ==
== Quintannic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 9805926501/9765625000
[[Comma list]]: 225/224, 9805926501/9765625000


[[Mapping]]: [{{val| 1 1 5 7 }}, {{val| 0 5 -23 -36 }}]
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
 
{{Multival|legend=1| 5 -23 -36 -48 -71 -19 }}


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


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


[[Badness]]: 0.150565
[[Badness]]: 0.150565
Line 1,074: Line 832:
Comma list: 225/224, 441/440, 43923/43750
Comma list: 225/224, 441/440, 43923/43750


Mapping: [{{val| 1 1 5 7 8 }}, {{val| 0 5 -23 -36 -39 }}]
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


POTE generator: ~320/297 = 139.827
Optimal tuning (POTE): ~2 = 1200.000, ~320/297 = 139.827


Optimal GPV sequence: {{Val list| 43, 60e, 103 }}
{{Optimal ET sequence|legend=1| 43, 60e, 103 }}


Badness: 0.052590
Badness: 0.052590
Line 1,087: Line 845:
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


Mapping: [{{val| 1 1 5 7 8 3 }}, {{val| 0 5 -23 -36 -39 6 }}]
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


POTE generator: ~13/12 = 139.812
Optimal tuning (POTE): ~2 = 1200.000, ~13/12 = 139.812


Optimal GPV sequence: {{Val list| 43, 60e, 103 }}
{{Optimal ET sequence|legend=1| 43, 60e, 103 }}


Badness: 0.032730
Badness: 0.032730
Line 1,100: Line 858:
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Mapping: [{{val| 1 1 5 7 8 3 7 }}, {{val| 0 5 -23 -36 -39 6 -25 }}]
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


POTE generator: ~13/12 = 139.815
Optimal tuning (POTE): ~2 = 1200.000, ~13/12 = 139.815


Optimal GPV sequence: {{Val list| 43, 60e, 103 }}
{{Optimal ET sequence|legend=1| 43, 60e, 103 }}


Badness: 0.023038
Badness: 0.023038


== Gwazy ==
== Gwazy ==
{{see also|Very high accuracy temperaments #Kwazy}}
{{See also| Very high accuracy temperaments #Kwazy }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 5971968/5764801
[[Comma list]]: 225/224, 5971968/5764801


[[Mapping]]: [{{val| 2 1 6 4 }}, {{val| 0 8 -5 6 }}]
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}


{{Multival|legend=1| 16 -10 12 -53 -26 56 }}
[[Optimal tuning]] ([[POTE]]): ~2401/1728 = 600.000, ~35/32 = 162.658


[[POTE generator]]: ~35/32 = 162.658
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}
 
{{Val list|legend=1| 22, 74, 96, 118d }}


[[Badness]]: 0.178826
[[Badness]]: 0.178826
Line 1,130: Line 886:
Comma list: 99/98, 176/175, 65536/65219
Comma list: 99/98, 176/175, 65536/65219


Mapping: [{{val| 2 1 6 4 8 }}, {{val| 0 8 -5 6 -4 }}]
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


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


Optimal GPV sequence: {{Val list| 22, 74, 96, 118d }}
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


Badness: 0.068410
Badness: 0.068410


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 14495514624/13841287201
[[Comma list]]: 225/224, 14495514624/13841287201


[[Mapping]]: [{{val| 3 7 5 9 }}, {{val| 0 -8 7 -2 }}]
{{Mapping|legend=1| 3 7 5 9 | 0 -8 7 -2 }}
 
{{Multival|legend=1| 24 -21 6 -89 -58 73 }}


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


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


[[Badness]]: 0.431636
[[Badness]]: 0.431636
Line 1,160: Line 914:
Comma list: 225/224, 3840/3773, 12005/11979
Comma list: 225/224, 3840/3773, 12005/11979


Mapping: [{{val| 3 7 5 9 9 }}, {{val| 0 -8 7 -2 5 }}]
Mapping: {{mapping| 3 7 5 9 9 | 0 -8 7 -2 5 }}


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


Optimal GPV sequence: {{Val list| 21, 54, 75e }}
{{Optimal ET sequence|legend=1| 21, 54, 75e }}


Badness: 0.173485
Badness: 0.173485


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Marvel]]
[[Category:Pages with mostly numerical content]]
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Marvel| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 13:05, 23 August 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This page discusses miscellaneous rank-2 temperaments tempering out [-5 2 2 -1⟩ = 225/224, the marvel comma or septimal kleisma.

Temperaments considered in families and clans are:

Considered below are wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, alphorn, tertiosec, gwazy, and gracecordial.

Since (5/4)2 = 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.

Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)

Wizard

Main article: Wizard
For the 5-limit version of this temperament, 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 22 & 72. Its ploidacot is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the perfect twelfth. 72edo, 94edo, and especially 166edo are good tunings for it.

Subgroup: 2.3.5.7

Comma list: 225/224, 118098/117649

Mapping: [⟨2 1 5 2], ⟨0 6 -1 10]]

mapping generators: ~1225/864, ~245/216

Optimal tunings:

  • CTE: ~1225/864 = 600.000, ~245/216 = 216.919
error map: ⟨0.000 -0.443 -3.232 +0.361]
  • POTE: ~1225/864 = 600.000, ~245/216 = 216.744
error map: ⟨0.000 -1.492 -3.057 -1.388]

Optimal ET sequence: 22, 50, 72, 166, 238c, 310c, 382c

Badness (Smith): 0.040846

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 4000/3993

Mapping: [⟨2 1 5 2 8], ⟨0 6 -1 10 -3]]

Optimal tunings:

  • CTE: ~99/70 = 600.000, ~25/22 = 216.900
  • POTE: ~99/70 = 600.000, ~25/22 = 216.768

Optimal ET sequence: 22, 50, 72, 166, 238c, 310c

Badness (Smith): 0.018539

Lizard

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 364/363, 385/384

Mapping: [⟨2 1 5 2 8 11], ⟨0 6 -1 10 -3 -10]]

Optimal tunings:

  • CTE: ~55/39 = 600.000, ~25/22 = 216.703
  • POTE: ~55/39 = 600.000, ~25/22 = 216.711

Optimal ET sequence: 22, 50, 72, 122, 194df

Badness (Smith): 0.021781

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 273/272, 289/288, 351/350, 375/374

Mapping: [⟨2 1 5 2 8 11 6], ⟨0 6 -1 10 -3 -10 6]]

Optimal tunings:

  • CTE: ~17/12 = 600.000, ~17/15 = 216.748
  • POTE: ~17/12 = 600.000, ~17/15 = 216.619

Optimal ET sequence: 22, 50, 72, 122g, 194dfg

Badness (Smith): 0.014536

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342

Mapping: [⟨2 1 5 2 8 11 6 2], ⟨0 6 -1 10 -3 -10 6 18]]

Optimal tunings:

  • CTE: ~17/12 = 600.000, ~17/15 = 216.677
  • POTE: ~17/12 = 600.000, ~17/15 = 216.523

Optimal ET sequence: 22h, 50, 72, 122g, 194dfg

Badness (Smith): 0.015702

Gizzard

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 1573/1568

Mapping: [⟨2 1 5 2 8 -2], ⟨0 6 -1 10 -3 26]]

Optimal tunings:

  • CTE: ~99/70 = 600.000, ~25/22 = 216.928
  • POTE: ~99/70 = 600.000, ~25/22 = 216.830

Optimal ET sequence: 22f, 72, 166, 238cf

Badness (Smith): 0.020252

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 289/288, 325/324, 375/374, 385/384

Mapping: [⟨2 1 5 2 8 -2 6], ⟨0 6 -1 10 -3 26 6]]

Optimal tunings:

  • CTE: ~17/12 = 600.000, ~17/15 = 216.943
  • POTE: ~17/12 = 600.000, ~17/15 = 216.825

Optimal ET sequence: 22f, 72, 166g, 238cfg

Badness (Smith): 0.013624

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594

Mapping: [⟨2 1 5 2 8 -2 6 15], ⟨0 6 -1 10 -3 26 6 -18]]

Optimal tunings:

  • CTE: ~17/12 = 600.000, ~17/15 = 216.918
  • POTE: ~17/12 = 600.000, ~17/15 = 216.862

Optimal ET sequence: 72, 94, 166g

Badness (Smith): 0.014810

Mage

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1331/1296

Mapping: [⟨2 1 5 2 4], ⟨0 6 -1 10 8]]

Optimal tunings:

  • CTE: ~77/54 = 600.000, ~55/48 = 217.247
  • POTE: ~77/54 = 600.000, ~55/48 = 216.876

Optimal ET sequence: 22, 50e, 72ee

Badness (Smith): 0.057799

Tritonic

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Tritonic.

Subgroup: 2.3.5.7

Comma list: 225/224, 50421/50000

Mapping: [⟨1 4 -3 -3], ⟨0 -5 11 12]]

Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.286

Optimal ET sequence: 29, 31, 60, 91, 122, 213bcd

Badness: 0.047578

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 441/440

Mapping: [⟨1 4 -3 -3 2], ⟨0 -5 11 12 3]]

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

Optimal ET sequence: 29, 31, 60e

Badness: 0.023659

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 196/195, 275/273

Mapping: [⟨1 4 -3 -3 2 -5], ⟨0 -5 11 12 3 18]]

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

Optimal ET sequence: 29, 31, 60e

Badness: 0.022993

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 121/120, 154/153, 196/195, 273/272

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 ET sequence: 29g, 31, 60e

Badness: 0.019093

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272

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 ET sequence: 29g, 31, 60e

Badness: 0.017010

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195

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 ET sequence: 29g, 31, 60e

Badness: 0.014471

Tritoni

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 27783/27500

Mapping: [⟨1 4 -3 -3 17], ⟨0 -5 11 12 -28]]

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

Optimal ET sequence: 31, 91, 122, 153d

Badness: 0.045456

Septimin

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Septimin.

Subgroup: 2.3.5.7

Comma list: 225/224, 84035/82944

Mapping: [⟨1 4 1 5], ⟨0 -11 6 -10]]

Optimal tuning (POTE): ~2 = 1200.000, ~7/6 = 263.632

Optimal ET sequence: 41, 91, 132d

Badness: 0.054502

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 2401/2376

Mapping: [⟨1 4 1 5 5], ⟨0 -11 6 -10 -7]]

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

Optimal ET sequence: 41, 91, 223cdef

Badness: 0.031309

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195, 245/242

Mapping: [⟨1 4 1 5 5 7], ⟨0 -11 6 -10 -7 -15]]

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

Optimal ET sequence: 41, 91

Badness: 0.023117

Merman

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Merman.

Subgroup: 2.3.5.7

Comma list: 225/224, 2500000/2470629

Mapping: [⟨1 5 -5 -5], ⟨0 -7 15 16]]

Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 585.585

Optimal ET sequence: 41, 84, 125

Badness: 0.055078

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 1344/1331

Mapping: [⟨1 5 -5 -5 2], ⟨0 -7 15 16 3]]

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

Optimal ET sequence: 41, 84, 125e

Badness: 0.036383

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 225/224, 364/363, 441/440

Mapping: [⟨1 5 -5 -5 2 12], ⟨0 -7 15 16 3 -17]]

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

Optimal ET sequence: 41, 84, 125e, 209ef, 293ef

Badness: 0.027544

Slender

Slender (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

Comma list: 225/224, 589824/588245

Mapping: [⟨1 2 2 3], ⟨0 -13 10 -6]]

Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 38.413

Optimal ET sequence: 31, 94, 125

Badness: 0.056934

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 1331/1323

Mapping: [⟨1 2 2 3 4], ⟨0 -13 10 -6 -17]]

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

Optimal ET sequence: 31, 63, 94, 125

Badness: 0.025342

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 385/384, 1331/1323

Mapping: [⟨1 2 2 3 4 3], ⟨0 -13 10 -6 -17 22]]

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

Optimal ET sequence: 31, 63, 94

Badness: 0.025913

Triton

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Stump.

Subgroup: 2.3.5.7

Comma list: 225/224, 1029/1000

Mapping: [⟨1 0 6 7], ⟨0 3 -7 -8]]

Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 568.865

Optimal ET sequence: 2, 17d, 19, 78bd, 97bd

Badness: 0.059245

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55, 1029/1000

Mapping: [⟨1 0 6 7 4], ⟨0 3 -7 -8 -1]]

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

Optimal ET sequence: 2, 17d, 19, 59bde, 78bde, 97bde

Badness: 0.045675

Marvolo

Subgroup: 2.3.5.7

Comma list: 225/224, 156250000/155649627

Mapping: [⟨1 2 1 1], ⟨0 -6 19 26]]

Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.348

Optimal ET sequence: 29, 43, 72, 619bcd, 691bcd

Badness: 0.083338

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 4000/3993

Mapping: [⟨1 2 1 1 2], ⟨0 -6 19 26 21]]

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

Optimal ET sequence: 29, 43, 72

Badness: 0.028965

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 364/363, 441/440

Mapping: [⟨1 2 1 1 2 3], ⟨0 -6 19 26 21 10]]

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

Optimal ET sequence: 29, 43, 72, 115f

Badness: 0.021470

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 225/224, 364/363, 441/440

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 ET sequence: 29g, 43, 72

Badness: 0.014924

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440

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 ET sequence: 29g, 43, 72

Badness: 0.014722

Enneaportent

Subgroup: 2.3.5.7

Comma list: 225/224, 40353607/40310784

Mapping: [⟨9 0 28 11], ⟨0 2 -1 2]]

Mapping generators: ~2592/2401, ~12005/6912

Optimal tuning (POTE): ~2592/2401 = 133.3333, ~12005/6912 = 950.1680 (~1728/1715 = 16.8347)

Optimal ET sequence: 9, 63, 72, 495bcd

Badness: 0.093679

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 12005/11979

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 ET sequence: 9, 63, 72, 423cd, 495bcd

Badness: 0.030426

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 364/363, 1716/1715

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 ET sequence: 9, 63, 72, 279cf

Badness: 0.022322

Gracecordial

For the 5-limit version of this temperament, see Schismic–Pythagorean equivalence continuum #Gracecordial.

Subgroup: 2.3.5.7

Comma list: 225/224, 781250000/771895089

Mapping: [⟨1 0 34 63], ⟨0 1 -20 -38]]

Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.824

Optimal ET sequence: 12, 113, 125, 238c, 363c

Badness: 0.096279

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 236328125/234365481

Mapping: [⟨1 0 34 63 -90], ⟨0 1 -20 -38 59]]

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

Optimal ET sequence: 12e, 101cde, 113, 125, 238c

Badness: 0.089588

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 831875/830466

Mapping: [⟨1 0 34 63 -90 -66], ⟨0 1 -20 -38 59 44]]

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

Optimal ET sequence: 12e, 101cde, 113, 125f, 238cf

Badness: 0.052235

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913

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 ET sequence: 12e, 101cde, 113, 125f, 238cf

Badness: 0.038565

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444

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 ET sequence: 12e, 101cde, 113, 125f, 238cf

Badness: 0.028165

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528

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 ET sequence: 12e, 101cde, 113, 238cfi

Badness: 0.021879

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493

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 ET sequence: 12e, 101cde, 113, 125f, 238cfi

Badness: 0.018011

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493

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 ET sequence: 12e, 101cdek, 113, 125f, 238cfi

Badness: 0.016007

Gracecord

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 109375/107811

Mapping: [⟨1 0 34 63 89], ⟨0 1 -20 -38 -54]]

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

Optimal ET sequence: 12, 101cd, 113

Badness: 0.066964

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 364/363, 441/440, 6125/6084

Mapping: [⟨1 0 34 63 89 113], ⟨0 1 -20 -38 -54 -69]]

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

Optimal ET sequence: 12f, 101cdf, 113

Badness: 0.044196

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989

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 ET sequence: 12f, 101cdf, 113

Badness: 0.036637

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663

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 ET sequence: 12f, 101cdf, 113

Badness: 0.027559

Alphorn

Subgroup: 2.3.5.7

Comma list: 225/224, 5764801/5668704

Mapping: [⟨1 9 0 13], ⟨0 -16 5 -22]]

Optimal tuning (POTE): ~2 = 1200.000, ~48/35 = 556.221

Optimal ET sequence: 28d, 41, 151cd, 192cd, 233cd

Badness: 0.129258

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 12250/11979

Mapping: [⟨1 9 0 13 3], ⟨0 -16 5 -22 1]]

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

Optimal ET sequence: 28d, 41, 315cde

Badness: 0.073459

Misneb

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Misneb.

Subgroup: 2.3.5.7

Comma list: 225/224, 4194304/4117715

Mapping: [⟨1 3 1 3], ⟨0 -15 14 -2]]

Optimal tuning (POTE): ~2 = 1200.000, ~16/15 = 113.235

Optimal ET sequence: 21, 32, 53

Badness: 0.140970

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1310720/1294139

Mapping: [⟨1 3 1 3 1], ⟨0 -15 14 -2 26]]

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

Optimal ET sequence: 21, 32e, 53, 127, 180de

Badness: 0.085390

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 640/637, 847/845

Mapping: [⟨1 3 1 3 1 2], ⟨0 -15 14 -2 26 18]]

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

Optimal ET sequence: 21, 32e, 53, 127, 180de

Badness: 0.045569

Musneb

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 66550/64827

Mapping: [⟨1 3 1 3 6], ⟨0 -15 14 -2 -27]]

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

Optimal ET sequence: 32, 53, 191de, 244cddee, 297cddee

Badness: 0.087333

Untriton

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Untriton.

Subgroup: 2.3.5.7

Comma list: 225/224, 125000000/121060821

Mapping: [⟨1 6 -7 -7], ⟨0 -9 19 20]]

Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 588.641

Optimal ET sequence: 51, 53, 316cd, 369cdd, 422cdd

Badness: 0.143976

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 22000/21609

Mapping: [⟨1 6 -7 -7 1], ⟨0 -9 19 20 5]]

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

Optimal ET sequence: 51, 53

Badness: 0.074295

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 225/224, 275/273, 1040/1029

Mapping: [⟨1 6 -7 -7 1 -12], ⟨0 -9 19 20 5 32]]

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

Optimal ET sequence: 51f, 53

Badness: 0.047441

Quintannic

Subgroup: 2.3.5.7

Comma list: 225/224, 9805926501/9765625000

Mapping: [⟨1 1 5 7], ⟨0 5 -23 -36]]

Optimal tuning (POTE): ~2 = 1200.000, ~10000/9261 = 139.838

Optimal ET sequence: 43, 60, 103, 266bcd, 369bcd

Badness: 0.150565

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 43923/43750

Mapping: [⟨1 1 5 7 8], ⟨0 5 -23 -36 -39]]

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

Optimal ET sequence: 43, 60e, 103

Badness: 0.052590

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 441/440, 1001/1000, 1188/1183

Mapping: [⟨1 1 5 7 8 3], ⟨0 5 -23 -36 -39 6]]

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

Optimal ET sequence: 43, 60e, 103

Badness: 0.032730

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 273/272, 375/374, 441/440, 891/884

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 ET sequence: 43, 60e, 103

Badness: 0.023038

Gwazy

See also: Very high accuracy temperaments § Kwazy

Subgroup: 2.3.5.7

Comma list: 225/224, 5971968/5764801

Mapping: [⟨2 1 6 4], ⟨0 8 -5 6]]

Optimal tuning (POTE): ~2401/1728 = 600.000, ~35/32 = 162.658

Optimal ET sequence: 22, 74, 96, 118d

Badness: 0.178826

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 65536/65219

Mapping: [⟨2 1 6 4 8], ⟨0 8 -5 6 -4]]

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

Optimal ET sequence: 22, 74, 96, 118d

Badness: 0.068410

Tertiosec

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Tertiosec.

Subgroup: 2.3.5.7

Comma list: 225/224, 14495514624/13841287201

Mapping: [⟨3 7 5 9], ⟨0 -8 7 -2]]

Optimal tuning (POTE): ~3072/2401 = 400.000, ~15/14 = 112.283

Optimal ET sequence: 21, 54, 75, 96, 171d

Badness: 0.431636

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 3840/3773, 12005/11979

Mapping: [⟨3 7 5 9 9], ⟨0 -8 7 -2 5]]

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

Optimal ET sequence: 21, 54, 75e

Badness: 0.173485

Retrieved from "https://en.xen.wiki/index.php?title=Marvel_temperaments&oldid=208108"