Marvel temperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Fredg999 category edits (talk | contribs)
autopatrolled, Bots, emailconfirmed
4,397 edits
m →Gracecordial: linking
 
(31 intermediate revisions by 12 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)
* ''[[Decic]]'', {225/224, 16807/16384} → [[Cloudy clan #Decic|Cloudy clan]]
* ''[[Quintapole]]'' → [[Quintaleap family #Quintapole|Quintaleap family]] (+7812500/7411887, fourth sliced in five)
* ''[[Amavil]]'', {225/224, 17496/16807} → [[Mabila family #Amavil|Mabila family]]
* ''[[Houborizic]]'' → [[Amity family #Houborizic|Amity family]] (+1250000/1240029, eleventh sliced in five)
* ''[[Escapade]]'', {225/224, 65625/65536} → [[Escapade family #Escapade|Escapade family]]
* ''[[Qintosec]]'' → [[Quintosec family #Qintosec|Quintosec family]] (+2560000/2470629, generated by the classical minor second with a 1/5-octave period)
* ''[[Fog]]'', {225/224, 156250/151263} → [[Misty family #Fog|Misty family]]
* [[Miracle]] → [[Gamelismic clan #Miracle|Gamelismic clan]] (+1029/1024, fifth sliced in six)
* ''[[Compton]]'', {225/224, 250047/250000} → [[Compton family #Compton|Compton family]]
* [[Catakleismic]] → [[Kleismic family #Catakleismic|Kleismic family]] (+4375/4374, twelfth sliced in six)
* ''[[Immune]]'', {225/224, 781250/750141} → [[Immunity family #Immune|Immunity family]]
* ''[[Marvo]]'' → [[Gravity family #Marvo|Gravity family]] (+78125000/78121827, two octaves and a fifth sliced in six)
* ''[[Betic]]'', {225/224, 1071875/1062882} → [[Sycamore family #Betic|Sycamore family]]
* [[Orwell]] → [[Semicomma family #Orwell|Semicomma family]] (+1728/1715, twelfth sliced in seven)
* ''[[Houborizic]]'' {225/224, 1250000/1240029} → [[Amity family #Houborizic|Amity family]]
* ''[[Snipes]]'' → [[Wesley family #Snipes|Wesley family]] (+6125/5832, two octaves and a fourth sliced in seven)
* ''[[Qintosec]]'', {225/224, 2560000/2470629} → [[Qintosec family #Qintosec|Qintosec family]]
* ''[[Submajor]]'' → [[Buzzardsmic clan #Submajor|Buzzardsmic clan]] (+65536/64827, two octaves and a fourth sliced in eight)
* ''[[Quintapole]]'', {225/224, 7812500/7411887} → [[Quintaleap family #Quintapole|Quintaleap family]]
* ''[[Escapist]]'' → [[Escapade family #Escapist|Escapade family]] (+65625/65536, fourth sliced in nine)
* ''[[Maquila]]'', {225/224, 30233088/28824005} → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Decic]]'' → [[Cloudy clan #Decic|Cloudy clan]] (+16807/16384, generated by the fifth with a 1/10-octave period)
* ''[[Marvo]]'', {225/224, 78125000/78121827} → [[Gravity family #Marvo|Gravity family]]
* ''[[Amavil]]'' → [[Mabila family #Amavil|Mabila family]] (+17496/16807, four octaves and a fourth sliced in ten)
* ''[[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.  
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.  


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16384
[[Comma list]]: 225/224, 118098/117649
 
[[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 }}
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}


[[POTE generator]]: ~15/14 = 125.608
: mapping generators: ~1225/864, ~245/216


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
[[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]]: 0.026483
{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}


==== Negra ====
[[Badness]] (Smith): 0.040846
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 ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 45/44, 49/48, 56/55
Comma list: 225/224, 385/384, 4000/3993


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}


POTE generator: ~15/14 = 126.474
Optimal tunings:
* CTE: ~99/70 = 600.000, ~25/22 = 216.900
* POTE: ~99/70 = 600.000, ~25/22 = 216.768


Optimal GPV sequence: {{Val list| 9, 10, 19 }}
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


Badness: 0.026190
Badness (Smith): 0.018539


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


Comma list: 45/44, 49/48, 56/55, 78/77
Comma list: 225/224, 351/350, 364/363, 385/384


Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}


POTE generator: ~14/13 = 126.431
Optimal tunings:
* CTE: ~55/39 = 600.000, ~25/22 = 216.703
* POTE: ~55/39 = 600.000, ~25/22 = 216.711


Optimal GPV sequence: {{Val list| 9, 10, 19 }}
{{Optimal ET sequence|legend=0| 22, 50, 72, 122, 194df }}


Badness: 0.017833
Badness (Smith): 0.021781


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


Comma list: 49/48, 100/99, 225/224
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}


POTE generator: ~15/14 = 124.767
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| 19, 29, 48d, 77cdd }}
{{Optimal ET sequence|legend=0| 22, 50, 72, 122g, 194dfg }}


Badness: 0.038679
Badness (Smith): 0.014536


==== 13-limit ====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 49/48, 65/64, 91/90, 875/858
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}


POTE generator: ~14/13 = 124.716
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| 19, 29, 48df, 77cddf }}
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


Badness: 0.024383
Badness (Smith): 0.015702


=== Negric ===
==== Gizzard ====
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
Subgroup: 2.3.5.7.11.13


Comma list: 33/32, 49/48, 65/64, 91/90
Comma list: 225/224, 325/324, 385/384, 1573/1568


Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}


POTE generator: ~14/13 = 127.039
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| 9, 19e }}
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}


Badness: 0.020205
Badness (Smith): 0.020252


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


Comma list: 49/48, 55/54, 225/224
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384


Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}


POTE generator: ~15/14 = 124.539
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| 10, 19e, 29, 77cddee }}
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}


Badness: 0.035296
Badness (Smith): 0.013624


==== 13-limit ====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 49/48, 55/54, 65/64, 91/90
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}


POTE generator: ~14/13 = 124.545
Optimal tunings:
* CTE: ~17/12 = 600.000, ~17/15 = 216.918
* POTE: ~17/12 = 600.000, ~17/15 = 216.862


Optimal GPV sequence: {{Val list| 10, 19e, 29, 77cddeef }}
{{Optimal ET sequence|legend=0| 72, 94, 166g }}


Badness: 0.021559
Badness (Smith): 0.014810


=== Wilsec ===
=== Mage ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 121/120, 225/224
Comma list: 99/98, 176/175, 1331/1296


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


POTE generator: ~11/8 = 537.186
Optimal tunings:
* CTE: ~77/54 = 600.000, ~55/48 = 217.247
* POTE: ~77/54 = 600.000, ~55/48 = 216.876


Optimal GPV sequence: {{Val list| 9, 20, 29, 38d, 67cdde }}
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


Badness: 0.041886
Badness (Smith): 0.057799


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


Comma list: 49/48, 65/64, 91/90, 121/120
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
[[Comma list]]: 225/224, 50421/50000


POTE generator: ~11/8 = 537.208
{{Mapping|legend=1| 1 4 -3 -3 | 0 -5 11 12 }}


Optimal GPV sequence: {{Val list| 9, 20, 29, 38df, 67cddef }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 580.286


Badness: 0.025192
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


==== 17-limit ====
[[Badness]]: 0.047578
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 ==
{{Main|Wizard}}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 225/224, 118098/117649
 
[[Mapping]]: [{{val| 2 1 5 2 }}, {{val| 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 }}
 
{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
 
[[Badness]]: 0.040846
 
Scales: [[wizard22]]


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


Comma list: 225/224, 385/384, 4000/3993
Comma list: 121/120, 225/224, 441/440


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
Mapping: {{mapping| 1 4 -3 -3 2 | 0 -5 11 12 3 }}


Mapping generators: ~99/70, ~25/22
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.267


POTE generator: ~5/4 = 383.232
{{Optimal ET sequence|legend=1| 29, 31, 60e }}


Optimal GPV sequence: {{Val list| 22, 50, 72, 166, 238c, 310c }}
Badness: 0.023659


Badness: 0.018539
==== 13-limit ====
 
Scales: [[wizard22]]
 
==== Lizard ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 351/350, 364/363, 385/384
Comma list: 105/104, 121/120, 196/195, 275/273


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
Mapping: {{mapping| 1 4 -3 -3 2 -5 | 0 -5 11 12 3 18 }}


Mapping generators: ~99/70, ~25/22
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.108


POTE generator: ~5/4 = 383.389
{{Optimal ET sequence|legend=1| 29, 31, 60e }}


Optimal GPV sequence: {{Val list| 22, 50, 72, 122, 194df }}
Badness: 0.022993


Badness: 0.021781
==== 17-limit ====
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
 
Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]


Mapping generators: ~17/12, ~17/15
Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 | 0 -5 11 12 3 18 25 }}


POTE generator: ~5/4 = 383.381
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.055


Optimal GPV sequence: {{Val list| 22, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=1| 29g, 31, 60e }}


Badness: 0.014536
Badness: 0.019093


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


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 -3 | 0 -5 11 12 3 18 25 15 }}


Mapping generators: ~17/12, ~17/15
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.026


POTE generator: ~5/4 = 383.477
{{Optimal ET sequence|legend=1| 29g, 31, 60e }}


Optimal GPV sequence: {{Val list| 22h, 50, 72, 122g, 194dfg }}
Badness: 0.017010


Badness: 0.015702
==== 23-limit ====
 
Subgroup: 2.3.5.7.11.13.17.19.23
==== Gizzard ====
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 325/324, 385/384, 1573/1568
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
Mapping: {{mapping| 1 4 -3 -3 2 -5 -8 -3 5 | 0 -5 11 12 3 18 25 15 -1 }}


Mapping generators: ~99/70, ~25/22
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.009


POTE generator: ~5/4 = 383.170
{{Optimal ET sequence|legend=1| 29g, 31, 60e }}


Optimal GPV sequence: {{Val list| 72, 166, 238cf }}
Badness: 0.014471


Badness: 0.020252
=== Tritoni ===
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
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 generators: ~17/12, ~17/15
 
POTE generator: ~5/4 = 383.175
 
Optimal GPV sequence: {{Val list| 72, 166g, 238cfg }}
 
Badness: 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: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
 
Mapping generators: ~17/12, ~17/15
 
POTE generator: ~5/4 = 383.138
 
Optimal GPV sequence: {{Val list| 72, 94, 166g }}
 
Badness: 0.014810
 
=== Mage ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 1331/1296
Comma list: 225/224, 385/384, 27783/27500


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
Mapping: {{mapping| 1 4 -3 -3 17 | 0 -5 11 12 -28 }}


Mapping generators: ~77/54, ~55/48
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 580.389


POTE generator: ~5/4 = 383.124
{{Optimal ET sequence|legend=1| 31, 91, 122, 153d }}


Optimal GPV sequence: {{Val list| 22, 50e, 72ee, 94ee }}
Badness: 0.045456


Badness: 0.057799
== Septimin ==
 
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Septimin]].''
== Triton ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 1029/1000
[[Comma list]]: 225/224, 84035/82944


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
{{Mapping|legend=1| 1 4 1 5 | 0 -11 6 -10 }}


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/6 = 263.632


[[POTE generator]]: ~7/5 = 568.865
{{Optimal ET sequence|legend=1| 41, 91, 132d }}


{{Val list|legend=1| 2, 17d, 19, 78bd, 97bd }}
[[Badness]]: 0.054502
 
[[Badness]]: 0.059245


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


Comma list: 45/44, 56/55, 1029/1000
Comma list: 225/224, 385/384, 2401/2376


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
Mapping: {{mapping| 1 4 1 5 5 | 0 -11 6 -10 -7 }}


POTE generator: ~7/5 = 569.144
Optimal tuning (POTE): ~2 = 1200.000, ~7/6 = 263.634


Optimal GPV sequence: {{Val list| 2, 17d, 19, 59bde, 78bde, 97bde }}
{{Optimal ET sequence|legend=1| 41, 91, 223cdef }}


Badness: 0.045675
Badness: 0.031309


== Tritonic ==
=== 13-limit ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 225/224, 50421/50000
 
[[Mapping]]: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
 
{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
 
[[POTE generator]]: ~7/5 = 580.286
 
{{Val list|legend=1| 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: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
 
POTE generator: ~7/5 = 580.267
 
Optimal GPV sequence: {{Val list| 29, 31, 60e }}
 
Badness: 0.023659
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 105/104, 121/120, 196/195, 275/273
Comma list: 105/104, 144/143, 196/195, 245/242


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
Mapping: {{mapping| 1 4 1 5 5 7 | 0 -11 6 -10 -7 -15 }}


POTE generator: ~7/5 = 580.108
Optimal tuning (POTE): ~2 = 1200.000, ~7/6 = 263.700


Optimal GPV sequence: {{Val list| 29, 31, 60e, 151cde }}
{{Optimal ET sequence|legend=1| 41, 91 }}


Badness: 0.022993
Badness: 0.023117
 
=== Tritoni ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 27783/27500
 
Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
 
POTE generator: ~7/5 = 580.389
 
Optimal GPV sequence: {{Val list| 31, 91, 122, 153d }}
 
Badness: 0.045456


== 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 493: 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 506: 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


== Untriton ==
== Slender ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Untriton]].''
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, 125000000/121060821
[[Comma list]]: 225/224, 589824/588245


[[Mapping]]: [{{Val|1 6 -7 -7}}, {{Val|0 -9 19 20}}]
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}


{{Multival|legend=1|9 -19 -20 -51 -57 7}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 38.413


[[POTE generator]]: ~7/5 = 588.641
{{Optimal ET sequence|legend=1| 31, 94, 125 }}


{{Val list|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}
[[Badness]]: 0.056934
 
[[Badness]]: 0.143976


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


Comma list: 121/120, 225/224, 22000/21609
Comma list: 225/224, 385/384, 1331/1323


Mapping: [{{Val|1 6 -7 -7 1}}, {{Val|0 -9 19 20 5}}]
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


POTE generator: ~7/5 = 588.626
Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 38.387


Optimal GPV sequence: {{Val list| 51, 53 }}
{{Optimal ET sequence|legend=1| 31, 63, 94, 125 }}


Badness: 0.074295
Badness: 0.025342


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


Comma list: 121/120, 225/224, 275/273, 1040/1029
Comma list: 225/224, 275/273, 385/384, 1331/1323


Mapping: [{{Val|1 6 -7 -7 1 -12}}, {{Val|0 -9 19 20 5 32}}]
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


POTE generator: ~7/5 = 588.654
Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 38.314


Optimal GPV sequence: {{Val list| 51f, 53 }}
{{Optimal ET sequence|legend=1| 31, 63, 94 }}


Badness: 0.047441
Badness: 0.025913


== Septimin ==
== Triton ==
: ''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 #Stump]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 84035/82944
[[Comma list]]: 225/224, 1029/1000


[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 568.865


[[POTE generator]]: ~7/6 = 263.632
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


{{Val list|legend=1| 41, 91, 132d }}
[[Badness]]: 0.059245
 
[[Badness]]: 0.054502


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


Comma list: 225/224, 385/384, 2401/2376
Comma list: 45/44, 56/55, 1029/1000


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


POTE generator: ~7/6 = 263.634
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 569.144


Optimal GPV sequence: {{Val list| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=1| 2, 17d, 19, 59bde, 78bde, 97bde }}


Badness: 0.031309
Badness: 0.045675


=== 13-limit ===
== Marvolo ==
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7


Comma list: 105/104, 144/143, 196/195, 245/242
[[Comma list]]: 225/224, 156250000/155649627


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}


POTE generator: ~7/6 = 263.700
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~21/20 = 83.348


Optimal GPV sequence: {{Val list| 41, 91 }}
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bcd, 691bcd }}


Badness: 0.023117
[[Badness]]: 0.083338


== Misneb ==
=== 11-limit ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Misneb]].''
Subgroup: 2.3.5.7.11


Subgroup: 2.3.5.7
Comma list: 225/224, 441/440, 4000/3993


[[Comma list]]: 225/224, 4194304/4117715
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


[[Mapping]]: [{{Val|1 3 1 3}}, {{Val|0 -15 14 -2}}]
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.340


{{Multival|legend=1|15 -14 2 -57 -39 44}}
{{Optimal ET sequence|legend=1| 29, 43, 72 }}


[[POTE generator]]: ~16/15 = 113.235
Badness: 0.028965


{{Val list|legend=1| 21, 32, 53 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.140970
Comma list: 169/168, 225/224, 364/363, 441/440


=== 11-limit ===
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 1310720/1294139
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.330


Mapping: [{{Val|1 3 1 3 1}}, {{Val|0 -15 14 -2 26}}]
{{Optimal ET sequence|legend=1| 29, 43, 72, 115f }}


POTE generator: ~16/15 = 113.323
Badness: 0.021470


Optimal GPV sequence: {{Val list| 21, 32e, 53, 127, 180de }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.085390
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


==== 13-limit ====
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
Subgroup: 2.3.5.7.11.13


Comma list: 99/98, 176/175, 640/637, 847/845
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.330


Mapping: [{{Val|1 3 1 3 1 2}}, {{Val|0 -15 14 -2 26 18}}]
{{Optimal ET sequence|legend=1| 29g, 43, 72 }}


POTE generator: ~16/15 = 113.323
Badness: 0.014924


Optimal GPV sequence: {{Val list| 21, 32e, 53, 127, 180de }}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.045569
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


=== Musneb ===
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 66550/64827
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 83.330


Mapping: [{{Val|1 3 1 3 6}}, {{Val|0 -15 14 -2 -27}}]
{{Optimal ET sequence|legend=1| 29g, 43, 72 }}


POTE generator: ~16/15 = 113.142
Badness: 0.014722


Optimal GPV sequence: {{Val list| 32, 53, 191de, 244cddee, 297cddee }}
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


Badness: 0.087333
[[Comma list]]: 225/224, 40353607/40310784


== Slender ==
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
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.


Subgroup: 2.3.5.7
: Mapping generators: ~2592/2401, ~12005/6912


[[Comma list]]: 225/224, 589824/588245
[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 133.3333, ~12005/6912 = 950.1680 (~1728/1715 = 16.8347)


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
{{Optimal ET sequence|legend=1| 9, 63, 72, 495bcd }}


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
[[Badness]]: 0.093679
 
[[POTE generator]]: ~49/48 = 38.413
 
{{Val list|legend=1| 31, 94, 125 }}
 
[[Badness]]: 0.056934


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


Comma list: 225/224, 385/384, 1331/1323
Comma list: 225/224, 385/384, 12005/11979


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


POTE generator: ~49/48 = 38.387
Optimal tuning (POTE): ~121/112 = 133.3333, ~210/121 = 950.1873 (~99/98 = 16.8540)


Optimal GPV sequence: {{Val list| 31, 63, 94, 125 }}
{{Optimal ET sequence|legend=1| 9, 63, 72, 423cd, 495bcd }}


Badness: 0.025342
Badness: 0.030426


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


Comma list: 225/224, 275/273, 385/384, 1331/1323
Comma list: 169/168, 225/224, 364/363, 1716/1715


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


POTE generator: ~49/48 = 38.314
Optimal tuning (POTE): ~14/13 = 133.3333, ~26/15 = 950.2867 (~105/104 = 16.9534)


Optimal GPV sequence: {{Val list| 31, 63, 94 }}
{{Optimal ET sequence|legend=1| 9, 63, 72, 279cf }}


Badness: 0.025913
Badness: 0.022322


== Marvolo ==
== Gracecordial ==
Subgroup: 2.3.5.7
: ''For the 5-limit version of this temperament, see [[Schismic–Pythagorean equivalence continuum #Gracecordial]].''


[[Comma list]]: 225/224, 156250000/155649627
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
[[Comma list]]: 225/224, 781250000/771895089


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}


[[POTE generator]]: ~21/20 = 83.348
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3/2 = 700.824


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
{{Optimal ET sequence|legend=1| 12, 113, 125, 238c, 363c }}


[[Badness]]: 0.083338
[[Badness]]: 0.096279


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


Comma list: 225/224, 441/440, 4000/3993
Comma list: 225/224, 385/384, 236328125/234365481


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


POTE generator: ~21/20 = 83.340
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.834


Optimal GPV sequence: {{Val list| 29, 43, 72 }}
{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125, 238c }}


Badness: 0.028965
Badness: 0.089588


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


Comma list: 169/168, 225/224, 364/363, 441/440
Comma list: 225/224, 325/324, 385/384, 831875/830466


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


POTE generator: ~21/20 = 83.330
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.841


Optimal GPV sequence: {{Val list| 29, 43, 72, 115f }}
{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}


Badness: 0.021470
Badness: 0.052235


== Enneaportent ==
==== 17-limit ====
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 225/224, 40353607/40310784
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


[[Mapping]]: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.841


[[POTE generator]]: ~5/4 = 383.165
{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}


{{Val list|legend=1| 9, 63, 72, 495bcd }}
Badness: 0.038565


[[Badness]]: 0.093679
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


=== 11-limit ===
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 12005/11979
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.842


POTE generator: ~5/4 = 383.146
{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}


Optimal GPV sequence: {{Val list| 9, 63, 72, 423cd, 495bcd }}
Badness: 0.028165


Badness: 0.030426
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


=== 13-limit ===
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 225/224, 364/363, 1716/1715
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.843


POTE generator: ~5/4 = 383.047
{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 238cfi }}


Optimal GPV sequence: {{Val list| 9, 63, 72, 279cf }}
Badness: 0.021879


Badness: 0.022322
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


== Submajor ==
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
Subgroup: 2.3.5


[[Comma list]]: 69198046875/68719476736
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.842


[[POTE generator]]: ~10125/8192 = 362.321
{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cfi }}


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
Badness: 0.018011


[[Badness]]: 0.130236
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


=== 7-limit ===
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 51200/50421
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]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.838


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
{{Optimal ET sequence|legend=1| 12e, 101cdek, 113, 125f, 238cfi }}


[[POTE generator]]: ~49/40 = 362.255
Badness: 0.016007


{{Val list|legend=1| 10, 33, 43, 53 }}
=== Gracecord ===
 
[[Badness]]: 0.060533
 
=== 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, 109375/107811


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


POTE generator: ~27/22 = 362.101
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.885


Optimal GPV sequence: {{Val list| 10, 43e, 53, 116, 169de, 285cde }}
{{Optimal ET sequence|legend=1| 12, 101cd, 113 }}


Badness: 0.050582
Badness: 0.066964


==== 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: 225/224, 364/363, 441/440, 6125/6084


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


POTE generator: ~16/13 = 362.105
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.890


Optimal GPV sequence: {{Val list| 10, 43e, 53, 116, 169de, 285cdef }}
{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}


Badness: 0.027689
Badness: 0.044196


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


POTE generator: ~49/40 = 362.418
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.870


POTE generator: ~49/40 = 362.418
{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}


Optimal GPV sequence: {{Val list| 43, 53, 96, 149d }}
Badness: 0.036637


Badness: 0.051806
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


==== 13-limit ====
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663
Subgroup: 2.3.5.7.11.13


Comma list: 99/98, 169/168, 176/175, 640/637
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


POTE generator: ~16/13 = 362.402
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 700.866


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}


POTE generator: ~16/13 = 362.402
Badness: 0.027559
 
Optimal GPV sequence: {{Val list| 43, 53, 96, 149d }}
 
Badness: 0.029680


== 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 885: 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


== Tertiosec ==
== Misneb ==
: ''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 #Misneb]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 14495514624/13841287201
[[Comma list]]: 225/224, 4194304/4117715


[[Mapping]]: [{{val| 3 7 5 9 }}, {{val| 0 -8 7 -2 }}]
{{Mapping|legend=1| 1 3 1 3 | 0 -15 14 -2 }}


{{Multival|legend=1| 24 -21 6 -89 -58 73 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16/15 = 113.235


[[POTE generator]]: ~15/14 = 112.283
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


{{Val list|legend=1| 21, 54, 75, 96, 171d }}
[[Badness]]: 0.140970
 
[[Badness]]: 0.431636


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


Comma list: 225/224, 3840/3773, 12005/11979
Comma list: 99/98, 176/175, 1310720/1294139


Mapping: [{{val| 3 7 5 9 9 }}, {{val| 0 -8 7 -2 5 }}]
Mapping: {{mapping| 1 3 1 3 1 | 0 -15 14 -2 26 }}


POTE generator: ~15/14 = 112.171
Optimal tuning (POTE): ~2 = 1200.000, ~16/15 = 113.323


Optimal GPV sequence: {{Val list| 21, 54, 75e }}
{{Optimal ET sequence|legend=1| 21, 32e, 53, 127, 180de }}


Badness: 0.173485
Badness: 0.085390


== Gwazy ==
==== 13-limit ====
{{see also|Very high accuracy temperaments #Kwazy}}
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5.7
Comma list: 99/98, 176/175, 640/637, 847/845


[[Comma list]]: 225/224, 5971968/5764801
Mapping: {{mapping| 1 3 1 3 1 2 | 0 -15 14 -2 26 18 }}


[[Mapping]]: [{{val| 2 1 6 4 }}, {{val| 0 8 -5 6 }}]
Optimal tuning (POTE): ~2 = 1200.000, ~16/15 = 113.323


{{Multival|legend=1| 16 -10 12 -53 -26 56 }}
{{Optimal ET sequence|legend=1| 21, 32e, 53, 127, 180de }}


[[POTE generator]]: ~35/32 = 162.658
Badness: 0.045569


{{Val list|legend=1| 22, 74, 96, 118d }}
=== Musneb ===
 
[[Badness]]: 0.178826
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 65536/65219
Comma list: 225/224, 385/384, 66550/64827


Mapping: [{{val| 2 1 6 4 8 }}, {{val| 0 8 -5 6 -4 }}]
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


POTE generator: ~11/10 = 162.592
Optimal tuning (POTE): ~2 = 1200.000, ~16/15 = 113.142


Optimal GPV sequence: {{Val list| 22, 74, 96, 118d }}
{{Optimal ET sequence|legend=1| 32, 53, 191de, 244cddee, 297cddee }}


Badness: 0.068410
Badness: 0.087333


== Gracecordial ==
== Untriton ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Gracecordial]].''
: ''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, 781250000/771895089
[[Comma list]]: 225/224, 125000000/121060821


[[Mapping]]: [{{val| 1 0 34 63 }}, {{val| 0 1 -20 -38 }}]
{{Mapping|legend=1| 1 6 -7 -7 | 0 -9 19 20 }}


{{Multival|legend=1| 1 -20 -38 -34 -63 -32 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/5 = 588.641


[[POTE generator]]: ~3/2 = 700.824
{{Optimal ET sequence|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}


{{Val list|legend=1| 12, 113, 125, 238c, 363c }}
[[Badness]]: 0.143976
 
[[Badness]]: 0.096279


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


Comma list: 225/224, 385/384, 236328125/234365481
Comma list: 121/120, 225/224, 22000/21609


Mapping: [{{val| 1 0 34 63 -90 }}, {{val| 0 1 -20 -38 59 }}]
Mapping: {{mapping| 1 6 -7 -7 1 | 0 -9 19 20 5 }}


POTE generator: ~3/2 = 700.834
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 588.626


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 125, 238c }}
{{Optimal ET sequence|legend=1| 51, 53 }}


Badness: 0.089588
Badness: 0.074295


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


Comma list: 225/224, 325/324, 385/384, 831875/830466
Comma list: 121/120, 225/224, 275/273, 1040/1029


Mapping: [{{val| 1 0 34 63 -90 -66 }}, {{val| 0 1 -20 -38 59 44 }}]
Mapping: {{mapping| 1 6 -7 -7 1 -12 | 0 -9 19 20 5 32 }}


POTE generator: ~3/2 = 700.841
Optimal tuning (POTE): ~2 = 1200.000, ~7/5 = 588.654


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cf }}
{{Optimal ET sequence|legend=1| 51f, 53 }}


Badness: 0.052235
Badness: 0.047441
 
== Quintannic ==
[[Subgroup]]: 2.3.5.7


==== 17-limit ====
[[Comma list]]: 225/224, 9805926501/9765625000
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}


Mapping: [{{val| 1 0 34 63 -90 -66 -7 }}, {{val| 0 1 -20 -38 59 44 7 }}]
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~10000/9261 = 139.838


POTE generator: ~3/2 = 700.841
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cf }}
[[Badness]]: 0.150565


Badness: 0.038565
=== 11-limit ===
Subgroup: 2.3.5.7.11


==== 19-limit ====
Comma list: 225/224, 441/440, 43923/43750
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 }}, {{val| 0 1 -20 -38 59 44 7 -3 }}]
Optimal tuning (POTE): ~2 = 1200.000, ~320/297 = 139.827


POTE generator: ~3/2 = 700.842
{{Optimal ET sequence|legend=1| 43, 60e, 103 }}


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cf }}
Badness: 0.052590


Badness: 0.028165
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==== 23-limit ====
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
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: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 -43 }}, {{val| 0 1 -20 -38 59 44 7 -3 30 }}]
Optimal tuning (POTE): ~2 = 1200.000, ~13/12 = 139.812


POTE generator: ~3/2 = 700.843
{{Optimal ET sequence|legend=1| 43, 60e, 103 }}


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 238cfi }}
Badness: 0.032730


Badness: 0.021879
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


==== 29-limit ====
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
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: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 -43 -49 }}, {{val| 0 1 -20 -38 59 44 7 -3 30 34 }}]
Optimal tuning (POTE): ~2 = 1200.000, ~13/12 = 139.815


POTE generator: ~3/2 = 700.842
{{Optimal ET sequence|legend=1| 43, 60e, 103 }}


Optimal GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cfi }}
Badness: 0.023038


Badness: 0.018011
== Gwazy ==
{{See also| Very high accuracy temperaments #Kwazy }}


==== 31-limit ====
[[Subgroup]]: 2.3.5.7
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
[[Comma list]]: 225/224, 5971968/5764801


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|legend=1| 2 1 6 4 | 0 8 -5 6 }}


POTE generator: ~3/2 = 700.838
[[Optimal tuning]] ([[POTE]]): ~2401/1728 = 600.000, ~35/32 = 162.658


Optimal GPV sequence: {{Val list| 12e, 101cdek, 113, 125f, 238cfi }}
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


Badness: 0.016007
[[Badness]]: 0.178826


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


Comma list: 225/224, 441/440, 109375/107811
Comma list: 99/98, 176/175, 65536/65219


Mapping: [{{val| 1 0 34 63 89 }}, {{val| 0 1 -20 -38 -54 }}]
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


POTE generator: ~3/2 = 700.885
Optimal tuning (POTE): ~363/256 = 600.000, ~11/10 = 162.592


Optimal GPV sequence: {{Val list| 12, 101cd, 113 }}
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


Badness: 0.066964
Badness: 0.068410


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


Comma list: 225/224, 364/363, 441/440, 6125/6084
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 0 34 63 89 113 }}, {{val| 0 1 -20 -38 -54 -69 }}]
[[Comma list]]: 225/224, 14495514624/13841287201


POTE generator: ~3/2 = 700.890
{{Mapping|legend=1| 3 7 5 9 | 0 -8 7 -2 }}


Optimal GPV sequence: {{Val list| 12f, 101cdf, 113 }}
[[Optimal tuning]] ([[POTE]]): ~3072/2401 = 400.000, ~15/14 = 112.283


Badness: 0.044196
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


==== 17-limit ====
[[Badness]]: 0.431636
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 0 34 63 89 113 -7 }}, {{val| 0 1 -20 -38 -54 -69 7 }}]
Comma list: 225/224, 3840/3773, 12005/11979


POTE generator: ~3/2 = 700.870
Mapping: {{mapping| 3 7 5 9 9 | 0 -8 7 -2 5 }}


Optimal GPV sequence: {{Val list| 12f, 101cdf, 113 }}
Optimal tuning (POTE): ~44/35 = 400.000, ~15/14 = 112.171


Badness: 0.036637
{{Optimal ET sequence|legend=1| 21, 54, 75e }}


==== 19-limit ====
Badness: 0.173485
Subgroup: 2.3.5.7.11.13.17.19
 
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 }}]
 
POTE generator: ~3/2 = 700.866
 
Optimal GPV sequence: {{Val list| 12f, 101cdf, 113 }}
 
Badness: 0.027559


[[Category:Regular temperament theory]]
[[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"