Xenharmonic Wiki

Marvel temperaments: Difference between revisions

← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,012 edits
mNo edit summary
Undo (see talk)
Tag: Undo
 
(73 intermediate revisions by 14 users not shown)
Line 1: Line 1:
This page discusses miscellaneous rank-2 temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the marvel comma or septimal kleisma.  
{{Technical data page}}
This page discusses miscellaneous [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] [[225/224]], the marvel comma or septimal kleisma.  


Temperaments considered in families and clans are:  
Temperaments considered in families and clans are:  
* ''[[sharp]]'', {25/24, 28/27} → [[Dicot family #Sharp]]
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* ''[[pelogic]]'', {21/20, 135/128} → [[Pelogic family #Pelogic]]
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[august]]'', {36/35, 128/125} → [[Augmented family #August]]
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[pajara]], {50/49, 64/63}, also in: [[Diaschismic family #Pajara|diaschismic family]] / [[Jubilismic clan #Pajara|jubilismic clan]] / [[Archytas clan #Pajara|archytas clan]]
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[septimal meantone]]'', {81/80, 126/125} → [[Meantone family #Septimal meantone]]
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* [[magic]], {225/224, 245/243}, also in: [[Magic family #Magic|magic family]]
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* [[miracle]], {225/224, 1029/1024}, also in: [[Gamelismic clan #Miracle|gamelismic clan]]
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* [[orwell]], {225/224, 1728/1715}, also in: [[Semicomma family #Orwell|semicomma family]]
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Garibaldi temperament|garibaldi]], {225/224, 3125/3087}, also in: [[Schismatic family #Garibaldi|schismatic family]]
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[catakleismic]], {225/224, 4375/4374}, also in: [[Kleismic family #Catakleismic|kleismic family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* ''[[decic]]'', {225/224, 16807/16384} → [[Cloudy clan #Decic]]
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[escapade]]'', {225/224, 65625/65536} → [[Escapade family #Escapade]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[fog]]'', {225/224, 156250/151263} → [[Misty family #Fog]]
* ''[[Quintapole]]'' (+7812500/7411887) → [[Quintaleap family #Quintapole|Quintaleap family]]
* ''[[compton]]'', {225/224, 250047/250000} → [[Compton family #Compton]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[houborizic]]'' {225/224, 1250000/1240029} → [[Amity family #Houborizic]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* ''[[qintosec]]'', {225/224, 2560000/2470629} → [[Qintosec family #Qintosec]]
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[maquila]]'', {225/224, 30233088/28824005} → [[Maquila family #Maquila]]
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
* ''[[Decic]]'' (+16807/16384) → [[Cloudy clan #Decic|Cloudy clan]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Hendeca]]'' (+122880/117649) → [[11th-octave temperaments #Hendeca|11th-octave temperaments]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]


Considered below are negri, wizard, tritonic, septimin, slender, triton, merman, marvo, marvolo, amavil, enneaportent, submajor, and alphorn.  
Considered below are wizard, tritonic, septimin, merman, slender, triton, marvolo, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, gwazy, and tertiosec, in the order of increasing [[badness]].  


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


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16384
[[Comma list]]: 225/224, 118098/117649


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216


{{Multival|legend=1| 4 -3 -14 }}
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}
: error map: {{val| 0.000 -1.169 -3.111 -0.849 }}


[[POTE generator]]: ~16/15 = 125.7549
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
[[Badness]] (Sintel): 1.03


[[Badness]]: 0.086856
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 7-limit ===
Comma list: 225/224, 385/384, 4000/3993
Subgroup: 2.3.5.7


[[Comma list]]: 49/48, 225/224
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
Optimal tunings:  
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


{{Multival|legend=1| 4 -3 2 -14 -8 13 }}
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


[[POTE generator]]: ~15/14 = 125.608
Badness (Sintel): 0.613


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
==== Lizard ====
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.026483
Comma list: 225/224, 351/350, 364/363, 385/384


==== Negra ====
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
This is the 2.3.5.7.13 extension of negri.


Subgroup: 2.3.5.7.13
Optimal tunings:  
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


[[Comma list]]: 49/48, 65/64, 91/90
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


[[Sval]] [[mapping]]: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -3 }}]
Badness (Sintel): 0.900


[[Gencom]] [[mapping]]: [{{val| 1 2 2 3 0 4 }}, {{val| 0 -4 3 -2 0 -3 }}]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


[[Gencom]]: [2 14/13; 49/48 65/64 91/90]
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}


[[POTE generator]]: ~14/13 = 125.567
Optimal tunings:  
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


=== 11-limit ===
Badness (Sintel): 0.741
Subgroup: 2.3.5.7.11
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 45/44, 49/48, 56/55
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


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


POTE generator: ~15/14 = 126.474
Optimal tunings:  
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


Vals: {{Val list| 9, 10, 19 }}
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


Badness: 0.026190
Badness (Sintel): 0.955


==== 13-limit ====
==== Gizzard ====
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, 325/324, 385/384, 1573/1568


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


POTE generator: ~14/13 = 126.431
Optimal tunings:  
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}


Vals: {{Val list| 9, 10, 19 }}
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}


Badness: 0.017833
Badness (Sintel): 0.837


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


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}


POTE generator: ~15/14 = 124.767
Optimal tunings:  
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}


Vals: {{Val list| 19, 29, 48d, 77cdd }}
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}


Badness: 0.038679
Badness (Sintel): 0.694


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


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}


POTE generator: ~14/13 = 124.716
Optimal tunings:  
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


Vals: {{Val list| 19, 29, 48df, 77cddf }}
{{Optimal ET sequence|legend=0| 72, 94, 166g }}


Badness: 0.024383
Badness (Sintel): 0.901


=== Negric ===
=== Mage ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 33/32, 49/48, 77/75
Comma list: 99/98, 176/175, 1331/1296
 
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}


Mapping: [{{val| 1 2 2 3 3 }}, {{val| 0 -4 3 -2 4 }}]
Optimal tunings:  
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


POTE generator: ~15/14 = 127.039
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


Vals: {{Val list| 9, 19e }}
Badness (Sintel): 1.91


Badness: 0.030617
== Tritonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


==== 13-limit ====
Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.  
Subgroup: 2.3.5.7.11.13


Comma list: 33/32, 49/48, 65/64, 91/90
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
[[Comma list]]: 225/224, 50421/50000


POTE generator: ~14/13 = 127.039
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


Vals: {{Val list| 9, 19e }}
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
: [[error map]]: {{val| +1.354 -1.243 -0.027 -1.598 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}}
: error map: {{val| 0.000 -3.566 -2.769 -4.959 }}


Badness: 0.020205
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


=== Negroni ===
[[Badness]] (Sintel): 1.20
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 55/54, 225/224
Comma list: 121/120, 225/224, 441/440


Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


POTE generator: ~15/14 = 124.539
Optimal tunings:  
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


Vals: {{Val list| 10, 19e, 29, 77cddee }}
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


Badness: 0.035296
Badness (Sintel): 0.782


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


Comma list: 49/48, 55/54, 65/64, 91/90
Comma list: 105/104, 121/120, 196/195, 275/273


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


POTE generator: ~14/13 = 124.545
Optimal tunings:  
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


Vals: {{Val list| 10, 19e, 29, 77cddeef }}
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


Badness: 0.021559
Badness (Sintel): 0.950


=== Wilsec ===
==== 17-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


Comma list: 49/48, 121/120, 225/224
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]
Optimal tunings:  
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


POTE generator: ~11/8 = 537.186
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Vals: {{Val list| 9, 20, 29, 38d, 67cdde }}
Badness (Sintel): 0.973


Badness: 0.041886
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


==== 13-limit ====
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 65/64, 91/90, 121/120
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


POTE generator: ~11/8 = 537.208
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Vals: {{Val list| 9, 20, 29, 38df, 67cddef }}
Badness (Sintel): 1.03


Badness: 0.025192
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


==== 17-limit ====
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
Subgroup: 2.3.5.7.11.13.17


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


POTE generator: ~11/8 = 537.230
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Vals: {{Val list| 9, 20g, 29g, 38df, 67cddefg }}
Badness (Sintel): 1.04


Badness: 0.021778
=== Tritoni ===
Subgroup: 2.3.5.7.11


==== 19-limit ====
Comma list: 225/224, 385/384, 27783/27500
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


POTE generator: ~11/8 = 537.214
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Vals: {{Val list| 9, 20g, 29g, 38df, 67cddefgh }}
Badness (Sintel): 1.50


Badness: 0.016828
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


== Passive ==
Septimin may be described as the {{nowrap| 41 & 50 }} temperament. It is generated by a septimal minor third ([[7/6]]), which gives rise to the name, but the generator can be taken to be the [[octave complement]], [[12/7]], such that eleven of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus eta-hendecacot. [[91edo]] may be recommended as a tuning.  
Passive (1&amp;11) tempers out the same 5-limit comma as the [[Archytas clan #Passion|passion temperament]] (12&amp;49), but with the marvel (126/125) rather than 64/63 tempered out.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 256/245
[[Comma list]]: 225/224, 84035/82944


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


[[POTE generator]]: ~16/15 = 98.809
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}
: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}


{{Val list|legend=1| 1, 11, 12, 49dd }}
{{Optimal ET sequence|legend=1| 41, 91, 132d }}


[[Badness]]: 0.075055
[[Badness]] (Sintel): 1.38


== Wizard ==
=== 11-limit ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
Subgroup: 2.3.5.7.11
{{See also|Wizard}}


Subgroup: 2.3.5.7
Comma list: 225/224, 245/242, 385/384


[[Comma list]]: 225/224, 118098/117649
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


[[Mapping]]: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
Optimal tunings:  
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


Mapping generators: ~1225/864, ~245/216
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


[[POTE generator]]: ~5/4 = 383.256
Badness (Sintel): 1.04


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
Comma list: 105/104, 144/143, 196/195, 245/242


[[Badness]]: 0.040846
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


Scales: [[wizard22]]
Optimal tunings:  
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


=== 11-limit ===
{{Optimal ET sequence|legend=0| 41, 91 }}
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 4000/3993
Badness (Sintel): 0.955


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Mapping generators: ~99/70, ~25/22
Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot.


POTE generator: ~5/4 = 383.232
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.  


Vals: {{Val list| 22, 50, 72, 166, 238c, 310c }}
[[Subgroup]]: 2.3.5.7


Badness: 0.018539
[[Comma list]]: 225/224, 2500000/2470629


Scales: [[wizard22]]
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


==== Lizard ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13
* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
: [[error map]]: {{val| +0.390 -0.435 -1.630 +1.634 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}}
: error map: {{val| 0.000 -1.104 -2.423 +0.657 }}


Comma list: 225/224, 351/350, 364/363, 385/384
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
[[Badness]] (Sintel): 1.39


Mapping generators: ~99/70, ~25/22
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~5/4 = 383.389
Comma list: 225/224, 441/440, 1344/1331


Vals: {{Val list| 22, 50, 72, 122, 194df }}
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


Badness: 0.021781
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


===== 17-limit =====
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
Badness (Sintel): 1.20


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping generators: ~17/12, ~17/15
Comma list: 144/143, 225/224, 364/363, 441/440


POTE generator: ~5/4 = 383.381
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Vals: {{Val list| 22, 50, 72, 122g, 194dfg }}
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Badness: 0.014536
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


===== 19-limit =====
Badness (Sintel): 1.14
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
=== Mermaid ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
Comma list: 225/224, 385/384, 532400/531441


Mapping generators: ~17/12, ~17/15
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


POTE generator: ~5/4 = 383.477
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


Vals: {{Val list| 22h, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


Badness: 0.015702
Badness (Sintel): 1.46


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


Comma list: 225/224, 325/324, 385/384, 1573/1568
Comma list: 225/224, 325/324, 385/384, 10648/10647


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


POTE generator: ~5/4 = 383.170
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


Vals: {{Val list| 72, 166, 238cf }}
Badness (Sintel): 1.47


Badness: 0.020252
== Slender ==
Slender tempers out the [[hewuermera comma]] in addition to the marvel comma, and may be described as the {{nowrap| 31 & 32 }} temperament. This temperament has a generator of [[49/48]], three of which equal marvel's [[16/15]][[~]][[15/14]], and ten generators give [[5/4]]. Its [[ploidacot]] is omega-13-cot.  


===== 17-limit =====
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
[[Comma list]]: 225/224, 589824/588245


Mapping generators: ~17/12, ~17/15
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


POTE generator: ~5/4 = 383.175
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
: [[error map]]: {{val| +0.382 -0.725 -1.295 +1.765 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}}
: error map: {{val| 0.000 -1.257 -2.235 +0.727 }}


Vals: {{Val list| 72, 166g, 238cfg }}
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Badness: 0.013624
[[Badness]] (Sintel): 1.44


===== 19-limit =====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11


Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
Comma list: 225/224, 385/384, 1331/1323


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


Mapping generators: ~17/12, ~17/15
Optimal tunings:
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


POTE generator: ~5/4 = 383.138
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Vals: {{Val list| 72, 94, 166g }}
Badness (Sintel): 0.838


Badness: 0.014810
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== Mage ===
Comma list: 225/224, 275/273, 385/384, 1331/1323
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 1331/1296
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Mapping generators: ~77/54, ~55/48
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


POTE generator: ~5/4 = 383.124
Badness (Sintel): 1.07


Vals: {{Val list| 22, 50e, 72ee, 94ee }}
== Triton ==
 
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''
Badness: 0.057799


== Triton ==
Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot.  
: ''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, 1029/1000


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
 
: mapping generators: ~2, ~10/7
{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}


[[POTE generator]]: ~7/5 = 568.865
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}
: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}


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


[[Badness]]: 0.059245
[[Badness]] (Sintel): 1.50


=== 11-limit ===
=== 11-limit ===
Line 416: Line 512:
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 tunings:  
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


Vals: {{Val list| 2, 17d, 19, 59bde, 78bde, 97bde }}
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


Badness: 0.045675
Badness (Sintel): 1.51


== Tritonic ==
== Marvolo ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
[[Subgroup]]: 2.3.5.7


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


[[Comma list]]: 225/224, 50421/50000
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


[[Mapping]]: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7714{{c}}, ~21/20 = 83.4014{{c}}
: [[error map]]: {{val| +0.772 -0.820 -0.916 +0.381 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}}
: error map: {{val| 0.000 -2.139 -2.398 -1.362 }}


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


[[POTE generator]]: ~7/5 = 580.286
[[Badness]] (Sintel): 2.11
 
{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
 
[[Badness]]: 0.047578


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


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


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


POTE generator: ~7/5 = 580.267
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


Vals: {{Val list| 29, 31, 60e }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.023659
Badness (Sintel): 0.958


==== 13-limit ====
=== 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: 169/168, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
 
Optimal tunings:
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}
 
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
Badness (Sintel): 0.887


POTE generator: ~7/5 = 580.108
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Vals: {{Val list| 29, 31, 60e, 151cde }}
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


Badness: 0.022993
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


=== Tritoni ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Comma list: 225/224, 385/384, 27783/27500
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
Badness (Sintel): 0.760


POTE generator: ~7/5 = 580.389
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Vals: {{Val list| 31, 91, 122, 153d }}
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Badness: 0.045456
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


== Merman ==
Optimal tunings:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


[[Comma list]]: 225/224, 2500000/2470629
Badness (Sintel): 0.895


[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
[[Comma list]]: 225/224, 40353607/40310784


[[POTE generator]]: ~7/5 = 585.585
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


{{Val list|legend=1| 41, 84, 125 }}
[[Optimal tuning]]s:
* [[WE]]: ~2592/2401 = 133.4174{{c}}, ~12005/6912 = 950.7667{{c}} (~1728/1715 = 16.8452{{c}})
: [[error map]]: {{val| +0.756 -0.422 -1.395 +0.298 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}})
: error map: {{val| 0.000 -1.361 -3.277 -1.565 }}


[[Badness]]: 0.055078
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}
 
[[Badness]] (Sintel): 2.37


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


Comma list: 225/224, 441/440, 1344/1331
Comma list: 225/224, 385/384, 12005/11979


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


POTE generator: ~7/5 = 585.606
Optimal tunings:  
* WE: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}})
* CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}})


Vals: {{Val list| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.036383
Badness (Sintel): 1.01


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


Comma list: 144/143, 225/224, 364/363, 441/440
Comma list: 169/168, 225/224, 364/363, 1716/1715


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


POTE generator: ~7/5 = 585.657
Optimal tunings:  
* WE: ~14/13 = 133.4245{{c}}, ~26/15 = 950.9362{{c}} (~105/104 = 16.9650{{c}})
* CWE: ~14/13 = 133.3333{{c}}, ~26/15 = 950.4364{{c}} (~99/98 = 17.1031{{c}})


Vals: {{Val list| 41, 84, 125e, 209ef, 293ef }}
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Badness: 0.027544
Badness (Sintel): 0.922


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 84035/82944
[[Comma list]]: 225/224, 781250000/771895089


[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


[[POTE generator]]: ~7/6 = 263.632
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


{{Val list|legend=1| 41, 91, 132d }}
[[Badness]] (Sintel): 2.44
 
[[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: 225/224, 385/384, 236328125/234365481


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


POTE generator: ~7/6 = 263.634
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Vals: {{Val list| 41, 91, 223cdef }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Badness: 0.031309
Badness (Sintel): 2.96


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


Comma list: 105/104, 144/143, 196/195, 245/242
Comma list: 225/224, 325/324, 385/384, 831875/830466
 
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


POTE generator: ~7/6 = 263.700
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Vals: {{Val list| 41, 91 }}
Badness (Sintel): 2.16


Badness: 0.023117
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


== Slender ==
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
Subgroup: 2.3.5.7
 
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
 
Optimal tunings:
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


[[Comma list]]: 225/224, 589824/588245
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
Badness (Sintel): 1.96


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


[[POTE generator]]: ~49/48 = 38.413
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


{{Val list|legend=1| 31, 94, 125 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


[[Badness]]: 0.056934
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


=== 11-limit ===
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 1331/1323
Badness (Sintel): 1.71


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


POTE generator: ~49/48 = 38.387
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


Vals: {{Val list| 31, 63, 94, 125 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Badness: 0.025342
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 275/273, 385/384, 1331/1323
Badness (Sintel): 1.57


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


POTE generator: ~49/48 = 38.314
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Vals: {{Val list| 31, 63, 94 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


Badness: 0.025913
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


== Marvo ==
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}
{{see also| Gravity family }}


Subgroup: 2.3.5.7
Badness (Sintel): 1.50


[[Comma list]]: 225/224, 78125000/78121827
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


[[Mapping]]: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


{{Multival|legend=1| 6 17 46 13 56 59 }}
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]]: ~27/20 = 516.694
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


{{Val list|legend=1| 65d, 72, 137, 209, 281, 569bcc }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


[[Badness]]: 0.097627
Badness (Sintel): 1.53


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


Comma list: 225/224, 243/242, 4000/3993
Comma list: 225/224, 441/440, 109375/107811


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


POTE generator: ~27/20 = 516.699
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


Vals: {{Val list| 65d, 72, 281, 353c, 425bc, 497bc }}
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Badness: 0.031685
Badness (Sintel): 2.21


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


Comma list: 225/224, 243/242, 351/350, 1625/1617
Comma list: 225/224, 364/363, 441/440, 6125/6084


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


POTE generator: ~27/20 = 516.730
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


Vals: {{Val list| 65d, 72, 137, 209, 281f, 490bcf }}
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Badness: 0.026882
Badness (Sintel): 1.83


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


[[Comma list]]: 225/224, 156250000/155649627
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
Optimal tunings:
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


[[POTE generator]]: ~21/20 = 83.348
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
Badness (Sintel): 1.87


[[Badness]]: 0.083338
==== 19-limit ====
 
Subgroup: 2.3.5.7.11.13.17.19
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 225/224, 441/440, 4000/3993
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


POTE generator: ~21/20 = 83.340
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


Vals: {{Val list| 29, 43, 72 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Badness: 0.028965
Badness (Sintel): 1.68


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


Comma list: 169/168, 225/224, 364/363, 441/440
[[Comma list]]: 225/224, 5764801/5668704


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


POTE generator: ~21/20 = 83.330
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.3004{{c}}, ~35/24 = 644.4767{{c}}
: [[error map]]: {{val| +1.300 +0.569 -2.195 -2.043 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~35/24 = 643.8137{{c}}
: error map: {{val| 0.000 -0.936 -5.382 -4.924 }}


Vals: {{Val list| 29, 43, 72, 115f }}
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Badness: 0.021470
[[Badness]] (Sintel): 3.27


== Amavil ==
=== 11-limit ===
=== 5-limit (Mabila) ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5


[[Comma list]]: 268435456/263671875
Comma list: 225/224, 385/384, 12250/11979


[[Mapping]]: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


[[POTE generator]]: ~512/375 = 529.6849
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


[[Badness]]: 0.232481
Badness (Sintel): 2.43


=== 7-limit ===
== Misneb ==
Subgroup: 2.3.5.7
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


[[Comma list]]: 225/224, 17496/16807
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
[[Comma list]]: 225/224, 4194304/4117715


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


[[POTE generator]]: ~48/35 = 529.979
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7642{{c}}, ~15/8 = 1086.5513{{c}}
: [[error map]]: {{val| -0.236 -0.856 -1.569 +4.041 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1086.7633{{c}}
: error map: {{val| 0.000 -0.506 -0.999 +4.701 }}


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


[[Badness]]: 0.109625
[[Badness]] (Sintel): 3.57


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


Comma list: 99/98, 176/175, 864/847
Comma list: 99/98, 176/175, 1310720/1294139


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


POTE generator: ~15/11 = 529.974
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


Vals: {{Val list| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.042649
Badness (Sintel): 2.82


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


Comma list: 78/77, 99/98, 144/143, 176/175
Comma list: 99/98, 176/175, 640/637, 847/845


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


POTE generator: ~15/11 = 529.951
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


Vals: {{Val list| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Badness: 0.025791
Badness (Sintel): 1.88


== Enneaportent ==
=== Musneb ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 66550/64827
 
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
 
Optimal tunings:
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}
 
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}
 
Badness (Sintel): 2.89
 
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''
 
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.


[[Comma list]]: 225/224, 40353607/40310784
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
[[Comma list]]: 225/224, 125000000/121060821


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


[[POTE generator]]: ~5/4 = 383.165
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
: [[error map]]: {{val| -0.172 +0.002 -2.533 +3.511 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}}
: error map: {{val| 0.000 +0.298 -2.181 +3.946 }}


{{Val list|legend=1| 9, 63, 72, 495bcd }}
{{Optimal ET sequence|legend=1| 51, 53 }}


[[Badness]]: 0.093679
[[Badness]] (Sintel): 3.64


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


Comma list: 225/224, 385/384, 12005/11979
Comma list: 121/120, 225/224, 22000/21609


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


POTE generator: ~5/4 = 383.146
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


Vals: {{Val list| 9, 63, 72, 423cd, 495bcd }}
{{Optimal ET sequence|legend=0| 51, 53 }}


Badness: 0.030426
Badness (Sintel): 2.46


=== 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, 1716/1715
Comma list: 121/120, 225/224, 275/273, 1040/1029
 
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}
 
Optimal tunings:
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}
 
{{Optimal ET sequence|legend=0| 51f, 53 }}
 
Badness (Sintel): 1.96
 
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 823543/800000


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


POTE generator: ~5/4 = 383.047
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1202.1198{{c}}, ~32/21 = 745.4675{{c}}
: [[error map]]: {{val| +2.120 -1.227 +0.459 -4.423 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~32/21 = 744.1318{{c}}
: error map: {{val| 0.000 -4.769 -4.159 -10.144 }}


Vals: {{Val list| 9, 63, 72, 279cf }}
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


Badness: 0.022322
[[Badness]] (Sintel): 3.67


== Submajor ==
=== 11-limit ===
Subgroup: 2.3.5
Subgroup: 2.3.5.7.11


[[Comma list]]: 69198046875/68719476736
Comma list: 225/224, 245/242, 1617/1600


[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


[[POTE generator]]: ~10125/8192 = 362.321
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


[[Badness]]: 0.130236
Badness (Sintel): 2.00


=== 7-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 225/224, 51200/50421
Comma list: 105/104, 196/195, 245/242, 1001/1000


[[Mapping]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
Optimal tunings:
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


[[POTE generator]]: ~49/40 = 362.255
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


{{Val list|legend=1| 10, 33, 43, 53 }}
Badness (Sintel): 1.43


[[Badness]]: 0.060533
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


=== 11-limit ===
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 6655/6561
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


POTE generator: ~27/22 = 362.101
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Vals: {{Val list| 10, 43e, 53, 116, 169de, 285cde }}
Badness (Sintel): 1.26


Badness: 0.050582
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.  


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


Comma list: 169/168, 225/224, 275/273, 385/384
[[Comma list]]: 225/224, 9805926501/9765625000


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


POTE generator: ~16/13 = 362.105
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.9803{{c}}, ~10000/9261 = 139.9522{{c}}
: [[error map]]: {{val| +0.980 -1.214 -0.313 -0.243 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10000/9261 = 139.8184{{c}}
: error map: {{val| 0.000 -2.863 -2.136 -2.287 }}


Vals: {{Val list| 10, 43e, 53, 116, 169de, 285cdef }}
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


Badness: 0.027689
[[Badness]] (Sintel): 3.81


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


Comma list: 99/98, 176/175, 51200/50421
Comma list: 225/224, 441/440, 43923/43750


POTE generator: ~49/40 = 362.418
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


POTE generator: ~49/40 = 362.418
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Vals: {{Val list| 43, 53, 96, 149d }}
Badness (Sintel): 1.74
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.051806
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


==== 13-limit ====
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
Subgroup: 2.3.5.7.11.13


Comma list: 99/98, 169/168, 176/175, 640/637
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


POTE generator: ~16/13 = 362.402
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
Badness (Sintel): 1.35


POTE generator: ~16/13 = 362.402
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Vals: {{Val list| 43, 53, 96, 149d }}
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Badness: 0.029680
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


== Alphorn ==
Optimal tunings:
Subgroup: 2.3.5.7
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


[[Comma list]]: 225/224, 5764801/5668704
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


[[Mapping]]: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
Badness (Sintel): 1.17


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


[[POTE generator]]: ~48/35 = 556.221
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.  


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.129258
[[Comma list]]: 225/224, 5971968/5764801


=== 11-limit ===
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
Subgroup: 2.3.5.7.11
: mapping generators: ~2401/1728, ~35/32


Comma list: 225/224, 385/384, 12250/11979
[[Optimal tuning]]s:  
* [[WE]]: ~2401/1728 = 599.7132{{c}}, ~35/32 = 162.5806{{c}}
: [[error map]]: {{val| -0.574 -1.597 -0.937 +5.510 }}
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~35/32 = 162.6388{{c}}
: error map: {{val| 0.000 -0.844 +0.492 +7.007 }}


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


POTE generator: ~11/8 = 556.144
[[Badness]] (Sintel): 4.53


Vals: {{Val list| 28d, 41, 315cde }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.073459
Comma list: 99/98, 176/175, 65536/65219


== Tertiosec ==
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}
Subgroup: 2.3.5


[[Comma list]]: {{monzo| -89 21 24 }}
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


[[Mapping]]: [{{val| 3 7 5 }}, {{val| 0 -8 7 }}]
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


[[POTE generator]]: ~16/15 = 112.281
Badness (Sintel): 2.26


{{Val list|legend=1| 21, 54, 75, 96, 171 }}
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


[[Badness]]: 0.851003
Tertiosec may be described as the {{nowrap| 21 & 75 }} temperament. It was initially named ''tertiomar'' by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, but was changed to ''tertiosec'' in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.  


=== 7-limit ===
[[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 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


{{Multival|legend=1| 24 -21 6 -89 -58 73 }}
[[Optimal tuning]]s:
* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
: [[error map]]: {{val| -0.523 -1.044 -1.549 +5.138 }}
* [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}}
: error map: {{val| 0.000 -0.284 -0.276 +6.592 }}


[[POTE generator]]: ~15/14 = 112.283
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


{{Val list|legend=1| 21, 54, 75, 96, 171d }}
[[Badness]] (Sintel): 10.9
 
[[Badness]]: 0.431636


=== 11-limit ===
=== 11-limit ===
Line 932: Line 1,177:
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 -1 12 7 14 | 0 8 -7 2 -5 }}
 
Optimal tunings:
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}


POTE generator: ~15/14 = 112.171
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


Vals: {{Val list| 21, 54, 75e }}
Badness (Sintel): 5.74


Badness: 0.173485
== References ==


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Marvel]]
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"