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Marvel temperaments: Difference between revisions

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This page discusses some of the temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the [[Marvel family|marvel]] comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton, merman and marvo. Considered elsewhere are [[Meantone family #Septimal meantone|meantone]], [[Gamelismic clan #Miracle|miracle]], [[Magic family|magic]], [[Diaschismic family #Pajara|pajara]], [[orwell]], [[Kleismic family #Catakleismic|catakleismic]], [[Schismatic family #Garibaldi|garibaldi]], [[Augmented family #August|august]], [[Pythagorean family #Compton|compton]], [[Dicot family #Sharp|sharp]], [[Escapade family #Escapade|escapade]] and [[Pelogic family #Mavila|mavila]].
{{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.  


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.
Temperaments considered in families and clans are:
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* [[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]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* [[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]]


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


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.)
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.


= Negri =
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.
{{main| Negri }}


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.  
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}}.)


== 5-limit ==
== Wizard ==
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''


Period: 1\1
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.


Optimal ([[POTE]]) generator: ~16/15 = 125.7549
[[Subgroup]]: 2.3.5.7


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
[[Comma list]]: 225/224, 118098/117649


Scales (Scala files):  
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Optimal tuning]]s:
<div style="line-height:1.6;">Technical data</div>
* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
<div class="mw-collapsible-content">
: [[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 }}


Subgroup: 2.3.5
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}


[[Comma list]]: 16875/16384
[[Badness]] (Sintel): 1.03


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Wedgie]]: {{wedgie| 4 -3 -14 }}
Comma list: 225/224, 385/384, 4000/3993


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}


</div></div>
Optimal tunings:
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


== 7-limit ==
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


Period: 1\1
Badness (Sintel): 0.613


Optimal ([[POTE]]) generator: ~15/14 = 125.608
==== Lizard ====
Subgroup: 2.3.5.7.11.13


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
Comma list: 225/224, 351/350, 364/363, 385/384


Scales (Scala files):  
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
<div class="mw-collapsible-content">
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


[[Comma list]]: 49/48, 225/224
Badness (Sintel): 0.900


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Wedgie]]: {{wedgie| 4 -3 2 -14 -8 13 }}
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}


</div></div>
Optimal tunings:
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


=== Negra ===
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


This is the 2.3.5.7.13 extension of negri.  
Badness (Sintel): 0.741


Period: 1\1
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Optimal ([[POTE]]) generator: ~14/13 = 125.567
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.13
Badness (Sintel): 0.955


[[Comma list]]: 49/48, 65/64, 91/90
==== Gizzard ====
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -3 }}]
Comma list: 225/224, 325/324, 385/384, 1573/1568


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


Gencom: [2 14/13; 49/48 65/64 91/90]
Optimal tunings:  
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}


</div></div>
Badness (Sintel): 0.837


== 11-limit ==
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Period: 1\1
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384


Optimal ([[POTE]]) generator: ~15/14 = 126.474
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
Optimal tunings:  
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.694
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


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


{{Val list|legend=1| 9, 10, 19 }}
Optimal tunings:
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


[[Badness]]: 0.0262
{{Optimal ET sequence|legend=0| 72, 94, 166g }}


</div></div>
Badness (Sintel): 0.901


=== 13-limit ===
=== Mage ===
Subgroup: 2.3.5.7.11


Period: 1\1
Comma list: 99/98, 176/175, 1331/1296
 
Optimal ([[POTE]]) generator: ~14/13 = 126.431
 
EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]


Scales (Scala files):  
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
<div class="mw-collapsible-content">
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


Comma list: 45/44, 49/48, 56/55, 78/77
Badness (Sintel): 1.91


[[POTE generator]]: ~14/13 = 126.431
== Tritonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


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


{{Val list|legend=1| 9, 10, 19 }}
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 50421/50000


== Negril ==
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


Period: 1\1
[[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 }}


Optimal ([[POTE]]) generator: ~15/14 = 124.767
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


EDO generators: [[19edo|2\19]], [[29edo|3\29]]
[[Badness]] (Sintel): 1.20
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 49/48, 100/99, 225/224
Comma list: 121/120, 225/224, 441/440


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


{{Val list|legend=1| 19, 29, 48d, 77cdd }}
Optimal tunings:
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


Badness: 0.0387
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


</div></div>
Badness (Sintel): 0.782


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


Period: 1\1
Comma list: 105/104, 121/120, 196/195, 275/273


Optimal ([[POTE]]) generator: ~14/13 = 124.716
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


EDO generators: [[19edo|2\19]], [[29edo|3\29]]
Optimal tunings:  
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.950
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Comma list: 49/48, 65/64, 91/90, 875/858
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


{{Val list|legend=1| 19, 29, 48df, 77cddf }}
Optimal tunings:
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


Badness: 0.0244
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


</div></div>
Badness (Sintel): 0.973


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


Period: 1\1
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


Optimal ([[POTE]]) generator: ~15/14 = 127.039
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


EDO generators: [[9edo|1\9]], [[19edo|2\19]]
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.03
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 33/32, 49/48, 77/75
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


Mapping: [{{val| 1 2 2 3 3 }}, {{val| 0 -4 3 -2 4 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
Optimal tunings:
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


Badness: 0.0306
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


</div></div>
Badness (Sintel): 1.04


=== 13-limit ===
=== Tritoni ===
Subgroup: 2.3.5.7.11


Period: 1\1
Comma list: 225/224, 385/384, 27783/27500


Optimal ([[POTE]]) generator: ~14/13 = 127.039
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


EDO generators: [[9edo|1\9]], [[19edo|2\19]]
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.50
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


Comma list: 33/32, 49/48, 65/64, 91/90
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.


Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
[[Comma list]]: 225/224, 84035/82944


Badness: 0.0202
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


</div></div>
[[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 }}


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


Period: 1\1
[[Badness]] (Sintel): 1.38
 
Optimal ([[POTE]]) generator: ~15/14 = 124.539
 
EDO generators: [[10edo|1\10]], [[19edo|2\19]], [[29edo|3\29]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 49/48, 55/54, 225/224
Comma list: 225/224, 245/242, 385/384


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


{{Val list|legend=1| 10, 19e, 29, 77cddee }}
Optimal tunings:
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


Badness: 0.0353
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


</div></div>
Badness (Sintel): 1.04


=== 13-limit ===
=== 13-limit ===
Period: 1\1
Optimal ([[POTE]]) generator: ~14/13 = 124.545
EDO generators: [[10edo|1\10]], [[19edo|2\19]], [[29edo|3\29]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
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, 144/143, 196/195, 245/242


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


{{Val list|legend=1| 10, 19e, 29, 77cddeef }}
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


Badness: 0.0216
{{Optimal ET sequence|legend=0| 41, 91 }}


</div></div>
Badness (Sintel): 0.955


== Wilsec ==
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Period: 1\1
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.


Optimal ([[POTE]]) generator: ~11/8 = 537.186
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.  


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
[[Subgroup]]: 2.3.5.7


Scales (Scala files):  
[[Comma list]]: 225/224, 2500000/2470629


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
<div style="line-height:1.6;">Technical data</div>
: mapping generators: ~2, ~10/7
<div class="mw-collapsible-content">


Comma list: 49/48, 121/120, 225/224
[[Optimal tuning]]s:  
* [[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 }}


Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


{{Val list|legend=1| 9, 20, 29, 38d, 67cdde }}
[[Badness]] (Sintel): 1.39


Badness: 0.0419
=== 11-limit ===
Subgroup: 2.3.5.7.11


</div></div>
Comma list: 225/224, 441/440, 1344/1331


=== 13-limit ===
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


Optimal ([[POTE]]) generator: ~11/8 = 537.208
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
Badness (Sintel): 1.20


Scales (Scala files):  
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 144/143, 225/224, 364/363, 441/440
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 49/48, 65/64, 91/90, 121/120
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


{{Val list|legend=1| 9, 20, 29, 38df, 67cddef }}
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Badness: 0.0252
Badness (Sintel): 1.14


</div></div>
=== Mermaid ===
Subgroup: 2.3.5.7.11


=== 17-limit ===
Comma list: 225/224, 385/384, 532400/531441


Period: 1\1
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


Optimal ([[POTE]]) generator: ~11/8 = 537.230
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


Scales (Scala files):  
Badness (Sintel): 1.46


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
==== 13-limit ====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13
<div class="mw-collapsible-content">


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
Comma list: 225/224, 325/324, 385/384, 10648/10647


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefg }}
Optimal tunings:
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


Badness: 0.0218
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


</div></div>
Badness (Sintel): 1.47


=== 19-limit ===
== 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.


Period: 1\1
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.


Optimal ([[POTE]]) generator: ~11/8 = 537.214
[[Subgroup]]: 2.3.5.7


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
[[Comma list]]: 225/224, 589824/588245


Scales (Scala files):  
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Optimal tuning]]s:
<div style="line-height:1.6;">Technical data</div>
* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
<div class="mw-collapsible-content">
: [[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 }}


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
[[Badness]] (Sintel): 1.44


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefgh }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0168
Comma list: 225/224, 385/384, 1331/1323


</div></div>
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


= Passive =
Optimal tunings:
{{see also | Archytas clan #Passion }}
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


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


Optimal ([[POTE]]) generator: ~16/15 = 98.809
Badness (Sintel): 0.838


EDO generators: [[12edo|1\12]], [[13edo|1\13]]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Scales (Scala files):  
Comma list: 225/224, 275/273, 385/384, 1331/1323


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 256/245
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


{{Val list|legend=1| 12, 49dd }}
Badness (Sintel): 1.07


Badness: 0.0751
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


</div></div>
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.


= Wizard =
[[Subgroup]]: 2.3.5.7


Period: 1\2
[[Comma list]]: 225/224, 1029/1000


Optimal ([[POTE]]) generator: ~5/4 = 383.256
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
[[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 }}


Scales (Scala files): [[wizard22]]
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 1.50
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 118098/117649
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
Comma list: 45/44, 56/55, 1029/1000


Mapping generators: ~1225/864, ~245/216
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


Wedgie: {{wedgie| 12 -2 20 -31 -2 52 }}
Optimal tunings:  
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


Badness: 0.0408
Badness (Sintel): 1.51


</div></div>
== Marvolo ==
[[Subgroup]]: 2.3.5.7


== 11-limit ==
[[Comma list]]: 225/224, 156250000/155649627


Period: 1\2
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


Optimal ([[POTE]]) generator: ~5/4 = 383.232
[[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 }}


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


Scales (Scala files): [[wizard22]]
[[Badness]] (Sintel): 2.11


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
=== 11-limit ===
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11
<div class="mw-collapsible-content">


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


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c }}
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Badness: 0.0185
Badness (Sintel): 0.958


</div></div>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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


Period: 1\2
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Optimal ([[POTE]]) generator: ~5/4 = 383.389
Optimal tunings:  
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


EDO generators: [[50edo|16\50]], [[72edo|23\72]]
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Scales (Scala files):  
Badness (Sintel): 0.887


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
=== 17-limit ===
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13.17
<div class="mw-collapsible-content">


Comma list: 225/224, 351/350, 364/363, 385/384
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


{{Val list|legend=1| 22, 50, 72, 122, 194df }}
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


Badness: 0.0218
Badness (Sintel): 0.760


</div></div>
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
==== 17-limit ====
 
Period: 1\2


Optimal ([[POTE]]) generator: ~5/4 = 383.381
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


EDO generators: [[50edo|16\50]], [[72edo|23\72]]
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
Badness (Sintel): 0.895


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


Mapping generators: ~17/12, ~17/15
Named by [[Xenllium]] in 2021, decic tempers out 16807/16384, the [[cloudy comma]], and {{monzo| 11 -10 -10 10 }}, the [[linus comma]], in addition to the marvel comma. It may be described as the {{nowrap| 50 & 60 }} temperament, with a period of 1/10 octave and a [[ploidacot]] signature of decaploid monocot. It is [[support]]ed by [[10edo|10-]], [[50edo|50-]], and [[60edo]].


{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
It can be extended to the 11-, 13-, and 17-limit by adding [[385/384]], [[105/104]], and [[170/169]] to the comma list in this order.


Badness: 0.0145
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 16807/16384


==== 19-limit ====
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


Period: 1\2
[[Optimal tuning]]s:  
* [[WE]]: ~15/14 = 120.1837{{c}}, ~3/2 = 699.7654{{c}} (~49/48 = 21.3366{{c}})
: [[error map]]: {{val| +1.837 -0.353 -0.753 -3.683 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 698.8236{{c}} (~49/48 = 21.1764{{c}})
: error map: {{val| 0.000 -3.131 -5.137 -8.826 }}


Optimal ([[POTE]]) generator: ~5/4 = 383.477
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


EDO generators: [[50edo|16\50]], [[72edo|23\72]]
[[Badness]] (Sintel): 2.26


Scales (Scala files):  
=== 11-limit ===
Subgroup: 2.3.5.7.11


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 225/224, 385/384, 3087/3025
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
Optimal tunings:  
* WE: ~15/14 = 120.1406{{c}}, ~3/2 = 697.6075{{c}} (~56/55 = 23.2360{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.0142{{c}} (~56/55 = 22.9858{{c}})


Mapping generators: ~17/12, ~17/15
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


{{Val list|legend=1| 22h, 50, 72, 122g, 194dfg }}
Badness (Sintel): 2.11


Badness: 0.0157
==== 13-limit ====
 
Subgroup: 2.3.5.7.11.13
</div></div>
 
=== Gizzard ===
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~5/4 = 383.170


EDO generators: [[72edo|23\72]], [[94edo|30\94]]
Comma list: 105/104, 144/143, 196/195, 2200/2197


Scales (Scala files):  
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~15/14 = 120.1166{{c}}, ~3/2 = 697.6705{{c}} (~78/77 = 23.0289{{c}})
<div class="mw-collapsible-content">
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.1492{{c}} (~78/77 = 22.8508{{c}})


Comma list: 225/224, 385/384, 325/324, 1573/1568
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
Badness (Sintel): 1.52
 
Mapping generators: ~99/70, ~25/22
 
{{Val list|legend=1| 72, 166, 238cf }}
 
Badness: 0.0203
 
</div></div>


==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Period: 1\2
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


Optimal ([[POTE]]) generator: ~5/4 = 383.175
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


EDO generators: [[72edo|23\72]], [[94edo|30\94]]
Optimal tunings:  
* WE: ~15/14 = 120.1262{{c}}, ~3/2 = 697.8185{{c}} (~78/77 = 22.9388{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.2757{{c}} (~78/77 = 22.7243{{c}})


Scales (Scala files):
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.28
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
=== Splendecic ===
Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[fantastic]] rank-3 temperament, its name is a portmanteau of ''splendid'' and ''decic''.


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
Subgroup: 2.3.5.7.11


Mapping generators: ~17/12, ~17/15
Comma list: 225/224, 1617/1600, 2401/2376


{{Val list|legend=1| 72, 166g, 238cfg }}
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}


Badness: 0.0136
Optimal tunings:  
* WE: ~15/14 = 120.1874{{c}}, ~3/2 = 699.6085{{c}} (~99/98 = 21.5156{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3531{{c}} (~99/98 = 21.6469{{c}})


</div></div>
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


==== 19-limit ====
Badness (Sintel): 1.98


Period: 1\2
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Optimal ([[POTE]]) generator: ~5/4 = 383.138
Comma list: 105/104, 196/195, 1001/1000, 1188/1183


EDO generators: [[72edo|23\72]], [[94edo|30\94]]
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


Scales (Scala files):  
Optimal tunings:
* WE: ~15/14 = 120.1565{{c}}, ~3/2 = 699.2756{{c}} (~91/90 = 21.6631{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.2480{{c}} (~91/90 = 21.7520{{c}})


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399
Badness (Sintel): 1.57


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Mapping generators: ~17/12, ~17/15
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374


{{Val list|legend=1| 72, 94, 166g }}
Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


Badness: 0.0148
Optimal tunings:  
* WE: ~15/14 = 120.1571{{c}}, ~3/2 = 699.2892{{c}} (~91/90 = 21.6536{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3144{{c}} (~91/90 = 21.6856{{c}})


</div></div>
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


== Mage ==
Badness (Sintel): 1.33


Period: 1\2
=== Prodecic ===
Prodecic (50e & 60e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the [[prodigy]] rank-3 temperament, its name is a portmanteau of ''prodigy'' and ''decic''.


Optimal ([[POTE]]) generator: ~5/4 = 383.124
Subgroup: 2.3.5.7.11


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
Comma list: 225/224, 441/440, 5929/5832


Scales (Scala files):  
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~15/14 = 120.2024{{c}}, ~3/2 = 701.3908{{c}} (~55/54 = 19.8237{{c}})
<div class="mw-collapsible-content">
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.5235{{c}} (~55/54 = 19.4765{{c}})


Comma list: 99/98, 176/175, 1331/1296
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
Badness (Sintel): 2.20


Mapping generators: ~77/54, ~55/48
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 22, 50e, 72ee, 94ee }}
Comma list: 105/104, 196/195, 275/273, 5929/5832


Badness: 0.0578
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


</div></div>
Optimal tunings:
* WE: ~15/14 = 120.1654{{c}}, ~3/2 = 701.4683{{c}} (~91/90 = 19.5242{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.7175{{c}} (~91/90 = 19.2825{{c}})


= Triton =
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Period: 1\1
Badness (Sintel): 1.73


Optimal ([[POTE]]) generator: ~7/5 = 568.865
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


EDO generators: [[19edo|9\19]], [[21edo|10\21]]
Comma list: 105/104, 154/153, 170/169, 196/195, 289/288


Scales (Scala files):  
Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~15/14 = 120.1577{{c}}, ~3/2 = 701.3950{{c}} (~91/90 = 19.5514{{c}})
<div class="mw-collapsible-content">
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.6932{{c}} (~91/90 = 19.3068{{c}})


Comma list: 225/224, 1029/1000
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Mapping: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
Badness (Sintel): 1.41


Wedgie: {{wedgie| 3 -7 -8 -18 -21 1 }}
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 19, 78bd, 97bd }}
[[Comma list]]: 225/224, 40353607/40310784


Badness: 0.0592
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


</div></div>
[[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 }}


== 11-limit ==
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


Period: 1\1
[[Badness]] (Sintel): 2.37


Optimal ([[POTE]]) generator: ~7/5 = 569.144
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDO generators: [[19edo|9\19]], [[21edo|10\21]]
Comma list: 225/224, 385/384, 12005/11979
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 45/44, 56/55, 1029/1000
 
Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]


{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


Badness: 0.0457
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}})


</div></div>
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


= Tritonic =
Badness (Sintel): 1.01
== 5-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~45/32 = 580.219
 
EDO generators: [[29edo|14\29]], [[31edo|15\31]], [[60edo|29\60]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 553584375/536870912
 
Mapping: [{{val| 1 4 -3 }}, {{val| 0 -5 11 }}]
 
{{Val list|legend=1| 29, 31, 60, 91 }}
 
Badness: 0.3785
 
</div></div>
 
== 7-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/5 = 580.286
 
EDO generators: [[29edo|14\29]], [[31edo|15\31]], [[60edo|29\60]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 225/224, 50421/50000
 
Mapping: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
 
Wedgie: {{wedgie| 5 -11 -12 -29 -33 3 }}
 
{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/5 = 580.267
 
EDO generators: [[29edo|14\29]], [[31edo|15\31]], [[60edo|29\60]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 121/120, 225/224, 441/440
 
Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
 
{{Val list|legend=1| 29, 31, 60e }}
 
Badness: 0.0237
 
</div></div>


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


Period: 1\1
Comma list: 169/168, 225/224, 364/363, 1716/1715


Optimal ([[POTE]]) generator: ~7/5 = 580.108
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


EDO generators: [[29edo|14\29]], [[31edo|15\31]], [[60edo|29\60]]
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}})


Scales (Scala files):
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.922
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 105/104, 121/120, 196/195, 275/273
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 29, 31, 60e, 151cde }}
[[Comma list]]: 225/224, 781250000/771895089


Badness: 0.0230
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


</div></div>
[[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 }}


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


Period: 1\1
[[Badness]] (Sintel): 2.44


Optimal ([[POTE]]) generator: ~7/5 = 580.389
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDO generators: [[31edo|15\31]], [[91edo|44\91]], [[122edo|59\122]]
Comma list: 225/224, 385/384, 236328125/234365481


Scales (Scala files):  
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Comma list: 225/224, 385/384, 27783/27500
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
Badness (Sintel): 2.96


{{Val list|legend=1| 31, 91, 122, 153d }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0455
Comma list: 225/224, 325/324, 385/384, 831875/830466


</div></div>
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


= Merman =
Optimal tunings:
== 5-limit ==
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


Period: 1\1
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Optimal ([[POTE]]) generator: ~45/32 = 585.634
Badness (Sintel): 2.16


EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Scales (Scala files):  
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 1121008359375/1099511627776
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


Mapping: [{{val| 1 5 -5 }}, {{val| 0 -7 15 }}]
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


{{Val list|legend=1| 41, 84, 125, 209, 334c, 543bc }}
Badness (Sintel): 1.96


Badness: 0.4538
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


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


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


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


Optimal ([[POTE]]) generator: ~7/5 = 585.585
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
Badness (Sintel): 1.71


Scales (Scala files):  
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 2500000/2470629
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Mapping: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Wedgie: {{wedgie| 7 -15 -16 -40 -45 5 }}
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


{{Val list|legend=1| 41, 84, 125 }}
Badness (Sintel): 1.57


Badness: 0.0551
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


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


== 11-limit ==
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Optimal ([[POTE]]) generator: ~7/5 = 585.606
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
Badness (Sintel): 1.50


Scales (Scala files):  
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 441/440, 1344/1331
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 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
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| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


Badness: 0.0364
Badness (Sintel): 1.53


</div></div>
=== Gracecord ===
Subgroup: 2.3.5.7.11


== 13-limit ==
Comma list: 225/224, 441/440, 109375/107811


Period: 1\1
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


Optimal ([[POTE]]) generator: ~7/5 = 585.657
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Scales (Scala files):  
Badness (Sintel): 2.21


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
==== 13-limit ====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13
<div class="mw-collapsible-content">


Comma list: 144/143, 225/224, 364/363, 441/440
Comma list: 225/224, 364/363, 441/440, 6125/6084


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
Optimal tunings:
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


Badness: 0.0275
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


</div></div>
Badness (Sintel): 1.83


= Septimin =
==== 17-limit ====
== 5-limit ==
Subgroup: 2.3.5.7.11.13.17


Period: 1\1
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Optimal ([[POTE]]) generator: ~75/64 = 263.658
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.87
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 35595703125/34359738368
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 4 1 }}, {{val| 0 -11 6 }}]
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


{{Val list|legend=1| 9, 23b, 32, 41, 91, 132, 223c, 355bc }}
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


Badness: 0.6709
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


== 7-limit ==
Badness (Sintel): 1.68


Period: 1\1
== Alphorn ==
[[Subgroup]]: 2.3.5.7


Optimal ([[POTE]]) generator: ~7/6 = 263.632
[[Comma list]]: 225/224, 5764801/5668704


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


Scales (Scala files):  
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 84035/82944
[[Badness]] (Sintel): 3.27


Mapping: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: {{wedgie| 11 -6 10 -35 -15 40 }}
Comma list: 225/224, 385/384, 12250/11979


{{Val list|legend=1| 41, 91, 132 }}
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Period: 1\1
Badness (Sintel): 2.43


Optimal ([[POTE]]) generator: ~7/6 = 263.634
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
[[Subgroup]]: 2.3.5.7


Scales (Scala files):  
[[Comma list]]: 225/224, 4194304/4117715


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
<div style="line-height:1.6;">Technical data</div>
: mapping generators: ~2, ~15/8
<div class="mw-collapsible-content">


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


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


{{Val list|legend=1| 41, 91, 223cdef }}
[[Badness]] (Sintel): 3.57


</div></div>
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 13-limit ==
Comma list: 99/98, 176/175, 1310720/1294139


Period: 1\1
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


Optimal ([[POTE]]) generator: ~7/6 = 263.700
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Scales (Scala files):  
Badness (Sintel): 2.82


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
==== 13-limit ====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13
<div class="mw-collapsible-content">
 
Comma list: 105/104, 144/143, 196/195, 245/242


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
Comma list: 99/98, 176/175, 640/637, 847/845


{{Val list|legend=1| 41, 91 }}
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


= Slender =
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Period: 1\1
Badness (Sintel): 1.88


Optimal ([[POTE]]) generator: ~49/48 = 38.413
=== Musneb ===
Subgroup: 2.3.5.7.11


EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
Comma list: 225/224, 385/384, 66550/64827


Scales (Scala files):  
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


Comma list: 225/224, 589824/588245
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
Badness (Sintel): 2.89


Wedgie: {{wedgie| 13 -10 6 -46 -27 42 }}
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


{{Val list|legend=1| 31, 94, 125 }}
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.


Badness: 0.0569
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 125000000/121060821


== 11-limit ==
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


Period: 1\1
[[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 }}


Optimal ([[POTE]]) generator: ~49/48 = 38.387
{{Optimal ET sequence|legend=1| 51, 53 }}


EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
[[Badness]] (Sintel): 3.64


Scales (Scala files):  
=== 11-limit ===
Subgroup: 2.3.5.7.11


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 121/120, 225/224, 22000/21609
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 225/224, 385/384, 1331/1323
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


{{Val list|legend=1| 31, 63, 94, 125 }}
{{Optimal ET sequence|legend=0| 51, 53 }}


Badness: 0.02534
Badness (Sintel): 2.46


</div></div>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 13-limit ==
Comma list: 121/120, 225/224, 275/273, 1040/1029


Period: 1\1
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


Optimal ([[POTE]]) generator: ~49/48 = 38.314
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
{{Optimal ET sequence|legend=0| 51f, 53 }}


Scales (Scala files):  
Badness (Sintel): 1.96


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
== Naiadical ==
<div style="line-height:1.6;">Technical data</div>
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.  
<div class="mw-collapsible-content">


Comma list: 225/224, 275/273, 385/384, 1331/1323
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
[[Comma list]]: 225/224, 823543/800000


{{Val list|legend=1| 31, 63, 94 }}
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


Badness: 0.02591
[[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 }}


</div></div>
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


= Marvo =
[[Badness]] (Sintel): 3.67
{{see also| Gravity family }}


Comma list: 225/224, 78125000/78121827
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~27/20 = 516.694
Comma list: 225/224, 245/242, 1617/1600


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


Wedgie: {{wedgie| 6 17 46 13 56 59 }}
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| 7d, 65d, 72, 137, 209, 281, 569bcc }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Badness: 0.0976
Badness (Sintel): 2.00


== 11-limit ==
=== 13-limit ===
Comma list: 225/224, 243/242, 4000/3993
Subgroup: 2.3.5.7.11.13


POTE generator: ~27/20 = 516.699
Comma list: 105/104, 196/195, 245/242, 1001/1000


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


{{Val list|legend=1| 7d, 65d, 72, 281, 353c, 425bc, 497bc }}
Optimal tunings:
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


Badness: 0.0317
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


== 13-limit ==
Badness (Sintel): 1.43
Comma list: 225/224, 243/242, 351/350, 1625/1617


POTE generator: ~27/20 = 516.730
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


{{Val list|legend=1| 72, 137, 209, 281f, 490bcf }}
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


Badness: 0.0269
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


= Marvolo =
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}
Comma list: 225/224, 156250000/155649627


POTE generator: ~21/20 = 83.348
Badness (Sintel): 1.26


Mapping: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


Wedgie: {{wedgie| 6 -19 -26 -44 -58 -7 }}
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
[[Comma list]]: 225/224, 9805926501/9765625000


Badness: 0.0833
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


== 11-limit ==
[[Optimal tuning]]s:
Comma list: 225/224, 441/440, 4000/3993
* [[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 }}


POTE generator: ~21/20 = 83.340
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
[[Badness]] (Sintel): 3.81


{{Val list|legend=1| 29, 43, 72 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0290
Comma list: 225/224, 441/440, 43923/43750


== 13-limit ==
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}
Comma list: 169/168, 225/224, 364/363, 441/440


POTE generator: ~21/20 = 83.330
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


{{Val list|legend=1| 29, 43, 72, 115f }}
Badness (Sintel): 1.74


Badness: 0.0215
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


= Amavil =
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
== 5-limit (mabila) ==
Comma list: 268435456/263671875


POTE generator: ~512/375 = 529.6849
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


Mapping: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 0.2325
Badness (Sintel): 1.35


== 7-limit ==
=== 17-limit ===
Comma list: 225/224, 17496/16807
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~48/35 = 529.979
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Mapping: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


Wedgie: {{wedgie| 10 -3 14 -28 -6 41 }}
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


{{Val list|legend=1| 9, 34d, 43, 77d }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 0.1096
Badness (Sintel): 1.17


== 11-limit ==
== Hendeca ==
Comma list: 99/98, 176/175, 864/847
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


POTE generator: ~15/11 = 529.974
Hendeca tempers out the same 5-limit comma as [[porwell temperaments #Hendecatonic|hendecatonic]], and has a period of 1/11 octave. However, in this temperament, nine periods represent [[7/4]], the same as [[keemic temperaments #Undeka|undeka]]. It can be tuned to [[22edo]] or [[33edo]] using the [[patent val]], or [[55edo]] using the 55d val. It was named by [[Xenllium]] in 2025 as a low-accuracy variant of hendecatonic.  


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 9, 34d, 43, 77de }}
[[Comma list]]: 225/224, 122880/117649


Badness: 0.0426
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


== 13-limit ==
[[Optimal tuning]]s:
Comma list: 78/77, 99/98, 144/143, 176/175
* [[WE]]: ~16/15 = 108.9526{{c}}, ~3/2 = 702.4215{{c}}
: [[error map]]: {{val| -1.521 -1.055 -2.252 +8.705 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 703.2071{{c}}
: error map: {{val| 0.000 +1.252 +1.388 +12.992 }}


POTE generator: ~15/11 = 529.951
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
[[Badness]] (Sintel): 4.37


{{Val list|legend=1| 9, 34d, 43, 77de }}
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
Badness: 0.0258


= Enneaportent =
Comma list: 121/120, 225/224, 352/343
Comma list: 225/224, 40353607/40310784


POTE generator: ~5/4 = 383.165
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}


Mapping: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
Optimal tunings:  
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


Wedgie: {{wedgie| 18 -9 18 -56 -22 67 }}
{{Optimal ET sequence|legend=0| 22, 55d }}


{{Val list|legend=1| 9, 54, 63, 72, 495bcd }}
Badness (Sintel): 2.46


Badness: 0.0937
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


== 11-limit ==
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.
Comma list: 225/224, 385/384, 12005/11979


POTE generator: ~5/4 = 383.146
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
[[Comma list]]: 225/224, 5971968/5764801


{{Val list|legend=1| 9, 54, 63, 72, 423cd, 495bcd }}
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


Badness: 0.0304
[[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 }}


== 13-limit ==
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}
Comma list: 169/168, 225/224, 364/363, 1716/1715


POTE generator: ~5/4 = 383.047
[[Badness]] (Sintel): 4.53


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 9, 54, 63, 72, 279cf }}
Comma list: 99/98, 176/175, 65536/65219


Badness: 0.0223
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


= Submajor =
Optimal tunings:
== 5-limit ==
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
Comma list: 69198046875/68719476736
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


POTE generator: ~10125/8192 = 362.321
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


Mapping: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
Badness (Sintel): 2.26


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


Badness: 0.1302
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
Comma list: 225/224, 51200/50421


POTE generator: ~49/40 = 362.255
[[Comma list]]: 225/224, 14495514624/13841287201


Mapping: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


Wedgie: {{wedgie| 8 -11 -6 -36 -32 17 }}
[[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 }}


{{Val list|legend=1| 10, 33, 43, 53 }}
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


Badness: 0.0605
[[Badness]] (Sintel): 10.9


== 11-limit ==
=== 11-limit ===
Comma list: 225/224, 385/384, 6655/6561
Subgroup: 2.3.5.7.11
 
POTE generator: ~27/22 = 362.101
 
Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
 
{{Val list|legend=1| 10, 53, 116, 169de, 285cde }}
 
Badness: 0.0506
 
=== 13-limit ===
Comma list: 169/168, 225/224, 275/273, 385/384


POTE generator: ~16/13 = 362.105
Comma list: 225/224, 3840/3773, 12005/11979


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
 
{{Val list|legend=1| 10, 53, 116, 169de, 285cdef }}
 
Badness: 0.0277
 
== Interpental ==
Comma list: 99/98, 176/175, 51200/50421
 
POTE generator: ~49/40 = 362.418
 
Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
 
{{Val list|legend=1| 43, 53, 96, 149d }}
 
Badness: 0.0518
 
=== 13-limit ===
Comma list: 99/98, 169/168, 176/175, 640/637
 
POTE generator: ~16/13 = 362.402
 
Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
 
{{Val list|legend=1| 43, 53, 96, 149d }}
 
Badness: 0.0297
 
= Alphorn =
Comma list: 225/224, 5764801/5668704
 
POTE generator: ~48/35 = 556.221
 
Mapping: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
 
Wedgie: {{wedgie| 16 -5 22 -45 -10 65 }}
 
{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
 
Badness: 0.1293
 
== 11-limit ==
Comma list: 225/224, 385/384, 12250/11979


POTE generator: ~11/8 = 556.144
Optimal tunings:  
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


{{Val list|legend=1| 41, 315cde }}
Badness (Sintel): 5.74


Badness: 0.0735
== References ==


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