Xenharmonic Wiki

Marvel temperaments: Difference between revisions

← Older edit
VisualWikitext
→Decic: remove breed in subtitle to facilitate linking; remove 10edo as it's trivial; apply new template for wedgie
→Hendeca: + history
 
(104 intermediate revisions by 15 users not shown)
Line 1: Line 1:
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.  


== 5-limit ==
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}}.)
Period: 1\1


Optimal ([[POTE]]) generator: ~16/15 = 125.7549
== Wizard ==
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
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.


Scales (Scala files):  
[[Subgroup]]: 2.3.5.7


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 225/224, 118098/117649
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216


[[Comma list]]: 16875/16384
[[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 }}


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}


[[Wedgie]]: {{wedgie| 4 -3 -14 }}
[[Badness]] (Sintel): 1.03
 
{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
 
</div></div>
 
== 7-limit ==
Period: 1\1
 
Optimal ([[POTE]]) generator: ~15/14 = 125.608
 
EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
 
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
 
[[Comma list]]: 49/48, 225/224
 
[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
 
[[Wedgie]]: {{wedgie| 4 -3 2 -14 -8 13 }}
 
{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
 
</div></div>
 
=== Negra ===
This is the 2.3.5.7.13 extension of negri.
 
Period: 1\1
 
Optimal (POTE) generator: ~14/13 = 125.567
 
EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
 
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: 2.3.5.7.13
 
[[Comma list]]: 49/48, 65/64, 91/90
 
[[Mapping]]: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -3 }}]
 
Gencom map: [{{val| 1 2 2 3 0 4 }}, {{val| 0 -4 3 -2 0 -3 }}]
 
Gencom: [2 14/13; 49/48 65/64 91/90]
 
{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
 
</div></div>
 
== 11-limit ==
Period: 1\1
 
Optimal ([[POTE]]) generator: ~15/14 = 126.474
 
EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
 
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: 45/44, 49/48, 56/55
Comma list: 225/224, 385/384, 4000/3993
 
Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
 
{{Val list|legend=1| 9, 10, 19 }}
 
[[Badness]]: 0.0262
 
</div></div>
 
=== 13-limit ===
Period: 1\1


Optimal ([[POTE]]) generator: ~14/13 = 126.431
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}


EDO generators: [[9edo|1\9]], [[10edo|1\10]], [[19edo|2\19]]
Optimal tunings:  
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.613
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


==== Lizard ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


[[POTE generator]]: ~14/13 = 126.431
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}


Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
Optimal tunings:  
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


{{Val list|legend=1| 9, 10, 19 }}
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


</div></div>
Badness (Sintel): 0.900


== Negril ==
===== 17-limit =====
Period: 1\1
Subgroup: 2.3.5.7.11.13.17


Optimal ([[POTE]]) generator: ~15/14 = 124.767
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


EDO generators: [[19edo|2\19]], [[29edo|3\29]]
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


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


Subgroup: 2.3.5.7.11
Badness (Sintel): 0.741


Comma list: 49/48, 100/99, 225/224
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
 
{{Val list|legend=1| 19, 29, 48d, 77cdd }}
 
Badness: 0.0387
 
</div></div>
 
=== 13-limit ===
Period: 1\1


Optimal ([[POTE]]) generator: ~14/13 = 124.716
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}


EDO generators: [[19edo|2\19]], [[29edo|3\29]]
Optimal tunings:  
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.955
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 49/48, 65/64, 91/90, 875/858
Comma list: 225/224, 325/324, 385/384, 1573/1568


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


{{Val list|legend=1| 19, 29, 48df, 77cddf }}
Optimal tunings:
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}


Badness: 0.0244
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}


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


== Negric ==
===== 17-limit =====
Period: 1\1
Subgroup: 2.3.5.7.11.13.17


Optimal ([[POTE]]) generator: ~15/14 = 127.039
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384


EDO generators: [[9edo|1\9]], [[19edo|2\19]]
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Badness (Sintel): 0.694


Comma list: 33/32, 49/48, 77/75
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 2 2 3 3 }}, {{val| 0 -4 3 -2 4 }}]
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
 
{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
 
Badness: 0.0306


</div></div>
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}


=== 13-limit ===
Optimal tunings:
Period: 1\1
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


Optimal ([[POTE]]) generator: ~14/13 = 127.039
{{Optimal ET sequence|legend=0| 72, 94, 166g }}


EDO generators: [[9edo|1\9]], [[19edo|2\19]]
Badness (Sintel): 0.901
 
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
 
Comma list: 33/32, 49/48, 65/64, 91/90
 
Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
 
{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
 
Badness: 0.0202
 
</div></div>
 
== Negroni ==
Period: 1\1
 
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">


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


Comma list: 49/48, 55/54, 225/224
Comma list: 99/98, 176/175, 1331/1296
 
Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
 
{{Val list|legend=1| 10, 19e, 29, 77cddee }}
 
Badness: 0.0353
 
</div></div>
 
=== 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):  
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


Comma list: 49/48, 55/54, 65/64, 91/90
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Badness (Sintel): 1.91


{{Val list|legend=1| 10, 19e, 29, 77cddeef }}
== Tritonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


Badness: 0.0216
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.  


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


== Wilsec ==
[[Comma list]]: 225/224, 50421/50000
Period: 1\1


Optimal ([[POTE]]) generator: ~11/8 = 537.186
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


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


Scales (Scala files):
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 1.20
<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, 121/120, 225/224
Comma list: 121/120, 225/224, 441/440
 
Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]
 
{{Val list|legend=1| 9, 20, 29, 38d, 67cdde }}
 
Badness: 0.0419
 
</div></div>
 
=== 13-limit ===
Period: 1\1


Optimal ([[POTE]]) generator: ~11/8 = 537.208
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


EDO generators: [[29edo|13\29]], [[38edo|17\38]]
Optimal tunings:  
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.782
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


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


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


{{Val list|legend=1| 9, 20, 29, 38df, 67cddef }}
Optimal tunings:
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


Badness: 0.0252
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


</div></div>
Badness (Sintel): 0.950
 
=== 17-limit ===
Period: 1\1
 
Optimal ([[POTE]]) generator: ~11/8 = 537.230
 
EDO generators: [[29edo|13\29]], [[38edo|17\38]]
 
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">


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


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


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefg }}
Optimal tunings:
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


Badness: 0.0218
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


</div></div>
Badness (Sintel): 0.973
 
=== 19-limit ===
Period: 1\1
 
Optimal ([[POTE]]) generator: ~11/8 = 537.214
 
EDO generators: [[29edo|13\29]], [[38edo|17\38]]
 
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">


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


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefgh }}
Optimal tunings:
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


Badness: 0.0168
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


</div></div>
Badness (Sintel): 1.03


= Passive =
==== 23-limit ====
{{see also | Archytas clan #Passion }}
Subgroup: 2.3.5.7.11.13.17.19.23


Period: 1\1
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


Optimal ([[POTE]]) generator: ~16/15 = 98.809
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


EDO generators: [[12edo|1\12]], [[13edo|1\13]]
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{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): 1.04
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
=== Tritoni ===
Subgroup: 2.3.5.7.11


Comma list: 225/224, 256/245
Comma list: 225/224, 385/384, 27783/27500


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


{{Val list|legend=1| 12, 49dd }}
Optimal tunings:
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


Badness: 0.0751
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


</div></div>
Badness (Sintel): 1.50


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


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


Period: 1\2
[[Subgroup]]: 2.3.5.7


Optimal ([[POTE]]) generator: ~5/4 = 383.256
[[Comma list]]: 225/224, 84035/82944


EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


Scales (Scala files): [[wizard22]]
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 41, 91, 132d }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 1.38
 
Comma list: 225/224, 118098/117649
 
Mapping: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
 
Mapping generators: ~1225/864, ~245/216
 
Wedgie: {{wedgie| 12 -2 20 -31 -2 52 }}
 
{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
 
Badness: 0.0408
 
</div></div>
 
== 11-limit ==
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~5/4 = 383.232
 
EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
 
Scales (Scala files): [[wizard22]]
 
<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: 225/224, 385/384, 4000/3993
Comma list: 225/224, 245/242, 385/384


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


Mapping generators: ~99/70, ~25/22
Optimal tunings:
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c }}
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


Badness: 0.0185
Badness (Sintel): 1.04
 
</div></div>
 
=== Lizard ===
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~5/4 = 383.389
 
EDO generators: [[50edo|16\50]], [[72edo|23\72]]
 
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">


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


Comma list: 225/224, 351/350, 364/363, 385/384
Comma list: 105/104, 144/143, 196/195, 245/242
 
Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
 
Mapping generators: ~99/70, ~25/22


{{Val list|legend=1| 22, 50, 72, 122, 194df }}
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


Badness: 0.0218
Optimal tunings:  
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 41, 91 }}


==== 17-limit ====
Badness (Sintel): 0.955


Period: 1\2
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Optimal ([[POTE]]) generator: ~5/4 = 383.381
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.  


EDO generators: [[50edo|16\50]], [[72edo|23\72]]
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.


Scales (Scala files):  
[[Subgroup]]: 2.3.5.7


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 225/224, 2500000/2470629
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
[[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| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


Mapping generators: ~17/12, ~17/15
[[Badness]] (Sintel): 1.39


{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0145
Comma list: 225/224, 441/440, 1344/1331


</div></div>
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


==== 19-limit ====
Optimal tunings:
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


Period: 1\2
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


Optimal ([[POTE]]) generator: ~5/4 = 383.477
Badness (Sintel): 1.20


EDO generators: [[50edo|16\50]], [[72edo|23\72]]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Scales (Scala files):  
Comma list: 144/143, 225/224, 364/363, 441/440


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17.19
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
Badness (Sintel): 1.14


Mapping generators: ~17/12, ~17/15
=== Mermaid ===
 
Subgroup: 2.3.5.7.11
{{Val list|legend=1| 22h, 50, 72, 122g, 194dfg }}
 
Badness: 0.0157
 
</div></div>


=== Gizzard ===
Comma list: 225/224, 385/384, 532400/531441


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


Optimal ([[POTE]]) generator: ~5/4 = 383.170
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


EDO generators: [[72edo|23\72]], [[94edo|30\94]]
{{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;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


==== 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}}


{{Val list|legend=1| 72, 166, 238cf }}
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


Badness: 0.0203
Badness (Sintel): 1.47


</div></div>
== 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.


Period: 1\2
[[Subgroup]]: 2.3.5.7


Optimal ([[POTE]]) generator: ~5/4 = 383.175
[[Comma list]]: 225/224, 589824/588245


EDO generators: [[72edo|23\72]], [[94edo|30\94]]
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


Scales (Scala files):  
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17
[[Badness]] (Sintel): 1.44
 
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
 
Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
 
Mapping generators: ~17/12, ~17/15
 
{{Val list|legend=1| 72, 166g, 238cfg }}
 
Badness: 0.0136
 
</div></div>
 
==== 19-limit ====
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~5/4 = 383.138
 
EDO generators: [[72edo|23\72]], [[94edo|30\94]]
 
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: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
 
Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
 
Mapping generators: ~17/12, ~17/15
 
{{Val list|legend=1| 72, 94, 166g }}
 
Badness: 0.0148
 
</div></div>
 
== Mage ==
 
Period: 1\2
 
Optimal ([[POTE]]) generator: ~5/4 = 383.124
 
EDO generators: [[22edo|7\22]], [[50edo|16\50]], [[72edo|23\72]], [[94edo|30\94]]
 
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: 99/98, 176/175, 1331/1296
Comma list: 225/224, 385/384, 1331/1323


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


Mapping generators: ~77/54, ~55/48
Optimal tunings:  
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


{{Val list|legend=1| 22, 50e, 72ee, 94ee }}
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Badness: 0.0578
Badness (Sintel): 0.838


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


= Triton =
Comma list: 225/224, 275/273, 385/384, 1331/1323
 
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/5 = 568.865
 
EDO generators: [[19edo|9\19]], [[21edo|10\21]]
 
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
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Comma list: 225/224, 1029/1000
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 0 6 7 }}, {{val| 0 3 -7 -8 }}]
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Wedgie: {{wedgie| 3 -7 -8 -18 -21 1 }}
Badness (Sintel): 1.07


{{Val list|legend=1| 19, 78bd, 97bd }}
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


Badness: 0.0592
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.  


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


== 11-limit ==
[[Comma list]]: 225/224, 1029/1000


Period: 1\1
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


Optimal ([[POTE]]) generator: ~7/5 = 569.144
[[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 }}


EDO generators: [[19edo|9\19]], [[21edo|10\21]]
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


Scales (Scala files):
[[Badness]] (Sintel): 1.50
 
<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: 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 }}


{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


Badness: 0.0457
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


</div></div>
Badness (Sintel): 1.51


= Tritonic =
== Marvolo ==
[[Subgroup]]: 2.3.5.7


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
[[Comma list]]: 225/224, 156250000/155649627


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


Optimal ([[POTE]]) generator: ~7/5 = 580.286
[[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: [[29edo|14\29]], [[31edo|15\31]], [[60edo|29\60]]
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


Scales (Scala files):
[[Badness]] (Sintel): 2.11
 
<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
 
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">


=== 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 }}


{{Val list|legend=1| 29, 31, 60e }}
Optimal tunings:
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


Badness: 0.0237
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


</div></div>
Badness (Sintel): 0.958


=== 13-limit ===
=== 13-limit ===
Period: 1\1
Optimal ([[POTE]]) generator: ~7/5 = 580.108
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">
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: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


{{Val list|legend=1| 29, 31, 60e, 151cde }}
Optimal tunings:
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


Badness: 0.0230
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


</div></div>
Badness (Sintel): 0.887


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


Period: 1\1
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


Optimal ([[POTE]]) generator: ~7/5 = 580.389
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


EDO generators: [[31edo|15\31]], [[91edo|44\91]], [[122edo|59\122]]
Optimal tunings:  
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.760
<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: 225/224, 385/384, 27783/27500


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


{{Val list|legend=1| 31, 91, 122, 153d }}
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Badness: 0.0455
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


= Merman =
Badness (Sintel): 0.895


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


Period: 1\1
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]].


Optimal ([[POTE]]) generator: ~7/5 = 585.585
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.


EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
[[Subgroup]]: 2.3.5.7


Scales (Scala files):
[[Comma list]]: 225/224, 16807/16384
 
<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
 
Comma list: 225/224, 2500000/2470629
 
Mapping: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
 
Wedgie: {{wedgie| 7 -15 -16 -40 -45 5 }}
 
{{Val list|legend=1| 41, 84, 125 }}
 
Badness: 0.0551
 
</div></div>
 
== 11-limit ==
 
Period: 1\1


Optimal ([[POTE]]) generator: ~7/5 = 585.606
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3


EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
[[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 2.26
<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: 225/224, 441/440, 1344/1331
Comma list: 225/224, 385/384, 3087/3025
 
Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
 
{{Val list|legend=1| 41, 84, 125e }}
 
Badness: 0.0364
 
</div></div>
 
== 13-limit ==


Period: 1\1
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}


Optimal ([[POTE]]) generator: ~7/5 = 585.657
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}})


EDO generators: [[41edo|20\41]], [[43edo|21\43]], [[84edo|41\84]]
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Scales (Scala files):  
Badness (Sintel): 2.11
 
<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">


==== 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: 105/104, 144/143, 196/195, 2200/2197


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
Optimal tunings:
* WE: ~15/14 = 120.1166{{c}}, ~3/2 = 697.6705{{c}} (~78/77 = 23.0289{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.1492{{c}} (~78/77 = 22.8508{{c}})


Badness: 0.0275
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


</div></div>
Badness (Sintel): 1.52


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


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220


Period: 1\1
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


Optimal ([[POTE]]) generator: ~7/6 = 263.632
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}})


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
{{Optimal ET sequence|legend=0| 10, 40, 50 }}


Scales (Scala files):  
Badness (Sintel): 1.28


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
=== Splendecic ===
<div style="line-height:1.6;">Technical data</div>
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''.
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7
 
Comma list: 225/224, 84035/82944
 
Mapping: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
 
Wedgie: {{wedgie| 11 -6 10 -35 -15 40 }}
 
{{Val list|legend=1| 41, 91, 132 }}
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~7/6 = 263.634
 
EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
 
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
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 2401/2376
Comma list: 225/224, 1617/1600, 2401/2376
 
Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
 
{{Val list|legend=1| 41, 91, 223cdef }}
 
</div></div>
 
== 13-limit ==


Period: 1\1
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}


Optimal ([[POTE]]) generator: ~7/6 = 263.700
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}})


EDO generators: [[41edo|9\41]], [[50edo|11\50]], [[91edo|20\91]]
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}


Scales (Scala files):
Badness (Sintel): 1.98
 
<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">


==== 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: 105/104, 196/195, 1001/1000, 1188/1183
 
Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]


{{Val list|legend=1| 41, 91 }}
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


</div></div>
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}})


= Slender =
{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}


Period: 1\1
Badness (Sintel): 1.57


Optimal ([[POTE]]) generator: ~49/48 = 38.413
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
Comma list: 105/104, 170/169, 196/195, 289/288, 375/374


Scales (Scala files):  
Mapping: {{mapping| 10 0 39 28 82 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 = 120.1571{{c}}, ~3/2 = 699.2892{{c}} (~91/90 = 21.6536{{c}})
<div class="mw-collapsible-content">
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3144{{c}} (~91/90 = 21.6856{{c}})


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}


Comma list: 225/224, 589824/588245
Badness (Sintel): 1.33


Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
=== 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''.
Wedgie: {{wedgie| 13 -10 6 -46 -27 42 }}
 
{{Val list|legend=1| 31, 94, 125 }}
 
Badness: 0.0569
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~49/48 = 38.387
 
EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
 
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
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 1331/1323
Comma list: 225/224, 441/440, 5929/5832
 
Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
 
{{Val list|legend=1| 31, 63, 94, 125 }}
 
Badness: 0.02534
 
</div></div>
 
== 13-limit ==


Period: 1\1
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}


Optimal ([[POTE]]) generator: ~49/48 = 38.314
Optimal tunings:
* WE: ~15/14 = 120.2024{{c}}, ~3/2 = 701.3908{{c}} (~55/54 = 19.8237{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.5235{{c}} (~55/54 = 19.4765{{c}})


EDO generators: [[31edo|1\31]], [[32edo|1\32]], [[63edo|2\63]], [[94edo|3\94]]
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Scales (Scala files):  
Badness (Sintel): 2.20
 
<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">


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


Comma list: 225/224, 275/273, 385/384, 1331/1323
Comma list: 105/104, 196/195, 275/273, 5929/5832
 
Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]


{{Val list|legend=1| 31, 63, 94 }}
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


Badness: 0.02591
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}})


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


= Marvo =
Badness (Sintel): 1.73
{{see also| Gravity family }}


Period: 1\1
==== 17-limit ====
 
Subgroup: 2.3.5.7.11.13.17
Optimal ([[POTE]]) generator: ~27/20 = 516.694


EDO generators: [[65edo|28\65]], [[72edo|31\72]], [[137edo|59\137]]
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}})


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 10, 50e, 60e }}


Comma list: 225/224, 78125000/78121827
Badness (Sintel): 1.41


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


Wedgie: {{wedgie| 6 17 46 13 56 59 }}
[[Comma list]]: 225/224, 40353607/40310784


{{Val list|legend=1| 65d, 72, 137, 209, 281, 569bcc }}
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


Badness: 0.0976
[[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 }}


</div></div>
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


== 11-limit ==
[[Badness]] (Sintel): 2.37
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~27/20 = 516.699
 
EDO generators: [[65edo|28\65]], [[72edo|31\72]], [[137edo|59\137]]
 
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: 225/224, 243/242, 4000/3993
Comma list: 225/224, 385/384, 12005/11979


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


{{Val list|legend=1| 65d, 72, 281, 353c, 425bc, 497bc }}
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}})


Badness: 0.0317
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


</div></div>
Badness (Sintel): 1.01
 
== 13-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~27/20 = 516.730
 
EDO generators: [[65edo|28\65]], [[72edo|31\72]], [[137edo|59\137]]
 
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">


=== 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: 169/168, 225/224, 364/363, 1716/1715


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


{{Val list|legend=1| 65d, 72, 137, 209, 281f, 490bcf }}
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}})


Badness: 0.0269
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


</div></div>
Badness (Sintel): 0.922


= Marvolo =
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


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


Optimal ([[POTE]]) generator: ~21/20 = 83.348
[[Comma list]]: 225/224, 781250000/771895089


EDO generators: [[29edo|2\29]], [[43edo|3\43]], [[72edo|5\72]]
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


Scales (Scala files):  
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 2.44
 
Comma list: 225/224, 156250000/155649627
 
Mapping: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
 
Wedgie: {{wedgie| 6 -19 -26 -44 -58 -7 }}
 
{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
 
Badness: 0.0833
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~21/20 = 83.340
 
EDO generators: [[29edo|2\29]], [[43edo|3\43]], [[72edo|5\72]]
 
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: 225/224, 441/440, 4000/3993
Comma list: 225/224, 385/384, 236328125/234365481


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


{{Val list|legend=1| 29, 43, 72 }}
Optimal tunings:
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


Badness: 0.0290
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


</div></div>
Badness (Sintel): 2.96
 
== 13-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~21/20 = 83.330
 
EDO generators: [[29edo|2\29]], [[43edo|3\43]], [[72edo|5\72]]
 
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">


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


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


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


{{Val list|legend=1| 29, 43, 72, 115f }}
Optimal tunings:
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


Badness: 0.0215
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


</div></div>
Badness (Sintel): 2.16


= Amavil =
==== 17-limit ====
== 5-limit (mabila) ==
Subgroup: 2.3.5.7.11.13.17


Period: 1\1
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


Optimal ([[POTE]]) generator: ~512/375 = 529.6849
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


EDO generators: [[34edo|15\34]], [[43edo|19\43]], [[77edo|34\77]], [[111edo|49\111]]
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.96
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5
==== 19-limit ====
 
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 268435456/263671875


Mapping: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


Badness: 0.2325
Optimal tunings:  
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


== 7-limit ==
Badness (Sintel): 1.71


Period: 1\1
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Optimal ([[POTE]]) generator: ~48/35 = 529.979
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


EDO generators: [[34edo|15\34]], [[43edo|19\43]]
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
Badness (Sintel): 1.57


Comma list: 225/224, 17496/16807
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Mapping: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


Wedgie: {{wedgie| 10 -3 14 -28 -6 41 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
Optimal tunings:
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Badness: 0.1096
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


</div></div>
Badness (Sintel): 1.50


== 11-limit ==
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Period: 1\1
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


Optimal ([[POTE]]) generator: ~15/11 = 529.974
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


EDO generators: [[34edo|15\34]], [[43edo|19\43]]
Optimal tunings:  
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.53
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== Gracecord ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 864/847
Comma list: 225/224, 441/440, 109375/107811
 
Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
 
{{Val list|legend=1| 9, 34d, 43, 77de }}
 
Badness: 0.0426
 
</div></div>
 
== 13-limit ==
 
Period: 1\1


Optimal ([[POTE]]) generator: ~15/11 = 529.951
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


EDO generators: [[34edo|15\34]], [[43edo|19\43]]
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 2.21
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


{{Val list|legend=1| 9, 34d, 43, 77de }}
Optimal tunings:
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


Badness: 0.0258
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


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


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


Period: 1\9
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Optimal ([[POTE]]) generator: ~5/4 = 383.165
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


EDO generators: [[63edo|20\63]], [[72edo|23\72]], [[81edo|26\81]]
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">


Subgroup: 2.3.5.7
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 225/224, 40353607/40310784
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


Mapping: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


Wedgie: {{wedgie| 18 -9 18 -56 -22 67 }}
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


{{Val list|legend=1| 63, 72, 495bcd }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Badness: 0.0937
Badness (Sintel): 1.68


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


== 11-limit ==
[[Comma list]]: 225/224, 5764801/5668704


Period: 1\9
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


Optimal ([[POTE]]) generator: ~5/4 = 383.146
[[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 }}


EDO generators: [[63edo|20\63]], [[72edo|23\72]], [[81edo|26\81]]
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Scales (Scala files):  
[[Badness]] (Sintel): 3.27
 
<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: 225/224, 385/384, 12005/11979
Comma list: 225/224, 385/384, 12250/11979


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


{{Val list|legend=1| 63, 72, 423cd, 495bcd }}
Optimal tunings:
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


Badness: 0.0304
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


</div></div>
Badness (Sintel): 2.43


== 13-limit ==
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


Period: 1\9
[[Subgroup]]: 2.3.5.7


Optimal ([[POTE]]) generator: ~5/4 = 383.047
[[Comma list]]: 225/224, 4194304/4117715


EDO generators: [[63edo|20\63]], [[72edo|23\72]], [[81edo|26\81]]
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


Scales (Scala files):  
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 21, 32, 53 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
[[Badness]] (Sintel): 3.57


Comma list: 169/168, 225/224, 364/363, 1716/1715
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
Comma list: 99/98, 176/175, 1310720/1294139


{{Val list|legend=1| 63, 72, 279cf }}
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


Badness: 0.0223
Optimal tunings:  
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


= Submajor =
Badness (Sintel): 2.82
== 5-limit ==


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


Optimal ([[POTE]]) generator: ~10125/8192 = 362.321
Comma list: 99/98, 176/175, 640/637, 847/845


EDO generators: [[43edo|13\43]], [[53edo|16\53]], [[63edo|19\63]], [[96edo|29\96]], [[116edo|35\116]]
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5
Badness (Sintel): 1.88


Comma list: 69198046875/68719476736
=== Musneb ===
 
Subgroup: 2.3.5.7.11
Mapping: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
 
{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
 
Badness: 0.1302
 
</div></div>
 
== 7-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~49/40 = 362.255
 
EDO generators: [[43edo|13\43]], [[53edo|16\53]], [[63edo|19\63]], [[96edo|29\96]], [[116edo|35\116]]
 
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
Comma list: 225/224, 385/384, 66550/64827


Comma list: 225/224, 51200/50421
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


Mapping: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


Wedgie: {{wedgie| 8 -11 -6 -36 -32 17 }}
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


{{Val list|legend=1| 10, 33, 43, 53 }}
Badness (Sintel): 2.89


Badness: 0.0605
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


</div></div>
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.


== 11-limit ==
[[Subgroup]]: 2.3.5.7


Period: 1\1
[[Comma list]]: 225/224, 125000000/121060821


Optimal ([[POTE]]) generator: ~27/22 = 362.101
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


EDO generators: [[53edo|16\53]], [[63edo|19\63]], [[116edo|35\116]]
[[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 51, 53 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 3.64
<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: 225/224, 385/384, 6655/6561
Comma list: 121/120, 225/224, 22000/21609


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


{{Val list|legend=1| 53, 116, 169de, 285cde }}
Optimal tunings:
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


Badness: 0.0506
{{Optimal ET sequence|legend=0| 51, 53 }}


</div></div>
Badness (Sintel): 2.46


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


Period: 1\1
Comma list: 121/120, 225/224, 275/273, 1040/1029
 
Optimal ([[POTE]]) generator: ~16/13 = 362.105
 
EDO generators: [[53edo|16\53]], [[63edo|19\63]], [[116edo|35\116]]
 
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


Comma list: 169/168, 225/224, 275/273, 385/384
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


{{Val list|legend=1| 53, 116, 169de, 285cdef }}
{{Optimal ET sequence|legend=0| 51f, 53 }}


Badness: 0.0277
Badness (Sintel): 1.96


</div></div>
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


== Interpental ==
[[Subgroup]]: 2.3.5.7


Period: 1\1
[[Comma list]]: 225/224, 823543/800000


Optimal ([[POTE]]) generator: ~49/40 = 362.418
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


EDO generators: [[43edo|13\43]], [[53edo|16\53]], [[96edo|29\96]]
[[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 3.67
<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: 99/98, 176/175, 51200/50421
Comma list: 225/224, 245/242, 1617/1600
 
POTE generator: ~49/40 = 362.418


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


{{Val list|legend=1| 43, 53, 96, 149d }}
Optimal tunings:
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


Badness: 0.0518
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


</div></div>
Badness (Sintel): 2.00


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


Period: 1\1
Comma list: 105/104, 196/195, 245/242, 1001/1000


Optimal ([[POTE]]) generator: ~16/13 = 362.402
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


EDO generators: [[43edo|13\43]], [[53edo|16\53]], [[96edo|29\96]]
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.43
<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: 99/98, 169/168, 176/175, 640/637
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


POTE generator: ~16/13 = 362.402
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


{{Val list|legend=1| 43, 53, 96, 149d }}
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Badness: 0.0297
Badness (Sintel): 1.26


</div></div>
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


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


Optimal ([[POTE]]) generator: ~48/35 = 556.221
[[Comma list]]: 225/224, 9805926501/9765625000


EDO generators: [[28edo|13\28]], [[41edo|19\41]]
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


Scales (Scala files):  
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 3.81


Comma list: 225/224, 5764801/5668704
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
Comma list: 225/224, 441/440, 43923/43750


Wedgie: {{wedgie| 16 -5 22 -45 -10 65 }}
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
Optimal tunings:
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Badness: 0.1293
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


</div></div>
Badness (Sintel): 1.74


== 11-limit ==
=== 13-limit ===
Period: 1\1
Subgroup: 2.3.5.7.11.13


Optimal ([[POTE]]) generator: ~11/8 = 556.144
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


EDO generators: [[28edo|13\28]], [[41edo|19\41]]
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Badness (Sintel): 1.35


Comma list: 225/224, 385/384, 12250/11979
=== 17-limit ===
 
Subgroup: 2.3.5.7.11.13.17
Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]


{{Val list|legend=1| 28d, 41, 315cde }}
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Badness: 0.0735
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


= Decic =
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
The ''decic'' temperament (10&amp;50, named by [[User:Xenllium|Xenllium]]) tempers out the [[cloudy comma]], 16807/16384 and the [[225/224|marvel comma]], 225/224 in the 7-limit. It can be extended naturally to the 11-, 13-, and 17-limit by adding 385/384, 105/104, and 170/169 to the comma list in this order.


== 7-limit ==
Badness (Sintel): 1.17
Period: 1\10


Optimal ([[POTE]]) generator: ~3/2 = 698.696
== Hendeca ==
{{Distinguish| Hendec }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''


EDO generators: [[50edo|1\50]], [[60edo|1\60]]
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.


Scales (Scala files):  
[[Subgroup]]: 2.3.5.7


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Comma list]]: 225/224, 122880/117649
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
: mapping generators: ~16/15, ~3


[[Comma list]]: 225/224, 16807/16384
[[Optimal tuning]]s:  
* [[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 }}


[[Mapping]]: [{{val| 10 0 39 28 }}, {{val| 0 1 -1 0 }}]
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}


{{Multival|legend=1| 10 -10 0 -39 -28 28 }}
[[Badness]] (Sintel): 4.37


{{Val list|legend=1| 10, 50, 60, 110d }}
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
[[Badness]]: 0.089135


</div></div>
Comma list: 121/120, 225/224, 352/343


== 11-limit ==
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }}
Period: 1\10


Optimal (POTE) generator: ~3/2 = 696.791
Optimal tunings:  
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}


EDO generators: [[50edo|1\50]], [[60edo|1\60]]
{{Optimal ET sequence|legend=0| 22, 55d }}


Scales (Scala files):  
Badness (Sintel): 2.46


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
== Gwazy ==
<div style="line-height:1.6;">Technical data</div>
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


<div class="mw-collapsible-content">
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7


Comma list: 225/224, 385/384, 3087/3025
[[Comma list]]: 225/224, 5971968/5764801


Mapping: [{{val| 10 0 39 28 3 }}, {{val| 0 1 -1 0 2 }}]
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


{{Val list|legend=1| 10, 50, 60 }}
[[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 }}


Badness: 0.063900
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


</div></div>
[[Badness]] (Sintel): 4.53


== 13-limit ==
=== 11-limit ===
Period: 1\10
Subgroup: 2.3.5.7.11


Optimal (POTE) generator: ~3/2 = 696.993
Comma list: 99/98, 176/175, 65536/65219


EDO generators: [[50edo|1\50]], [[60edo|1\60]]
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 22, 74, 96 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Badness (Sintel): 2.26


Comma list: 105/104, 144/143, 196/195, 2200/2197
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


Mapping: [{{val| 10 0 39 28 3 37 }}, {{val| 0 1 -1 0 2 0 }}]
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>.


{{Val list|legend=1| 10, 50, 60 }}
[[Subgroup]]: 2.3.5.7


Badness: 0.036880
[[Comma list]]: 225/224, 14495514624/13841287201


</div></div>
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


== 17-limit ==
[[Optimal tuning]]s:
Period: 1\10
* [[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 }}


Optimal (POTE) generator: ~3/2 = 697.085
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


EDO generators: [[50edo|1\50]], [[60edo|1\60]]
[[Badness]] (Sintel): 10.9


Scales (Scala files):
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
<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: 2.3.5.7.11.13.17
Comma list: 225/224, 3840/3773, 12005/11979


Comma list: 105/104, 144/143, 170/169, 196/195, 221/220
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


Mapping: [{{val| 10 0 39 28 3 37 25 }}, {{val| 0 1 -1 0 2 0 1 }}]
Optimal tunings:  
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}


{{Val list|legend=1| 10, 50, 60 }}
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


Badness: 0.025064
Badness (Sintel): 5.74


</div></div>
== 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"