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Marvel temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This page discusses miscellaneous [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] [[225/224]], the marvel comma or septimal kleisma.  
: This revision was by author [[User:x31eq|x31eq]] and made on <tt>2013-01-26 07:22:39 UTC</tt>.<br>
: The original revision id was <tt>401591066</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc]]
This page discusses some of the temperaments tempering out |-5 2 2 -1&gt; =[[225_224| 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|meantone]], [[Miracle|miracle]], [[Magic family|magic]], [[Pajara|pajara]], [[Orwell|orwell]], [[Kleismic family|catakleismic]], [[Garibaldi temperament|garibaldi]], [[Augmented family|august]], [[Pythagorean family|compton]], sharp [where?], [[Escapade family|escapade]] and [[Pelogic family|mavila]].


Since (5/4)^2 = 225/224 * 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess 5/4-9/7 versions of the augmented triad. 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]]
* ''[[Quintapole]]'' (+7812500/7411887) → [[Quintaleap family #Quintapole|Quintaleap 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]]
* ''[[Decic]]'' (+16807/16384) → [[Cloudy clan #Decic|Cloudy clan]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Hendeca]]'' (+122880/117649) → [[11th-octave temperaments #Hendeca|11th-octave temperaments]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]


=Negri=
Considered below are wizard, tritonic, septimin, merman, slender, triton, marvolo, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, gwazy, and tertiosec, in the order of increasing [[badness]].
[[Comma]]s: 49/48, 225/224


[[POTE tuning|POTE temperament]]: ~15/14 = 125.608
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.


[[Map]]: [&lt;1 2 2 3|, &lt;0 -4 3 -2|]
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.
[[Wedgie]]: &lt;&lt;4 -3 2 -14 -8 13||
EDOs: [[9edo|9]], [[10edo|10]], [[19edo|19]], [[86edo|86]]


==11-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}}.)
Commas: 45/44, 49/48, 56/55


[[POTE tuning|POTE generator]]: ~15/14 = 126.474
== Wizard ==
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''


Map: [&lt;1 2 2 3 4|, &lt;0 -4 3 -2 -5|]
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.  
EDOs: 9, 10, 19
[[Badness]]: 0.0262


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 45/44, 49/48, 56/55, 78/77


[[POTE tuning|POTE generator]]: ~14/13 = 126.431
[[Comma list]]: 225/224, 118098/117649


Map: [&lt;1 2 2 3 4 4|, &lt;0 -4 3 -2 -5 -3|]
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
EDOs: 9, 10, 19
: mapping generators: ~1225/864, ~245/216


=Negril=  
[[Optimal tuning]]s:
Commas: 49/48, 100/99, 225/224
* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}
: error map: {{val| 0.000 -1.169 -3.111 -0.849 }}


POTE generator: ~15/14 = 124.767
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}


Map: [&lt;1 2 2 3 2|, &lt;0 -4 3 -2 14|]
[[Badness]] (Sintel): 1.03
EDOs: 19, [[29edo|29]], [[48edo|48df]], [[77edo|77cdf]]
Badness: 0.0387


==13-limit==  
=== 11-limit ===
Commas: 49/48, 65/64, 91/90, 875/858
Subgroup: 2.3.5.7.11


POTE generator: ~15/14 = 124.716
Comma list: 225/224, 385/384, 4000/3993


Map: [&lt;1 2 2 3 2 4|, &lt;0 -4 3 -2 14 -3|]
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}
EDOs: 9, 10, 19, 29, 48, 77
Badness: 0.0244


=Negric=  
Optimal tunings:
Commas: 33/32, 49/48, 77/75
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


POTE generator: ~15/14 = 127.039
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


Map: [&lt;1 2 2 3 3|, &lt;0 -4 3 -2 4|]
Badness (Sintel): 0.613
EDOs: 9, 19, [[85edo|85]]
Badness: 0.0306


==13-limit==  
==== Lizard ====
Commas: 33/32, 49/48, 65/64, 91/90
Subgroup: 2.3.5.7.11.13


POTE generator: ~15/14 = 127.039
Comma list: 225/224, 351/350, 364/363, 385/384


Map: [&lt;1 2 2 3 3 4|, &lt;0 -4 3 -2 4 -3|]
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
EDOs: 9, 19, 85
Badness: 0.0202


=Negroni=  
Optimal tunings:
Commas: 49/48, 55/54, 225/224
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


POTE generator: ~15/14 = 124.539
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


Map: [&lt;1 2 2 3 5|, &lt;0 -4 3 -2 -15|]
Badness (Sintel): 0.900
EDOs: 10, 19e, 29, 77cde
Badness: 0.0353


==13-limit==  
===== 17-limit =====
Commas: 49/48, 55/54, 65/64, 91/90
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~15/14 = 124.545
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


Map: [&lt;1 2 2 3 5 4|, &lt;0 -4 3 -2 -15 -3|]
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
EDOs: 10, 19e, 29, 77cdef
Badness: 0.0216


=Wizard=  
Optimal tunings:
Comma: 2197265625/2147483648
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


POTE generator: ~5/4 = 383.212
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


Map: [&lt;2 1 5|, &lt;0 6 -1|]
Badness (Sintel): 0.741
EDOs: 22, 50, 72, 166, 238c, 310c, 548bc
Badness: 0.3864


==7-limit==  
===== 19-limit =====
Commas: 225/224, 118098/117649
Subgroup: 2.3.5.7.11.13.17.19


[[POTE tuning|POTE temperament]]: 383.256
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


Map: [&lt;2 1 5 2|, &lt;0 6 -1 10|]
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
Wedgie: &lt;&lt;12 -2 20 -31 -2 52||
EDOs: [[22edo|22]], [[50edo|50]], [[72edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238c]], [[310edo|310c]], [[382edo|382c]]
Badness: 0.0408


==11-limit==
Optimal tunings:
Commas: 225/224, 385/384, 4000/3993
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


[[POTE tuning|POTE temperament]]: 383.232
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


Map: [&lt;2 1 5 2 8|, &lt;0 6 -1 10 -3|]
Badness (Sintel): 0.955
EDOs: 22, 50, 72, 94, [[166edo|166]], [[238edo|238c]], [[310edo|310c]]
Badness: 0.0185


==Lizard==  
==== Gizzard ====
Commas: 225/224, 351/350, 364/363, 385/384
Subgroup: 2.3.5.7.11.13


[[POTE tuning|POTE temperament]]: 383.389
Comma list: 225/224, 325/324, 385/384, 1573/1568


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
EDOs: 22, 50, 72, 122, 194df
Badness: 0.0218


===17-limit===
Optimal tunings:
Commas: 221/220, 273/272, 289/288, 351/350, 375/374
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}


[[POTE tuning|POTE temperament]]: 383.381
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}


Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
Badness (Sintel): 0.837
EDOs: 22, 50, 72, 122g, 194dfg
Badness: 0.0145


===19-limit===  
===== 17-limit =====
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Subgroup: 2.3.5.7.11.13.17


[[POTE tuning|POTE generator]]: ~5/4 = 383.477
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
EDOs: 22h, 50, 72, [[122edo|122g]], [[194edo|194dfg]]
Badness: 0.0157


==Gizzard==
Optimal tunings:
Commas: 225/224, 385/384, 325/324, 1573/1568
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}


POTE generator: ~5/4 = 383.170
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}


Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]
Badness (Sintel): 0.694
EDOs: [[22edo|22]], [[26edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238cf]]
Badness: 0.0203


===17-limit===  
===== 19-limit =====
Commas: 225/224, 289/288, 325/324, 375/374, 385/384
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~5/4 = 383.175
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594


Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
EDOs: 22f, 72, 94, [[166edo|166g]], [[238edo|238cfg]]
Badness: 0.0136


===19-limit===
Optimal tunings:
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


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


Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]
Badness (Sintel): 0.901
EDOs: 22f, 72, 94, 166g
Badness: 0.0148


==Mage==  
=== Mage ===
Commas: 99/98, 176/175, 1331/1296
Subgroup: 2.3.5.7.11


POTE generator: ~55/48 = 216.876
Comma list: 99/98, 176/175, 1331/1296


Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
EDOs: 22, 72e, 94e
Badness: 0.0578


=Triton=  
Optimal tunings:
Commas: 225/224, 1029/1000
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


[[POTE tuning|POTE temperament]]: ~7/5 = 568.865
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
Badness (Sintel): 1.91
Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
EDOs: 17d, 19, 78bd, 97bd
Badness: 0.0592


==11-limit==  
== Tritonic ==
Commas: 45/44, 56/55, 1029/1000
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


POTE generator: ~7/5 = 569.144
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.  


Map: [&lt;1 0 6 7 4|, &lt;0 3 -7 -8 -1|]
[[Subgroup]]: 2.3.5.7
EDOs: 19, 59bde, 78bde, 97bde
Badness: 0.0457


=Tritonic=
[[Comma list]]: 225/224, 50421/50000
Comnma: 553584375/536870912


POTE generator: ~45/32 = 580.219
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


Map: [&lt;1 4 -3|, &lt;0 -5 11|]
[[Optimal tuning]]s:
EDOs: 29, 31, 60, 91
* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
Badness: 0.3785
: [[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 }}


==7-limit==
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
Commas: 225/224, 50421/50000


[[POTE tuning|POTE temperament]]: 580.286
[[Badness]] (Sintel): 1.20


Map: [&lt;1 4 -3 -3|, &lt;0 -5 11 12|]
=== 11-limit ===
Wedgie: &lt;&lt;5 -11 -12 -29 -33 3||
Subgroup: 2.3.5.7.11
EDOs: [[29edo|29]], [[31edo|31]], [[60edo|60]], [[91edo|91]], [[122edo|122]], 213bcd


==11-limit==
Comma list: 121/120, 225/224, 441/440
Commas: 121/120, 225/224, 441/440


[[POTE tuning|POTE generator]]: ~7/5 = 580.267
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
Optimal tunings:  
EDOs: 29, 31, 60e
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
Badness: 0.0237
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
Commas: 105/104, 121/120, 196/195, 275/273


POTE generator: ~7/5 = 580.108
Badness (Sintel): 0.782


Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
==== 13-limit ====
EDOs: 29, 31, 60e, 151cde
Subgroup: 2.3.5.7.11.13
Badness: 0.0230


==Tritoni==
Comma list: 105/104, 121/120, 196/195, 275/273
Commas: 225/224, 385/384, 27783/27500


POTE generator: ~7/5 = 580.389
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
Optimal tunings:  
EDOs: 31, 91, 122
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
Badness: 0.0455
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


=Merman=
{{Optimal ET sequence|legend=0| 29, 31, 60e }}
Comma: 1121008359375/1099511627776


POTE generator: ~45/32 = 585.634
Badness (Sintel): 0.950


Map: [&lt;1 5 -5|, &lt;0 -7 15|]
==== 17-limit ====
EDOs: 41, 84, 125, 209, 334c, 543bc
Subgroup: 2.3.5.7.11.13.17
Badness: 0.4538


==7-limit==
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
Commas: 225/224, 2500000/2470629


POTE generator: ~7/5 = 585.585
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


Map: [&lt;1 5 -5 -5|, &lt;0 -7 15 16|]
Optimal tunings:  
Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
EDOs: 41, 84, 125
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}
Badness: 0.0551


==11-limit==
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Commas: 225/224, 441/440, 1344/1331


POTE generator: ~7/5 = 585.606
Badness (Sintel): 0.973


Map: [&lt;1 5 -5 -5 2|, &lt;0 -7 15 16 3]]
==== 19-limit ====
EDOs: 41, 84, 125e
Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.0364


==13-limit==
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
Commas: 144/143, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 585.657
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


Map: [&lt;1 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
Optimal tunings:  
EDOs: 41, 84, 125e, 209ef, 293ef
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
Badness: 0.0275
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


=Septimin=
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Comma: 35595703125/34359738368


POTE generator: ~75/64 = 263.658
Badness (Sintel): 1.03


Map: [&lt;1 4 1|, &lt;0 -11 6|]
==== 23-limit ====
EDOs: 9, 32, 41, 91, 132, 223c, 355bc
Subgroup: 2.3.5.7.11.13.17.19.23
Badness: 0.6709


==7-limit==
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
Commas: 225/224, 84035/82944


[[POTE tuning|POTE temperament]]: 263.632
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


Map: [&lt;1 4 1 5|, &lt;0 -11 6 -10|]
Optimal tunings:  
Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Commas: 225/224, 385/384, 2401/2376


[[POTE tuning|POTE temperament]]: 263.634
Badness (Sintel): 1.04


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
=== Tritoni ===
EDOs: 9, 32, 41, 91, 223cdef
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 225/224, 385/384, 27783/27500
Commas: 105/104, 144/143, 196/195, 245/242


[[POTE tuning|POTE temperament]]: 263.700
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
Optimal tunings:  
EDOs: 9, 41, 91
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


=Slender=
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
Commas: 225/225, 589824/588245


[[POTE tuning|POTE temperament]]: ~49/48 = 38.413
Badness (Sintel): 1.50


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
== Septimin ==
Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''
EDOs: 31, 94, 125
Badness: 0.0569


==11-limit==
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.
Commas: 225/224, 385/384, 1331/1323


[[POTE tuning|POTE temperament]]: ~49/48 = 38.387
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
[[Comma list]]: 225/224, 84035/82944
EDOs: 31, 63, 94, 125
Badness: 25.342


==13-limit==
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
Commas: 225/224, 275/273, 385/384, 1331/1323
: mapping generators: ~2, ~12/7


POTE generator: ~49/48 = 38.314
[[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 }}


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
{{Optimal ET sequence|legend=1| 41, 91, 132d }}
EDOs: 31, 63, 94
Badness: 25.913


=Marvo=
[[Badness]] (Sintel): 1.38
Commas: 225/224, 78125000/78121827


[[POTE tuning|POTE temperament]]: ~27/20 = 516.694
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
Comma list: 225/224, 245/242, 385/384
Wedgie: &lt;&lt;6, 17, 46, 13, 56 59||
EDOs: 65, 72


==11-limit==
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
Commas: 225/224, 243/242, 4000/3993


[[POTE tuning|POTE temperament]]: ~27/20 = 516.699
Optimal tunings:  
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}
EDOs: 65, 72, 281, 497


==13-limit==
Badness (Sintel): 1.04
Commas: 243/242, 225/224, 351/350, 1625/1617


POTE generator: ~27/20 = 516.730
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
Comma list: 105/104, 144/143, 196/195, 245/242
EDOs: 72, 137, 209, 281f, 490bcf
Badness: 0.0269


=Marvolo=
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
Commas: 225/224, 156250000/155649627


POTE generator: ~21/20 = 83.348
Optimal tunings:  
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


Map: [&lt;1 2 1 1|, &lt;0 -6 19 26|]
{{Optimal ET sequence|legend=0| 41, 91 }}
Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||
EDOs: 29, 43, 72, 619bcd, 691bcd
Badness: 0.0833


==11-limit==
Badness (Sintel): 0.955
Commas: 225/224, 441/440, 4000/3993


POTE generator: ~21/20 = 83.340
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Map: [&lt;1 2 1 1 2|, &lt;0 -6 19 26 21|]
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.  
EDOs: 29, 43, 72
Badness: 0.0290


==13-limit==
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.
Commas: 169/168, 225/224, 364/363, 441/440


POTE generator: ~21/20 = 83.330
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
[[Comma list]]: 225/224, 2500000/2470629
EDOs: 29, 43, 72
Badness: 0.0215


=Amavil=
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
Commas: 225/224, 17496/16807
: mapping generators: ~2, ~10/7


POTE generator: ~48/35 = 529.979
[[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 }}


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
{{Optimal ET sequence|legend=1| 41, 84, 125 }}
Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
EDOs: 9, 34d, 43, 77d
Badness: 0.1096


==11-limit==
[[Badness]] (Sintel): 1.39
Commas: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
Comma list: 225/224, 441/440, 1344/1331
EDOs: 9, 34d, 43, 77de
Badness: 0.0426


==13-limit==
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
Commas: 78/77, 99/98, 144/143, 176/175


POTE generator: ~15/11 = 529.951
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
EDOs: 9, 34d, 43, 77de
Badness: 0.0258


=Enneaportent=
Badness (Sintel): 1.20
Commas: 225/224, 40353607/40310784


POTE generator: ~5/4 = 383.165
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
Comma list: 144/143, 225/224, 364/363, 441/440
Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
EDOs: 9, 54, 63, 72, 495bcd
Badness: 0.0937


==11-limit==
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
Commas: 225/224, 385/384, 12005/11979


POTE generator: ~5/4 = 383.146
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
EDOs: 9, 54, 63, 72, 423cd, 495bcd
Badness: 0.0304


==13-limit==
Badness (Sintel): 1.14
Commas: 169/168 225/224 364/363 1716/1715


POTE generator: ~5/4 = 383.047
=== Mermaid ===
Subgroup: 2.3.5.7.11


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
Comma list: 225/224, 385/384, 532400/531441
EDOs: 9, 54, 63, 72, 279cf
Badness: 0.0223


=Submajor=
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
Comma: 69198046875/68719476736


POTE generator: ~10125/8192 = 362.321
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


Map: [&lt;1 4 -1|, &lt;0 -8 11|]
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}
EDOs: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc
Badness: 0.1302


==7-limit==
Badness (Sintel): 1.46
Commas: 225/224, 51200/50421


POTE generator: ~49/40 = 362.255
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 4 -1 1|, &lt;0 -8 11 6|]
Comma list: 225/224, 325/324, 385/384, 10648/10647
Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
EDOs: 10, 33, 43, 53
Badness: 0.0605


==11-limit==
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
Commas: 225/224, 385/384, 6655/6561


POTE generator: ~27/22 = 362.101
Optimal tunings:  
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


Map: [&lt;1 4 -1 1 11|, &lt;0 -8 11 6 -25|]
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
EDOs: 10, 53, 116, 169de, 285cde
Badness: 0.0506


==13-limit==
Badness (Sintel): 1.47
Commas: 169/168, 225/224, 275/273, 385/384


POTE generator: ~16/13 = 362.105
== 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.  


Map: [&lt;1 4 -1 1 11 4|, &lt;0 -8 11 6 -25 -1|]
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  
EDOs: 10, 53, 116, 169de, 285cdef
Badness: 0.0277


==Interpental==
[[Subgroup]]: 2.3.5.7
Commas: 99/98, 176/175, 51200/50421


POTE generator: ~49/40 = 362.418
[[Comma list]]: 225/224, 589824/588245


Map: [&lt;1 4 -1 1 -5|, &lt;0 -8 11 6 28|]
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
EDOs: 43, 53, 96, 149d
: mapping generators: ~2, ~49/48
Badness: 0.0518


===13-limit===  
[[Optimal tuning]]s:
Commas: 99/98, 169/168, 176/175, 640/637
* [[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 }}


POTE generator: ~16/13 = 362.402
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Map: [&lt;1 4 -1 1 -5 4|, &lt;0 -8 11 6 28 -1|]
[[Badness]] (Sintel): 1.44
EDOs: 43, 53, 96, 149d
Badness: 0.0297


=Alphorn=  
=== 11-limit ===
Commas: 225/224, 5764801/5668704
Subgroup: 2.3.5.7.11


POTE generator: ~48/35 = 556.221
Comma list: 225/224, 385/384, 1331/1323


Map: [&lt;1 9 0 13|, &lt;0 -16 5 -22|]
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
Wedgie: &lt;&lt;16 -5 22 -45 -10 65||
EDOs: 28d, 41, 151cd, 192cd, 233cd
Badness: 0.1293


==11-limit==
Optimal tunings:
Commas: 225/224, 385/384, 12250/11979
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


POTE generator: ~11/8 = 556.144
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Map: [&lt;1 9 0 13 3|, &lt;0 -16 5 -22 1|]
Badness (Sintel): 0.838
EDOs: 41, 315cde
 
Badness: 0.0735</pre></div>
=== 13-limit ===
<h4>Original HTML content:</h4>
Subgroup: 2.3.5.7.11.13
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Marvel temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:114:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Negri"&gt;Negri&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:115 --&gt;&lt;!-- ws:start:WikiTextTocRule:116: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Negri-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
Comma list: 225/224, 275/273, 385/384, 1331/1323
&lt;!-- ws:end:WikiTextTocRule:116 --&gt;&lt;!-- ws:start:WikiTextTocRule:117: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Negri-13-limit"
 
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
 
Optimal tunings:
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}
 
{{Optimal ET sequence|legend=0| 31, 63, 94 }}
 
Badness (Sintel): 1.07
 
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''
 
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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 1029/1000
 
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
 
[[Badness]] (Sintel): 1.50
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 56/55, 1029/1000
 
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
 
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}
 
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
 
Badness (Sintel): 1.51
 
== Marvolo ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 156250000/155649627
 
{
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