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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:genewardsmith|genewardsmith]] and made on <tt>2012-04-26 23:39:05 UTC</tt>.<br>
: The original revision id was <tt>325865272</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 comma]] or [[septimal kleisma]]. These include negri, wizard, tritonic, septimin, slender, triton and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august, compton, sharp, escapade and mavila.
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]]
* [[Wizard]] (+118098/117649) → [[Stearnsmic clan #Wizard|Stearnsmic clan]]
* ''[[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]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]


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


=Negri=
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.
[[Comma]]s: 49/48, 225/224


[[POTE tuning|POTE temperament]]: 125.608
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.


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


==11-limit==  
== Tritonic ==
Commas: 45/44, 49/48, 56/55
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


[[POTE tuning|POTE generator]]: 126.474
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 2 2 3 4|, &lt;0 -4 3 -2 -5|]
[[Subgroup]]: 2.3.5.7
EDOs: 9, 10, 19
[[Badness]]: 0.0262


==13-limit==
[[Comma list]]: 225/224, 50421/50000
Commas: 45/44, 49/48, 56/55, 78/77


[[POTE tuning|POTE generator]]: 126.431
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


Map: [&lt;1 2 2 3 4 4|, &lt;0 -4 3 -2 -5 -3|]
[[Optimal tuning]]s:  
EDOs: 9, 10, 19
* [[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 }}


=Negril=
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
Commas: 49/48, 100/99, 225/224


POTE generator: ~15/14 = 124.767
[[Badness]] (Sintel): 1.20


Map: [&lt;1 2 2 3 2|, &lt;0 -4 3 -2 14|]
=== 11-limit ===
EDOs: 19, [[29edo|29]], [[48edo|48df]], [[77edo|77cdf]]
Subgroup: 2.3.5.7.11
Badness: 0.0387


==13-limit==
Comma list: 121/120, 225/224, 441/440
Commas: 49/48, 65/64, 91/90, 875/858


POTE generator: ~15/14 = 124.716
{{Mapping|legend=0| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


Map: [&lt;1 2 2 3 2 4|, &lt;0 -4 3 -2 14 -3|]
Optimal tunings:  
EDOs: 9, 10, 19, 29, 48, 77
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
Badness: 0.0244
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


=Negric=
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
Commas: 33/32, 49/48, 77/75


POTE generator: ~15/14 = 127.039
Badness (Sintel): 0.782


Map: [&lt;1 2 2 3 3|, &lt;0 -4 3 -2 4|]
==== 13-limit ====
EDOs: 9, 19, [[85edo|85]]
Subgroup: 2.3.5.7.11.13
Badness: 0.0306


==13-limit==
Comma list: 105/104, 121/120, 196/195, 275/273
Commas: 33/32, 49/48, 65/64, 91/90


POTE generator: ~15/14 = 127.039
{{Mapping|legend=0| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


Map: [&lt;1 2 2 3 3 4|, &lt;0 -4 3 -2 4 -3|]
Optimal tunings:  
EDOs: 9, 19, 85
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
Badness: 0.0202
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


=Negroni=
{{Optimal ET sequence|legend=0| 29, 31, 60e }}
Commas: 49/48, 55/54, 225/224


POTE generator: ~15/14 = 124.539
Badness (Sintel): 0.950


Map: [&lt;1 2 2 3 5|, &lt;0 -4 3 -2 -15|]
==== 17-limit ====
EDOs: 10, 19e, 29, 77cde
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0353


=Wizard=
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
Commas: 225/224, 118098/117649


[[POTE tuning|POTE temperament]]: 383.256
{{Mapping|legend=0| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


Map: [&lt;2 1 5 2|, &lt;0 6 -1 10|]
Optimal tunings:  
Wedgie: &lt;&lt;12 -2 20 -31 -2 52||
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
EDOs: [[22edo|22]], [[50edo|50]], [[72edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238c]], [[310edo|310c]], [[382edo|382c]]
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}
Badness: 0.0408


==11-limit==
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Commas: 225/224, 385/384, 4000/3993


[[POTE tuning|POTE temperament]]: 383.232
Badness (Sintel): 0.973


Map: [&lt;2 1 5 2 8|, &lt;0 6 -1 10 -3|]
==== 19-limit ====
EDOs: 22, 50, 72, 94, [[166edo|166]], [[238edo|238c]], [[310edo|310c]]
Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.0185


==Lizard==
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
Commas: 225/224, 351/350, 364/363, 385/384


[[POTE tuning|POTE temperament]]: 383.389
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
Optimal tunings:  
EDOs: 22, 50, 72, 122, 194df
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
Badness: 0.0218
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


===17-limit===
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Commas: 221/220, 273/272, 289/288, 351/350, 375/374


[[POTE tuning|POTE temperament]]: 383.381
Badness (Sintel): 1.03


Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
==== 23-limit ====
EDOs: 22, 50, 72, 122g, 194dfg
Subgroup: 2.3.5.7.11.13.17.19.23
Badness: 0.0145


===19-limit===
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


[[POTE tuning|POTE generator]]: ~5/4 = 383.477
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
Optimal tunings:  
EDOs: 22h, 50, 72, [[122edo|122g]], [[194edo|194dfg]]
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
Badness: 0.0157
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


==Gizzard==
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Commas: 225/224, 385/384, 325/324, 1573/1568


POTE generator: ~5/4 = 383.170
Badness (Sintel): 1.04


Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]
=== Tritoni ===
EDOs: [[22edo|22]], [[26edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238cf]]
Subgroup: 2.3.5.7.11
Badness: 0.0203


===17-limit===
Comma list: 225/224, 385/384, 27783/27500
Commas: 225/224, 289/288, 325/324, 375/374, 385/384


POTE generator: ~5/4 = 383.175
{{Mapping|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
Optimal tunings:  
EDOs: 22f, 72, 94, [[166edo|166g]], [[238edo|238cfg]]
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
Badness: 0.0136
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


===19-limit===
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399


POTE generator: ~5/4 = 383.138
Badness (Sintel): 1.50


Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]
== Septimin ==
EDOs: 22f, 72, 94, 166g
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''
Badness: 0.0148


==Mage==
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: 99/98, 176/175, 1331/1296


POTE generator: ~55/48 = 216.876
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
[[Comma list]]: 225/224, 84035/82944
EDOs: 22, 72e, 94e
Badness: 0.0578


=Triton=
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
Commas: 225/224, 1029/1000
: mapping generators: ~2, ~12/7


[[POTE tuning|POTE temperament]]: ~7/5 = 568.865
[[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 0 6 7|, &lt;0 3 -7 -8|]
{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}
Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
EDOs: 17d, 19, 78bd, 97bd
Badness: 0.0592


==11-limit==
[[Badness]] (Sintel): 1.38
Commas: 45/44, 56/55, 1029/1000


POTE generator: ~7/5 = 569.144
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 6 7 4|, &lt;0 3 -7 -8 -1|]
Comma list: 225/224, 245/242, 385/384
EDOs: 19, 59bde, 78bde, 97bde
Badness: 0.0457


=Tritonic=
{{Mapping|legend=0| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
Commas: 225/224, 50421/50000


[[POTE tuning|POTE temperament]]: 580.286
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 4 -3 -3|, &lt;0 -5 11 12|]
{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}
Wedgie: &lt;&lt;5 -11 -12 -29 -33 3||
EDOs: [[29edo|29]], [[31edo|31]], [[60edo|60]], [[91edo|91]], [[122edo|122]], 213bcd


==11-limit==
Badness (Sintel): 1.04
Commas: 121/120, 225/224, 441/440


[[POTE tuning|POTE generator]]: ~7/5 = 580.267
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
Comma list: 105/104, 144/143, 196/195, 245/242
EDOs: 29, 31, 60e
Badness: 0.0237


==13-limit==
{{Mapping|legend=0| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
Commas: 105/104, 121/120, 196/195, 275/273


POTE generator: ~7/5 = 580.108
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 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}
EDOs: 29, 31, 60e, 151cde
Badness: 0.0230


==Tritoni==
Badness (Sintel): 0.955
Commas: 225/224, 385/384, 27783/27500


POTE generator: ~7/5 = 580.389
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
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: 31, 91, 122
Badness: 0.0455


=Merman=
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.
Commas: 225/224, 2500000/2470629


POTE generator: ~7/5 = 585.585
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 5 -5 -5|, &lt;0 -7 15 16|]
[[Comma list]]: 225/224, 2500000/2470629
Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
EDOs: 41, 84, 125
Badness: 0.0551


==11-limit==
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
Commas: 225/224, 441/440, 1344/1331
: mapping generators: ~2, ~10/7


POTE generator: ~7/5 = 585.606
[[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 5 -5 -5 2|, &lt;0 -7 15 16 3]]
{{Optimal ET sequence|legend=1| 41, 84, 125 }}
EDOs: 41, 84, 125e
Badness: 0.0364


==13-limit==
[[Badness]] (Sintel): 1.39
Commas: 144/143, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 585.657
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
Comma list: 225/224, 441/440, 1344/1331
EDOs: 41, 84, 125e, 209ef, 293ef


=Septimin=
{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
Commas: 225/224, 84035/82944


[[POTE tuning|POTE temperament]]: 263.632
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 4 1 5|, &lt;0 -11 6 -10|]
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]


==11-limit==
Badness (Sintel): 1.20
Commas: 225/224, 385/384, 2401/2376


[[POTE tuning|POTE temperament]]: 263.634
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
Comma list: 144/143, 225/224, 364/363, 441/440
EDOs: 9, 32, 41, 91, 223cdef


==13-limit==
{{Mapping|legend=0| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
Commas: 105/104, 144/143, 196/195, 245/242


[[POTE tuning|POTE temperament]]: 263.700
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
EDOs: 9, 41, 91


=Slender=
Badness (Sintel): 1.14
Commas: 225/225, 589824/588245


[[POTE tuning|POTE temperament]]: ~49/48 = 38.413
=== Mermaid ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
Comma list: 225/224, 385/384, 532400/531441
Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
EDOs: 31, 94, 125
Badness: 0.0569


==11-limit==
{{Mapping|legend=0| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
Commas: 225/224, 385/384, 1331/1323


[[POTE tuning|POTE temperament]]: ~49/48 = 38.387
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 2 2 3 4|, &lt;0 -13 10 -6 -17|]
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}
EDOs: 31, 63, 94, 125
Badness: 25.342


==13-limit==
Badness (Sintel): 1.46
Commas: 225/224, 275/273, 385/384, 1331/1323


POTE generator: ~49/48 = 38.314
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
Comma list: 225/224, 325/324, 385/384, 10648/10647
EDOs: 31, 63, 94
Badness: 25.913


=Marvo=
{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
Commas: 225/224, 78125000/78121827


[[POTE tuning|POTE temperament]]: ~27/20 = 516.694
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 5 12 29|, &lt;0 -6 -17 -46|]
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
Wedgie: &lt;&lt;6, 17, 46, 13, 56 59||
EDOs: 65, 72


==11-limit==
Badness (Sintel): 1.47
Commas: 225/224, 243/242, 4000/3993


[[POTE tuning|POTE temperament]]: ~27/20 = 516.699
== 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 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
EDOs: 65, 72, 281, 497


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 243/242, 225/224, 351/350, 1625/1617


POTE generator: ~27/20 = 516.730
[[Comma list]]: 225/224, 589824/588245


Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
EDOs: 72, 137, 209, 281f, 490bcf
: mapping generators: ~2, ~49/48
Badness: 0.0269


=Marvolo=  
[[Optimal tuning]]s:
Commas: 225/224, 156250000/155649627
* [[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: ~21/20 = 83.348
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Map: [&lt;1 2 1 1|, &lt;0 -6 19 26|]
[[Badness]] (Sintel): 1.44
Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||
EDOs: 29, 43, 72, 619bcd, 691bcd
Badness: 0.0833


==11-limit==  
=== 11-limit ===
Commas: 225/224, 441/440, 4000/3993
Subgroup: 2.3.5.7.11


POTE generator: ~21/20 = 83.340
Comma list: 225/224, 385/384, 1331/1323


Map: [&lt;1 2 1 1 2|, &lt;0 -6 19 26 21|]
{{Mapping|legend=0| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
EDOs: 29, 43, 72
Badness: 0.0290


==13-limit==
Optimal tunings:
Commas: 169/168, 225/224, 364/363, 441/440
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


POTE generator: ~21/20 = 83.330
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
Badness (Sintel): 0.838
EDOs: 29, 43, 72
Badness: 0.0215


=Amavil=  
=== 13-limit ===
Commas: 225/224, 17496/16807
Subgroup: 2.3.5.7.11.13


POTE generator: ~48/35 = 529.979
Comma list: 225/224, 275/273, 385/384, 1331/1323


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
EDOs: 9, 34d, 43, 77d
Badness: 0.1096


==11-limit==
Optimal tunings:
Commas: 99/98, 176/175, 864/847
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


POTE generator: ~15/11 = 529.974
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
Badness (Sintel): 1.07
EDOs: 9, 34d, 43, 77de
Badness: 0.0426


==13-limit==  
== Triton ==
Commas: 78/77, 99/98, 144/143, 176/175
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


POTE generator: ~15/11 = 529.951
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.  


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
[[Subgroup]]: 2.3.5.7
EDOs: 9, 34d, 43, 77de
Badness: 0.0258


=Enneaportent=
[[Comma list]]: 225/224, 1029/1000
Commas: 225/224, 40353607/40310784


POTE generator: ~5/4 = 383.165
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
EDOs: 9, 54, 63, 72, 495bcd
: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
Badness: 0.0937
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}
: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}


==11-limit==
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
Commas: 225/224, 385/384, 12005/11979


POTE generator: ~5/4 = 383.146
[[Badness]] (Sintel): 1.50


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
=== 11-limit ===
EDOs: 9, 54, 63, 72, 423cd, 495bcd
Subgroup: 2.3.5.7.11
Badness: 0.0304


==13-limit==
Comma list: 45/44, 56/55, 1029/1000
Commas: 169/168 225/224 364/363 1716/1715


POTE generator: ~5/4 = 383.047
{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
Optimal tunings:
EDOs: 9, 54, 63, 72, 279cf
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
Badness: 0.0223</pre></div>
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}
<h4>Original HTML 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;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:88:&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:88 --&gt;&lt;!-- ws:start:WikiTextTocRule:89: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Negri"&gt;Negri&lt;/a&gt;&lt;/div&gt;
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
&lt;!-- ws:end:WikiTextTocRule:89 --&gt;&lt
 
Badness (Sintel): 1.51
 
== Marvolo ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 156250000/155649627
 
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}
 
[[Badness]] (Sintel): 2.11
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 4000/3993
 
{{Mapping|legend=0| 1 2 1 1 2 | 0 -6 19 26 21 }}
 
Optimal tunings:
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}
 
{{Optimal ET sequence|legend=0| 29, 43, 72 }}
 
Badness (Sintel): 0.958
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 364/363, 441/440
 
{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
 
Optimal tunings:
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}
 
{{Optimal ET sequence|legend=0| 29, 43, 72 }}
 
Badness (Sintel): 0.887
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
 
{{Mapping|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
 
Optimal tunings:
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
 
Badness (Sintel): 0.760
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
 
 
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"