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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>2012-05-20 19:21:43 UTC</tt>.<br>
: The original revision id was <tt>337659208</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, merman 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]]
* ''[[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]]


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 wizard, tritonic, septimin, merman, slender, triton, marvolo, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, 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==  
== Wizard ==
Commas: 45/44, 49/48, 56/55
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''


[[POTE tuning|POTE generator]]: 126.474
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.  


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, 118098/117649
Commas: 45/44, 49/48, 56/55, 78/77


[[POTE tuning|POTE generator]]: 126.431
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216


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


=Negril=
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}
Commas: 49/48, 100/99, 225/224


POTE generator: ~15/14 = 124.767
[[Badness]] (Sintel): 1.03


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: 225/224, 385/384, 4000/3993
Commas: 49/48, 65/64, 91/90, 875/858


POTE generator: ~15/14 = 124.716
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -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: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
Badness: 0.0244
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


=Negric=
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
Commas: 33/32, 49/48, 77/75


POTE generator: ~15/14 = 127.039
Badness (Sintel): 0.613


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


==13-limit==
Comma list: 225/224, 351/350, 364/363, 385/384
Commas: 33/32, 49/48, 65/64, 91/90


POTE generator: ~15/14 = 127.039
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}


Map: [&lt;1 2 2 3 3 4|, &lt;0 -4 3 -2 4 -3|]
Optimal tunings:  
EDOs: 9, 19, 85
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
Badness: 0.0202
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


=Negroni=
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
Commas: 49/48, 55/54, 225/224


POTE generator: ~15/14 = 124.539
Badness (Sintel): 0.900


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: 221/220, 273/272, 289/288, 351/350, 375/374
Comma: 2197265625/2147483648


POTE generator: ~5/4 = 383.212
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}


Map: [&lt;2 1 5|, &lt;0 6 -1|]
Optimal tunings:  
EDOs: 22, 50, 72, 166, 238c, 310c, 548bc
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
Badness: 0.3864
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


==7-limit==
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
Commas: 225/224, 118098/117649


[[POTE tuning|POTE temperament]]: 383.256
Badness (Sintel): 0.741


Map: [&lt;2 1 5 2|, &lt;0 6 -1 10|]
===== 19-limit =====
Wedgie: &lt;&lt;12 -2 20 -31 -2 52||
Subgroup: 2.3.5.7.11.13.17.19
EDOs: [[22edo|22]], [[50edo|50]], [[72edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238c]], [[310edo|310c]], [[382edo|382c]]
Badness: 0.0408


==11-limit==
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Commas: 225/224, 385/384, 4000/3993


[[POTE tuning|POTE temperament]]: 383.232
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}


Map: [&lt;2 1 5 2 8|, &lt;0 6 -1 10 -3|]
Optimal tunings:  
EDOs: 22, 50, 72, 94, [[166edo|166]], [[238edo|238c]], [[310edo|310c]]
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
Badness: 0.0185
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


==Lizard==
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
Commas: 225/224, 351/350, 364/363, 385/384


[[POTE tuning|POTE temperament]]: 383.389
Badness (Sintel): 0.955


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
==== Gizzard ====
EDOs: 22, 50, 72, 122, 194df
Subgroup: 2.3.5.7.11.13
Badness: 0.0218


===17-limit===
Comma list: 225/224, 325/324, 385/384, 1573/1568
Commas: 221/220, 273/272, 289/288, 351/350, 375/374


[[POTE tuning|POTE temperament]]: 383.381
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}


Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
Optimal tunings:  
EDOs: 22, 50, 72, 122g, 194dfg
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
Badness: 0.0145
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}


===19-limit===
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


[[POTE tuning|POTE generator]]: ~5/4 = 383.477
Badness (Sintel): 0.837


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
===== 17-limit =====
EDOs: 22h, 50, 72, [[122edo|122g]], [[194edo|194dfg]]
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0157


==Gizzard==
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Commas: 225/224, 385/384, 325/324, 1573/1568


POTE generator: ~5/4 = 383.170
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}


Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]
Optimal tunings:  
EDOs: [[22edo|22]], [[26edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238cf]]
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
Badness: 0.0203
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}


===17-limit===
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
Commas: 225/224, 289/288, 325/324, 375/374, 385/384


POTE generator: ~5/4 = 383.175
Badness (Sintel): 0.694


Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
===== 19-limit =====
EDOs: 22f, 72, 94, [[166edo|166g]], [[238edo|238cfg]]
Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.0136


===19-limit===
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399


POTE generator: ~5/4 = 383.138
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}


Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]
Optimal tunings:  
EDOs: 22f, 72, 94, 166g
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
Badness: 0.0148
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


==Mage==
{{Optimal ET sequence|legend=0| 72, 94, 166g }}
Commas: 99/98, 176/175, 1331/1296


POTE generator: ~55/48 = 216.876
Badness (Sintel): 0.901


Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
=== Mage ===
EDOs: 22, 72e, 94e
Subgroup: 2.3.5.7.11
Badness: 0.0578


=Triton=
Comma list: 99/98, 176/175, 1331/1296
Commas: 225/224, 1029/1000


[[POTE tuning|POTE temperament]]: ~7/5 = 568.865
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
Optimal tunings:  
Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
EDOs: 17d, 19, 78bd, 97bd
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}
Badness: 0.0592


==11-limit==
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
Commas: 45/44, 56/55, 1029/1000


POTE generator: ~7/5 = 569.144
Badness (Sintel): 1.91


Map: [&lt;1 0 6 7 4|, &lt;0 3 -7 -8 -1|]
== Tritonic ==
EDOs: 19, 59bde, 78bde, 97bde
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''
Badness: 0.0457


=Tritonic=
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.
Comnma: 553584375/536870912


POTE generator: ~45/32 = 580.219
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 4 -3|, &lt;0 -5 11|]
[[Comma list]]: 225/224, 50421/50000
EDOs: 29, 31, 60, 91
Badness: 0.3785


==7-limit==
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
Commas: 225/224, 50421/50000
: mapping generators: ~2, ~10/7


[[POTE tuning|POTE temperament]]: 580.286
[[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 }}


Map: [&lt;1 4 -3 -3|, &lt;0 -5 11 12|]
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
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): 1.20
Commas: 121/120, 225/224, 441/440


[[POTE tuning|POTE generator]]: ~7/5 = 580.267
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
Comma list: 121/120, 225/224, 441/440
EDOs: 29, 31, 60e
Badness: 0.0237


==13-limit==
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}
Commas: 105/104, 121/120, 196/195, 275/273


POTE generator: ~7/5 = 580.108
Optimal tunings:  
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
EDOs: 29, 31, 60e, 151cde
Badness: 0.0230


==Tritoni==
Badness (Sintel): 0.782
Commas: 225/224, 385/384, 27783/27500


POTE generator: ~7/5 = 580.389
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
Comma list: 105/104, 121/120, 196/195, 275/273
EDOs: 31, 91, 122
Badness: 0.0455


=Merman=
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}
Comma: 1121008359375/1099511627776


POTE generator: ~45/32 = 585.634
Optimal tunings:  
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


Map: [&lt;1 5 -5|, &lt;0 -7 15|]
{{Optimal ET sequence|legend=0| 29, 31, 60e }}
EDOs: 41, 84, 125, 209, 334c, 543bc
Badness: 0.4538


==7-limit==
Badness (Sintel): 0.950
Commas: 225/224, 2500000/2470629


POTE generator: ~7/5 = 585.585
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 5 -5 -5|, &lt;0 -7 15 16|]
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
EDOs: 41, 84, 125
Badness: 0.0551


==11-limit==
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
Commas: 225/224, 441/440, 1344/1331


POTE generator: ~7/5 = 585.606
Optimal tunings:  
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


Map: [&lt;1 5 -5 -5 2|, &lt;0 -7 15 16 3]]
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
EDOs: 41, 84, 125e
Badness: 0.0364


==13-limit==
Badness (Sintel): 0.973
Commas: 144/143, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 585.657
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
EDOs: 41, 84, 125e, 209ef, 293ef


=Septimin=
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
Comma: 35595703125/34359738368


POTE generator: ~75/64 = 263.658
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


Map: [&lt;1 4 1|, &lt;0 -11 6|]
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
EDOs: 9, 32, 41, 91, 132, 223c, 355bc
Badness: 0.6709


==7-limit==
Badness (Sintel): 1.03
Commas: 225/224, 84035/82944


[[POTE tuning|POTE temperament]]: 263.632
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Map: [&lt;1 4 1 5|, &lt;0 -11 6 -10|]
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]


==11-limit==
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}
Commas: 225/224, 385/384, 2401/2376


[[POTE tuning|POTE temperament]]: 263.634
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
EDOs: 9, 32, 41, 91, 223cdef


==13-limit==
Badness (Sintel): 1.04
Commas: 105/104, 144/143, 196/195, 245/242


[[POTE tuning|POTE temperament]]: 263.700
=== Tritoni ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
Comma list: 225/224, 385/384, 27783/27500
EDOs: 9, 41, 91


=Slender=
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}
Commas: 225/225, 589824/588245


[[POTE tuning|POTE temperament]]: ~49/48 = 38.413
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
EDOs: 31, 94, 125
Badness: 0.0569


==11-limit==
Badness (Sintel): 1.50
Commas: 225/224, 385/384, 1331/1323


[[POTE tuning|POTE temperament]]: ~49/48 = 38.387
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
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.  
EDOs: 31, 63, 94, 125
Badness: 25.342


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 225/224, 275/273, 385/384, 1331/1323


POTE generator: ~49/48 = 38.314
[[Comma list]]: 225/224, 84035/82944


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
EDOs: 31, 63, 94
: mapping generators: ~2, ~12/7
Badness: 25.913


=Marvo=  
[[Optimal tuning]]s:
Commas: 225/224, 78125000/78121827
* [[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 }}


[[POTE tuning|POTE temperament]]: ~27/20 = 516.694
{{Optimal ET sequence|legend=1| 41, 91, 132d }}


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
[[Badness]] (Sintel): 1.38
Wedgie: &lt;&lt;6, 17, 46, 13, 56 59||
EDOs: 65, 72


==11-limit==  
=== 11-limit ===
Commas: 225/224, 243/242, 4000/3993
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE temperament]]: ~27/20 = 516.699
Comma list: 225/224, 245/242, 385/384


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
EDOs: 65, 72, 281, 497


==13-limit==
Optimal tunings:
Commas: 243/242, 225/224, 351/350, 1625/1617
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


POTE generator: ~27/20 = 516.730
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}


Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
Badness (Sintel): 1.04
EDOs: 72, 137, 209, 281f, 490bcf
Badness: 0.0269


=Marvolo=  
=== 13-limit ===
Commas: 225/224, 156250000/155649627
Subgroup: 2.3.5.7.11.13


POTE generator: ~21/20 = 83.348
Comma list: 105/104, 144/143, 196/195, 245/242


Map: [&lt;1 2 1 1|, &lt;0 -6 19 26|]
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||
EDOs: 29, 43, 72, 619bcd, 691bcd
Badness: 0.0833


==11-limit==
Optimal tunings:
Commas: 225/224, 441/440, 4000/3993
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


POTE generator: ~21/20 = 83.340
{{Optimal ET sequence|legend=0| 41, 91 }}


Map: [&lt;1 2 1 1 2|, &lt;0 -6 19 26 21|]
Badness (Sintel): 0.955
EDOs: 29, 43, 72
Badness: 0.0290


==13-limit==  
== Merman ==
Commas: 169/168, 225/224, 364/363, 441/440
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


POTE generator: ~21/20 = 83.330
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.  


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.  
EDOs: 29, 43, 72
Badness: 0.0215


=Amavil=
[[Subgroup]]: 2.3.5.7
Commas: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
[[Comma list]]: 225/224, 2500000/2470629


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
: mapping generators: ~2, ~10/7
EDOs: 9, 34d, 43, 77d
Badness: 0.1096


==11-limit==  
[[Optimal tuning]]s:
Commas: 99/98, 176/175, 864/847
* [[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 }}


POTE generator: ~15/11 = 529.974
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
[[Badness]] (Sintel): 1.39
EDOs: 9, 34d, 43, 77de
Badness: 0.0426


==13-limit==  
=== 11-limit ===
Commas: 78/77, 99/98, 144/143, 176/175
Subgroup: 2.3.5.7.11


POTE generator: ~15/11 = 529.951
Comma list: 225/224, 441/440, 1344/1331


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
EDOs: 9, 34d, 43, 77de
Badness: 0.0258


=Enneaportent=  
Optimal tunings:
Commas: 225/224, 40353607/40310784
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


POTE generator: ~5/4 = 383.165
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
Badness (Sintel): 1.20
Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
EDOs: 9, 54, 63, 72, 495bcd
Badness: 0.0937


==11-limit==  
==== 13-limit ====
Commas: 225/224, 385/384, 12005/11979
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 383.146
Comma list: 144/143, 225/224, 364/363, 441/440


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
EDOs: 9, 54, 63, 72, 423cd, 495bcd
Badness: 0.0304


==13-limit==
Optimal tunings:
Commas: 169/168 225/224 364/363 1716/1715
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


POTE generator: ~5/4 = 383.047
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
Badness (Sintel): 1.14
EDOs: 9, 54, 63, 72, 279cf
Badness: 0.0223


=Submajor=
=== Mermaid ===
Comma: 69198046875/68719476736
Subgroup: 2.3.5.7.11


POTE generator: ~10125/8192 = 362.321
Comma list: 225/224, 385/384, 532400/531441


Map: [&lt;1 4 -1|, &lt;0 -8 11|]
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
EDOs: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc
Badness: 0.1302


==7-limit==
Optimal tunings:
Commas: 225/224, 51200/50421
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


POTE generator: ~49/40 = 362.255
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


Map: [&lt;1 4 -1 1|, &lt;0 -8 11 6|]
Badness (Sintel): 1.46
Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
EDOs: 10, 33, 43, 53
Badness: 0.0605


==11-limit==
==== 13-limit ====
Commas: 225/224, 385/384, 6655/6561
Subgroup: 2.3.5.7.11.13


POTE generator: ~27/22 = 362.101
Comma list: 225/224, 325/324, 385/384, 10648/10647


Map: [&lt;1 4 -1 1 11|, &lt;0 -8 11 6 -25|]
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
EDOs: 10, 53, 116, 169de, 285cde
Badness: 0.0506


==13-limit==
Optimal tunings:
Commas: 169/168, 225/224, 275/273, 385/384
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


POTE generator: ~16/13 = 362.105
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


Map: [&lt;1 4 -1 1 11 4|, &lt;0 -8 11 6 -25 -1|]
Badness (Sintel): 1.47
EDOs: 10, 53, 116, 169de, 285cdef
Badness: 0.0277


==Interpental==
== Slender ==
Commas: 99/98, 176/175, 51200/50421
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.


POTE generator: ~49/40 = 362.418
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  


Map: [&lt;1 4 -1 1 -5|, &lt;0 -8 11 6 28|]
[[Subgroup]]: 2.3.5.7
EDOs: 43, 53, 96, 149d
Badness: 0.0518


===13-limit===
[[Comma list]]: 225/224, 589824/588245
Commas: 99/98, 169/168, 176/175, 640/637


POTE generator: ~16/13 = 362.402
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


Map: [&lt;1 4 -1 1 -5 4|, &lt;0 -8 11 6 28 -1|]
[[Optimal tuning]]s:  
EDOs: 43, 53, 96, 149d
* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
Badness: 0.0297
: [[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 }}


=Alphorn=
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}
Commas: 225/224, 5764801/5668704


POTE generator: ~48/35 = 556.221
[[Badness]] (Sintel): 1.44


Map: [&lt;1 9 0 13|, &lt;0 -16 5 -22|]
=== 11-limit ===
Wedgie: &lt;&lt;16 -5 22 -45 -10 65||
Subgroup: 2.3.5.7.11
EDOs: 28d, 41, 151cd, 192cd, 233cd
Badness: 0.1293


==11-limit==
Comma list: 225/224, 385/384, 1331/1323
Commas: 225/224, 385/384, 12250/11979


POTE generator: ~11/8 = 556.144
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


Map: [&lt;1 9 0 13 3|, &lt;0 -16 5 -22 1|]
Optimal tunings:
EDOs: 41, 315cde
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
Badness: 0.0735
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}
</pre></div>
 
<h4>Original HTML content:</h4>
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
<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:112:&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:112 --&gt;&lt;!-- ws:start:WikiTextTocRule:113: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Negri"&gt;Negri&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:113 --&gt;&lt;!-- ws:start:WikiTextTocRule:114: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Negri-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
Badness (Sintel): 0.838
&lt;!-- ws:end:WikiTextTocRule:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 275/273, 385/384, 1331/1323
 
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
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"