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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>2012-03-10 13:28:51 UTC</tt>.<br>
: The original revision id was <tt>309659520</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]]
* ''[[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:


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


=Wizard=
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
Commas: 225/224, 118098/117649


[[POTE tuning|POTE temperament]]: 383.256
Optimal tunings:  
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


Map: [&lt;2 1 5 2|, &lt;0 6 -1 10|]
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
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==
Badness (Sintel): 0.741
Commas: 225/224, 385/384, 4000/3993


[[POTE tuning|POTE temperament]]: 383.232
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;2 1 5 2 8|, &lt;0 6 -1 10 -3|]
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
EDOs: 22, 50, 72, 94, [[166edo|166]], [[238edo|238c]], [[310edo|310c]]
Badness: 0.0185


==Lizard==
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
Commas: 225/224, 351/350, 364/363, 385/384


[[POTE tuning|POTE temperament]]: 383.389
Optimal tunings:  
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
EDOs: 22, 50, 72, 122, 194df
Badness: 0.0218


===17-limit===
Badness (Sintel): 0.955
Commas: 221/220, 273/272, 289/288, 351/350, 375/374


[[POTE tuning|POTE temperament]]: 383.381
==== Gizzard ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
Comma list: 225/224, 325/324, 385/384, 1573/1568
EDOs: 22, 50, 72, 122g, 194dfg
Badness: 0.0145


===19-limit===
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


[[POTE tuning|POTE generator]]: ~5/4 = 383.477
Optimal tunings:  
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
EDOs: 22h, 50, 72, [[122edo|122g]], [[194edo|194dfg]]
Badness: 0.0157


==Gizzard==
Badness (Sintel): 0.837
Commas: 225/224, 385/384, 325/324, 1573/1568


POTE generator: ~5/4 = 383.170
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
EDOs: [[22edo|22]], [[26edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238cf]]
Badness: 0.0203


===17-limit===
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
Commas: 225/224, 289/288, 325/324, 375/374, 385/384


POTE generator: ~5/4 = 383.175
Optimal tunings:  
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}


Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
EDOs: 22f, 72, 94, [[166edo|166g]], [[238edo|238cfg]]
Badness: 0.0136


===19-limit===
Badness (Sintel): 0.694
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399


POTE generator: ~5/4 = 383.138
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
EDOs: 22f, 72, 94, 166g
Badness: 0.0148


==Mage==
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
Commas: 99/98, 176/175, 1331/1296


POTE generator: ~55/48 = 216.876
Optimal tunings:  
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
{{Optimal ET sequence|legend=0| 72, 94, 166g }}
EDOs: 22, 72e, 94e
Badness: 0.0578


=Triton=
Badness (Sintel): 0.901
Commas: 225/224, 1029/1000


[[POTE tuning|POTE temperament]]: 568.865
=== Mage ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
Comma list: 99/98, 176/175, 1331/1296
Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
EDOs: 19, 135, 289


=Tritonic=
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
Commas: 225/224, 50421/50000


[[POTE tuning|POTE temperament]]: 580.286
Optimal tunings:  
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


Map: [&lt;1 4 -3 -3|, &lt;0 -5 11 12|]
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
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): 1.91
Commas: 121/120, 225/224, 441/440


[[POTE tuning|POTE generator]]: ~7/5 = 580.267
== Tritonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.  
EDOs: 29, 31, 60e
Badness: 0.0237


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 105/104, 121/120, 196/195, 275/273


POTE generator: ~7/5 = 580.108
[[Comma list]]: 225/224, 50421/50000


Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
EDOs: 29, 31, 60e, 151cde
: mapping generators: ~2, ~10/7
Badness: 0.0230


==Tritoni==  
[[Optimal tuning]]s:
Commas: 225/224, 385/384, 27783/27500
* [[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 }}


POTE generator: ~7/5 = 580.389
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
[[Badness]] (Sintel): 1.20
EDOs: 31, 91, 122
Badness: 0.0455


=Merman=  
=== 11-limit ===
Commas: 225/224, 2500000/2470629
Subgroup: 2.3.5.7.11


POTE generator: ~7/5 = 585.585
Comma list: 121/120, 225/224, 441/440


Map: [&lt;1 5 -5 -5|, &lt;0 -7 15 16|]
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}
Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
EDOs: 41, 84, 125
Badness: 0.0551


==11-limit==
Optimal tunings:
Commas: 225/224, 441/440, 1344/1331
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


POTE generator: ~7/5 = 585.606
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


Map: [&lt;1 5 -5 -5 2|, &lt;0 -7 15 16 3]]
Badness (Sintel): 0.782
EDOs: 41, 84, 125e
Badness: 0.0364


==13-limit==
==== 13-limit ====
Commas: 144/143, 225/224, 364/363, 441/440
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 585.657
Comma list: 105/104, 121/120, 196/195, 275/273


Map: [&lt;1 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}
EDOs: 41, 84, 125e, 209ef, 293ef


=Septimin=  
Optimal tunings:
Commas: 225/224, 84035/82944
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


[[POTE tuning|POTE temperament]]: 263.632
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


Map: [&lt;1 4 1 5|, &lt;0 -11 6 -10|]
Badness (Sintel): 0.950
Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]


==11-limit==  
==== 17-limit ====
Commas: 225/224, 385/384, 2401/2376
Subgroup: 2.3.5.7.11.13.17


[[POTE tuning|POTE temperament]]: 263.634
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
EDOs: 9, 32, 41, 91, 223cdef


==13-limit==
Optimal tunings:
Commas: 105/104, 144/143, 196/195, 245/242
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


[[POTE tuning|POTE temperament]]: 263.700
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
Badness (Sintel): 0.973
EDOs: 9, 41, 91


=Slender=  
==== 19-limit ====
Commas: 225/225, 589824/588245
Subgroup: 2.3.5.7.11.13.17.19


[[POTE tuning|POTE temperament]]: ~49/48 = 38.413
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
EDOs: 31, 94, 125
Badness: 0.0569


==11-limit==
Optimal tunings:
Commas: 225/224, 385/384, 1331/1323
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


[[POTE tuning|POTE temperament]]: ~49/48 = 38.387
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
Badness (Sintel): 1.03
EDOs: 31, 63, 94, 125
Badness: 25.342


==13-limit==  
==== 23-limit ====
Commas: 225/224, 275/273, 385/384, 1331/1323
Subgroup: 2.3.5.7.11.13.17.19.23


POTE generator: ~49/48 = 38.314
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}
EDOs: 31, 63, 94
Badness: 25.913


=Marvo=  
Optimal tunings:
Commas: 225/224, 78125000/78121827
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


[[POTE tuning|POTE temperament]]: ~27/20 = 516.694
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
Badness (Sintel): 1.04
Wedgie: &lt;&lt;6, 17, 46, 13, 56 59||
EDOs: 65, 72


==11-limit==  
=== Tritoni ===
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, 385/384, 27783/27500


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}
EDOs: 65, 72, 281, 497


==13-limit==
Optimal tunings:
Commas: 243/242, 225/224, 351/350, 1625/1617
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


POTE generator: ~27/20 = 516.730
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
Badness (Sintel): 1.50
EDOs: 72, 137, 209, 281f, 490bcf
Badness: 0.0269


=Marvolo=  
== Septimin ==
Commas: 225/224, 156250000/155649627
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


POTE generator: ~21/20 = 83.348
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.  


Map: [&lt;1 2 1 1|, &lt;0 -6 19 26|]
[[Subgroup]]: 2.3.5.7
Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||
EDOs: 29, 43, 72, 619bcd, 691bcd
Badness: 0.0833


==11-limit==
[[Comma list]]: 225/224, 84035/82944
Commas: 225/224, 441/440, 4000/3993


POTE generator: ~21/20 = 83.340
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7


Map: [&lt;1 2 1 1 2|, &lt;0 -6 19 26 21|]
[[Optimal tuning]]s:  
EDOs: 29, 43, 72
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
Badness: 0.0290
: [[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 }}


==13-limit==
{{Optimal ET sequence|legend=1| 41, 91, 132d }}
Commas: 169/168, 225/224, 364/363, 441/440


POTE generator: ~21/20 = 83.330
[[Badness]] (Sintel): 1.38


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
=== 11-limit ===
EDOs: 29, 43, 72
Subgroup: 2.3.5.7.11
Badness: 0.0215


=Amavil=
Comma list: 225/224, 245/242, 385/384
Commas: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
Optimal tunings:  
Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
EDOs: 9, 34d, 43, 77d
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}
Badness: 0.1096


==11-limit==
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}
Commas: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
Badness (Sintel): 1.04


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
=== 13-limit ===
EDOs: 9, 34d, 43, 77de
Subgroup: 2.3.5.7.11.13
Badness: 0.0426


==13-limit==
Comma list: 105/104, 144/143, 196/195, 245/242
Commas: 78/77, 99/98, 144/143, 176/175


POTE generator: ~15/11 = 529.951
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
Optimal tunings:  
EDOs: 9, 34d, 43, 77de
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
Badness: 0.0258
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


=Enneaportent=
{{Optimal ET sequence|legend=0| 41, 91 }}
Commas: 225/224, 40353607/40310784


POTE generator: ~5/4 = 383.165
Badness (Sintel): 0.955


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
== Merman ==
Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''
EDOs: 9, 54, 63, 72, 495bcd
Badness: 0.0937


==11-limit==
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.
Commas: 225/224, 385/384, 12005/11979


POTE generator: ~5/4 = 383.146
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.  


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
[[Subgroup]]: 2.3.5.7
EDOs: 9, 54, 63, 72, 423cd, 495bcd
Badness: 0.0304


==13-limit==
[[Comma list]]: 225/224, 2500000/2470629
Commas: 169/168 225/224 364/363 1716/1715


POTE generator: ~5/4 = 383.047
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
[[Optimal tuning]]s:
EDOs: 9, 54, 63, 72, 279cf
* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
Badness: 0.0223</pre></div>
: [[error map]]: {{val| +0.390 -0.435 -1.630 +1.634 }}
<h4>Original HTML content:</h4>
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}}
<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:86:&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:86 --&gt;&lt;!-- ws:start:WikiTextTocRule:87: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Negri"&gt;Negri&lt;/a&gt;&lt;/div&gt;
: error map: {{val| 0.000 -1.104 -2.423 +0.657 }}
&lt;!-- ws:end:
 
{{Optimal ET sequence|legend=1| 41, 84, 125 }}
 
[[Badness]] (Sintel): 1.39
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 1344/1331
 
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
 
Badness (Sintel): 1.20
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
 
Optimal tunings:
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
 
Badness (Sintel): 1.14
 
=== Mermaid ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 532400/531441
 
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
 
Optimal tunings:
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}
 
Badness (Sintel): 1.46
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 10648/10647
 
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
 
Optimal tunings:
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
 
Badness (Sintel): 1.47
 
== Slender ==
Slender tempers out the [[hewuermera
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"