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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>2012-05-18 13:59:23 UTC</tt>.<br>
: The original revision id was <tt>337131926</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]]
* ''[[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
Comma: 2197265625/2147483648


POTE generator: ~5/4 = 383.212
{{Mapping|legend=0| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


Map: [&lt;2 1 5|, &lt;0 6 -1|]
Optimal tunings:  
EDOs: 22, 50, 72, 166, 238c, 310c, 548bc
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
Badness: 0.3864
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


==7-limit==
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Commas: 225/224, 118098/117649


[[POTE tuning|POTE temperament]]: 383.256
Badness (Sintel): 0.973


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: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
Commas: 225/224, 385/384, 4000/3993


[[POTE tuning|POTE temperament]]: 383.232
{{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|, &lt;0 6 -1 10 -3|]
Optimal tunings:  
EDOs: 22, 50, 72, 94, [[166edo|166]], [[238edo|238c]], [[310edo|310c]]
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
Badness: 0.0185
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


==Lizard==
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Commas: 225/224, 351/350, 364/363, 385/384


[[POTE tuning|POTE temperament]]: 383.389
Badness (Sintel): 1.03


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
==== 23-limit ====
EDOs: 22, 50, 72, 122, 194df
Subgroup: 2.3.5.7.11.13.17.19.23
Badness: 0.0218


===17-limit===
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
Commas: 221/220, 273/272, 289/288, 351/350, 375/374


[[POTE tuning|POTE temperament]]: 383.381
{{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|, &lt;0 6 -1 10 -3 -10 6|]
Optimal tunings:  
EDOs: 22, 50, 72, 122g, 194dfg
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
Badness: 0.0145
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


===19-limit===
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


[[POTE tuning|POTE generator]]: ~5/4 = 383.477
Badness (Sintel): 1.04


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
=== Tritoni ===
EDOs: 22h, 50, 72, [[122edo|122g]], [[194edo|194dfg]]
Subgroup: 2.3.5.7.11
Badness: 0.0157


==Gizzard==
Comma list: 225/224, 385/384, 27783/27500
Commas: 225/224, 385/384, 325/324, 1573/1568


POTE generator: ~5/4 = 383.170
{{Mapping|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


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: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
Badness: 0.0203
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


===17-limit===
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
Commas: 225/224, 289/288, 325/324, 375/374, 385/384


POTE generator: ~5/4 = 383.175
Badness (Sintel): 1.50


Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
== Septimin ==
EDOs: 22f, 72, 94, [[166edo|166g]], [[238edo|238cfg]]
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''
Badness: 0.0136


===19-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, 325/324, 595/594, 375/374, 400/399


POTE generator: ~5/4 = 383.138
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]
[[Comma list]]: 225/224, 84035/82944
EDOs: 22f, 72, 94, 166g
Badness: 0.0148


==Mage==
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
Commas: 99/98, 176/175, 1331/1296
: mapping generators: ~2, ~12/7


POTE generator: ~55/48 = 216.876
[[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;2 1 5 2 4|, &lt;0 6 -1 10 8|]
{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}
EDOs: 22, 72e, 94e
Badness: 0.0578


=Triton=
[[Badness]] (Sintel): 1.38
Commas: 225/224, 1029/1000


[[POTE tuning|POTE temperament]]: ~7/5 = 568.865
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
Comma list: 225/224, 245/242, 385/384
Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
EDOs: 17d, 19, 78bd, 97bd
Badness: 0.0592


==11-limit==
{{Mapping|legend=0| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
Commas: 45/44, 56/55, 1029/1000


POTE generator: ~7/5 = 569.144
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 0 6 7 4|, &lt;0 3 -7 -8 -1|]
{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}
EDOs: 19, 59bde, 78bde, 97bde
Badness: 0.0457


=Tritonic=
Badness (Sintel): 1.04
Comnma: 553584375/536870912


POTE generator: ~45/32 = 580.219
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 4 -3|, &lt;0 -5 11|]
Comma list: 105/104, 144/143, 196/195, 245/242
EDOs: 29, 31, 60, 91
Badness: 0.3785


==7-limit==
{{Mapping|legend=0| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
Commas: 225/224, 50421/50000


[[POTE tuning|POTE temperament]]: 580.286
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|, &lt;0 -5 11 12|]
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}
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): 0.955
Commas: 121/120, 225/224, 441/440


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


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
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, 31, 60e
Badness: 0.0237


==13-limit==
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.
Commas: 105/104, 121/120, 196/195, 275/273


POTE generator: ~7/5 = 580.108
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
[[Comma list]]: 225/224, 2500000/2470629
EDOs: 29, 31, 60e, 151cde
Badness: 0.0230


==Tritoni==
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
Commas: 225/224, 385/384, 27783/27500
: mapping generators: ~2, ~10/7


POTE generator: ~7/5 = 580.389
[[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 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
{{Optimal ET sequence|legend=1| 41, 84, 125 }}
EDOs: 31, 91, 122
Badness: 0.0455


=Merman=
[[Badness]] (Sintel): 1.39
Comma: 1121008359375/1099511627776


POTE generator: ~45/32 = 585.634
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 5 -5|, &lt;0 -7 15|]
Comma list: 225/224, 441/440, 1344/1331
EDOs: 41, 84, 125, 209, 334c, 543bc
Badness: 0.4538


==7-limit==
{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
Commas: 225/224, 2500000/2470629


POTE generator: ~7/5 = 585.585
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 5 -5 -5|, &lt;0 -7 15 16|]
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
EDOs: 41, 84, 125
Badness: 0.0551


==11-limit==
Badness (Sintel): 1.20
Commas: 225/224, 441/440, 1344/1331


POTE generator: ~7/5 = 585.606
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 5 -5 -5 2|, &lt;0 -7 15 16 3]]
Comma list: 144/143, 225/224, 364/363, 441/440
EDOs: 41, 84, 125e
Badness: 0.0364


==13-limit==
{{Mapping|legend=0| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
Commas: 144/143, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 585.657
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 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
EDOs: 41, 84, 125e, 209ef, 293ef


=Septimin=
Badness (Sintel): 1.14
Commas: 225/224, 84035/82944


[[POTE tuning|POTE temperament]]: 263.632
=== Mermaid ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 4 1 5|, &lt;0 -11 6 -10|]
Comma list: 225/224, 385/384, 532400/531441
Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]


==11-limit==
{{Mapping|legend=0| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
Commas: 225/224, 385/384, 2401/2376


[[POTE tuning|POTE temperament]]: 263.634
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 5 5|, &lt;0 -11 6 -10 -7|]
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}
EDOs: 9, 32, 41, 91, 223cdef


==13-limit==
Badness (Sintel): 1.46
Commas: 105/104, 144/143, 196/195, 245/242


[[POTE tuning|POTE temperament]]: 263.700
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
Comma list: 225/224, 325/324, 385/384, 10648/10647
EDOs: 9, 41, 91


=Slender=
{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
Commas: 225/225, 589824/588245


[[POTE tuning|POTE temperament]]: ~49/48 = 38.413
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 2 2 3|, &lt;0 -13 10 -6|]
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
EDOs: 31, 94, 125
Badness: 0.0569


==11-limit==
Badness (Sintel): 1.47
Commas: 225/224, 385/384, 1331/1323


[[POTE tuning|POTE temperament]]: ~49/48 = 38.387
== 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 2 2 3 4|, &lt;0 -13 10 -6 -17|]
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  
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, 589824/588245


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
EDOs: 31, 63, 94
: mapping generators: ~2, ~49/48
Badness: 25.913


=Marvo=  
[[Optimal tuning]]s:
Commas: 225/224, 78125000/78121827
* [[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 tuning|POTE temperament]]: ~27/20 = 516.694
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
[[Badness]] (Sintel): 1.44
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, 385/384, 1331/1323


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
{{Mapping|legend=0| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
EDOs: 65, 72, 281, 497


==13-limit==
Optimal tunings:
Commas: 243/242, 225/224, 351/350, 1625/1617
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


POTE generator: ~27/20 = 516.730
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
Badness (Sintel): 0.838
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: 225/224, 275/273, 385/384, 1331/1323


Map: [&lt;1 2 1 1|, &lt;0 -6 19 26|]
{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
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 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


POTE generator: ~21/20 = 83.340
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Map: [&lt;1 2 1 1 2|, &lt;0 -6 19 26 21|]
Badness (Sintel): 1.07
EDOs: 29, 43, 72
Badness: 0.0290


==13-limit==  
== Triton ==
Commas: 169/168, 225/224, 364/363, 441/440
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


POTE generator: ~21/20 = 83.330
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 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
[[Subgroup]]: 2.3.5.7
EDOs: 29, 43, 72
Badness: 0.0215


=Amavil=
[[Comma list]]: 225/224, 1029/1000
Commas: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
EDOs: 9, 34d, 43, 77d
: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
Badness: 0.1096
* [[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: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
[[Badness]] (Sintel): 1.50


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
=== 11-limit ===
EDOs: 9, 34d, 43, 77de
Subgroup: 2.3.5.7.11
Badness: 0.0426


==13-limit==
Comma list: 45/44, 56/55, 1029/1000
Commas: 78/77, 99/98, 144/143, 176/175


POTE generator: ~15/11 = 529.951
{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
Optimal tunings:  
EDOs: 9, 34d, 43, 77de
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
Badness: 0.0258
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


=Enneaportent=
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
Commas: 225/224, 40353607/40310784


POTE generator: ~5/4 = 383.165
Badness (Sintel): 1.51


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
== Marvolo ==
Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
[[Subgroup]]: 2.3.5.7
EDOs: 9, 54, 63, 72, 495bcd
Badness: 0.0937


==11-limit==
[[Comma list]]: 225/224, 156250000/155649627
Commas: 225/224, 385/384, 12005/11979


POTE generator: ~5/4 = 383.146
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
[[Optimal tuning]]s:  
EDOs: 9, 54, 63, 72, 423cd, 495bcd
* [[WE]]: ~2 = 1200.7714{{c}}, ~21/20 = 83.4014{{c}}
Badness: 0.0304
: [[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 }}


==13-limit==
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}
Commas: 169/168 225/224 364/363 1716/1715


POTE generator: ~5/4 = 383.047
[[Badness]] (Sintel): 2.11


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
=== 11-limit ===
EDOs: 9, 54, 63, 72, 279cf
Subgroup: 2.3.5.7.11
Badness: 0.0223


=Submajor=
Comma list: 225/224, 441/440, 4000/3993
Comma: 69198046875/68719476736


POTE generator: ~10125/8192 = 362.321
{{Mapping|legend=0| 1 2 1 1 2 | 0 -6 19 26 21 }}


Map: [&lt;1 4 -1|, &lt;0 -8 11|]
Optimal tunings:  
EDOs: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
Badness: 0.1302
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


==7-limit==
{{Optimal ET sequence|legend=0| 29, 43, 72 }}
Commas:  225/224, 51200/50421


POTE generator: ~49/40 = 362.255
Badness (Sintel): 0.958


Map: [&lt;1 4 -1 1|, &lt;0 -8 11 6|]
=== 13-limit ===
Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
Subgroup: 2.3.5.7.11.13
EDOs: 10, 33, 43, 53
Badness: 0.0605


==11-limit==
Comma list: 169/168, 225/224, 364/363, 441/440
Commas: 225/224, 385/384, 6655/6561


POTE generator: ~27/22 = 362.101
{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


Map: [&lt;1 4 -1 1 11|, &lt;0 -8 11 6 -25|]
Optimal tunings:  
EDOs: 10, 53, 116, 169de, 285cde
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
Badness: 0.0506
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 29, 43, 72 }}
Commas: 169/168, 225/224, 275/273, 385/384


POTE generator: ~16/13 = 362.105
Badness (Sintel): 0.887


Map: [&lt;1 4 -1 1 11 4|, &lt;0 -8 11 6 -25 -1|]
=== 17-limit ===
EDOs: 10, 53, 116, 169de, 285cdef
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0277


==Interpental==
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
Commas: 99/98, 176/175, 51200/50421


POTE generator: ~49/40 = 362.418
{{Mapping|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


Map: [&lt;1 4 -1 1 -5|, &lt;0 -8 11 6 28|]
Optimal tunings:  
EDOs: 43, 53, 96, 149d
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
Badness: 0.0518
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


===13-limit===
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
Commas: 99/98, 169/168, 176/175, 640/637


POTE generator: ~16/13 = 362.402
Badness (Sintel): 0.760


Map: [&lt;1 4 -1 1 -5 4|, &lt;0 -8 11 6 28 -1|]
=== 19-limit ===
EDOs: 43, 53, 96, 149d
Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.0297</pre></div>
 
<h4>Original HTML content:</h4>
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
<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:106:&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:106 --&gt;&lt;!-- ws:start:WikiTextTocRule:107: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Negri"&gt;Negri&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:107 --&gt;&lt;!-- ws:start:WikiTextTocRule:108: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Negri-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
{{Mapping|legend=0| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
&lt;!-- ws:end:WikiTextTocRule:108 --&gt;&
 
Optimal tunings:
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
 
Badness (Sintel): 0.895
 
== Decic ==
{{Main| Decic }}
 
Named by [[Xenllium]] in 2021, decic tempers out 16807/16384, the [[cloudy comma]], and {{monzo| 11 -10 -10 10 }}, the [[linus comma]], in addition to the marvel comma. It may be described as the {{nowrap| 50 & 60 }} temperament, with a period of 1/10 octave and a [[ploidacot]] signature of decaploid monocot. It is [[support]]ed by [[10edo|10-]], [[50edo|50-]], and [[60edo]].
 
It can be extended to the 11-, 13-, and 17-limit by adding [[385/384]], [[105/104]], and [[170/169]] to the comma list in this order.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 16807/16384
 
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 120.1837{{c}}, ~3/2 = 699.7654{{c}} (~49/48 = 21.3366{{c}})
: [[error map]]: {{val| +1.837 -0.353 -0.753 -3.683 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 698.8236{{c}} (~49/48 = 21.1764{{c}})
: error map: {{val| 0.000 -3.131 -5.137 -8.826 }}
 
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}
 
[[Badness]] (Sintel): 2.26
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 3087/3025
 
{{Mapping|legend=0| 10 0 39 28 3 | 0 1 -1 0 2 }}
 
Optimal tunings:
* WE: ~15/14 = 120.1406{{c}}, ~3/2 = 697.6075{{c}} (~56/55 = 23.2360{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.0142{{c}} (~56/55 = 22.9858{{c}})
 
{{
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