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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]]
* ''[[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:


Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


=Wizard=
{{Mapping|legend=0| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
Commas: 225/224, 118098/117649


[[POTE tuning|POTE temperament]]: 383.256
Optimal tunings:  
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


Map: [&lt;2 1 5 2|, &lt;0 6 -1 10|]
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
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): 0.973
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: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
EDOs: 22, 50, 72, 94, [[166edo|166]], [[238edo|238c]], [[310edo|310c]]
Badness: 0.0185


==Lizard==
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
Commas: 225/224, 351/350, 364/363, 385/384


[[POTE tuning|POTE temperament]]: 383.389
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
EDOs: 22, 50, 72, 122, 194df
Badness: 0.0218


===17-limit===
Badness (Sintel): 1.03
Commas: 221/220, 273/272, 289/288, 351/350, 375/374


[[POTE tuning|POTE temperament]]: 383.381
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
EDOs: 22, 50, 72, 122g, 194dfg
Badness: 0.0145


===19-limit===
{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


[[POTE tuning|POTE generator]]: ~5/4 = 383.477
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{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| 29g, 31, 60e }}
EDOs: 22h, 50, 72, [[122edo|122g]], [[194edo|194dfg]]
Badness: 0.0157


==Gizzard==
Badness (Sintel): 1.04
Commas: 225/224, 385/384, 325/324, 1573/1568


POTE generator: ~5/4 = 383.170
=== Tritoni ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]
Comma list: 225/224, 385/384, 27783/27500
EDOs: [[22edo|22]], [[26edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238cf]]
Badness: 0.0203


===17-limit===
{{Mapping|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}
Commas: 225/224, 289/288, 325/324, 375/374, 385/384


POTE generator: ~5/4 = 383.175
Optimal tunings:  
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
EDOs: 22f, 72, 94, [[166edo|166g]], [[238edo|238cfg]]
Badness: 0.0136


===19-limit===
Badness (Sintel): 1.50
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399


POTE generator: ~5/4 = 383.138
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


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


==Mage==
[[Subgroup]]: 2.3.5.7
Commas: 99/98, 176/175, 1331/1296


POTE generator: ~55/48 = 216.876
[[Comma list]]: 225/224, 84035/82944


Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
EDOs: 22, 72e, 94e
: mapping generators: ~2, ~12/7
Badness: 0.0578


=Triton=  
[[Optimal tuning]]s:
Commas: 225/224, 1029/1000
* [[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]]: 568.865
{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
[[Badness]] (Sintel): 1.38
Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
EDOs: 19, 135, 289


=Tritonic=  
=== 11-limit ===
Commas: 225/224, 50421/50000
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE temperament]]: 580.286
Comma list: 225/224, 245/242, 385/384


Map: [&lt;1 4 -3 -3|, &lt;0 -5 11 12|]
{{Mapping|legend=0| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
Wedgie: &lt;&lt;5 -11 -12 -29 -33 3||
EDOs: [[29edo|29]], [[31edo|31]], [[60edo|60]], [[91edo|91]], [[122edo|122]], 213bcd


==11-limit==
Optimal tunings:
Commas: 121/120, 225/224, 441/440
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


[[POTE tuning|POTE generator]]: ~7/5 = 580.267
{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
Badness (Sintel): 1.04
EDOs: 29, 31, 60e
Badness: 0.0237


==13-limit==  
=== 13-limit ===
Commas: 105/104, 121/120, 196/195, 275/273
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 580.108
Comma list: 105/104, 144/143, 196/195, 245/242


Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
{{Mapping|legend=0| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
EDOs: 29, 31, 60e, 151cde
Badness: 0.0230


==Tritoni==
Optimal tunings:
Commas: 225/224, 385/384, 27783/27500
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


POTE generator: ~7/5 = 580.389
{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
Badness (Sintel): 0.955
EDOs: 31, 91, 122
Badness: 0.0455


=Merman=  
== Merman ==
Commas: 225/224, 2500000/2470629
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


POTE generator: ~7/5 = 585.585
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 5 -5 -5|, &lt;0 -7 15 16|]
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.  
Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
EDOs: 41, 84, 125
Badness: 0.0551


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 225/224, 441/440, 1344/1331


POTE generator: ~7/5 = 585.606
[[Comma list]]: 225/224, 2500000/2470629


Map: [&lt;1 5 -5 -5 2|, &lt;0 -7 15 16 3]]
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
EDOs: 41, 84, 125e
: mapping generators: ~2, ~10/7
Badness: 0.0364


==13-limit==
[[Optimal tuning]]s:
Commas: 144/143, 225/224, 364/363, 441/440
* [[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: ~7/5 = 585.657
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


Map: [&lt;1 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
[[Badness]] (Sintel): 1.39
EDOs: 41, 84, 125e, 209ef, 293ef


=Septimin=  
=== 11-limit ===
Commas: 225/224, 84035/82944
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE temperament]]: 263.632
Comma list: 225/224, 441/440, 1344/1331


Map: [&lt;1 4 1 5|, &lt;0 -11 6 -10|]
{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]


==11-limit==
Optimal tunings:
Commas: 225/224, 385/384, 2401/2376
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


[[POTE tuning|POTE temperament]]: 263.634
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
Badness (Sintel): 1.20
EDOs: 9, 32, 41, 91, 223cdef


==13-limit==  
==== 13-limit ====
Commas: 105/104, 144/143, 196/195, 245/242
Subgroup: 2.3.5.7.11.13


[[POTE tuning|POTE temperament]]: 263.700
Comma list: 144/143, 225/224, 364/363, 441/440


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
{{Mapping|legend=0| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
EDOs: 9, 41, 91


=Slender=  
Optimal tunings:
Commas: 225/225, 589824/588245
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


[[POTE tuning|POTE temperament]]: ~49/48 = 38.413
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
Badness (Sintel): 1.14
Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
EDOs: 31, 94, 125
Badness: 0.0569


==11-limit==  
=== Mermaid ===
Commas: 225/224, 385/384, 1331/1323
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE temperament]]: ~49/48 = 38.387
Comma list: 225/224, 385/384, 532400/531441


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
{{Mapping|legend=0| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
EDOs: 31, 63, 94, 125
Badness: 25.342


==13-limit==
Optimal tunings:
Commas: 225/224, 275/273, 385/384, 1331/1323
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


POTE generator: ~49/48 = 38.314
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
Badness (Sintel): 1.46
EDOs: 31, 63, 94
Badness: 25.913


=Marvo=  
==== 13-limit ====
Commas: 225/224, 78125000/78121827
Subgroup: 2.3.5.7.11.13


[[POTE tuning|POTE temperament]]: ~27/20 = 516.694
Comma list: 225/224, 325/324, 385/384, 10648/10647


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
Wedgie: &lt;&lt;6, 17, 46, 13, 56 59||
EDOs: 65, 72


==11-limit==
Optimal tunings:
Commas: 225/224, 243/242, 4000/3993
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


[[POTE tuning|POTE temperament]]: ~27/20 = 516.699
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
Badness (Sintel): 1.47
EDOs: 65, 72, 281, 497


==13-limit==  
== Slender ==
Commas: 243/242, 225/224, 351/350, 1625/1617
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: ~27/20 = 516.730
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  


Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
[[Subgroup]]: 2.3.5.7
EDOs: 72, 137, 209, 281f, 490bcf
Badness: 0.0269


=Marvolo=
[[Comma list]]: 225/224, 589824/588245
Commas: 225/224, 156250000/155649627


POTE generator: ~21/20 = 83.348
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


Map: [&lt;1 2 1 1|, &lt;0 -6 19 26|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||
* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
EDOs: 29, 43, 72, 619bcd, 691bcd
: [[error map]]: {{val| +0.382 -0.725 -1.295 +1.765 }}
Badness: 0.0833
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}}
: error map: {{val| 0.000 -1.257 -2.235 +0.727 }}


==11-limit==
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}
Commas: 225/224, 441/440, 4000/3993


POTE generator: ~21/20 = 83.340
[[Badness]] (Sintel): 1.44


Map: [&lt;1 2 1 1 2|, &lt;0 -6 19 26 21|]
=== 11-limit ===
EDOs: 29, 43, 72
Subgroup: 2.3.5.7.11
Badness: 0.0290


==13-limit==
Comma list: 225/224, 385/384, 1331/1323
Commas: 169/168, 225/224, 364/363, 441/440


POTE generator: ~21/20 = 83.330
{{Mapping|legend=0| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
Optimal tunings:  
EDOs: 29, 43, 72
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
Badness: 0.0215
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


=Amavil=
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
Commas: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
Badness (Sintel): 0.838


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
=== 13-limit ===
Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
Subgroup: 2.3.5.7.11.13
EDOs: 9, 34d, 43, 77d
Badness: 0.1096


==11-limit==
Comma list: 225/224, 275/273, 385/384, 1331/1323
Commas: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
Optimal tunings:  
EDOs: 9, 34d, 43, 77de
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
Badness: 0.0426
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 31, 63, 94 }}
Commas: 78/77, 99/98, 144/143, 176/175


POTE generator: ~15/11 = 529.951
Badness (Sintel): 1.07


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
== Triton ==
EDOs: 9, 34d, 43, 77de
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''
Badness: 0.0258


=Enneaportent=
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.
Commas: 225/224, 40353607/40310784


POTE generator: ~5/4 = 383.165
[[Subgroup]]: 2.3.5.7


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
[[Comma list]]: 225/224, 1029/1000
Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
EDOs: 9, 54, 63, 72, 495bcd
Badness: 0.0937


==11-limit==
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
Commas: 225/224, 385/384, 12005/11979
: mapping generators: ~2, ~10/7


POTE generator: ~5/4 = 383.146
[[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 }}


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
EDOs: 9, 54, 63, 72, 423cd, 495bcd
Badness: 0.0304


==13-limit==
[[Badness]] (Sintel): 1.50
Commas: 169/168 225/224 364/363 1716/1715


POTE generator: ~5/4 = 383.047
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
Comma list: 45/44, 56/55, 1029/1000
EDOs: 9, 54, 63, 72, 279cf
 
Badness: 0.0223</pre></div>
{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
<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: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;
Optimal tunings:
&lt;!-- ws:end:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}
 
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
 
Badness (Sintel): 1.51
 
== Marvolo ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 156250000/155649627
 
{{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
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"