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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:phylingual|phylingual]] and made on <tt>2012-07-17 21:38:37 UTC</tt>.<br>
: The original revision id was <tt>353613468</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.


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.
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]]


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


[[POTE tuning|POTE temperament]]: ~15/14 = 125.608
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.


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


==11-limit==
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}}.)
Commas: 45/44, 49/48, 56/55


[[POTE tuning|POTE generator]]: ~15/14 = 126.474
== Wizard ==
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''


Map: [&lt;1 2 2 3 4|, &lt;0 -4 3 -2 -5|]
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.  
EDOs: 9, 10, 19
[[Badness]]: 0.0262


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 45/44, 49/48, 56/55, 78/77


[[POTE tuning|POTE generator]]: ~14/13 = 126.431
[[Comma list]]: 225/224, 118098/117649


Map: [&lt;1 2 2 3 4 4|, &lt;0 -4 3 -2 -5 -3|]
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
EDOs: 9, 10, 19
: mapping generators: ~1225/864, ~245/216


=Negril=  
[[Optimal tuning]]s:
Commas: 49/48, 100/99, 225/224
* [[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 }}


POTE generator: ~15/14 = 124.767
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}


Map: [&lt;1 2 2 3 2|, &lt;0 -4 3 -2 14|]
[[Badness]] (Sintel): 1.03
EDOs: 19, [[29edo|29]], [[48edo|48df]], [[77edo|77cdf]]
Badness: 0.0387


==13-limit==  
=== 11-limit ===
Commas: 49/48, 65/64, 91/90, 875/858
Subgroup: 2.3.5.7.11


POTE generator: ~15/14 = 124.716
Comma list: 225/224, 385/384, 4000/3993


Map: [&lt;1 2 2 3 2 4|, &lt;0 -4 3 -2 14 -3|]
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}
EDOs: 9, 10, 19, 29, 48, 77
Badness: 0.0244


=Negric=  
Optimal tunings:
Commas: 33/32, 49/48, 77/75
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


POTE generator: ~15/14 = 127.039
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


Map: [&lt;1 2 2 3 3|, &lt;0 -4 3 -2 4|]
Badness (Sintel): 0.613
EDOs: 9, 19, [[85edo|85]]
Badness: 0.0306


==13-limit==  
==== Lizard ====
Commas: 33/32, 49/48, 65/64, 91/90
Subgroup: 2.3.5.7.11.13


POTE generator: ~15/14 = 127.039
Comma list: 225/224, 351/350, 364/363, 385/384


Map: [&lt;1 2 2 3 3 4|, &lt;0 -4 3 -2 4 -3|]
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
EDOs: 9, 19, 85
Badness: 0.0202


=Negroni=  
Optimal tunings:
Commas: 49/48, 55/54, 225/224
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


POTE generator: ~15/14 = 124.539
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


Map: [&lt;1 2 2 3 5|, &lt;0 -4 3 -2 -15|]
Badness (Sintel): 0.900
EDOs: 10, 19e, 29, 77cde
Badness: 0.0353


=Wizard=  
===== 17-limit =====
Comma: 2197265625/2147483648
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~5/4 = 383.212
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


Map: [&lt;2 1 5|, &lt;0 6 -1|]
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
EDOs: 22, 50, 72, 166, 238c, 310c, 548bc
Badness: 0.3864


==7-limit==
Optimal tunings:
Commas: 225/224, 118098/117649
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


[[POTE tuning|POTE temperament]]: 383.256
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


Map: [&lt;2 1 5 2|, &lt;0 6 -1 10|]
Badness (Sintel): 0.741
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==  
===== 19-limit =====
Commas: 225/224, 385/384, 4000/3993
Subgroup: 2.3.5.7.11.13.17.19


[[POTE tuning|POTE temperament]]: 383.232
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


Map: [&lt;2 1 5 2 8|, &lt;0 6 -1 10 -3|]
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
EDOs: 22, 50, 72, 94, [[166edo|166]], [[238edo|238c]], [[310edo|310c]]
Badness: 0.0185


==Lizard==
Optimal tunings:
Commas: 225/224, 351/350, 364/363, 385/384
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


[[POTE tuning|POTE temperament]]: 383.389
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
Badness (Sintel): 0.955
EDOs: 22, 50, 72, 122, 194df
Badness: 0.0218


===17-limit===  
==== Gizzard ====
Commas: 221/220, 273/272, 289/288, 351/350, 375/374
Subgroup: 2.3.5.7.11.13


[[POTE tuning|POTE temperament]]: 383.381
Comma list: 225/224, 325/324, 385/384, 1573/1568


Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
EDOs: 22, 50, 72, 122g, 194dfg
Badness: 0.0145


===19-limit===
Optimal tunings:
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}


[[POTE tuning|POTE generator]]: ~5/4 = 383.477
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
Badness (Sintel): 0.837
EDOs: 22h, 50, 72, [[122edo|122g]], [[194edo|194dfg]]
Badness: 0.0157


==Gizzard==  
===== 17-limit =====
Commas: 225/224, 385/384, 325/324, 1573/1568
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~5/4 = 383.170
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384


Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
EDOs: [[22edo|22]], [[26edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238cf]]
Badness: 0.0203


===17-limit===  
Optimal tunings:
Commas: 225/224, 289/288, 325/324, 375/374, 385/384
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}


POTE generator: ~5/4 = 383.175
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}


Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
Badness (Sintel): 0.694
EDOs: 22f, 72, 94, [[166edo|166g]], [[238edo|238cfg]]
Badness: 0.0136


===19-limit===  
===== 19-limit =====
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~5/4 = 383.138
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594


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


==Mage==
Optimal tunings:
Commas: 99/98, 176/175, 1331/1296
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


POTE generator: ~55/48 = 216.876
{{Optimal ET sequence|legend=0| 72, 94, 166g }}


Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
Badness (Sintel): 0.901
EDOs: 22, 72e, 94e
Badness: 0.0578


=Triton=  
=== Mage ===
Commas: 225/224, 1029/1000
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE temperament]]: ~7/5 = 568.865
Comma list: 99/98, 176/175, 1331/1296


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
EDOs: 17d, 19, 78bd, 97bd
Badness: 0.0592


==11-limit==
Optimal tunings:
Commas: 45/44, 56/55, 1029/1000
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


POTE generator: ~7/5 = 569.144
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


Map: [&lt;1 0 6 7 4|, &lt;0 3 -7 -8 -1|]
Badness (Sintel): 1.91
EDOs: 19, 59bde, 78bde, 97bde
Badness: 0.0457


=Tritonic=  
== Tritonic ==
Comnma: 553584375/536870912
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


POTE generator: ~45/32 = 580.219
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 4 -3|, &lt;0 -5 11|]
[[Subgroup]]: 2.3.5.7
EDOs: 29, 31, 60, 91
Badness: 0.3785


==7-limit==
[[Comma list]]: 225/224, 50421/50000
Commas: 225/224, 50421/50000


[[POTE tuning|POTE temperament]]: 580.286
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


Map: [&lt;1 4 -3 -3|, &lt;0 -5 11 12|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;5 -11 -12 -29 -33 3||
* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
EDOs: [[29edo|29]], [[31edo|31]], [[60edo|60]], [[91edo|91]], [[122edo|122]], 213bcd
: [[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 }}


==11-limit==
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
Commas: 121/120, 225/224, 441/440


[[POTE tuning|POTE generator]]: ~7/5 = 580.267
[[Badness]] (Sintel): 1.20


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
=== 11-limit ===
EDOs: 29, 31, 60e
Subgroup: 2.3.5.7.11
Badness: 0.0237


==13-limit==
Comma list: 121/120, 225/224, 441/440
Commas: 105/104, 121/120, 196/195, 275/273


POTE generator: ~7/5 = 580.108
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
Optimal tunings:  
EDOs: 29, 31, 60e, 151cde
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
Badness: 0.0230
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


==Tritoni==
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
Commas: 225/224, 385/384, 27783/27500


POTE generator: ~7/5 = 580.389
Badness (Sintel): 0.782


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
==== 13-limit ====
EDOs: 31, 91, 122
Subgroup: 2.3.5.7.11.13
Badness: 0.0455


=Merman=
Comma list: 105/104, 121/120, 196/195, 275/273
Comma: 1121008359375/1099511627776


POTE generator: ~45/32 = 585.634
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


Map: [&lt;1 5 -5|, &lt;0 -7 15|]
Optimal tunings:  
EDOs: 41, 84, 125, 209, 334c, 543bc
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
Badness: 0.4538
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


==7-limit==
{{Optimal ET sequence|legend=0| 29, 31, 60e }}
Commas: 225/224, 2500000/2470629


POTE generator: ~7/5 = 585.585
Badness (Sintel): 0.950


Map: [&lt;1 5 -5 -5|, &lt;0 -7 15 16|]
==== 17-limit ====
Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
Subgroup: 2.3.5.7.11.13.17
EDOs: 41, 84, 125
Badness: 0.0551


==11-limit==
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
Commas: 225/224, 441/440, 1344/1331


POTE generator: ~7/5 = 585.606
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


Map: [&lt;1 5 -5 -5 2|, &lt;0 -7 15 16 3]]
Optimal tunings:  
EDOs: 41, 84, 125e
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
Badness: 0.0364
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Commas: 144/143, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 585.657
Badness (Sintel): 0.973


Map: [&lt;1 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
==== 19-limit ====
EDOs: 41, 84, 125e, 209ef, 293ef
Subgroup: 2.3.5.7.11.13.17.19


=Septimin=
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
Comma: 35595703125/34359738368


POTE generator: ~75/64 = 263.658
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


Map: [&lt;1 4 1|, &lt;0 -11 6|]
Optimal tunings:  
EDOs: 9, 32, 41, 91, 132, 223c, 355bc
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
Badness: 0.6709
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


==7-limit==
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Commas: 225/224, 84035/82944


[[POTE tuning|POTE temperament]]: 263.632
Badness (Sintel): 1.03


Map: [&lt;1 4 1 5|, &lt;0 -11 6 -10|]
==== 23-limit ====
Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
Subgroup: 2.3.5.7.11.13.17.19.23
EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]


==11-limit==
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
Commas: 225/224, 385/384, 2401/2376


[[POTE tuning|POTE temperament]]: 263.634
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
Optimal tunings:  
EDOs: 9, 32, 41, 91, 223cdef
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
Commas: 105/104, 144/143, 196/195, 245/242


[[POTE tuning|POTE temperament]]: 263.700
Badness (Sintel): 1.04


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
=== Tritoni ===
EDOs: 9, 41, 91
Subgroup: 2.3.5.7.11


=Slender=
Comma list: 225/224, 385/384, 27783/27500
Commas: 225/225, 589824/588245


[[POTE tuning|POTE temperament]]: ~49/48 = 38.413
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
Optimal tunings:  
Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
EDOs: 31, 94, 125
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}
Badness: 0.0569


==11-limit==
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
Commas: 225/224, 385/384, 1331/1323


[[POTE tuning|POTE temperament]]: ~49/48 = 38.387
Badness (Sintel): 1.50


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
== Septimin ==
EDOs: 31, 63, 94, 125
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''
Badness: 25.342


==13-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, 275/273, 385/384, 1331/1323


POTE generator: ~49/48 = 38.314
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
[[Comma list]]: 225/224, 84035/82944
EDOs: 31, 63, 94
Badness: 25.913


=Marvo=
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
Commas: 225/224, 78125000/78121827
: mapping generators: ~2, ~12/7


[[POTE tuning|POTE temperament]]: ~27/20 = 516.694
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}
: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
{{Optimal ET sequence|legend=1| 41, 91, 132d }}
Wedgie: &lt;&lt;6, 17, 46, 13, 56 59||
EDOs: 65, 72


==11-limit==
[[Badness]] (Sintel): 1.38
Commas: 225/224, 243/242, 4000/3993


[[POTE tuning|POTE temperament]]: ~27/20 = 516.699
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
Comma list: 225/224, 245/242, 385/384
EDOs: 65, 72, 281, 497


==13-limit==
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
Commas: 243/242, 225/224, 351/350, 1625/1617


POTE generator: ~27/20 = 516.730
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 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}
EDOs: 72, 137, 209, 281f, 490bcf
Badness: 0.0269


=Marvolo=
Badness (Sintel): 1.04
Commas: 225/224, 156250000/155649627


POTE generator: ~21/20 = 83.348
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 1 1|, &lt;0 -6 19 26|]
Comma list: 105/104, 144/143, 196/195, 245/242
Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||
EDOs: 29, 43, 72, 619bcd, 691bcd
Badness: 0.0833


==11-limit==
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
Commas: 225/224, 441/440, 4000/3993


POTE generator: ~21/20 = 83.340
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 2 1 1 2|, &lt;0 -6 19 26 21|]
{{Optimal ET sequence|legend=0| 41, 91 }}
EDOs: 29, 43, 72
Badness: 0.0290


==13-limit==
Badness (Sintel): 0.955
Commas: 169/168, 225/224, 364/363, 441/440


POTE generator: ~21/20 = 83.330
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
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, 43, 72
Badness: 0.0215


=Amavil=
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.
Commas: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
[[Comma list]]: 225/224, 2500000/2470629
Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
EDOs: 9, 34d, 43, 77d
Badness: 0.1096


==11-limit==
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
Commas: 99/98, 176/175, 864/847
: mapping generators: ~2, ~10/7


POTE generator: ~15/11 = 529.974
[[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 6 1 9 7|, &lt;0 -10 3 -14 -8|]
{{Optimal ET sequence|legend=1| 41, 84, 125 }}
EDOs: 9, 34d, 43, 77de
Badness: 0.0426


==13-limit==
[[Badness]] (Sintel): 1.39
Commas: 78/77, 99/98, 144/143, 176/175


POTE generator: ~15/11 = 529.951
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
Comma list: 225/224, 441/440, 1344/1331
EDOs: 9, 34d, 43, 77de
Badness: 0.0258


=Enneaportent=
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
Commas: 225/224, 40353607/40310784


POTE generator: ~5/4 = 383.165
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
EDOs: 9, 54, 63, 72, 495bcd
Badness: 0.0937


==11-limit==
Badness (Sintel): 1.20
Commas: 225/224, 385/384, 12005/11979


POTE generator: ~5/4 = 383.146
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
Comma list: 144/143, 225/224, 364/363, 441/440
EDOs: 9, 54, 63, 72, 423cd, 495bcd
Badness: 0.0304


==13-limit==
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
Commas: 169/168 225/224 364/363 1716/1715


POTE generator: ~5/4 = 383.047
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
EDOs: 9, 54, 63, 72, 279cf
Badness: 0.0223


=Submajor=
Badness (Sintel): 1.14
Comma: 69198046875/68719476736


POTE generator: ~10125/8192 = 362.321
=== Mermaid ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 4 -1|, &lt;0 -8 11|]
Comma list: 225/224, 385/384, 532400/531441
EDOs: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc
Badness: 0.1302


==7-limit==
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
Commas: 225/224, 51200/50421


POTE generator: ~49/40 = 362.255
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 1|, &lt;0 -8 11 6|]
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}
Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
EDOs: 10, 33, 43, 53
Badness: 0.0605


==11-limit==
Badness (Sintel): 1.46
Commas: 225/224, 385/384, 6655/6561


POTE generator: ~27/22 = 362.101
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 4 -1 1 11|, &lt;0 -8 11 6 -25|]
Comma list: 225/224, 325/324, 385/384, 10648/10647
EDOs: 10, 53, 116, 169de, 285cde
Badness: 0.0506


==13-limit==
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
Commas: 169/168, 225/224, 275/273, 385/384


POTE generator: ~16/13 = 362.105
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 4 -1 1 11 4|, &lt;0 -8 11 6 -25 -1|]
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
EDOs: 10, 53, 116, 169de, 285cdef
Badness: 0.0277


==Interpental==
Badness (Sintel): 1.47
Commas: 99/98, 176/175, 51200/50421


POTE generator: ~49/40 = 362.418
== 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 4 -1 1 -5|, &lt;0 -8 11 6 28|]
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  
EDOs: 43, 53, 96, 149d
Badness: 0.0518


===13-limit===
[[Subgroup]]: 2.3.5.7
Commas: 99/98, 169/168, 176/175, 640/637


POTE generator: ~16/13 = 362.402
[[Comma list]]: 225/224, 589824/588245


Map: [&lt;1 4 -1 1 -5 4|, &lt;0 -8 11 6 28 -1|]
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
EDOs: 43, 53, 96, 149d
: mapping generators: ~2, ~49/48
Badness: 0.0297


=Alphorn=  
[[Optimal tuning]]s:
Commas: 225/224, 5764801/5668704
* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
: [[error map]]: {{val| +0.382 -0.725 -1.295 +1.765 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}}
: error map: {{val| 0.000 -1.257 -2.235 +0.727 }}


POTE generator: ~48/35 = 556.221
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Map: [&lt;1 9 0 13|, &lt;0 -16 5 -22|]
[[Badness]] (Sintel): 1.44
Wedgie: &lt;&lt;16 -5 22 -45 -10 65||
EDOs: 28d, 41, 151cd, 192cd, 233cd
Badness: 0.1293


==11-limit==  
=== 11-limit ===
Commas: 225/224, 385/384, 12250/11979
Subgroup: 2.3.5.7.11


POTE generator: ~11/8 = 556.144
Comma list: 225/224, 385/384, 1331/1323


Map: [&lt;1 9 0 13 3|, &lt;0 -16 5 -22 1|]
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
EDOs: 41, 315cde
 
Badness: 0.0735</pre></div>
Optimal tunings:
<h4>Original HTML content:</h4>
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{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: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;
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}
&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;
 
&lt;!-- ws:end:WikiTextTocRule:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
 
Badness (Sintel): 0.838
 
=== 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
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"