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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>2011-10-23 15:26:25 UTC</tt>.<br>
: The original revision id was <tt>267652626</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|flat]]
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.


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]]: 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]]: 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]]: 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: 9, 10, 19, [[29edo|29]], [[48edo|48]], [[77edo|77]]
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


=Wizard=  
Optimal tunings:
Commas: 225/224, 118098/117649
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{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): 0.900
Wedgie: &lt;&lt;12 -2 20 -31 -2 52||
EDOs: [[22edo|22]], [[50edo|50]], [[72edo|72]], [[382edo|382]], [[454edo|454]]
Badness: 0.0408


==11-limit==  
===== 17-limit =====
Commas: 225/224, 385/384, 4000/3993
Subgroup: 2.3.5.7.11.13.17


[[POTE tuning|POTE temperament]]: 383.232
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


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


==Lizard==
Optimal tunings:
Commas: 225/224, 351/350, 364/363, 385/384
* 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]]: 383.389
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
Badness (Sintel): 0.741
EDOs: 22, 50, 72
Badness: 0.0218


===17-limit===  
===== 19-limit =====
Commas: 221/220, 273/272, 289/288, 351/350, 375/374
Subgroup: 2.3.5.7.11.13.17.19


[[POTE tuning|POTE temperament]]: 383.381
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
EDOs: 22, 50, 72
Badness: 0.0145


===19-limit===
Optimal tunings:
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
* 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 generator]]: ~5/4 = 383.477
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
Badness (Sintel): 0.955
EDOs: 22, 50, 72, [[194edo|194]], [[460edo|460]]
Badness: 0.0157


==Gizzard==  
==== Gizzard ====
Commas: 225/224, 385/384, 325/324, 1573/1568
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 383.170
Comma list: 225/224, 325/324, 385/384, 1573/1568


Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
EDOs: [[28edo|22]], [[28edo|28]], [[50edo|50]], [[72edo|72]], [[94edo|94]], [[166edo|166]]
Badness: 0.0203


===17-limit===
Optimal tunings:
Commas: 225/224, 289/288, 325/324, 375/374, 385/384
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}


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


Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
Badness (Sintel): 0.837
EDOs: 22, 72, 94, 166, [[238edo|238]]
Badness: 0.0136


===19-limit===  
===== 17-limit =====
Commas: 225/224, 289/28, 325/324, 385/384, 494/493, 969/968
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~5/4 = 383.138
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384


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 | 0 6 -1 10 -3 26 6 }}
EDOs: 22, 72, 94, 166
Badness: 0.0148


=Tritonic=  
Optimal tunings:
Commas: 225/224, 50421/50000
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}


[[POTE tuning|POTE temperament]]: 580.286
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}


Map: [&lt;1 4 -3 -3|, &lt;0 -5 11 12|]
Badness (Sintel): 0.694
Wedgie: &lt;&lt;5 -11 -12 -29 -33 3||
EDOs: [[29edo|29]], [[31edo|31]], [[60edo|60]], [[91edo|91]], [[122edo|122]], 213bcd


==11-limit==  
===== 19-limit =====
Commas: 121/120, 225/224, 441/440
Subgroup: 2.3.5.7.11.13.17.19


[[POTE tuning|POTE generator]]: ~7/5 = 580.267
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
EDOs: 29, 31, 60e
Badness: 0.0237


==13-limit==
Optimal tunings:
Commas: 105/104, 121/120, 196/195, 275/273
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


POTE generator: ~7/5 = 580.108
{{Optimal ET sequence|legend=0| 72, 94, 166g }}


Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
Badness (Sintel): 0.901
EDOs: 29, 31, 60e, 151cde
Badness: 0.0230


=== Mage ===
Subgroup: 2.3.5.7.11


==Tritoni==
Comma list: 99/98, 176/175, 1331/1296
Commas: 225/224, 385/384, 27783/27500


POTE generator: ~7/5 = 580.389
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
Optimal tunings:  
EDOs: 31, 91, 122
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
Badness: 0.0455
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


=Septimin=
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
Commas: 225/224, 84035/82944


[[POTE tuning|POTE temperament]]: 263.632
Badness (Sintel): 1.91


Map: [&lt;1 4 1 5|, &lt;0 -11 6 -10|]
== Tritonic ==
Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''
EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]


==11-limit==
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.
Commas: 225/224, 385/384, 2401/2376


[[POTE tuning|POTE temperament]]: 263.634
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
[[Comma list]]: 225/224, 50421/50000
EDOs: 9, 32, 41, 91


==13-limit==
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
Commas: 105/104, 144/143, 196/195, 245/242
: mapping generators: ~2, ~10/7


[[POTE tuning|POTE temperament]]: 263.700
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
: [[error map]]: {{val| +1.354 -1.243 -0.027 -1.598 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}}
: error map: {{val| 0.000 -3.566 -2.769 -4.959 }}


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
EDOs: 9, 41, 91


=Slender=
[[Badness]] (Sintel): 1.20
Commas: 225/225, 589824/588245


[[POTE tuning|POTE temperament]]: ~49/48 = 38.413
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
Comma list: 121/120, 225/224, 441/440
Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
EDOs: 31, 94, 125
Badness: 0.0569


==11-limit==
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}
Commas: 225/224, 385/384, 1331/1323


[[POTE tuning|POTE temperament]]: ~49/48 = 38.387
Optimal tunings:  
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
EDOs: 31, 63, 94, 125
Badness: 25.342


==13-limit==
Badness (Sintel): 0.782
Commas: 225/224, 275/273, 385/384, 1331/1323


POTE generator: ~49/48 = 38.314
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
Comma list: 105/104, 121/120, 196/195, 275/273
EDOs: 31, 63, 94
Badness: 25.913


=Triton=
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}
Commas: 225/224, 1029/1000


[[POTE tuning|POTE temperament]]: 568.865
Optimal tunings:  
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
{{Optimal ET sequence|legend=0| 29, 31, 60e }}
Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
EDOs: 19, 135, 289


=Marvo=
Badness (Sintel): 0.950
Commas: 225/224, 78125000/78121827


[[POTE tuning|POTE temperament]]: (~27/20) 516.694
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
Wedgie: &lt;&lt;6, 17, 46, 13, 56 59||
EDOs: 65, 72


==11-limit==
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
Commas: 225/224, 243/242, 4000/3993


[[POTE tuning|POTE temperament]]: 516.699
Optimal tunings:  
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
EDOs: 65, 72, 281, 497</pre></div>
 
<h4>Original HTML content:</h4>
Badness (Sintel): 0.973
<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:56:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
 
Optimal tunings:
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
Badness (Sintel): 1.03
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}
 
Optimal tunings:
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
Badness (Sintel): 1.04
 
=== Tritoni ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 27783/27500
 
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}
 
Optimal tunings:
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}
 
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
 
Badness (Sintel): 1.50
 
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''
 
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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 84035/82944
 
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
: mapping generators: ~2, ~12/7
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 41, 91, 132d }}
 
[[Badness]] (Sintel): 1
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