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>2015-12-21 12:27:40 UTC</tt>.<br>
: The original revision id was <tt>570593167</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 family|marvel]] comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton, merman and marvo. Considered elsewhere are [[Meantone family|meantone]], [[Miracle|miracle]], [[Magic family|magic]], [[Pajara|pajara]], [[Orwell|orwell]], [[Kleismic family|catakleismic]], [[Garibaldi temperament|garibaldi]], [[Augmented family|august]], [[Pythagorean family|compton]], [[Dicot family|sharp]], [[Escapade family|escapade]] and [[Pelogic family|mavila]].


Since (5/4)^2 = 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.
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]]
 
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]].
 
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.


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.
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.


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 (16/9)/(5/4)=64/45. Setting this equal to 10/7 gives (10/7)/(64/45)=225/224.)
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}}.)
 
== Wizard ==
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''
 
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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 118098/117649
 
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216
 
[[Optimal tuning]]s:
* [[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 }}
 
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}
 
[[Badness]] (Sintel): 1.03
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 4000/3993
 
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}
 
Optimal tunings:
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
 
Badness (Sintel): 0.613
 
==== Lizard ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 351/350, 364/363, 385/384
 
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
 
Optimal tunings:
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
 
Badness (Sintel): 0.900
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
 
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
 
Badness (Sintel): 0.741
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
 
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}
 
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
 
Badness (Sintel): 0.955
 
==== Gizzard ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 1573/1568
 
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
 
Optimal tunings:
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
 
Badness (Sintel): 0.837
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
 
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
 
Badness (Sintel): 0.694
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
 
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}
 
{{Optimal ET sequence|legend=0| 72, 94, 166g }}
 
Badness (Sintel): 0.901
 
=== Mage ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 1331/1296
 
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
 
Optimal tunings:
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
 
Badness (Sintel): 1.91
 
== Tritonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''
 
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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 50421/50000
 
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
 
[[Badness]] (Sintel): 1.20
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 225/224, 441/440
 
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}
 
Optimal tunings:
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}
 
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
 
Badness (Sintel): 0.782
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 121/120, 196/195, 275/273
 
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}
 
Optimal tunings:
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}
 
{{Optimal ET sequence|legend=0| 29, 31, 60e }}
 
Badness (Sintel): 0.950
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
 
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
 
Optimal tunings:
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
 
Badness (Sintel): 0.973
 
==== 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): 1.38
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 245/242, 385/384
 
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
 
Optimal tunings:
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}
 
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}
 
Badness (Sintel): 1.04
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 196/195, 245/242
 
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
 
Optimal tunings:
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}
 
{{Optimal ET sequence|legend=0| 41, 91 }}
 
Badness (Sintel): 0.955
 
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''
 
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.
 
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 2500000/2470629
 
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 41, 84, 125 }}
 
[[Badness]] (Sintel): 1.39
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 1344/1331
 
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84, 125e }}
 
Badness (Sintel): 1.20
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
 
Optimal tunings:
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
 
Badness (Sintel): 1.14
 
=== Mermaid ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 532400/531441
 
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
 
Optimal tunings:
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}
 
Badness (Sintel): 1.46
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 10648/10647
 
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
 
Optimal tunings:
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}
 
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
 
Badness (Sintel): 1.47
 
== Slender ==
Slender tempers out the [[hewuermera 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.
 
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 589824/588245
 
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48
 
[[Optimal tuning]]s:
* [[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 }}
 
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}
 
[[Badness]] (Sintel): 1.44
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 1331/1323
 
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
 
Optimal tunings:
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}
 
{{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}}
* 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: 2.3.5.7.11.13.17.19
 
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
 
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
 
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
 
== Enneaportent ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 40353607/40310784
 
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912
 
[[Optimal tuning]]s:
* [[WE]]: ~2592/2401 = 133.4174{{c}}, ~12005/6912 = 950.7667{{c}} (~1728/1715 = 16.8452{{c}})
: [[error map]]: {{val| +0.756 -0.422 -1.395 +0.298 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}})
: error map: {{val| 0.000 -1.361 -3.277 -1.565 }}
 
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}
 
[[Badness]] (Sintel): 2.37
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 12005/11979
 
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}
 
Optimal tunings:
* WE: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}})
* CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}})
 
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
 
Badness (Sintel): 1.01
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 364/363, 1716/1715
 
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}
 
Optimal tunings:
* WE: ~14/13 = 133.4245{{c}}, ~26/15 = 950.9362{{c}} (~105/104 = 16.9650{{c}})
* CWE: ~14/13 = 133.3333{{c}}, ~26/15 = 950.4364{{c}} (~99/98 = 17.1031{{c}})
 
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
 
Badness (Sintel): 0.922
 
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 781250000/771895089
 
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}
 
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}
 
[[Badness]] (Sintel): 2.44
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 236328125/234365481
 
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
 
Optimal tunings:
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}
 
Badness (Sintel): 2.96
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 831875/830466
 
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
 
Optimal tunings:
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
 
Badness (Sintel): 2.16
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
 
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
 
Optimal tunings:
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
 
Badness (Sintel): 1.96
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
 
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}
 
Optimal tunings:
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
 
Badness (Sintel): 1.71
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528
 
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
 
Optimal tunings:
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}
 
Badness (Sintel): 1.57
 
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


=Negri=
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
[[Comma]]s: 49/48, 225/224


[[POTE tuning|POTE temperament]]: ~15/14 = 125.608
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


[[Map]]: [&lt;1 2 2 3|, &lt;0 -4 3 -2|]
Optimal tunings:  
[[Wedgie]]: &lt;&lt;4 -3 2 -14 -8 13||
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
EDOs: [[9edo|9]], [[10edo|10]], [[19edo|19]], [[86edo|86]]
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}
Commas: 45/44, 49/48, 56/55


[[POTE tuning|POTE generator]]: ~15/14 = 126.474
Badness (Sintel): 1.50


Map: [&lt;1 2 2 3 4|, &lt;0 -4 3 -2 -5|]
==== 31-limit ====
EDOs: 9, 10, 19
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
[[Badness]]: 0.0262


==13-limit==
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
Commas: 45/44, 49/48, 56/55, 78/77


[[POTE tuning|POTE generator]]: ~14/13 = 126.431
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


Map: [&lt;1 2 2 3 4 4|, &lt;0 -4 3 -2 -5 -3|]
Optimal tunings:  
EDOs: 9, 10, 19
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


=Negril=
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}
Commas: 49/48, 100/99, 225/224


POTE generator: ~15/14 = 124.767
Badness (Sintel): 1.53


Map: [&lt;1 2 2 3 2|, &lt;0 -4 3 -2 14|]
=== Gracecord ===
EDOs: 19, [[29edo|29]], [[48edo|48df]], [[77edo|77cdf]]
Subgroup: 2.3.5.7.11
Badness: 0.0387


==13-limit==
Comma list: 225/224, 441/440, 109375/107811
Commas: 49/48, 65/64, 91/90, 875/858


POTE generator: ~15/14 = 124.716
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


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 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
Badness: 0.0244
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


=Negric=
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}
Commas: 33/32, 49/48, 77/75


POTE generator: ~15/14 = 127.039
Badness (Sintel): 2.21


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: 225/224, 364/363, 441/440, 6125/6084
Commas: 33/32, 49/48, 65/64, 91/90


POTE generator: ~15/14 = 127.039
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


Map: [&lt;1 2 2 3 3 4|, &lt;0 -4 3 -2 4 -3|]
Optimal tunings:  
EDOs: 9, 19, 85
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
Badness: 0.0202
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


=Negroni=
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}
Commas: 49/48, 55/54, 225/224


POTE generator: ~15/14 = 124.539
Badness (Sintel): 1.83


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


==13-limit==
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
Commas: 49/48, 55/54, 65/64, 91/90


POTE generator: ~15/14 = 124.545
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Map: [&lt;1 2 2 3 5 4|, &lt;0 -4 3 -2 -15 -3|]
Optimal tunings:  
EDOs: 10, 19e, 29, 77cdef
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
Badness: 0.0216
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


=Wizard=
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
Comma: 2197265625/2147483648


POTE generator: ~5/4 = 383.212
Badness (Sintel): 1.87


Map: [&lt;2 1 5|, &lt;0 6 -1|]
==== 19-limit ====
EDOs: 22, 50, 72, 166, 238c, 310c, 548bc
Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.3864


==7-limit==
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663
Commas: 225/224, 118098/117649


[[POTE tuning|POTE temperament]]: 383.256
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


Map: [&lt;2 1 5 2|, &lt;0 6 -1 10|]
Optimal tunings:  
Wedgie: &lt;&lt;12 -2 20 -31 -2 52||
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
EDOs: [[22edo|22]], [[50edo|50]], [[72edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238c]], [[310edo|310c]], [[382edo|382c]]
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}
Badness: 0.0408


==11-limit==
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
Commas: 225/224, 385/384, 4000/3993


[[POTE tuning|POTE temperament]]: 383.232
Badness (Sintel): 1.68


Map: [&lt;2 1 5 2 8|, &lt;0 6 -1 10 -3|]
== Alphorn ==
EDOs: 22, 50, 72, 94, [[166edo|166]], [[238edo|238c]], [[310edo|310c]]
[[Subgroup]]: 2.3.5.7
Badness: 0.0185


==Lizard==
[[Comma list]]: 225/224, 5764801/5668704
Commas: 225/224, 351/350, 364/363, 385/384


[[POTE tuning|POTE temperament]]: 383.389
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


Map: [&lt;2 1 5 2 8 11|, &lt;0 6 -1 10 -3 -10|]
[[Optimal tuning]]s:  
EDOs: 22, 50, 72, 122, 194df
* [[WE]]: ~2 = 1201.3004{{c}}, ~35/24 = 644.4767{{c}}
Badness: 0.0218
: [[error map]]: {{val| +1.300 +0.569 -2.195 -2.043 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~35/24 = 643.8137{{c}}
: error map: {{val| 0.000 -0.936 -5.382 -4.924 }}


===17-limit===
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}
Commas: 221/220, 273/272, 289/288, 351/350, 375/374


[[POTE tuning|POTE temperament]]: 383.381
[[Badness]] (Sintel): 3.27


Map: [&lt;2 1 5 2 8 11 6|, &lt;0 6 -1 10 -3 -10 6|]
=== 11-limit ===
EDOs: 22, 50, 72, 122g, 194dfg
Subgroup: 2.3.5.7.11
Badness: 0.0145


===19-limit===
Comma list: 225/224, 385/384, 12250/11979
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


[[POTE tuning|POTE generator]]: ~5/4 = 383.477
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


Map: [&lt;2 1 5 2 8 11 6 2|, &lt;0 6 -1 10 -3 -10 6 18|]
Optimal tunings:  
EDOs: 22h, 50, 72, [[122edo|122g]], [[194edo|194dfg]]
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
Badness: 0.0157
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


==Gizzard==
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}
Commas: 225/224, 385/384, 325/324, 1573/1568


POTE generator: ~5/4 = 383.170
Badness (Sintel): 2.43


Map: [&lt;2 1 5 2 8 -2|, &lt;0 6 -1 10 -3 26|]
== Misneb ==
EDOs: [[22edo|22]], [[26edo|72]], [[94edo|94]], [[166edo|166]], [[238edo|238cf]]
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''
Badness: 0.0203


===17-limit===
[[Subgroup]]: 2.3.5.7
Commas: 225/224, 289/288, 325/324, 375/374, 385/384


POTE generator: ~5/4 = 383.175
[[Comma list]]: 225/224, 4194304/4117715


Map: [&lt;2 1 5 2 8 -2 6|, &lt;0 6 -1 10 -3 26 6|]
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
EDOs: 22f, 72, 94, [[166edo|166g]], [[238edo|238cfg]]
: mapping generators: ~2, ~15/8
Badness: 0.0136


===19-limit===  
[[Optimal tuning]]s:
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399
* [[WE]]: ~2 = 1199.7642{{c}}, ~15/8 = 1086.5513{{c}}
: [[error map]]: {{val| -0.236 -0.856 -1.569 +4.041 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1086.7633{{c}}
: error map: {{val| 0.000 -0.506 -0.999 +4.701 }}


POTE generator: ~5/4 = 383.138
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


Map: [&lt;2 1 5 2 8 -2 6 15|, &lt;0 6 -1 10 -3 26 6 -18|]
[[Badness]] (Sintel): 3.57
EDOs: 22f, 72, 94, 166g
Badness: 0.0148


==Mage==  
=== 11-limit ===
Commas: 99/98, 176/175, 1331/1296
Subgroup: 2.3.5.7.11


POTE generator: ~55/48 = 216.876
Comma list: 99/98, 176/175, 1310720/1294139


Map: [&lt;2 1 5 2 4|, &lt;0 6 -1 10 8|]
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}
EDOs: 22, 72e, 94e
Badness: 0.0578


=Triton=  
Optimal tunings:
Commas: 225/224, 1029/1000
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


[[POTE tuning|POTE temperament]]: ~7/5 = 568.865
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Map: [&lt;1 0 6 7|, &lt;0 3 -7 -8|]
Badness (Sintel): 2.82
Wedgie: &lt;&lt;3 -7 -8 -18 -21 1||
EDOs: 17d, 19, 78bd, 97bd
Badness: 0.0592


==11-limit==  
==== 13-limit ====
Commas: 45/44, 56/55, 1029/1000
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 569.144
Comma list: 99/98, 176/175, 640/637, 847/845


Map: [&lt;1 0 6 7 4|, &lt;0 3 -7 -8 -1|]
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}
EDOs: 19, 59bde, 78bde, 97bde
Badness: 0.0457


=Tritonic=  
Optimal tunings:
Comnma: 553584375/536870912
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


POTE generator: ~45/32 = 580.219
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


Map: [&lt;1 4 -3|, &lt;0 -5 11|]
Badness (Sintel): 1.88
EDOs: 29, 31, 60, 91
Badness: 0.3785


==7-limit==  
=== Musneb ===
Commas: 225/224, 50421/50000
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE temperament]]: 580.286
Comma list: 225/224, 385/384, 66550/64827


Map: [&lt;1 4 -3 -3|, &lt;0 -5 11 12|]
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
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 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


[[POTE tuning|POTE generator]]: ~7/5 = 580.267
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


Map: [&lt;1 4 -3 -3 2|, &lt;0 -5 11 12 3|]
Badness (Sintel): 2.89
EDOs: 29, 31, 60e
Badness: 0.0237


==13-limit==  
== Untriton ==
Commas: 105/104, 121/120, 196/195, 275/273
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


POTE generator: ~7/5 = 580.108
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.  


Map: [&lt;1 4 -3 -3 2 -5|, &lt;0 -5 11 12 3 18|]
[[Subgroup]]: 2.3.5.7
EDOs: 29, 31, 60e, 151cde
Badness: 0.0230


==Tritoni==
[[Comma list]]: 225/224, 125000000/121060821
Commas: 225/224, 385/384, 27783/27500


POTE generator: ~7/5 = 580.389
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


Map: [&lt;1 4 -3 -3 17|, &lt;0 -5 11 12 -28|]
[[Optimal tuning]]s:  
EDOs: 31, 91, 122
* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
Badness: 0.0455
: [[error map]]: {{val| -0.172 +0.002 -2.533 +3.511 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}}
: error map: {{val| 0.000 +0.298 -2.181 +3.946 }}


=Merman=
{{Optimal ET sequence|legend=1| 51, 53 }}
Comma: 1121008359375/1099511627776


POTE generator: ~45/32 = 585.634
[[Badness]] (Sintel): 3.64


Map: [&lt;1 5 -5|, &lt;0 -7 15|]
=== 11-limit ===
EDOs: 41, 84, 125, 209, 334c, 543bc
Subgroup: 2.3.5.7.11
Badness: 0.4538


==7-limit==
Comma list: 121/120, 225/224, 22000/21609
Commas: 225/224, 2500000/2470629


POTE generator: ~7/5 = 585.585
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


Map: [&lt;1 5 -5 -5|, &lt;0 -7 15 16|]
Optimal tunings:  
Wedgie: &lt;&lt;7 -15 -16 -40 -45 5||
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
EDOs: 41, 84, 125
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}
Badness: 0.0551


==11-limit==
{{Optimal ET sequence|legend=0| 51, 53 }}
Commas: 225/224, 441/440, 1344/1331


POTE generator: ~7/5 = 585.606
Badness (Sintel): 2.46


Map: [&lt;1 5 -5 -5 2|, &lt;0 -7 15 16 3]]
=== 13-limit ===
EDOs: 41, 84, 125e
Subgroup: 2.3.5.7.11.13
Badness: 0.0364


==13-limit==
Comma list: 121/120, 225/224, 275/273, 1040/1029
Commas: 144/143, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 585.657
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


Map: [&lt;1 5 -5 -5 2 12|, &lt;0 -7 15 16 3 -17|]
Optimal tunings:  
EDOs: 41, 84, 125e, 209ef, 293ef
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
Badness: 0.0275
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


=Septimin=
{{Optimal ET sequence|legend=0| 51f, 53 }}
Comma: 35595703125/34359738368


POTE generator: ~75/64 = 263.658
Badness (Sintel): 1.96


Map: [&lt;1 4 1|, &lt;0 -11 6|]
== Naiadical ==
EDOs: 9, 32, 41, 91, 132, 223c, 355bc
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.  
Badness: 0.6709


==7-limit==
[[Subgroup]]: 2.3.5.7
Commas: 225/224, 84035/82944


[[POTE tuning|POTE temperament]]: 263.632
[[Comma list]]: 225/224, 823543/800000


Map: [&lt;1 4 1 5|, &lt;0 -11 6 -10|]
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
Wedgie: &lt;&lt;11 -6 10 -35 -15 40||
: mapping generators: ~2, ~32/21
EDOs: 9, [[32edo|32]], 41, 91, [[132edo|132]]


==11-limit==  
[[Optimal tuning]]s:
Commas: 225/224, 385/384, 2401/2376
* [[WE]]: ~2 = 1202.1198{{c}}, ~32/21 = 745.4675{{c}}
: [[error map]]: {{val| +2.120 -1.227 +0.459 -4.423 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~32/21 = 744.1318{{c}}
: error map: {{val| 0.000 -4.769 -4.159 -10.144 }}


[[POTE tuning|POTE temperament]]: 263.634
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


Map: [&lt;1 4 1 5 5|, &lt;0 -11 6 -10 -7|]
[[Badness]] (Sintel): 3.67
EDOs: 9, 32, 41, 91, 223cdef


==13-limit==  
=== 11-limit ===
Commas: 105/104, 144/143, 196/195, 245/242
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE temperament]]: 263.700
Comma list: 225/224, 245/242, 1617/1600


Map: [&lt;1 4 1 5 5 7|, &lt;0 -11 6 -10 -7 -15|]
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}
EDOs: 9, 41, 91


=Slender=  
Optimal tunings:
Commas: 225/224, 589824/588245
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


[[POTE tuning|POTE temperament]]: ~49/48 = 38.413
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Map: [&lt;1 2 2 3|, &lt;0 -13 10 -6|]
Badness (Sintel): 2.00
Wedgie: &lt;&lt;13 -10 6 -46 -27 42||
EDOs: 31, 94, 125
Badness: 0.0569


==11-limit==  
=== 13-limit ===
Commas: 225/224, 385/384, 1331/1323
Subgroup: 2.3.5.7.11.13


[[POTE tuning|POTE temperament]]: ~49/48 = 38.387
Comma list: 105/104, 196/195, 245/242, 1001/1000


Map: [&lt;1 2 2 3 4|, &lt;0 -13 10 -6 -17|]
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}
EDOs: 31, 63, 94, 125
Badness: 25.342


==13-limit==  
Optimal tunings:
Commas: 225/224, 275/273, 385/384, 1331/1323
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


POTE generator: ~49/48 = 38.314
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Map: [&lt;1 2 2 3 4 3|, &lt;0 -13 10 -6 -17 22|]
Badness (Sintel): 1.43
EDOs: 31, 63, 94
Badness: 25.913


=Marvo=  
=== 17-limit ===
Commas: 225/224, 78125000/78121827
Subgroup: 2.3.5.7.11.13.17


[[POTE tuning|POTE temperament]]: ~27/20 = 516.694
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


Map: [&lt;1 5 12 29|, &lt;0 -6 -17 -46|]
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}
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 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


[[POTE tuning|POTE temperament]]: ~27/20 = 516.699
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Map: [&lt;1 5 12 29 12|, &lt;0 -6 -17 -46 -15|]
Badness (Sintel): 1.26
EDOs: 65, 72, 281, 497


==13-limit==  
== Quintannic ==
Commas: 243/242, 225/224, 351/350, 1625/1617
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


POTE generator: ~27/20 = 516.730
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 5 12 29 12 39|, &lt;0 -6 -17 -46 -15 -62|]
[[Comma list]]: 225/224, 9805926501/9765625000
EDOs: 72, 137, 209, 281f, 490bcf
Badness: 0.0269


=Marvolo=
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
Commas: 225/224, 156250000/155649627
: mapping generators: ~2, ~10000/9261


POTE generator: ~21/20 = 83.348
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.9803{{c}}, ~10000/9261 = 139.9522{{c}}
: [[error map]]: {{val| +0.980 -1.214 -0.313 -0.243 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10000/9261 = 139.8184{{c}}
: error map: {{val| 0.000 -2.863 -2.136 -2.287 }}


Map: [&lt;1 2 1 1|, &lt;0 -6 19 26|]
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}
Wedgie: &lt;&lt;6 -19 -26 -44 -58 -7||
EDOs: 29, 43, 72, 619bcd, 691bcd
Badness: 0.0833


==11-limit==
[[Badness]] (Sintel): 3.81
Commas: 225/224, 441/440, 4000/3993


POTE generator: ~21/20 = 83.340
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 1 1 2|, &lt;0 -6 19 26 21|]
Comma list: 225/224, 441/440, 43923/43750
EDOs: 29, 43, 72
Badness: 0.0290


==13-limit==
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}
Commas: 169/168, 225/224, 364/363, 441/440


POTE generator: ~21/20 = 83.330
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


Map: [&lt;1 2 1 1 2 3|, &lt;0 -6 19 26 21 10|]
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}
EDOs: 29, 43, 72, 115f
Badness: 0.0215


=Amavil=
Badness (Sintel): 1.74
Commas: 225/224, 17496/16807


POTE generator: ~48/35 = 529.979
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 6 1 9|, &lt;0 -10 3 -14|]
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
Wedgie: &lt;&lt;10 -3 14 -28 -6 41||
EDOs: 9, 34d, 43, 77d
Badness: 0.1096


==11-limit==
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
Commas: 99/98, 176/175, 864/847


POTE generator: ~15/11 = 529.974
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


Map: [&lt;1 6 1 9 7|, &lt;0 -10 3 -14 -8|]
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
EDOs: 9, 34d, 43, 77de
Badness: 0.0426


==13-limit==
Badness (Sintel): 1.35
Commas: 78/77, 99/98, 144/143, 176/175


POTE generator: ~15/11 = 529.951
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 6 1 9 7 9|, &lt;0 -10 3 -14 -8 -12|]
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
EDOs: 9, 34d, 43, 77de
Badness: 0.0258


=Enneaportent=
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
Commas: 225/224, 40353607/40310784


POTE generator: ~5/4 = 383.165
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


Map: [&lt;9 0 28 11|, &lt;0 2 -1 2|]
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
Wedgie: &lt;&lt;18 -9 18 -56 -22 67||
EDOs: 9, 54, 63, 72, 495bcd
Badness: 0.0937


==11-limit==
Badness (Sintel): 1.17
Commas: 225/224, 385/384, 12005/11979


POTE generator: ~5/4 = 383.146
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Map: [&lt;9 0 28 11 24|, &lt;0 2 -1 2 1|]
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.  
EDOs: 9, 54, 63, 72, 423cd, 495bcd
Badness: 0.0304


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 169/168 225/224 364/363 1716/1715


POTE generator: ~5/4 = 383.047
[[Comma list]]: 225/224, 5971968/5764801


Map: [&lt;9 0 28 11 24 19|, &lt;0 2 -1 2 1 2|]
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
EDOs: 9, 54, 63, 72, 279cf
: mapping generators: ~2401/1728, ~35/32
Badness: 0.0223


=Submajor=  
[[Optimal tuning]]s:
Comma: 69198046875/68719476736
* [[WE]]: ~2401/1728 = 599.7132{{c}}, ~35/32 = 162.5806{{c}}
: [[error map]]: {{val| -0.574 -1.597 -0.937 +5.510 }}
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~35/32 = 162.6388{{c}}
: error map: {{val| 0.000 -0.844 +0.492 +7.007 }}


POTE generator: ~10125/8192 = 362.321
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


Map: [&lt;1 4 -1|, &lt;0 -8 11|]
[[Badness]] (Sintel): 4.53
EDOs: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc
Badness: 0.1302


==7-limit==  
=== 11-limit ===
Commas: 225/224, 51200/50421
Subgroup: 2.3.5.7.11


POTE generator: ~49/40 = 362.255
Comma list: 99/98, 176/175, 65536/65219


Map: [&lt;1 4 -1 1|, &lt;0 -8 11 6|]
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}
Wedgie: &lt;&lt;8 -11 -6 -36 -32 17||
EDOs: 10, 33, 43, 53
Badness: 0.0605


==11-limit==  
Optimal tunings:
Commas: 225/224, 385/384, 6655/6561
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


POTE generator: ~27/22 = 362.101
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


Map: [&lt;1 4 -1 1 11|, &lt;0 -8 11 6 -25|]
Badness (Sintel): 2.26
EDOs: 10, 53, 116, 169de, 285cde
Badness: 0.0506


==13-limit==  
== Tertiosec ==
Commas: 169/168, 225/224, 275/273, 385/384
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


POTE generator: ~16/13 = 362.105
Tertiosec may be described as the {{nowrap| 21 & 75 }} temperament. It was initially named ''tertiomar'' by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, but was changed to ''tertiosec'' in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.  


Map: [&lt;1 4 -1 1 11 4|, &lt;0 -8 11 6 -25 -1|]
[[Subgroup]]: 2.3.5.7
EDOs: 10, 53, 116, 169de, 285cdef
Badness: 0.0277


==Interpental==
[[Comma list]]: 225/224, 14495514624/13841287201
Commas: 99/98, 176/175, 51200/50421


POTE generator: ~49/40 = 362.418
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


Map: [&lt;1 4 -1 1 -5|, &lt;0 -8 11 6 28|]
[[Optimal tuning]]s:  
EDOs: 43, 53, 96, 149d
* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
Badness: 0.0518
: [[error map]]: {{val| -0.523 -1.044 -1.549 +5.138 }}
* [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}}
: error map: {{val| 0.000 -0.284 -0.276 +6.592 }}


===13-limit===
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}
Commas: 99/98, 169/168, 176/175, 640/637


POTE generator: ~16/13 = 362.402
[[Badness]] (Sintel): 10.9


Map: [&lt;1 4 -1 1 -5 4|, &lt;0 -8 11 6 28 -1|]
=== 11-limit ===
EDOs: 43, 53, 96, 149d
Subgroup: 2.3.5.7.11
Badness: 0.0297


=Alphorn=
Comma list: 225/224, 3840/3773, 12005/11979
Commas: 225/224, 5764801/5668704


POTE generator: ~48/35 = 556.221
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


Map: [&lt;1 9 0 13|, &lt;0 -16 5 -22|]
Optimal tunings:  
Wedgie: &lt;&lt;16 -5 22 -45 -10 65||
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
EDOs: 28d, 41, 151cd, 192cd, 233cd
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}
Badness: 0.1293


==11-limit==
{{Optimal ET sequence|legend=0| 21, 54, 75e }}
Commas: 225/224, 385/384, 12250/11979


POTE generator: ~11/8 = 556.144
Badness (Sintel): 5.74


Map: [&lt;1 9 0 13 3|, &lt;0 -16 5 -22 1|]
== References ==
EDOs: 41, 315cde
Badness: 0.0735


</pre></div>
[[Category:Temperament collections]]
<h4>Original HTML content:</h4>
[[Category:Marvel temperaments| ]] <!-- main article -->
<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:114:&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:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Negri"&gt;Negri&lt;/a&gt;&lt;/div&gt;
[[Category:Rank 2]]
&lt;!-- ws:end:WikiTextTocRule:115 --&gt;&lt;!-- ws:start:WikiTextTocRule:116: --&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:116 --&gt;&lt;!-- ws:start:WikiTextTocRule:117: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Negri-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:117 --&gt;&lt;!-- ws:start:WikiTextTocRule:118: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Negril"&gt;Negril&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:118 --&gt;&lt;!-- ws:start:WikiTextTocRule:119: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Negril-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:119 --&gt;&lt;!-- ws:start:WikiTextTocRule:120: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Negric"&gt;Negric&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:120 --&gt;&lt;!-- ws:start:WikiTextTocRule:121: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Negric-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:121 --&gt;&lt;!-- ws:start:WikiTextTocRule:122: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Negroni"&gt;Negroni&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:122 --&gt;&lt;!-- ws:start:WikiTextTocRule:123: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Negroni-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:123 --&gt;&lt;!-- ws:start:WikiTextTocRule:124: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Wizard"&gt;Wizard&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:124 --&gt;&lt;!-- ws:start:WikiTextTocRule:125: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Wizard-7-limit"&gt;7-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:125 --&gt;&lt;!-- ws:start:WikiTextTocRule:126: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Wizard-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:126 --&gt;&lt;!-- ws:start:WikiTextTocRule:127: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Wizard-Lizard"&gt;Lizard&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:127 --&gt;&lt;!-- ws:start:WikiTextTocRule:128: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Wizard-Lizard-17-limit"&gt;17-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:128 --&gt;&lt;!-- ws:start:WikiTextTocRule:129: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Wizard-Lizard-19-limit"&gt;19-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:129 --&gt;&lt;!-- ws:start:WikiTextTocRule:130: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Wizard-Gizzard"&gt;Gizzard&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:130 --&gt;&lt;!-- ws:start:WikiTextTocRule:131: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Wizard-Gizzard-17-limit"&gt;17-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:131 --&gt;&lt;!-- ws:start:WikiTextTocRule:132: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Wizard-Gizzard-19-limit"&gt;19-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:132 --&gt;&lt;!-- ws:start:WikiTextTocRule:133: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Wizard-Mage"&gt;Mage&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:133 --&gt;&lt;!-- ws:start:WikiTextTocRule:134: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Triton"&gt;Triton&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:134 --&gt;&lt;!-- ws:start:WikiTextTocRule:135: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Triton-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:135 --&gt;&lt;!-- ws:start:WikiTextTocRule:136: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Tritonic"&gt;Tritonic&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:136 --&gt;&lt;!-- ws:start:WikiTextTocRule:137: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Tritonic-7-limit"&gt;7-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:137 --&gt;&lt;!-- ws:start:WikiTextTocRule:138: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Tritonic-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:138 --&gt;&lt;!-- ws:start:WikiTextTocRule:139: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Tritonic-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:139 --&gt;&lt;!-- ws:start:WikiTextTocRule:140: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Tritonic-Tritoni"&gt;Tritoni&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:140 --&gt;&lt;!-- ws:start:WikiTextTocRule:141: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Merman"&gt;Merman&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:141 --&gt;&lt;!-- ws:start:WikiTextTocRule:142: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Merman-7-limit"&gt;7-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:142 --&gt;&lt;!-- ws:start:WikiTextTocRule:143: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Merman-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:143 --&gt;&lt;!-- ws:start:WikiTextTocRule:144: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Merman-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:144 --&gt;&lt;!-- ws:start:WikiTextTocRule:145: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Septimin"&gt;Septimin&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:145 --&gt;&lt;!-- ws:start:WikiTextTocRule:146: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Septimin-7-limit"&gt;7-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:146 --&gt;&lt;!-- ws:start:WikiTextTocRule:147: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Septimin-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:147 --&gt;&lt;!-- ws:start:WikiTextTocRule:148: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Septimin-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:148 --&gt;&lt;!-- ws:start:WikiTextTocRule:149: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Slender"&gt;Slender&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:149 --&gt;&lt;!-- ws:start:WikiTextTocRule:150: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Slender-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:150 --&gt;&lt;!-- ws:start:WikiTextTocRule:151: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Slender-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:151 --&gt;&lt;!-- ws:start:WikiTextTocRule:152: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Marvo"&gt;Marvo&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:152 --&gt;&lt;!-- ws:start:WikiTextTocRule:153: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Marvo-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:153 --&gt;&lt;!-- ws:start:WikiTextTocRule:154: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Marvo-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:154 --&gt;&lt;!-- ws:start:WikiTextTocRule:155: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Marvolo"&gt;Marvolo&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:155 --&gt;&lt;!-- ws:start:WikiTextTocRule:156: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Marvolo-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:156 --&gt;&lt;!-- ws:start:WikiTextTocRule:157: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Marvolo-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:157 --&gt;&lt;!-- ws:start:WikiTextTocRule:158: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Amavil"&gt;Amavil&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:158 --&gt;&lt;!-- ws:start:WikiTextTocRule:159: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Amavil-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:159 --&gt;&lt;!-- ws:start:WikiTextTocRule:160: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Amavil-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:160 --&gt;&lt;!-- ws:start:WikiTextTocRule:161: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Enneaportent"&gt;Enneaportent&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:161 --&gt;&lt;!-- ws:start:WikiTextTocRule:162: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Enneaportent-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:162 --&gt;&lt;!-- ws:start:WikiTextTocRule:163: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Enneaportent-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:163 --&gt;&lt;!-- ws:start:WikiTextTocRule:164: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Submajor"&gt;Submajor&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:164 --&gt;&lt;!-- ws:start:WikiTextTocRule:165: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Submajor-7-limit"&gt;7-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:165 --&gt;&lt;!-- ws:start:WikiTextTocRule:166: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Submajor-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:166 --&gt;&lt;!-- ws:start:WikiTextTocRule:167: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Submajor-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:167 --&gt;&lt;!-- ws:start:WikiTextTocRule:168: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Submajor-Interpental"&gt;Interpental&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:168 --&gt;&lt;!-- ws:start:WikiTextTocRule:169: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Submajor-Interpental-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:169 --&gt;&lt;!-- ws:start:WikiTextTocRule:170: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Alphorn"&gt;Alphorn&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:170 --&gt;&lt;!-- ws:start:WikiTextTocRule:171: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Alphorn-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:171 --&gt;&lt;!-- ws:start:WikiTextTocRule:172: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:172 --&gt;This page discusses some of the temperaments tempering out |-5 2 2 -1&amp;gt; =&lt;a class="wiki_link" href="/225_224"&gt; 225/224&lt;/a&gt;, the &lt;a class="wiki_link" href="/Marvel%20family"&gt;marvel&lt;/a&gt; comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton, merman and marvo. Considered elsewhere are &lt;a class="wiki_link" href="/Meantone%20family"&gt;meantone&lt;/a&gt;, &lt;a class="wiki_link" href="/Miracle"&gt;miracle&lt;/a&gt;, &lt;a class="wiki_link" href="/Magic%20family"&gt;magic&lt;/a&gt;, &lt;a class="wiki_link" href="/Pajara"&gt;pajara&lt;/a&gt;, &lt;a class="wiki_link" href="/Orwell"&gt;orwell&lt;/a&gt;, &lt;a class="wiki_link" href="/Kleismic%20family"&gt;catakleismic&lt;/a&gt;, &lt;a class="wiki_link" href="/Garibaldi%20temperament"&gt;garibaldi&lt;/a&gt;, &lt;a class="wiki_link" href="/Augmented%20family"&gt;august&lt;/a&gt;, &lt;a class="wiki_link" href="/Pythagorean%20family"&gt;compton&lt;/a&gt;, &lt;a class="wiki_link" href="/Dicot%20family"&gt;sharp&lt;/a&gt;, &lt;a class="wiki_link" href="/Escapade%20family"&gt;escapade&lt;/a&gt; and &lt;a class="wiki_link" href="/Pelogic%20family"&gt;mavila&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
Since (5/4)^2 = 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.&lt;br /&gt;
&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
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 (16/9)/(5/4)=64/45. Setting this equal to 10/7 gives (10/7)/(64/45)=225/224.)&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Negri"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Negri&lt;/h1&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;s: 49/48, 225/224&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: ~15/14 = 125.608&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Map"&gt;Map&lt;/a&gt;: [&amp;lt;1 2 2 3|, &amp;lt;0 -4 3 -2|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Wedgie"&gt;Wedgie&lt;/a&gt;: &amp;lt;&amp;lt;4 -3 2 -14 -8 13||&lt;br /&gt;
EDOs: &lt;a class="wiki_link" href="/9edo"&gt;9&lt;/a&gt;, &lt;a class="wiki_link" href="/10edo"&gt;10&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/86edo"&gt;86&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Negri-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
Commas: 45/44, 49/48, 56/55&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/14 = 126.474&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 2 3 4|, &amp;lt;0 -4 3 -2 -5|]&lt;br /&gt;
EDOs: 9, 10, 19&lt;br /&gt;
&lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0262&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Negri-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
Commas: 45/44, 49/48, 56/55, 78/77&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~14/13 = 126.431&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 2 3 4 4|, &amp;lt;0 -4 3 -2 -5 -3|]&lt;br /&gt;
EDOs: 9, 10, 19&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Negril"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Negril&lt;/h1&gt;
Commas: 49/48, 100/99, 225/224&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~15/14 = 124.767&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 2 3 2|, &amp;lt;0 -4 3 -2 14|]&lt;br /&gt;
EDOs: 19, &lt;a class="wiki_link" href="/29edo"&gt;29&lt;/a&gt;, &lt;a class="wiki_link" href="/48edo"&gt;48df&lt;/a&gt;, &lt;a class="wiki_link" href="/77edo"&gt;77cdf&lt;/a&gt;&lt;br /&gt;
Badness: 0.0387&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Negril-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;13-limit&lt;/h2&gt;
Commas: 49/48, 65/64, 91/90, 875/858&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~15/14 = 124.716&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 2 3 2 4|, &amp;lt;0 -4 3 -2 14 -3|]&lt;br /&gt;
EDOs: 9, 10, 19, 29, 48, 77&lt;br /&gt;
Badness: 0.0244&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Negric"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Negric&lt;/h1&gt;
Commas: 33/32, 49/48, 77/75&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~15/14 = 127.039&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 2 3 3|, &amp;lt;0 -4 3 -2 4|]&lt;br /&gt;
EDOs: 9, 19, &lt;a class="wiki_link" href="/85edo"&gt;85&lt;/a&gt;&lt;br /&gt;
Badness: 0.0306&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Negric-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit&lt;/h2&gt;
Commas: 33/32, 49/48, 65/64, 91/90&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~15/14 = 127.039&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 2 3 3 4|, &amp;lt;0 -4 3 -2 4 -3|]&lt;br /&gt;
EDOs: 9, 19, 85&lt;br /&gt;
Badness: 0.0202&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Negroni"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Negroni&lt;/h1&gt;
Commas: 49/48, 55/54, 225/224&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~15/14 = 124.539&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 2 3 5|, &amp;lt;0 -4 3 -2 -15|]&lt;br /&gt;
EDOs: 10, 19e, 29, 77cde&lt;br /&gt;
Badness: 0.0353&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Negroni-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;13-limit&lt;/h2&gt;
Commas: 49/48, 55/54, 65/64, 91/90&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~15/14 = 124.545&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 2 3 5 4|, &amp;lt;0 -4 3 -2 -15 -3|]&lt;br /&gt;
EDOs: 10, 19e, 29, 77cdef&lt;br /&gt;
Badness: 0.0216&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Wizard"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Wizard&lt;/h1&gt;
Comma: 2197265625/2147483648&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~5/4 = 383.212&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 1 5|, &amp;lt;0 6 -1|]&lt;br /&gt;
EDOs: 22, 50, 72, 166, 238c, 310c, 548bc&lt;br /&gt;
Badness: 0.3864&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Wizard-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;7-limit&lt;/h2&gt;
Commas: 225/224, 118098/117649&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 383.256&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 1 5 2|, &amp;lt;0 6 -1 10|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;12 -2 20 -31 -2 52||&lt;br /&gt;
EDOs: &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/50edo"&gt;50&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/94edo"&gt;94&lt;/a&gt;, &lt;a class="wiki_link" href="/166edo"&gt;166&lt;/a&gt;, &lt;a class="wiki_link" href="/238edo"&gt;238c&lt;/a&gt;, &lt;a class="wiki_link" href="/310edo"&gt;310c&lt;/a&gt;, &lt;a class="wiki_link" href="/382edo"&gt;382c&lt;/a&gt;&lt;br /&gt;
Badness: 0.0408&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Wizard-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;11-limit&lt;/h2&gt;
Commas: 225/224, 385/384, 4000/3993&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 383.232&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 1 5 2 8|, &amp;lt;0 6 -1 10 -3|]&lt;br /&gt;
EDOs: 22, 50, 72, 94, &lt;a class="wiki_link" href="/166edo"&gt;166&lt;/a&gt;, &lt;a class="wiki_link" href="/238edo"&gt;238c&lt;/a&gt;, &lt;a class="wiki_link" href="/310edo"&gt;310c&lt;/a&gt;&lt;br /&gt;
Badness: 0.0185&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Wizard-Lizard"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Lizard&lt;/h2&gt;
Commas: 225/224, 351/350, 364/363, 385/384&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 383.389&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 1 5 2 8 11|, &amp;lt;0 6 -1 10 -3 -10|]&lt;br /&gt;
EDOs: 22, 50, 72, 122, 194df&lt;br /&gt;
Badness: 0.0218&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Wizard-Lizard-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;17-limit&lt;/h3&gt;
Commas: 221/220, 273/272, 289/288, 351/350, 375/374&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 383.381&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 1 5 2 8 11 6|, &amp;lt;0 6 -1 10 -3 -10 6|]&lt;br /&gt;
EDOs: 22, 50, 72, 122g, 194dfg&lt;br /&gt;
Badness: 0.0145&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Wizard-Lizard-19-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;19-limit&lt;/h3&gt;
Commas: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~5/4 = 383.477&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 1 5 2 8 11 6 2|, &amp;lt;0 6 -1 10 -3 -10 6 18|]&lt;br /&gt;
EDOs: 22h, 50, 72, &lt;a class="wiki_link" href="/122edo"&gt;122g&lt;/a&gt;, &lt;a class="wiki_link" href="/194edo"&gt;194dfg&lt;/a&gt;&lt;br /&gt;
Badness: 0.0157&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Wizard-Gizzard"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Gizzard&lt;/h2&gt;
Commas: 225/224, 385/384, 325/324, 1573/1568&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~5/4 = 383.170&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 1 5 2 8 -2|, &amp;lt;0 6 -1 10 -3 26|]&lt;br /&gt;
EDOs: &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/26edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/94edo"&gt;94&lt;/a&gt;, &lt;a class="wiki_link" href="/166edo"&gt;166&lt;/a&gt;, &lt;a class="wiki_link" href="/238edo"&gt;238cf&lt;/a&gt;&lt;br /&gt;
Badness: 0.0203&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="Wizard-Gizzard-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;17-limit&lt;/h3&gt;
Commas: 225/224, 289/288, 325/324, 375/374, 385/384&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~5/4 = 383.175&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 1 5 2 8 -2 6|, &amp;lt;0 6 -1 10 -3 26 6|]&lt;br /&gt;
EDOs: 22f, 72, 94, &lt;a class="wiki_link" href="/166edo"&gt;166g&lt;/a&gt;, &lt;a class="wiki_link" href="/238edo"&gt;238cfg&lt;/a&gt;&lt;br /&gt;
Badness: 0.0136&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc17"&gt;&lt;a name="Wizard-Gizzard-19-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;19-limit&lt;/h3&gt;
Commas: 225/224, 385/384, 325/324, 595/594, 375/374, 400/399&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~5/4 = 383.138&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 1 5 2 8 -2 6 15|, &amp;lt;0 6 -1 10 -3 26 6 -18|]&lt;br /&gt;
EDOs: 22f, 72, 94, 166g&lt;br /&gt;
Badness: 0.0148&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Wizard-Mage"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Mage&lt;/h2&gt;
Commas: 99/98, 176/175, 1331/1296&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~55/48 = 216.876&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 1 5 2 4|, &amp;lt;0 6 -1 10 8|]&lt;br /&gt;
EDOs: 22, 72e, 94e&lt;br /&gt;
Badness: 0.0578&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc19"&gt;&lt;a name="Triton"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;Triton&lt;/h1&gt;
Commas: 225/224, 1029/1000&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: ~7/5 = 568.865&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 6 7|, &amp;lt;0 3 -7 -8|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;3 -7 -8 -18 -21 1||&lt;br /&gt;
EDOs: 17d, 19, 78bd, 97bd&lt;br /&gt;
Badness: 0.0592&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Triton-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;11-limit&lt;/h2&gt;
Commas: 45/44, 56/55, 1029/1000&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~7/5 = 569.144&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 6 7 4|, &amp;lt;0 3 -7 -8 -1|]&lt;br /&gt;
EDOs: 19, 59bde, 78bde, 97bde&lt;br /&gt;
Badness: 0.0457&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc21"&gt;&lt;a name="Tritonic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;Tritonic&lt;/h1&gt;
Comnma: 553584375/536870912&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~45/32 = 580.219&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 -3|, &amp;lt;0 -5 11|]&lt;br /&gt;
EDOs: 29, 31, 60, 91&lt;br /&gt;
Badness: 0.3785&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Tritonic-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;7-limit&lt;/h2&gt;
Commas: 225/224, 50421/50000&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 580.286&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 -3 -3|, &amp;lt;0 -5 11 12|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;5 -11 -12 -29 -33 3||&lt;br /&gt;
EDOs: &lt;a class="wiki_link" href="/29edo"&gt;29&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/60edo"&gt;60&lt;/a&gt;, &lt;a class="wiki_link" href="/91edo"&gt;91&lt;/a&gt;, &lt;a class="wiki_link" href="/122edo"&gt;122&lt;/a&gt;, 213bcd&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc23"&gt;&lt;a name="Tritonic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;11-limit&lt;/h2&gt;
Commas: 121/120, 225/224, 441/440&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~7/5 = 580.267&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 -3 -3 2|, &amp;lt;0 -5 11 12 3|]&lt;br /&gt;
EDOs: 29, 31, 60e&lt;br /&gt;
Badness: 0.0237&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc24"&gt;&lt;a name="Tritonic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;13-limit&lt;/h2&gt;
Commas: 105/104, 121/120, 196/195, 275/273&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~7/5 = 580.108&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 -3 -3 2 -5|, &amp;lt;0 -5 11 12 3 18|]&lt;br /&gt;
EDOs: 29, 31, 60e, 151cde&lt;br /&gt;
Badness: 0.0230&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc25"&gt;&lt;a name="Tritonic-Tritoni"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;Tritoni&lt;/h2&gt;
Commas: 225/224, 385/384, 27783/27500&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~7/5 = 580.389&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 -3 -3 17|, &amp;lt;0 -5 11 12 -28|]&lt;br /&gt;
EDOs: 31, 91, 122&lt;br /&gt;
Badness: 0.0455&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc26"&gt;&lt;a name="Merman"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;Merman&lt;/h1&gt;
Comma: 1121008359375/1099511627776&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~45/32 = 585.634&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 5 -5|, &amp;lt;0 -7 15|]&lt;br /&gt;
EDOs: 41, 84, 125, 209, 334c, 543bc&lt;br /&gt;
Badness: 0.4538&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc27"&gt;&lt;a name="Merman-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;7-limit&lt;/h2&gt;
Commas: 225/224, 2500000/2470629&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~7/5 = 585.585&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 5 -5 -5|, &amp;lt;0 -7 15 16|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;7 -15 -16 -40 -45 5||&lt;br /&gt;
EDOs: 41, 84, 125&lt;br /&gt;
Badness: 0.0551&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc28"&gt;&lt;a name="Merman-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;11-limit&lt;/h2&gt;
Commas: 225/224, 441/440, 1344/1331&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~7/5 = 585.606&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 5 -5 -5 2|, &amp;lt;0 -7 15 16 3]]&lt;br /&gt;
EDOs: 41, 84, 125e&lt;br /&gt;
Badness: 0.0364&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc29"&gt;&lt;a name="Merman-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;13-limit&lt;/h2&gt;
Commas: 144/143, 225/224, 364/363, 441/440&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~7/5 = 585.657&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 5 -5 -5 2 12|, &amp;lt;0 -7 15 16 3 -17|]&lt;br /&gt;
EDOs: 41, 84, 125e, 209ef, 293ef&lt;br /&gt;
Badness: 0.0275&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc30"&gt;&lt;a name="Septimin"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;Septimin&lt;/h1&gt;
Comma: 35595703125/34359738368&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~75/64 = 263.658&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 1|, &amp;lt;0 -11 6|]&lt;br /&gt;
EDOs: 9, 32, 41, 91, 132, 223c, 355bc&lt;br /&gt;
Badness: 0.6709&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc31"&gt;&lt;a name="Septimin-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;7-limit&lt;/h2&gt;
Commas: 225/224, 84035/82944&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 263.632&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 1 5|, &amp;lt;0 -11 6 -10|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;11 -6 10 -35 -15 40||&lt;br /&gt;
EDOs: 9, &lt;a class="wiki_link" href="/32edo"&gt;32&lt;/a&gt;, 41, 91, &lt;a class="wiki_link" href="/132edo"&gt;132&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc32"&gt;&lt;a name="Septimin-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt;11-limit&lt;/h2&gt;
Commas: 225/224, 385/384, 2401/2376&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 263.634&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 1 5 5|, &amp;lt;0 -11 6 -10 -7|]&lt;br /&gt;
EDOs: 9, 32, 41, 91, 223cdef&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:66:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc33"&gt;&lt;a name="Septimin-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:66 --&gt;13-limit&lt;/h2&gt;
Commas: 105/104, 144/143, 196/195, 245/242&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: 263.700&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 1 5 5 7|, &amp;lt;0 -11 6 -10 -7 -15|]&lt;br /&gt;
EDOs: 9, 41, 91&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:68:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc34"&gt;&lt;a name="Slender"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;Slender&lt;/h1&gt;
Commas: 225/224, 589824/588245&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: ~49/48 = 38.413&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 2 3|, &amp;lt;0 -13 10 -6|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;13 -10 6 -46 -27 42||&lt;br /&gt;
EDOs: 31, 94, 125&lt;br /&gt;
Badness: 0.0569&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:70:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc35"&gt;&lt;a name="Slender-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:70 --&gt;11-limit&lt;/h2&gt;
Commas: 225/224, 385/384, 1331/1323&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: ~49/48 = 38.387&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 2 3 4|, &amp;lt;0 -13 10 -6 -17|]&lt;br /&gt;
EDOs: 31, 63, 94, 125&lt;br /&gt;
Badness: 25.342&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:72:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc36"&gt;&lt;a name="Slender-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;13-limit&lt;/h2&gt;
Commas: 225/224, 275/273, 385/384, 1331/1323&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~49/48 = 38.314&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 2 3 4 3|, &amp;lt;0 -13 10 -6 -17 22|]&lt;br /&gt;
EDOs: 31, 63, 94&lt;br /&gt;
Badness: 25.913&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:74:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc37"&gt;&lt;a name="Marvo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:74 --&gt;Marvo&lt;/h1&gt;
Commas: 225/224, 78125000/78121827&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: ~27/20 = 516.694&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 5 12 29|, &amp;lt;0 -6 -17 -46|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;6, 17, 46, 13, 56 59||&lt;br /&gt;
EDOs: 65, 72&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:76:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc38"&gt;&lt;a name="Marvo-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:76 --&gt;11-limit&lt;/h2&gt;
Commas: 225/224, 243/242, 4000/3993&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE temperament&lt;/a&gt;: ~27/20 = 516.699&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 5 12 29 12|, &amp;lt;0 -6 -17 -46 -15|]&lt;br /&gt;
EDOs: 65, 72, 281, 497&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:78:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc39"&gt;&lt;a name="Marvo-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:78 --&gt;13-limit&lt;/h2&gt;
Commas: 243/242, 225/224, 351/350, 1625/1617&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~27/20 = 516.730&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 5 12 29 12 39|, &amp;lt;0 -6 -17 -46 -15 -62|]&lt;br /&gt;
EDOs: 72, 137, 209, 281f, 490bcf&lt;br /&gt;
Badness: 0.0269&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:80:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc40"&gt;&lt;a name="Marvolo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:80 --&gt;Marvolo&lt;/h1&gt;
Commas: 225/224, 156250000/155649627&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~21/20 = 83.348&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 1 1|, &amp;lt;0 -6 19 26|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;6 -19 -26 -44 -58 -7||&lt;br /&gt;
EDOs: 29, 43, 72, 619bcd, 691bcd&lt;br /&gt;
Badness: 0.0833&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:82:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc41"&gt;&lt;a name="Marvolo-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:82 --&gt;11-limit&lt;/h2&gt;
Commas: 225/224, 441/440, 4000/3993&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~21/20 = 83.340&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 1 1 2|, &amp;lt;0 -6 19 26 21|]&lt;br /&gt;
EDOs: 29, 43, 72&lt;br /&gt;
Badness: 0.0290&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:84:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc42"&gt;&lt;a name="Marvolo-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:84 --&gt;13-limit&lt;/h2&gt;
Commas: 169/168, 225/224, 364/363, 441/440&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~21/20 = 83.330&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 1 1 2 3|, &amp;lt;0 -6 19 26 21 10|]&lt;br /&gt;
EDOs: 29, 43, 72, 115f&lt;br /&gt;
Badness: 0.0215&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:86:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc43"&gt;&lt;a name="Amavil"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:86 --&gt;Amavil&lt;/h1&gt;
Commas: 225/224, 17496/16807&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~48/35 = 529.979&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 6 1 9|, &amp;lt;0 -10 3 -14|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;10 -3 14 -28 -6 41||&lt;br /&gt;
EDOs: 9, 34d, 43, 77d&lt;br /&gt;
Badness: 0.1096&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:88:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc44"&gt;&lt;a name="Amavil-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:88 --&gt;11-limit&lt;/h2&gt;
Commas: 99/98, 176/175, 864/847&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~15/11 = 529.974&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 6 1 9 7|, &amp;lt;0 -10 3 -14 -8|]&lt;br /&gt;
EDOs: 9, 34d, 43, 77de&lt;br /&gt;
Badness: 0.0426&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:90:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc45"&gt;&lt;a name="Amavil-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:90 --&gt;13-limit&lt;/h2&gt;
Commas: 78/77, 99/98, 144/143, 176/175&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~15/11 = 529.951&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 6 1 9 7 9|, &amp;lt;0 -10 3 -14 -8 -12|]&lt;br /&gt;
EDOs: 9, 34d, 43, 77de&lt;br /&gt;
Badness: 0.0258&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:92:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc46"&gt;&lt;a name="Enneaportent"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:92 --&gt;Enneaportent&lt;/h1&gt;
Commas: 225/224, 40353607/40310784&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~5/4 = 383.165&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;9 0 28 11|, &amp;lt;0 2 -1 2|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;18 -9 18 -56 -22 67||&lt;br /&gt;
EDOs: 9, 54, 63, 72, 495bcd&lt;br /&gt;
Badness: 0.0937&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:94:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc47"&gt;&lt;a name="Enneaportent-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:94 --&gt;11-limit&lt;/h2&gt;
Commas: 225/224, 385/384, 12005/11979&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~5/4 = 383.146&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;9 0 28 11 24|, &amp;lt;0 2 -1 2 1|]&lt;br /&gt;
EDOs: 9, 54, 63, 72, 423cd, 495bcd&lt;br /&gt;
Badness: 0.0304&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:96:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc48"&gt;&lt;a name="Enneaportent-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:96 --&gt;13-limit&lt;/h2&gt;
Commas: 169/168 225/224 364/363 1716/1715&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~5/4 = 383.047&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;9 0 28 11 24 19|, &amp;lt;0 2 -1 2 1 2|]&lt;br /&gt;
EDOs: 9, 54, 63, 72, 279cf&lt;br /&gt;
Badness: 0.0223&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:98:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc49"&gt;&lt;a name="Submajor"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:98 --&gt;Submajor&lt;/h1&gt;
Comma: 69198046875/68719476736&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~10125/8192 = 362.321&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 -1|, &amp;lt;0 -8 11|]&lt;br /&gt;
EDOs: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc&lt;br /&gt;
Badness: 0.1302&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:100:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc50"&gt;&lt;a name="Submajor-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:100 --&gt;7-limit&lt;/h2&gt;
Commas: 225/224, 51200/50421&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~49/40 = 362.255&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 -1 1|, &amp;lt;0 -8 11 6|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;8 -11 -6 -36 -32 17||&lt;br /&gt;
EDOs: 10, 33, 43, 53&lt;br /&gt;
Badness: 0.0605&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:102:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc51"&gt;&lt;a name="Submajor-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:102 --&gt;11-limit&lt;/h2&gt;
Commas: 225/224, 385/384, 6655/6561&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~27/22 = 362.101&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 -1 1 11|, &amp;lt;0 -8 11 6 -25|]&lt;br /&gt;
EDOs: 10, 53, 116, 169de, 285cde&lt;br /&gt;
Badness: 0.0506&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:104:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc52"&gt;&lt;a name="Submajor-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:104 --&gt;13-limit&lt;/h2&gt;
Commas: 169/168, 225/224, 275/273, 385/384&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~16/13 = 362.105&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 -1 1 11 4|, &amp;lt;0 -8 11 6 -25 -1|]&lt;br /&gt;
EDOs: 10, 53, 116, 169de, 285cdef&lt;br /&gt;
Badness: 0.0277&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:106:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc53"&gt;&lt;a name="Submajor-Interpental"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:106 --&gt;Interpental&lt;/h2&gt;
Commas: 99/98, 176/175, 51200/50421&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~49/40 = 362.418&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 -1 1 -5|, &amp;lt;0 -8 11 6 28|]&lt;br /&gt;
EDOs: 43, 53, 96, 149d&lt;br /&gt;
Badness: 0.0518&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:108:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc54"&gt;&lt;a name="Submajor-Interpental-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:108 --&gt;13-limit&lt;/h3&gt;
Commas: 99/98, 169/168, 176/175, 640/637&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~16/13 = 362.402&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 4 -1 1 -5 4|, &amp;lt;0 -8 11 6 28 -1|]&lt;br /&gt;
EDOs: 43, 53, 96, 149d&lt;br /&gt;
Badness: 0.0297&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:110:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc55"&gt;&lt;a name="Alphorn"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:110 --&gt;Alphorn&lt;/h1&gt;
Commas: 225/224, 5764801/5668704&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~48/35 = 556.221&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 9 0 13|, &amp;lt;0 -16 5 -22|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;16 -5 22 -45 -10 65||&lt;br /&gt;
EDOs: 28d, 41, 151cd, 192cd, 233cd&lt;br /&gt;
Badness: 0.1293&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:112:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc56"&gt;&lt;a name="Alphorn-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:112 --&gt;11-limit&lt;/h2&gt;
Commas: 225/224, 385/384, 12250/11979&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~11/8 = 556.144&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 9 0 13 3|, &amp;lt;0 -16 5 -22 1|]&lt;br /&gt;
EDOs: 41, 315cde&lt;br /&gt;
Badness: 0.0735&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 10:20, 2 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This page discusses miscellaneous rank-2 temperaments tempering out 225/224, the marvel comma or septimal kleisma.

Temperaments considered in families and clans are:

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.

Since (5/4)2 = (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.

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.

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 (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)

Wizard

Main article: Wizard
For the 5-limit version, see Miscellaneous 5-limit temperaments #Wizard.

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 22 & 72. Its ploidacot is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the perfect twelfth. 72edo, 94edo, and especially 166edo are good tunings for it.

Subgroup: 2.3.5.7

Comma list: 225/224, 118098/117649

Mapping: [⟨2 1 5 2], ⟨0 6 -1 10]]

mapping generators: ~1225/864, ~245/216

Optimal tunings:

  • WE: ~1225/864 = 600.3438 ¢, ~245/216 = 216.8680 ¢
error map: ⟨+0.688 -0.403 -1.463 +0.541]
  • CWE: ~1225/864 = 600.0000 ¢, ~245/216 = 216.7977 ¢
error map: ⟨0.000 -1.169 -3.111 -0.849]

Optimal ET sequence: 22, 50, 72, 238c, 310c, 382c, 454bccd

Badness (Sintel): 1.03

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 4000/3993

Mapping: [⟨2 1 5 2 8], ⟨0 6 -1 10 -3]]

Optimal tunings:

  • WE: ~99/70 = 600.3051 ¢, ~25/22 = 216.8782 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~25/22 = 216.7961 ¢

Optimal ET sequence: 22, 50, 72, 166, 238c, 310c

Badness (Sintel): 0.613

Lizard

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 364/363, 385/384

Mapping: [⟨2 1 5 2 8 11], ⟨0 6 -1 10 -3 -10]]

Optimal tunings:

  • WE: ~55/39 = 600.4824 ¢, ~25/22 = 216.7852 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~25/22 = 216.6247 ¢

Optimal ET sequence: 22, 50, 72

Badness (Sintel): 0.900

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨2 1 5 2 8 11 6], ⟨0 6 -1 10 -3 -10 6]]

Optimal tunings:

  • WE: ~17/12 = 600.5032 ¢, ~17/15 = 216.8002 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.6361 ¢

Optimal ET sequence: 22, 50, 72

Badness (Sintel): 0.741

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342

Mapping: [⟨2 1 5 2 8 11 6 2], ⟨0 6 -1 10 -3 -10 6 18]]

Optimal tunings:

  • WE: ~17/12 = 600.4698 ¢, ~17/15 = 216.6925 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.5434 ¢

Optimal ET sequence: 22h, 50, 72, 122g, 194dfg

Badness (Sintel): 0.955

Gizzard

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 1573/1568

Mapping: [⟨2 1 5 2 8 -2], ⟨0 6 -1 10 -3 26]]

Optimal tunings:

  • WE: ~99/70 = 600.2896 ¢, ~25/22 = 216.9343 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~25/22 = 216.8501 ¢

Optimal ET sequence: 22f, 72, 166, 238cf

Badness (Sintel): 0.837

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 289/288, 325/324, 375/374, 385/384

Mapping: [⟨2 1 5 2 8 -2 6], ⟨0 6 -1 10 -3 26 6]]

Optimal tunings:

  • WE: ~17/12 = 600.3227 ¢, ~17/15 = 216.9414 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.8469 ¢

Optimal ET sequence: 22f, 72, 166g, 238cfg

Badness (Sintel): 0.694

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594

Mapping: [⟨2 1 5 2 8 -2 6 15], ⟨0 6 -1 10 -3 26 6 -18]]

Optimal tunings:

  • WE: ~17/12 = 600.2637 ¢, ~17/15 = 216.9570 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.8687 ¢

Optimal ET sequence: 72, 94, 166g

Badness (Sintel): 0.901

Mage

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1331/1296

Mapping: [⟨2 1 5 2 4], ⟨0 6 -1 10 8]]

Optimal tunings:

  • WE: ~77/54 = 600.6486 ¢, ~55/48 = 217.1099 ¢
  • CWE: ~77/54 = 600.0000 ¢, ~55/48 = 216.9841 ¢

Optimal ET sequence: 22, 50e, 72ee

Badness (Sintel): 1.91

Tritonic

For the 5-limit version, see Miscellaneous 5-limit temperaments #Tritonic.

Tritonic tempers out 50421/50000 and may be described as the 29 & 31 temperament. It splits the 6th harmonic into five generators of ~10/7 tritones, hence the name. Its ploidacot is beta-pentacot. 60edo may be used as a tuning, which in the 11-limit entails the 60e val.

Subgroup: 2.3.5.7

Comma list: 225/224, 50421/50000

Mapping: [⟨1 -1 8 9], ⟨0 5 -11 -12]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1201.3539 ¢, ~10/7 = 620.4131 ¢
error map: ⟨+1.354 -1.243 -0.027 -1.598]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6778 ¢
error map: ⟨0.000 -3.566 -2.769 -4.959]

Optimal ET sequence: 29, 31, 60, 91, 122, 213bcd

Badness (Sintel): 1.20

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 441/440

Mapping: [⟨1 -1 8 9 5], ⟨0 5 -11 -12 -3]]

Optimal tunings:

  • WE: ~2 = 1201.7116 ¢, ~10/7 = 620.6166 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6890 ¢

Optimal ET sequence: 29, 31, 60e, 91e, 213bcdeee

Badness (Sintel): 0.782

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 196/195, 275/273

Mapping: [⟨1 -1 8 9 5 13], ⟨0 5 -11 -12 -3 -18]]

Optimal tunings:

  • WE: ~2 = 1201.5355 ¢, ~10/7 = 620.6855 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.8469 ¢

Optimal ET sequence: 29, 31, 60e

Badness (Sintel): 0.950

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -1 8 9 5 13 17], ⟨0 5 -11 -12 -3 -18 -25]]

Optimal tunings:

  • WE: ~2 = 1201.5260 ¢, ~10/7 = 620.7330 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.8986 ¢

Optimal ET sequence: 29g, 31, 60e

Badness (Sintel): 0.973

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: [⟨1 -1 8 9 5 13 17 12], ⟨0 5 -11 -12 -3 -18 -25 -15]]

Optimal tunings:

  • WE: ~2 = 1201.3100 ¢, ~10/7 = 620.6509 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.9328 ¢

Optimal ET sequence: 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: [⟨1 -1 8 9 5 13 17 12 4], ⟨0 5 -11 -12 -3 -18 -25 -15 1]]

Optimal tunings:

  • WE: ~2 = 1201.4074 ¢, ~10/7 = 620.7185 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.9548 ¢

Optimal ET sequence: 29g, 31, 60e

Badness (Sintel): 1.04

Tritoni

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 27783/27500

Mapping: [⟨1 -1 8 9 -11], ⟨0 5 -11 -12 28]]

Optimal tunings:

  • WE: ~2 = 1201.0888 ¢, ~10/7 = 620.1733 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6146 ¢

Optimal ET sequence: 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 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 reduced give the 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: [⟨1 -7 7 -5], ⟨0 11 -6 10]]

mapping generators: ~2, ~12/7

Optimal tunings:

  • WE: ~2 = 1201.2452 ¢, ~12/7 = 937.3394 ¢
error map: ⟨+1.245 +0.062 -1.633 -1.658]
  • CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.4036 ¢
error map: ⟨0.000 -1.516 -4.735 -4.790]

Optimal ET sequence: 41, 91, 132d

Badness (Sintel): 1.38

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/242, 385/384

Mapping: [⟨1 -7 7 -5 -2], ⟨0 11 -6 10 7]]

Optimal tunings:

  • WE: ~2 = 1200.8059 ¢, ~12/7 = 936.9952 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.3906 ¢

Optimal ET sequence: 41, 91, 223cdef

Badness (Sintel): 1.04

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195, 245/242

Mapping: [⟨1 -7 7 -5 -2 -8], ⟨0 11 -6 10 7 15]]

Optimal tunings:

  • WE: ~2 = 1200.5990 ¢, ~12/7 = 936.7670 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.3196 ¢

Optimal ET sequence: 41, 91

Badness (Sintel): 0.955

Merman

For the 5-limit version, see Miscellaneous 5-limit temperaments #Merman.

Merman may be described as the 41 & 43 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, seven generator steps give the interval class of 3. The ploidacot for this temperament is gamma-heptacot.

The name was likely derived from Triton, which was in turn derived from tritonic.

Subgroup: 2.3.5.7

Comma list: 225/224, 2500000/2470629

Mapping: [⟨1 -2 10 11], ⟨0 7 -15 -16]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1200.3898 ¢, ~10/7 = 614.6413 ¢
error map: ⟨+0.390 -0.435 -1.630 +1.634]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4073 ¢
error map: ⟨0.000 -1.104 -2.423 +0.657]

Optimal ET sequence: 41, 84, 125

Badness (Sintel): 1.39

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 1344/1331

Mapping: [⟨1 -2 10 11 5], ⟨0 7 -15 -16 -3]]

Optimal tunings:

  • WE: ~2 = 1199.9578 ¢, ~10/7 = 614.3720 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.3943 ¢

Optimal ET sequence: 41, 84, 125e

Badness (Sintel): 1.20

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 225/224, 364/363, 441/440

Mapping: [⟨1 -2 10 11 5 -5], ⟨0 7 -15 -16 -3 17]]

Optimal tunings:

  • WE: ~2 = 1199.7422 ¢, ~10/7 = 614.2110 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.3442 ¢

Optimal ET sequence: 41, 84, 125e, 209ef, 293ef

Badness (Sintel): 1.14

Mermaid

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 532400/531441

Mapping: [⟨1 -2 10 11 -16], ⟨0 7 -15 -16 38]]

Optimal tunings:

  • WE: ~2 = 1199.4973 ¢, ~10/7 = 614.7004 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4470 ¢

Optimal ET sequence: 41, 84e, 125, 166

Badness (Sintel): 1.46

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 10648/10647

Mapping: [⟨1 -2 10 11 22 32], ⟨0 7 -15 -16 38 58]]

Optimal tunings:

  • WE: ~2 = 1200.5126 ¢, ~10/7 = 614.7152 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4562 ¢

Optimal ET sequence: 41, 84ef, 125f, 166

Badness (Sintel): 1.47

Slender

Slender tempers out the hewuermera comma in addition to the marvel comma, and may be described as the 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.

The name was likely derived from slendro diesis, one of the names for the interval 49/48.

Subgroup: 2.3.5.7

Comma list: 225/224, 589824/588245

Mapping: [⟨1 2 2 3], ⟨0 -13 10 -6]]

mapping generators: ~2, ~49/48

Optimal tunings:

  • WE: ~2 = 1200.3816 ¢, ~49/48 = 38.4256 ¢
error map: ⟨+0.382 -0.725 -1.295 +1.765]
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.4079 ¢
error map: ⟨0.000 -1.257 -2.235 +0.727]

Optimal ET sequence: 31, 94, 125, 406c

Badness (Sintel): 1.44

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 1331/1323

Mapping: [⟨1 2 2 3 4], ⟨0 -13 10 -6 -17]]

Optimal tunings:

  • WE: ~2 = 1199.4983 ¢, ~49/48 = 38.4030 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.3775 ¢

Optimal ET sequence: 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: [⟨1 2 2 3 4 3], ⟨0 -13 10 -6 -17 22]]

Optimal tunings:

  • WE: ~2 = 1200.1728 ¢, ~49/48 = 38.3192 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.3129 ¢

Optimal ET sequence: 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 19 & 21 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, three generator steps give the interval class of 3. The ploidacot for this temperament is alpha-tricot.

Subgroup: 2.3.5.7

Comma list: 225/224, 1029/1000

Mapping: [⟨1 0 6 7], ⟨0 3 -7 -8]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1203.3828 ¢, ~10/7 = 632.9137 ¢
error map: ⟨+3.383 -3.214 +3.587 -8.457]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 630.9827 ¢
error map: ⟨0.000 -9.007 -3.192 -16.687]

Optimal ET sequence: 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: [⟨1 0 6 7 4], ⟨0 3 -7 -8 -1]]

Optimal tunings:

  • WE: ~2 = 1201.3875 ¢, ~10/7 = 631.5852 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 630.8007 ¢

Optimal ET sequence: 2, 17d, 19

Badness (Sintel): 1.51

Marvolo

Subgroup: 2.3.5.7

Comma list: 225/224, 156250000/155649627

Mapping: [⟨1 2 1 1], ⟨0 -6 19 26]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • WE: ~2 = 1200.7714 ¢, ~21/20 = 83.4014 ¢
error map: ⟨+0.772 -0.820 -0.916 +0.381]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3640 ¢
error map: ⟨0.000 -2.139 -2.398 -1.362]

Optimal ET sequence: 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: [⟨1 2 1 1 2], ⟨0 -6 19 26 21]]

Optimal tunings:

  • WE: ~2 = 1200.7075 ¢, ~21/20 = 83.3888 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3564 ¢

Optimal ET sequence: 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: [⟨1 2 1 1 2 3], ⟨0 -6 19 26 21 10]]

Optimal tunings:

  • WE: ~2 = 1200.9467 ¢, ~21/20 = 83.3956 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3516 ¢

Optimal ET sequence: 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: [⟨1 2 1 1 2 3 2], ⟨0 -6 19 26 21 10 30]]

Optimal tunings:

  • WE: ~2 = 1200.9606 ¢, ~21/20 = 83.4030 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3594 ¢

Optimal ET sequence: 29g, 43, 72

Badness (Sintel): 0.760

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440

Mapping: [⟨1 2 1 1 2 3 2 3], ⟨0 -6 19 26 21 10 30 18]]

Optimal tunings:

  • WE: ~2 = 1200.7625 ¢, ~21/20 = 83.3895 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3551 ¢

Optimal ET sequence: 29g, 43, 72

Badness (Sintel): 0.895

Enneaportent

Subgroup: 2.3.5.7

Comma list: 225/224, 40353607/40310784

Mapping: [⟨9 0 28 11], ⟨0 2 -1 2]]

mapping generators: ~2592/2401, ~12005/6912

Optimal tunings:

  • WE: ~2592/2401 = 133.4174 ¢, ~12005/6912 = 950.7667 ¢ (~1728/1715 = 16.8452 ¢)
error map: ⟨+0.756 -0.422 -1.395 +0.298]
  • CWE: ~2592/2401 = 133.3333 ¢, ~12005/6912 = 950.2969 ¢ (~1728/1715 = 16.9636 ¢)
error map: ⟨0.000 -1.361 -3.277 -1.565]

Optimal ET sequence: 9, 54, 63, 72, 495bccd, 567bcccd

Badness (Sintel): 2.37

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 12005/11979

Mapping: [⟨9 0 28 11 24], ⟨0 2 -1 2 1]]

Optimal tunings:

  • WE: ~121/112 = 133.4071 ¢, ~210/121 = 950.7131 ¢ (~99/98 = 16.8633 ¢)
  • CWE: ~121/112 = 133.3333 ¢, ~210/121 = 950.2994 ¢ (~99/98 = 16.9661 ¢)

Optimal ET sequence: 9, 54, 63, 72

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 364/363, 1716/1715

Mapping: [⟨9 0 28 11 24 19], ⟨0 2 -1 2 1 2]]

Optimal tunings:

  • WE: ~14/13 = 133.4245 ¢, ~26/15 = 950.9362 ¢ (~105/104 = 16.9650 ¢)
  • CWE: ~14/13 = 133.3333 ¢, ~26/15 = 950.4364 ¢ (~99/98 = 17.1031 ¢)

Optimal ET sequence: 9, 54, 63, 72

Badness (Sintel): 0.922

Gracecordial

For the 5-limit version, see Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit).

Subgroup: 2.3.5.7

Comma list: 225/224, 781250000/771895089

Mapping: [⟨1 0 34 63], ⟨0 1 -20 -38]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1200.4904 ¢, ~3/2 = 701.1103 ¢
error map: ⟨+0.490 -0.354 -1.655 +1.241]
  • CWE: ~2 = 1200.3333 ¢, ~3/2 = 700.8112 ¢
error map: ⟨0.000 -1.144 -2.537 +0.349]

Optimal ET sequence: 12, …, 113, 125, 238c, 363c

Badness (Sintel): 2.44

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 236328125/234365481

Mapping: [⟨1 0 34 63 -90], ⟨0 1 -20 -38 59]]

Optimal tunings:

  • WE: ~2 = 1200.5571 ¢, ~3/2 = 701.1589 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8328 ¢

Optimal ET sequence: 12e, 113, 125, 238c

Badness (Sintel): 2.96

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 831875/830466

Mapping: [⟨1 0 34 63 -90 -66], ⟨0 1 -20 -38 59 44]]

Optimal tunings:

  • WE: ~2 = 1200.6282 ¢, ~3/2 = 701.2080 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8421 ¢

Optimal ET sequence: 12e, 113, 125f, 238cf

Badness (Sintel): 2.16

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913

Mapping: [⟨1 0 34 63 -90 -66 -7], ⟨0 1 -20 -38 59 44 7]]

Optimal tunings:

  • WE: ~2 = 1200.5058 ¢, ~3/2 = 701.1360 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8414 ¢

Optimal ET sequence: 12e, 113, 125f, 238cf

Badness (Sintel): 1.96

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444

Mapping: [⟨1 0 34 63 -90 -66 -7 9], ⟨0 1 -20 -38 59 44 7 -3]]

Optimal tunings:

  • WE: ~2 = 1200.4418 ¢, ~3/2 = 701.0999 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8425 ¢

Optimal ET sequence: 12e, 113, 125f, 238cf

Badness (Sintel): 1.71

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528

Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43], ⟨0 1 -20 -38 59 44 7 -3 30]]

Optimal tunings:

  • WE: ~2 = 1200.4641 ¢, ~3/2 = 701.1145 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8444 ¢

Optimal ET sequence: 12e, 113, 238cfi

Badness (Sintel): 1.57

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493

Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43 -49], ⟨0 1 -20 -38 59 44 7 -3 30 34]]

Optimal tunings:

  • WE: ~2 = 1200.4400 ¢, ~3/2 = 701.0986 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8428 ¢

Optimal ET sequence: 12e, 113, 125f, 238cfi

Badness (Sintel): 1.50

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493

Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43 -49 -79], ⟨0 1 -20 -38 59 44 7 -3 30 34 53]]

Optimal tunings:

  • WE: ~2 = 1200.4178 ¢, ~3/2 = 701.0822 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8396 ¢

Optimal ET sequence: 12e, 113, 125f, 238cfi

Badness (Sintel): 1.53

Gracecord

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 109375/107811

Mapping: [⟨1 0 34 63 89], ⟨0 1 -20 -38 -54]]

Optimal tunings:

  • WE: ~2 = 1200.6064 ¢, ~3/2 = 701.2398 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8718 ¢

Optimal ET sequence: 12, …, 101cd, 113

Badness (Sintel): 2.21

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 364/363, 441/440, 6125/6084

Mapping: [⟨1 0 34 63 89 113], ⟨0 1 -20 -38 -54 -69]]

Optimal tunings:

  • WE: ~2 = 1200.6225 ¢, ~3/2 = 701.2539 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8781 ¢

Optimal ET sequence: 12f, …, 101cdf, 113

Badness (Sintel): 1.83

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989

Mapping: [⟨1 0 34 63 89 113 -7], ⟨0 1 -20 -38 -54 -69 7]]

Optimal tunings:

  • WE: ~2 = 1200.3308 ¢, ~3/2 = 701.0632 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8654 ¢

Optimal ET sequence: 12f, 101cdf, 113

Badness (Sintel): 1.87

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663

Mapping: [⟨1 0 34 63 89 113 -7 9], ⟨0 1 -20 -38 -54 -69 7 -3]]

Optimal tunings:

  • WE: ~2 = 1200.2658 ¢, ~3/2 = 701.0213 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8629 ¢

Optimal ET sequence: 12f, 101cdf, 113

Badness (Sintel): 1.68

Alphorn

Subgroup: 2.3.5.7

Comma list: 225/224, 5764801/5668704

Mapping: [⟨1 -7 5 -9], ⟨0 16 -5 22]]

mapping generators: ~2, ~35/24

Optimal tunings:

  • WE: ~2 = 1201.3004 ¢, ~35/24 = 644.4767 ¢
error map: ⟨+1.300 +0.569 -2.195 -2.043]
  • CWE: ~2 = 1200.3333 ¢, ~35/24 = 643.8137 ¢
error map: ⟨0.000 -0.936 -5.382 -4.924]

Optimal ET sequence: 13d, 28d, 41, 151cd, 192cdd, 233ccdd

Badness (Sintel): 3.27

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 12250/11979

Mapping: [⟨1 -7 5 -9 4], ⟨0 16 -5 22 -1]]

Optimal tunings:

  • WE: ~2 = 1200.5123 ¢, ~16/11 = 644.1307 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 643.8662 ¢

Optimal ET sequence: 13d, 28d, 41

Badness (Sintel): 2.43

Misneb

For the 5-limit version, see Miscellaneous 5-limit temperaments #Misneb.

Subgroup: 2.3.5.7

Comma list: 225/224, 4194304/4117715

Mapping: [⟨1 -12 15 1], ⟨0 15 -14 2]]

mapping generators: ~2, ~15/8

Optimal tunings:

  • WE: ~2 = 1199.7642 ¢, ~15/8 = 1086.5513 ¢
error map: ⟨-0.236 -0.856 -1.569 +4.041]
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.7633 ¢
error map: ⟨0.000 -0.506 -0.999 +4.701]

Optimal ET sequence: 21, 32, 53

Badness (Sintel): 3.57

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1310720/1294139

Mapping: [⟨1 -12 15 1 27], ⟨0 15 -14 2 -26]]

Optimal tunings:

  • WE: ~2 = 1200.1654 ¢, ~15/8 = 1086.8269 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.6766 ¢

Optimal ET sequence: 21, 32e, 53, 127

Badness (Sintel): 2.82

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 640/637, 847/845

Mapping: [⟨1 -12 15 1 27 20], ⟨0 15 -14 2 -26 -18]]

Optimal tunings:

  • WE: ~2 = 1200.1687 ¢, ~15/8 = 1086.8295 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.6757 ¢

Optimal ET sequence: 21, 32e, 53, 127

Badness (Sintel): 1.88

Musneb

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 66550/64827

Mapping: [⟨1 3 1 3 6], ⟨0 -15 14 -2 -27]]

Optimal tunings:

  • WE: ~2 = 1200.0839 ¢, ~15/8 = 1086.9343 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.8593 ¢

Optimal ET sequence: 21e, 32, 53

Badness (Sintel): 2.89

Untriton

For the 5-limit version, see Miscellaneous 5-limit temperaments #Untriton.

Named by Petr Pařízek in 2011[1], untriton may be described as the 51 & 53 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, nine generator steps give the interval class of 3. The ploidacot for this temperament is delta-enneacot.

Subgroup: 2.3.5.7

Comma list: 225/224, 125000000/121060821

Mapping: [⟨1 -3 12 13], ⟨0 9 -19 -20]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1199.8275 ¢, ~10/7 = 611.2710 ¢
error map: ⟨-0.172 +0.002 -2.533 +3.511]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3614 ¢
error map: ⟨0.000 +0.298 -2.181 +3.946]

Optimal ET sequence: 51, 53

Badness (Sintel): 3.64

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 22000/21609

Mapping: [⟨1 -3 12 13 6], ⟨0 9 -19 -20 -5]]

Optimal tunings:

  • WE: ~2 = 1200.3591 ¢, ~10/7 = 611.5569 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3690 ¢

Optimal ET sequence: 51, 53

Badness (Sintel): 2.46

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 225/224, 275/273, 1040/1029

Mapping: [⟨1 -3 12 13 6 20], ⟨0 9 -19 -20 -5 -32]]

Optimal tunings:

  • WE: ~2 = 1200.4078 ¢, ~10/7 = 611.5536 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3392 ¢

Optimal ET sequence: 51f, 53

Badness (Sintel): 1.96

Naiadical

Named by Xenllium in 2026, naiadical may be described as the 21 & 29 temperament.

Subgroup: 2.3.5.7

Comma list: 225/224, 823543/800000

Mapping: [⟨1 -4 11 9], ⟨0 9 -14 -10]]

mapping generators: ~2, ~32/21

Optimal tunings:

  • WE: ~2 = 1202.1198 ¢, ~32/21 = 745.4675 ¢
error map: ⟨+2.120 -1.227 +0.459 -4.423]
  • CWE: ~2 = 1200.0000 ¢, ~32/21 = 744.1318 ¢
error map: ⟨0.000 -4.769 -4.159 -10.144]

Optimal ET sequence: 21, 29, 50, 79d, 129cdd, 179bcddd

Badness (Sintel): 3.67

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/242, 1617/1600

Mapping: [⟨1 -4 11 9 14], ⟨0 9 -14 -10 -17]]

Optimal tunings:

  • WE: ~2 = 1201.9008 ¢, ~21/16 = 745.3867 ¢
  • CWE: ~2 = 1200.0000 ¢, ~32/21 = 744.1777 ¢

Optimal ET sequence: 21, 29, 50, 79d

Badness (Sintel): 2.00

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 245/242, 1001/1000

Mapping: [⟨1 -4 11 9 14 13], ⟨0 9 -14 -10 -17 -15]]

Optimal tunings:

  • WE: ~2 = 1201.7863 ¢, ~20/13 = 745.3344 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.1931 ¢

Optimal ET sequence: 21, 29, 50, 79d

Badness (Sintel): 1.43

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 170/169, 196/195, 221/220, 245/242

Mapping: [⟨1 -4 11 9 14 13 14], ⟨0 9 -14 -10 -17 -15 -16]]

Optimal tunings:

  • WE: ~2 = 1201.9208 ¢, ~20/13 = 745.3976 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.1669 ¢

Optimal ET sequence: 21, 29g, 50, 79dg

Badness (Sintel): 1.26

Quintannic

Named by Scott Dakota, quintannic may be described as the 43 & 60 temperament.

Subgroup: 2.3.5.7

Comma list: 225/224, 9805926501/9765625000

Mapping: [⟨1 1 5 7], ⟨0 5 -23 -36]]

mapping generators: ~2, ~10000/9261

Optimal tunings:

  • WE: ~2 = 1200.9803 ¢, ~10000/9261 = 139.9522 ¢
error map: ⟨+0.980 -1.214 -0.313 -0.243]
  • CWE: ~2 = 1200.0000 ¢, ~10000/9261 = 139.8184 ¢
error map: ⟨0.000 -2.863 -2.136 -2.287]

Optimal ET sequence: 43, 60, 103, 266bcd, 369bcd

Badness (Sintel): 3.81

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 43923/43750

Mapping: [⟨1 1 5 7 8], ⟨0 5 -23 -36 -39]]

Optimal tunings:

  • WE: ~2 = 1201.0031 ¢, ~320/297 = 139.9435 ¢
  • CWE: ~2 = 1200.0000 ¢, ~320/297 = 139.8053 ¢

Optimal ET sequence: 43, 60e, 103, 369bcdeee, 472bbcddeee

Badness (Sintel): 1.74

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 441/440, 1001/1000, 1188/1183

Mapping: [⟨1 1 5 7 8 3], ⟨0 5 -23 -36 -39 6]]

Optimal tunings:

  • WE: ~2 = 1200.8354 ¢, ~13/12 = 139.9095 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.7997 ¢

Optimal ET sequence: 43, 60e, 103

Badness (Sintel): 1.35

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 273/272, 375/374, 441/440, 891/884

Mapping: [⟨1 1 5 7 8 3 7], ⟨0 5 -23 -36 -39 6 -25]]

Optimal tunings:

  • WE: ~2 = 1200.7402 ¢, ~13/12 = 139.9015 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.8038 ¢

Optimal ET sequence: 43, 60e, 103

Badness (Sintel): 1.17

Gwazy

For the 5-limit version, see Very high accuracy temperaments #Kwazy.

Named by Petr Pařízek in 2011[1], gwazy may be described as the 22 & 74 temperament.

Subgroup: 2.3.5.7

Comma list: 225/224, 5971968/5764801

Mapping: [⟨2 1 6 4], ⟨0 8 -5 6]]

mapping generators: ~2401/1728, ~35/32

Optimal tunings:

  • WE: ~2401/1728 = 599.7132 ¢, ~35/32 = 162.5806 ¢
error map: ⟨-0.574 -1.597 -0.937 +5.510]
  • CWE: ~2401/1728 = 600.0000 ¢, ~35/32 = 162.6388 ¢
error map: ⟨0.000 -0.844 +0.492 +7.007]

Optimal ET sequence: 22, 74, 96, 118d

Badness (Sintel): 4.53

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 65536/65219

Mapping: [⟨2 1 6 4 8], ⟨0 8 -5 6 -4]]

Optimal tunings:

  • WE: ~363/256 = 599.8517 ¢, ~11/10 = 162.5518 ¢
  • CWE: ~363/256 = 600.0000 ¢, ~11/10 = 162.5863 ¢

Optimal ET sequence: 22, 74, 96

Badness (Sintel): 2.26

Tertiosec

For the 5-limit version, see Miscellaneous 5-limit temperaments #Tertiosec.

Tertiosec may be described as the 21 & 75 temperament. It was initially named tertiomar by Petr Pařízek in 2011[1], but was changed to tertiosec in 2012[2].

Subgroup: 2.3.5.7

Comma list: 225/224, 14495514624/13841287201

Mapping: [⟨3 -1 12 7], ⟨0 8 -7 2]]

mapping generators: ~3072/2401, ~2048/1715

Optimal tunings:

  • WE: ~3072/2401 = 399.8257 ¢, ~2048/1715 = 287.5920 ¢
error map: ⟨-0.523 -1.044 -1.549 +5.138]
  • CWE: ~3072/2401 = 400.0000 ¢, ~2048/1715 = 287.7088 ¢
error map: ⟨0.000 -0.284 -0.276 +6.592]

Optimal ET sequence: 21, 54, 75, 96, 171d

Badness (Sintel): 10.9

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 3840/3773, 12005/11979

Mapping: [⟨3 -1 12 7 14], ⟨0 8 -7 2 -5]]

Optimal tunings:

  • WE: ~44/35 = 399.6550 ¢, ~33/28 = 287.5803 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~33/28 = 287.8224 ¢

Optimal ET sequence: 21, 54, 75e

Badness (Sintel): 5.74

References

  1. ↑ 1.0 1.1 1.2 Yahoo! Tuning Group | Suggested names for the unclasified temperaments
  2. ↑ Yahoo! Tuning Group | 2D temperament names, part I -- reclassified temperaments from message #101780
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