Ragismic microtemperaments: Difference between revisions

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→Oviminor: due to being less accurate than egads at higher complexity, this temperament should be deleted.
 
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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 is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the ragisma, [[4375/4374]] ({{monzo| -1 -7 4 1 }}). The ragisma is the smallest [[7-limit]] [[superparticular ratio]].  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-08-22 00:54:11 UTC</tt>.<br>
: The original revision id was <tt>519260570</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]]
The ragisma is 4375/4374 with a monzo of |-1 -7 4 1&gt;, the smallest 7-limit superparticular ratio. Since (10/9)^4=4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.


=Ennealimmal=
Since {{nowrap|(10/9)<sup>4</sup> {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)<sup>2</sup> }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
Ennealimmal temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the ennealimma comma, |1 -27 18&gt;, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is &lt;&lt;18 27 18 1 -22 -34||.


Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 equal, though its hardly likely anyone could tell the difference.
Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, ragitritonic, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]
* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]
* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]]
* [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]
* [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]]
* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]
* ''[[Zarvo]]'' (+33075/32768) → [[Gravity family #Zarvo|Gravity family]]
* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortonic family #Mitonic|Minortonic family]]
* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnuzmic family #Septimal vishnu|Vishnuzmic family]]
* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]
* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]]
* ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]]
* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]]
* ''[[Chlorine]]'' (+{{monzo| -52 -17 34}}) → [[17th-octave temperaments #Chlorine|17th-octave temperaments]]
* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]
* ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments #Dzelic|37th-octave temperaments]]


If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example.) In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS.
== Supermajor ==
The generator for supermajor temperament is a supermajor third, [[9/7]], tuned about 0.002 cents flat. Note that in the data that follow, the generator is given as its [[octave complement]]. 37 of these give 3/2<sup>22</sup>, 46 give 5/2<sup>27</sup>, and 75 give 7/2<sup>45</sup>. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: [[1106edo]] or [[1277edo]] can be used as tunings, leading to accuracy even greater than that of [[ennealimmal]]. The 80-note generator chain is presumably the place to start, and if that is not enough notes for you, there is always the 171-note generator chain.


[[Tuning Ranges of Regular Temperaments|valid range]]: [26.667, 66.667] (45bcd to 18bcd)
[[Subgroup]]: 2.3.5.7
nice range: [48.920, 49.179]
strict range: [48.920, 49.179]


Commas: 2401/2400, 4375/4374
[[Comma list]]: 4375/4374, 52734375/52706752


POTE generators: 36/35: 49.0205; 10/9: 182.354; 6/5: 315.687; 49/40: 350.980
{{Mapping|legend=1| 1 -22 -27 -45 | 0 37 46 75 }}
: mapping generators: ~2, ~14/9


Map: [&lt;9 1 1 2|, &lt;0 2 3 2|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;18 27 18 1 -22 -34||
* [[WE]]: ~2 = 1200.0067{{c}}, ~14/9 = 764.9222{{c}}
EDOs: [[27edo|27]], [[45edo|45]], [[72edo|72]], [[99edo|99]], [[171edo|171]], [[270edo|270]], [[441edo|441]], [[612edo|612]], [[3600edo|3600]]
: [[error map]]: {{val| +0.007 +0.019 -0.074 +0.037 }}
Badness: 0.00361
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 764.9181{{c}}
: error map: {{val| 0.000 +0.013 -0.083 +0.029 }}


==11 limit hemiennealimmal==
{{Optimal ET sequence|legend=1| 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214 }}
Commas: 2401/2400, 4375/4374, 3025/3024


valid range: [13.333, 22.222] (90bcd, 54c)
[[Badness]] (Sintel): 0.274
nice range: [17.304, 17.985]
strict range:  [17.304, 17.985]


POTE generator: 99/98: 17.6219 or 6/5: 315.7114
=== Semisupermajor ===
Subgroup: 2.3.5.7.11


Map: [&lt;18 0 -1 22 48|, &lt;0 2 3 2 1|]
Comma list: 3025/3024, 4375/4374, 35156250/35153041
EDOs: 72, 198, 270, 342, 612, 954, 1566
Badness: 0.00628


==13 limit hemiennealimmal==
Mapping: {{mapping| 2 -7 -8 -15 -6 | 0 37 46 75 47 }}
Commas: 676/675, 1001/1000, 1716/1715, 3025/3024
: mapping generators: ~99/70, ~11/10


POTE generator ~99/98 = 17.7504
Optimal tunings:
* WE: ~99/70 = 600.0103{{c}}, ~11/10 = 164.9205{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 164.9180{{c}}


Map: [&lt;18 0 -1 22 48 -19|, &lt;0 2 3 2 1 6|]
{{Optimal ET sequence|legend=0| 80, 262d, 342, 764, 1106, 1448, 2554, 4002e, 6556cee }}
EDOs: 72, 198, 270
Badness: 0.0125


==Semiennealimmal==
Badness (Sintel): 0.422
Commas: 2401/2400, 4375/4374, 4000/3993


POTE generator: ~140/121 = 250.3367
== Enneadecal ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Enneadecal (5-limit)]].''


Map: [&lt;9 3 4 14 18|, &lt;0 6 9 6 7|]
Enneadecal tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen [[6/5|just minor thirds]] fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out [[703125/702464]], the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones.  
EDOs: 72, 369, 441
Badness: 0.0342


===13 limit semiennealimmal===
[[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.
Commas: 1575/1573, 2080/2079, 2401/2400, 4375/4374


POTE generator: ~140/121 = 250.3375
[[Subgroup]]: 2.3.5.7


Map: [&lt;9 3 4 14 18 -8|, &lt;0 6 9 6 7 22|]
[[Comma list]]: 4375/4374, 703125/702464
EDOs: 72, 441
Badness: 0.0261


==Quadraennealimmal==
{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}
Commas: 2401/2400 4375/4374 234375/234256
: mapping generators: ~28/27, ~3


POTE generator: ~77/75 = 45.595
[[Optimal tuning]]s:
* [[WE]]: ~28/27 = 63.1599{{c}}, ~3/2 = 701.9027{{c}} (~225/224 = 7.1437{{c}})
: [[error map]]: {{val| +0.038 -0.014 -0.134 +0.080 }}
* [[CWE]]: ~28/27 = 63.1579{{c}}, ~3/2 = 701.9002{{c}} (~225/224 = 7.1634{{c}})
: error map: {{val| 0.000 -0.055 -0.203 +0.033 }}


Map: [&lt;9 1 1 12 -7|, [0 8 12 8 23]]
{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}
EDOs: 342, 1053, 1395, 1737, 4869d, 6606cd
Badness: 0.0213


==Ennealimnic==
[[Badness]] (Sintel): 0.277
Commas: 243/242, 441/440, 4375/4356


POTE generator: ~36/35 = 49.395
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;9 1 1 12 -2|, &lt;0 2 3 2 5|]
Comma list: 540/539, 4375/4374, 16384/16335
EDOs: 72, 171, 243
Badness: 0.0203


===13 limit ennealimnic===
Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}
Commas: 243/242, 364/363, 441/440, 625/624


POTE generator: ~36/35 = 49.341
Optimal tunings:  
* WE: ~28/27 = 63.1431{{c}}, ~3/2 = 702.1956{{c}} (~225/224 = 7.6216{{c}})
* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.3164{{c}} (~225/224 = 7.5795{{c}})


Map: [&lt;9 1 1 12 -2 -33|, &lt;0 2 3 2 5 10|]
{{Optimal ET sequence|legend=0| 19, 133d, 152, 323e, 475de, 627de }}
EDOs: 72, 171, 243
Badness: 0.0233


====17 limit ennealimnic====
Badness (Sintel): 1.45
Commas: 243/242, 364/363, 375/374, 441/440, 595/594


POTE generator: ~36/35 = 49.335
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;9 1 1 12 -2 -33 -3|, &lt;0 2 3 2 5 10 6|]
Comma list: 540/539, 625/624, 729/728, 2205/2197
EDOs: 72, 171, 243
Badness: 0.0146


===Ennealim===
Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}
Commas: 169/168, 243/242, 325/324, 441/440


POTE generator: ~36/35 = 49.708
Optimal tunings:  
* WE: ~28/27 = 63.1406{{c}}, ~3/2 = 702.0192{{c}} (~225/224 = 7.4730{{c}})
* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.1539{{c}} (~225/224 = 7.4171{{c}})


Map: [&lt;9 1 1 12 -2 20|, &lt;0 2 3 2 5 2|]
{{Optimal ET sequence|legend=0| 19, 133df, 152f, 323ef }}
EDOs: 27e, 45f, 72, 315ff, 387cff, 459cdfff
Badness: 0.0207


==Ennealiminal==
Badness (Sintel): 1.39
Commas: 385/384, 1375/1372, 4375/4374


POTE generator: ~36/35 = 49.504
=== Hemienneadecal ===
Subgroup: 2.3.5.7.11


Map: [&lt;9 1 1 12 51|, &lt;0 2 3 2 -3|]
Comma list: 3025/3024, 4375/4374, 234375/234256
EDOs: 27, 45, 72, 171e, 243e, 315e
Badness: 0.0311


===13-limit===
Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}
Commas: 169/168, 325/324, 385/384, 1375/1372
: mapping generators: ~55/54, ~3


POTE generator: ~36/35 = 49.486
Optimal tunings:  
* WE: ~55/54 = 31.5800{{c}}, ~3/2 = 701.9053{{c}} (~243/242 = 7.1448{{c}})
* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9034{{c}} (~243/242 = 7.1666{{c}})


Map: [&lt;9 1 1 12 51 20|, &lt;0 2 3 2 -3 2|]
{{Optimal ET sequence|legend=0| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}
EDOs: 27, 45f, 72, 171ef, 243ef
Badness: 0.0303


==Trinealimmal==
Badness (Sintel): 0.330
Commas: 2401/2400, 4375/4374, 2097152/2096325


POTE generator: ~6/5 = 315.644
==== Hemienneadecalis ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;27 1 0 34 177|, &lt;0 2 3 2 -4|]
Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256
EDOs: 27, 243, 270, 783, 1053, 1323, 10854bcde
Badness: 0.0298


==Semihemiennealimmal==
Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}
Commas: 2401/2400, 4375/4374, 3025/3024, 4225/4224


POTE generator: ~39/32 = 342.139
Optimal tunings:  
* WE: ~55/54 = 31.5785{{c}}, ~3/2 = 701.9995{{c}} (~243/242 = 7.2727{{c}})
* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 702.0053{{c}} (~243/242 = 7.2685{{c}})


Map: [&lt;18 0 -1 22 48 88|, &lt;0 4 6 4 2 -3|]
{{Optimal ET sequence|legend=0| 152f, 342f, 494 }}
EDOs: 126, 144, 270, 684, 954
Badness: 0.0131


=Gamera=
Badness (Sintel): 0.859
Commas: 4375/4374, 589824/588245


POTE generator ~8/7 = 230.336
==== Hemienneadec ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 6 10 3|, &lt;0 -23 -40 -1|]
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
EDOs: 26, 73, 99, 224, 323, 422, 735
Badness: 0.0376


==Hemigamera==
Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}
Commas: 3025/3024, 4375/4374, 202397184/201768035


POTE generator: ~8/7 = 230.337
Optimal tunings:  
* WE: ~55/54 = 31.5784{{c}}, ~3/2 = 701.9736{{c}} (~243/242 = 7.2493{{c}})
* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9855{{c}} (~243/242 = 7.2487{{c}})


Map: [&lt;2 12 20 6 5|, &lt;0 -23 -40 -1 5|]
{{Optimal ET sequence|legend=0| 152, 342, 494, 1330, 1824, 2318d }}
EDOs: 26, 198, 224, 422, 646, 1068d
Badness: 0.0410


===13-limit===
Badness (Sintel): 1.26
Commas: 1716/1715 2080/2079 2200/2197 3025/3024


Map: [&lt;2 12 20 6 5 17|, &lt;0 -23 -40 -1 5 -25|]
==== Semihemienneadecal ====
EDOs: 26, 198, 224, 422, 646f, 1068df
Subgroup: 2.3.5.7.11.13
Badness: 0.0204


=Supermajor=
Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.0002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of &lt;&lt;37 46 75 -13 15 45||. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.


Commas: 4375/4374, 52734375/52706752
Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}
: mapping generators: ~55/54, ~429/250


POTE generator: ~9/7 = 435.082
Optimal tunings:  
* WE: ~55/54 = 31.5799{{c}}, ~429/250 = 935.1824{{c}} (~144/143 = 12.2152{{c}})
* CWE: ~55/54 = 31.5789{{c}}, ~429/250 = 935.1617{{c}} (~144/143 = 12.2067{{c}})


Map: [&lt;1 15 19 30|, &lt;0 -37 -46 -75|]
{{Optimal ET sequence|legend=0| 190, 304d, 494, 684, 1178, 2850, 4028ce }}
EDOs: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214
Badness: 0.0108


==Semisupermajor==
Badness (Sintel): 0.607
Commas: 3025/3024, 4375/4374, 35156250/35153041


POTE generator: ~9/7 = 435.082
=== Kalium ===
Named after the 19th element, potassium, and after an archaic variant of the element's name to resolve a name conflict. [[19/16]] can be used as a generator. Since it is enfactored in the 17-limit and lower, it makes no sense to name it for the lower subgroups.


Map: [&lt;2 30 38 60 41|, &lt;0 -37 -46 -75 -47|]
Subgroup: 2.3.5.7.11.13.17.19
EDOs: 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf
Badness: 0.0128


=Enneadecal=
Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344
Enndedecal temperament tempers out the enneadeca, |-14 -19 19&gt;, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5 or 7 limits, and [[494edo]] shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.


Commas: 4375/4374, 703125/702464
Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}


POTE generator: ~3/2 = 701.880
Optimal tunings:  
* WE: ~28/27 = 63.1582{{c}}, ~6545/5928 = 171.2448{{c}}
* CWE: ~28/27 = 63.1579{{c}}, ~6545/5928 = 171.2439{{c}}


Map: [&lt;19 0 14 -37|, &lt;0 1 1 3|]
{{Optimal ET sequence|legend=0| 855, 988, 1843 }}
Generators: 28/27, 3
EDOs: 19, 152, 171, 665, 836, 1007, 2185
Badness: 0.0110


=Deca=
Badness (Sintel): 3.15
Commas: 4375/4374, 165288374272/164794921875


POTE generator: ~460992/390625 = 284.423
== Semidimi ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimi]].''


Map: [&lt;10 4 2 9|, &lt;0 5 6 11|]
The generator of semidimi is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374.
EDOs: 80, 190, 270, 1270, 1540, 1810, 2080
Badness: 0.0806


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 3025/3024, 4375/4374, 422576/421875


POTE generator: ~33/28 = 284.418
[[Comma list]]: 4375/4374, 3955078125/3954653486


Map: [&lt;10 4 2 9 18|, &lt;0 5 6 11 7|]
{{Mapping|legend=1| 1 -19 -25 -32 | 0 55 73 93 }}
EDOs: 80, 190, 270, 1000, 1270
: mapping generators: ~2, ~35/27
Badness: 0.0243


==13-limit==  
[[Optimal tuning]]s:
Commas: 1001/1000, 3025/3024, 4225/4224, 4375/4374
* [[WE]]: ~2 = 1200.0018{{c}}, ~35/27 = 449.1277{{c}}
: [[error map]]: {{val| +0.002 +0.031 -0.040 -0.012 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 449.1270{{c}}
: error map: {{val| 0.000 +0.030 -0.043 -0.015 }}


POTE generator: ~33/28 = 284.398
{{Optimal ET sequence|legend=1| 8d, …, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}


Map: [&lt;10 4 2 9 18 37|, &lt;0 5 6 11 7 0|]
[[Badness]] (Sintel): 0.382
EDOs: 80, 190, 270, 730, 1000
Badness: 0.0168


=Mitonic=  
== Brahmagupta ==
Commas: 4375/4374, 2100875/2097152
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}), and may be described as the {{nowrap| 217 & 224 }} temperament.


POTE generator: ~10/9 = 182.458
Early in the design of the [[Sagittal]] notation system, [[George Secor|Secor]] and [[Dave Keenan|Keenan]] found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4{{c}} many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of brahmagupta that has pure octaves and pure fifths, which can also be described as a 17-limit extension having a 1/7-octave period (171.4286{{c}}) and 1/21-apotome generator (5.4136{{c}}).


Map: [&lt;1 16 32 -15|, &lt;0 -17 -35 21|]
[[Subgroup]]: 2.3.5.7
EDOs: 46, 125, 171
Badness: 0.0252


=Abigail=
[[Comma list]]: 4375/4374, {{monzo| 46 -14 -3 -6 }}
Commas: 4375/4374, 2147483648/2144153025


[[POTE tuning|POTE generator]]: 208.899
{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}
: mapping generators: ~1157625/1048576, ~27/20


Map: [&lt;2 7 13 -1|, &lt;0 -11 -24 19|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;22 48 -38 25 -122 -223||
* [[WE]]: ~1157625/1048576 = 171.4275{{c}}, ~27/20 = 519.7125{{c}}
EDOs: 46, 132, 178, 224, 270, 494, 764, 1034, 1798
: [[error map]]: {{val| -0.007 +0.037 -0.034 -0.004 }}
Badness: 0.0370
* [[CWE]]: ~1157625/1048576 = 171.4286{{c}}, ~27/20 = 519.7156{{c}}
: error map: {{val| 0.000 +0.049 -0.018 +0.017 }}


==11-limit==
{{Optimal ET sequence|legend=1| 7, …, 217, 224, 441, 1106, 1547 }}
Comma: 3025/3024, 4375/4374, 20614528/20588575


[[POTE tuning|POTE generator]]: 208.901
[[Badness]] (Sintel): 0.737


Map: [&lt;2 7 13 -1 1|, &lt;0 -11 -24 19 17|]
=== 11-limit ===
EDOs: 46, 132, 178, 224, 270, 494, 764
Subgroup: 2.3.5.7.11
Badness: 0.0129


==13-limit==
Comma list: 4000/3993, 4375/4374, 131072/130977
Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095


[[POTE tuning|POTE generator]]: 208.903
Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}


Map: [&lt;2 7 13 -1 1 -2|, &lt;0 -11 -24 19 17 27|]
Optimal tunings:  
EDOs: 46, 178, 224, 270, 494, 764, 1258
* WE: ~243/220 = 171.4208{{c}}, ~27/20 = 519.6807{{c}}
Badness: 0.00886
* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7034{{c}}


=Semidimi=
{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665 }}
Commas: 4375/4374, 3955078125/3954653486


POTE generator: ~35/27 = 449.127
Badness (Sintel): 1.73


Map: [&lt;1 36 48 61|, &lt;0 -55 -73 -93|]
=== 13-limit ===
Wedgie: &lt;&lt;55 73 93 -12 -7 11||
Subgroup: 2.3.5.7.11.13
EDOs: 171, 863, 8419, 1205, 1376, 1547, 1718, 4983, 6701, 8419
Badness: 0.0151


=Brahmagupta=
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374
Commas: 4375/4374, 70368744177664/70338939985125


POTE generator: ~27/20 = 519.716
Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}


Map: [&lt;7 2 -8 53|, &lt;0 3 8 -11|]
Optimal tunings:  
Wedgie: &lt;&lt;21 56 -77 40 -181 -336||
* WE: ~243/220 = 171.4197{{c}}, ~27/20 = 519.6789{{c}}
EDOs: 217, 224, 441, 1106, 1547
* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7052{{c}}
Badness: 0.0291


==11-limit==
{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1106e }}
Commas: 4000/3993, 4375/4374, 131072/130977


POTE generator: ~27/20 = 519.704
Badness (Sintel): 0.956


Map: [&lt;7 2 -8 53 3|, &lt;0 3 8 -11 7|]
== Abigail ==
EDOs: 217, 224, 441, 665, 1771e
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Abigail]].''
Badness: 0.0522


=Neusec=
Abigail tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit, and may be described as the {{nowrap| 46 & 224 }} temperament, with a [[ploidacot]] signature of diploid wau-hendecacot. It extends into a very strong 11- and 13-limit temperament. [[494edo]], [[764edo]] and [[1258edo]] are among the possible tunings.
Commas: 3025/3024, 4375/4374, 235298/234375


POTE generator: ~12/11 = 151.547
Abigail was named by [[Gene Ward Smith]] in 2010 after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930 Yahoo! Tuning Group | ''11-limit rank 2 using only wedgies''] "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things." —Gene Ward Smith</ref>


Map: [&lt;2 11 15 19 15|, &lt;0 -31 -41 -53 -32|]
[[Subgroup]]: 2.3.5.7
EDOs: 190, 388
Badness: 0.0591


==13-limit==
[[Comma list]]: 4375/4374, 2147483648/2144153025
Commas: 847/845, 1001/1000, 3025/3024, 4375/4374


POTE generator: ~12/11 = 151.545
{{Mapping|legend=1| 2 -4 -11 18 | 0 11 24 -19 }}
: mapping generators: ~46305/32768, ~1536/1225


Map: [&lt;2 11 15 19 15 17|, &lt;0 -31 -41 -53 -32 -38|]
[[Optimal tuning]]s:  
EDOs: 190, 198, 388
* [[WE]]: ~46305/32768 = 599.9699{{c}}, ~1536/1225 = 391.0818{{c}}
Badness: 0.0309
: [[error map]]: {{val| -0.060 +0.065 -0.021 +0.079 }}
* [[CWE]]: ~46305/32768 = 600.0000{{c}}, ~1536/1225 = 391.1007{{c}}
: error map: {{val| 0.000 +0.152 +0.102 +0.262 }}


=Quasithird=
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd }}
Commas: 4375/4374, 1153470752371588581/1152921504606846976


POTE generator: ~5103/4096 = 380.388
[[Badness]] (Sintel): 0.936


Map: [&lt;4 0 -11 48|, &lt;0 5 16 -29|]
=== 11-limit ===
Wedgie: &lt;&lt;20 64 -116 55 -240 -449||
Subgroup: 2.3.5.7.11
EDOs: 164, 224, 388, 612, 1448, 2060
Badness: 0.0618


==11-limit==
Comma list: 3025/3024, 4375/4374, 131072/130977
Commas: 3025/3024, 4375/4374, 4296700485/4294967296


POTE generator: ~5103/4096 = 380.387
Mapping: {{mapping| 2 -4 -11 18 18 | 0 11 24 -19 -17 }}


Map: [&lt;4 0 -11 48 43|, &lt;0 5 16 -29 -23|]
Optimal tunings:  
EDOs: 164, 224, 388, 612, 836, 1448
* WE: ~99/70 = 599.9782{{c}}, ~1536/1225 = 391.0852{{c}}
Badness: 0.0211
* CWE: ~99/70 = 600.0000{{c}}, ~1536/1225 = 391.0992{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 46, 132, 178, 224, 270, 494, 764 }}
Commas: 2200/2197, 3025/3024, 4375/4374, 468512/468195


POTE generator: ~5103/4096 = 380.385
Badness (Sintel): 0.425


Map: [&lt;4 0 -11 48 43 11|, &lt;0 5 16 -29 -23 3|]
=== 13-limit ===
EDOs: 164, 224, 388, 612, 836, 1448f, 2284f
Subgroup: 2.3.5.7.11.13
Badness: 0.0295


=Semidimfourth=
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
Commas: 4375/4374, 235298/234375


POTE generator: ~35/27 = 448.457
Mapping: {{mapping| 2 -4 -11 18 18 25 | 0 11 24 -19 -17 -27 }}


Map: [&lt;1 21 28 36|, &lt;0 -31 -41 -53|]
Optimal tunings:  
Wedgie: &lt;&lt;31 41 53 -7 -3 8||
* WE: ~99/70 = 599.9862{{c}}, ~351/280 = 391.0879{{c}}
EDOs: 91, 99, 289, 388, 875, 1263d, 1651d
* CWE: ~99/70 = 600.0000{{c}}, ~351/280 = 391.0969{{c}}
Badness: 0.0552


=Acrokleismic=
{{Optimal ET sequence|legend=0| 46, 178, 224, 270, 494, 764, 1258 }}
Commas: 4375/4374, 2202927104/2197265625


POTE generator: ~6/5 = 315.557
Badness (Sintel): 0.366


Map: [&lt;1 10 11 27|, &lt;0 -32 -33 -92|]
== Gamera ==
Wedgie: &lt;&lt;32 33 92 -22 56 121||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Gamera]].''
EDOs: 19, 251, 270
Badness: 0.0562


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 4375/4374, 41503/41472, 172032/171875


POTE generator: ~6/5 = 315.558
[[Comma list]]: 4375/4374, 589824/588245


Map: [&lt;1 10 11 27 -16|, &lt;0 -32 -33 -92 74|]
{{Mapping|legend=1| 1 -17 -30 2 | 0 23 40 1 }}
EDOs: 19, 251, 270, 829, 1099, 1369, 1639
: mapping generators: ~2, ~7/4
Badness: 0.0369


==13-limit==
[[Optimal tuning]]s:
Commas: 676/675, 1001/1000, 4375/4374, 10985/10976
* [[WE]]: ~2 = 1199.8483{{c}}, ~7/4 = 969.5415{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 969.6608{{c}}


POTE generator: ~6/5 = 315.557
{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}


Map: [&lt;1 10 11 27 -16 25|, &lt;0 -32 -33 -92 74 -81|]
[[Badness]] (Sintel): 0.953
EDOs: 19, 251, 270
Badness: 0.0268


==Counteracro==  
=== Hemigamera ===
Commas: 4375/4374, 5632/5625, 117649/117612
Subgroup: 2.3.5.7.11


POTE generator: ~6/5 = 315.553
Comma list: 3025/3024, 4375/4374, 589824/588245


Map: [&lt;1 10 11 27 55|, &lt;0 -32 -33 -92 -196|]
Mapping: {{mapping| 2 -11 -20 5 10 | 0 23 40 1 -5 }}
EDOs: 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde
: mapping generators: ~99/70, ~99/80
Badness: 0.0426


===13-limit===
Optimal tunings:
Commas: 676/675, 1716/1715, 4225/4224, 4375/4374
* WE: ~99/70 = 599.9323{{c}}, ~99/80 = 369.6212{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~99/80 = 369.6610{{c}}


POTE generator: ~6/5 = 315.554
{{Optimal ET sequence|legend=0| 26, 172c, 198, 224, 422, 646, 1068d }}


Map: [&lt;1 10 11 27 55 25|, &lt;0 -32 -33 -92 -196 -81|]
Badness (Sintel): 1.35
EDOs: 270, 1331c, 1601c, 1871bcf, 2141bcf
Badness: 0.0260


=Seniority=
==== 13-limit ====
Commas: 4375/4374 201768035/201326592
Subgroup: 2.3.5.7.11.13


POTE generator: ~3087/2560 = 322.804
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024


Map: [&lt;1 11 19 2|, &lt;0 -35 -62 3|]
Mapping: {{mapping| 2 -11 -20 5 10 -8 | 0 23 40 1 -5 25 }}
Wedgie: &lt;&lt;35 62 -3 17 -103 -181||
EDOs: 26, 145, 171, 2710d
Badness: 0.0449


=Orga=
Optimal tunings:
Commas: 4375/4374 54975581388800/54936068900769
* WE: ~99/70 = 599.9207{{c}}, ~26/21 = 369.6139{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~26/21 = 369.6603{{c}}


POTE generator: ~8/7 = 231.104
{{Optimal ET sequence|legend=0| 26, 172cf, 198, 224, 422, 646f, 1068df }}


Map: [&lt;2 21 36 5|, &lt;0 -29 -51 1|]
Badness (Sintel): 0.844
Wedgie: &lt;&lt;58 102 -2 27 -166 -291||
EDOs: 26, 244, 270, 836, 1106, 1376, 2482, 19856bd, 23714bd
Badness: 0.0402


==11-limit==
=== Semigamera ===
Commas: 3025/3024 4375/4374 5767168/5764801
Subgroup: 2.3.5.7.11


POTE generator: ~8/7 = 231.103
Comma list: 4375/4374, 14641/14580, 15488/15435


Map: [&lt;2 21 36 5 2|, &lt;0 -29 -51 1 8|]
Mapping: {{mapping| 1 -40 -70 1 -77 | 0 46 80 2 89 }}
EDOs: 26, 244, 270, 566, 836, 1106, 7472e, 8578de, 9684cde, 10790cde, 11896cde
: mapping generators: ~2, ~144/77
Badness: 0.0162


=Quatracot=
Optimal tunings:
Commas: 4375/4374, 1483154296875/1473173782528
* WE: ~2 = 1199.8845{{c}}, ~144/77 = 1084.7314{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8345{{c}}


POTE generator: ~448/405 = 176.805
{{Optimal ET sequence|legend=0| 73, 125, 198, 323, 521 }}


Map: [&lt;2 7 7 23|, &lt;0 -13 -8 -59|]
Badness (Sintel): 2.59
Wedgie: &lt;&lt;26 16 118 -35 114 229||
EDOs: 190, 224, 414, 638, 1052c, 1690bc
Badness: 0.1760


==11-limit==
==== 13-limit ====
Commas: 3025/3024, 4375/4374, 1265625/1261568
Subgroup: 2.3.5.7.11.13


POTE generator: ~448/405 = 176.806
Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580


Map: [&lt;2 7 7 23 19|, &lt;0 -13 -8 -59 -41|]
Mapping: {{mapping| 1 -40 -70 1 -77 -131 | 0 46 80 2 89 149 }}
EDOs: 190, 224, 414, 638, 1052c
Badness: 0.0410


==13-limit==
Optimal tunings:
Commas: 625/624, 729/728, 1575/1573, 2200/2197
* WE: ~2 = 1199.8726{{c}}, ~144/77 = 1084.7220{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8359{{c}}


POTE generator: ~448/405 = 176.804
{{Optimal ET sequence|legend=0| 73f, 125f, 198, 323, 521 }}


Map: [&lt;2 7 7 23 19 13|, &lt;0 -13 -8 -59 -41 -19|]
Badness (Sintel): 1.82
EDOs: 190, 224, 414, 638, 1690bc, 2328bcde
Badness: 0.0226


=Nearly Micro=  
== Crazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


=Octoid=
Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament, with a [[ploidacot]] of diploid alpha-octacot. [[1106edo]] gives a strong tuning.
Commas: 4375/4374, 16875/16807


POTE generator: ~7/5 = 583.940
Crazy was named by [[Flora Canou]] in 2025 by removing the mutation from ''kwazy'', the name for the 5-limit microtemperament.  


Map: [&lt;8 1 3 3|, &lt;0 3 4 5|]
[[Subgroup]]: 2.3.5.7
Generators: 49/45, 7/5
EDOs: 72, 152, 224
Badness: 0.0427


==11-limit==
[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}
Commas: 540/539, 1375/1372, 4000/3993


POTE generator: ~7/5 = 583.692
{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}
: mapping generators: ~332150625/234881024, ~1125/1024


Map: [&lt;8 1 3 3 16|, &lt;0 3 4 5 3|]
[[Optimal tuning]]s:
EDOs: 72, 152, 224
* [[WE]]: ~332150625/234881024 = 600.0019{{c}}, ~1125/1024 = 162.7479{{c}}
Badness: 0.0141
: [[error map]]: {{val| +0.004 +0.030 -0.042 -0.014 }}
* [[CWE]]: ~332150625/234881024 = 600.0000{{c}}, ~1125/1024 = 162.7474{{c}}
: error map: {{val| 0.000 +0.024 -0.051 -0.022 }}


==13-limit==
{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}
Commas: 540/539, 1375/1372, 4000/3993, 625/624


POTE generator: ~7/5 = 583.905
[[Badness]] (Sintel): 0.998


Map: [&lt;8 1 3 3 16 -21|, &lt;0 3 4 5 3 13|]
=== 11-limit ===
EDOs: 72, 224
Subgroup: 2.3.5.7.11
Badness: 0.0153


==Music==
Comma list: 3025/3024, 4375/4374, 2791309312/2790703125
http://www.archive.org/details/Dreyfus
[[http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3|play]]


==Octopus==
Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}
Commas: 169/168, 325/324, 364/363, 540/539


POTE generator: ~7/5 = 583.892
Optimal tunings:
* WE: ~99/70 = 600.0047{{c}}, ~1125/1024 = 162.7493{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~1125/1024 = 162.7481{{c}}


Map: [&lt;8 1 3 3 16 14|, &lt;0 3 4 5 3 4|]
{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}
EDOs: 72, 152, 224f
Badness: 0.0217


=Amity=
Badness (Sintel): 0.562
The generator for amity temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&amp;53 temperament, or by its wedgie, &lt;&lt;5 13 -17 9 -41 -76||. [[99edo]] is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.


In the 5-limit amity is a genuine microtemperament, with 58/205 being a possible tuning. Another good choice is (64/5)^(1/13), which gives pure major thirds.
== Orga ==
Orga may be described as the {{nowrap| 26 & 270 }} temperament, and [[1106edo]] gives a strong tuning.  


==5-limit==
[[Subgroup]]: 2.3.5.7
Comma: 1600000/1594323


POTE generator: ~243/200 = 339.519
[[Comma list]]: 4375/4374, {{monzo| 41 -4 2 -14 }}


Map: [&lt;1 3 6|, &lt;0 -5 -13|]
{{Mapping|legend=1| 2 -8 -15 6 | 0 29 51 -1 }}
EDOs: 7, 39, 46, 53, 152, 205, 463, 668, 873
: mapping generators: ~7411887/5242880, ~8/7
Badness: 0.0220


==7-limit==
[[Optimal tuning]]s:
Commas: 4375/4374, 5120/5103
* [[WE]]: ~7411887/5242880 = 599.9927{{c}}, ~8/7 = 231.1012{{c}}
: [[error map]]: {{val| -0.015 +0.037 -0.045 +0.029 }}
* [[CWE]]: ~7411887/5242880 = 600.0000{{c}}, ~8/7 = 231.1037{{c}}
: error map: {{val| 0.000 +0.053 -0.023 +0.070 }}


POTE generator: ~243/200 = 339.432
{{Optimal ET sequence|legend=1| 26, …, 244, 270, 836, 1106, 1376, 2482 }}


Map: [&lt;1 3 6 -2|, &lt;0 -5 -13 17|]
[[Badness]] (Sintel): 1.02
Wedgie: &lt;&lt;5 13 -17 9 -41 -76||
EDOs: 7, 39, 46, 53, 99, 251, 350
Badness: 0.0236


==11-limit==  
=== 11-limit ===
Commas: 540/539, 4375/4374, 5120/5103
Subgroup: 2.3.5.7.11


POTE generator: ~243/200 = 339.464
Comma list: 3025/3024, 4375/4374, 5767168/5764801


Map: [&lt;1 3 6 -2 21|, &lt;0 -5 -13 17 -62|]
Mapping: {{mapping| 2 -8 -15 6 10 | 0 29 51 -1 -8 }}
EDOs: 53, 99e, 152, 555de, 707de, 859bde
Badness: 0.0315


==13-limit==
Optimal tunings:
Commas: 352/351, 540/539, 625/624, 847/845
* WE: ~99/70 = 600.0025{{c}}, ~8/7 = 231.1039{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1030{{c}}


POTE generator: ~243/200 = 339.481
{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836, 1106 }}


Map: [&lt;1 3 6 -2 21 17|, &lt;0 -5 -13 17 -62 -47|]
Badness (Sintel): 0.535
EDOS: 53, 99ef, 152f, 205
Badness: 0.0280


==Accord==
=== 13-limit ===
Commas: 126/125, 100352/98415
Subgroup: 2.3.5.7.11.13


POTE generator: ~243/200 = 338.993
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360


Map: [&lt;1 3 6 11|, &lt;0 -5 -13 -29|]
Mapping: {{mapping| 2 -8 -15 6 10 -3 | 0 29 51 -1 -8 27 }}
Wedgie: &lt;&lt;5 13 29 9 32 31||
EDOs: 46, 131c, 177c
Badness: 0.0956


===11-limit===
Optimal tunings:
Commas: 121/120, 126/125, 896/891
* WE: ~99/70 = 600.0192{{c}}, ~8/7 = 231.1102{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1033{{c}}


POTE generator: ~11/9 = 339.047
{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836f, 1106f }}


Map: [&lt;1 3 6 11 6|, &lt;0 -5 -13 -29 -9|]
Badness (Sintel): 0.899
EDOs: 46, 177c, 223bc, 269bce
Badness: 0.0425


==Hitchcock==  
== Seniority ==
Commas: 121/120, 176/175, 2200/2187
: ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]].


POTE generator: ~11/9 = 339.340
Aside from the ragisma, the seniority temperament tempers out the [[wadisma]], 201768035/201326592, and may be described as {{nowrap| 26 & 145 }}. It is so named because the [[senior comma]] ({{monzo| -17 62 -35 }}) is tempered out.


Map: [&lt;1 3 6 -2 6|, &lt;0 -5 -13 17 -9|]
[[Subgroup]]: 2.3.5.7
EDOs: 7, 39, 46, 53, 99
Badness: 0.0352


===13-limit===
[[Comma list]]: 4375/4374, 201768035/201326592
Commas: 121/120, 169/168, 176/175, 325/324


POTE generator: ~11/9 = 339.419
{{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }}
: mapping generators: ~2, ~5120/3087


Map: [&lt;1 3 6 -2 6 2|, &lt;0 -5 -13 17 -9 6|]
[[Optimal tuning]]s:  
EDOs: 7, 39, 46, 53, 99
* [[WE]]: ~2 = 1200.0745{{c}}, ~5120/3087 = 877.2500{{c}}
Badness: 0.0224
: [[error map]]: {{val| +0.075 +0.008 -0.016 -0.203 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5120/3087 = 877.1965{{c}}
: error map: {{val| 0.000 -0.077 -0.130 -0.415 }}


==Hemiamity==
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d }}
Commas: 4375/4374, 5120/5103, 3025/3024


POTE generator: ~ 243/200 = 339.493
[[Badness]] (Sintel): 1.14


Map: [&lt;2 1 -1 13 13|, &lt;0 5 13 -17 -14|]
=== Senator ===
EDOs: 14, 46, 106, 152, 350
Senator (26 & 145) extends seniority by tempering out [[441/440]] and [[65536/65219]], and can be extended to the 13- and 17-limit immediately by adding [[364/363]] and [[595/594]] to the comma list in this order.


=Parakleismic=
Subgroup: 2.3.5.7.11
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13&gt;, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being &lt;&lt;13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie &lt;&lt;13 14 35 -36 ...|| adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.


Comma: 124440064/1220703125
Comma list: 441/440, 4375/4374, 65536/65219


POTE generator: ~6/5 = 315.240
Mapping: {{mapping| 1 -24 -43 5 2 | 0 35 62 -3 2 }}


Map: [&lt;1 5 6|, &lt;0 -13 -14|]
Optimal tunings:  
EDOs: 19, 61, 80, 99, 118, 453, 571, 689, 1496
* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}}
Badness: 0.0433
* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}}


==7-limit==
{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e }}
Commas: 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.181
Badness (Sintel): 3.05


Map: [&lt;1 5 6 12|, &lt;0 -13 -14 -35|]
==== 13-limit ====
EDOs: 19, 80, 99, 217, 316, 415
Subgroup: 2.3.5.7.11.13
Badness: 0.0274


==11-limit==
Comma list: 364/363, 441/440, 2200/2197, 4375/4374
Commas: 385/384, 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.251
Mapping: {{mapping| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}


Map: [&lt;1 5 6 12 -6|, &lt;0 -13 -14 -35 36|]
Optimal tunings:  
EDOs: 19, 99, 118
* WE: ~2 = 1199.7136{{c}}, ~108/65 = 877.9974{{c}}
Badness: 0.0497
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2038{{c}}


==Parkleismic==
{{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }}
Commas: 176/175, 1375/1372, 2200/2187


POTE generator: ~6/5 = 315.060
Badness (Sintel): 1.85


Map: [&lt;1 5 6 12 20|, &lt;0 -13 -14 -35 -63|]
==== 17-limit ====
EDOs: 80, 179, 259cd
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0559


===13-limit===
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197
Commas: 169/168, 176/175, 325/324, 1375/1372


POTE generator: ~6/5 = 315.075
Mapping: {{mapping| 1 -24 -43 5 2 -27 -31 | 0 35 62 -3 2 42 48 }}


Map: [&lt;1 5 6 12 20 10|, &lt;0 -13 -14 -35 -63 -24|]
Optimal tunings:  
EDOs: 15, 19, 80, 179
* WE: ~2 = 1199.7195{{c}}, ~108/65 = 877.0018{{c}}
Badness: 0.0366
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}}


==Paradigmic==
{{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }}
Commas: 540/539, 896/891, 3136/3125


POTE generator: ~6/5 = 315.096
Badness (Sintel): 1.35


Map: [&lt;1 5 6 12 -1|, &lt;0 -13 -14 -35 17|]
== Monzismic ==
EDOs: 19, 80, 99e, 179e
: ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]].  
Badness: 0.0417


===13-limit===
Monzismic tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]]. It may be described as the {{nowrap| 53 & 612 }} temperament, with a [[ploidacot]] signature of alpha-dicot. A notable tuning not appearing on the optimal ET sequence is [[665edo]], which is nearly equivalent to the pure-3's tuning.
Commas: 169/168, 325/324, 540/539, 832/825


POTE generator: ~6/5 = 315.080
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 5 6 12 -1 10|, &lt;0 -13 -14 -35 17 -24|]
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}
EDOs: 19, 80, 99e, 179e
Badness: 0.0358


=Semiparakleismic=
{{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }}
Commas: 3025/3024, 3136/3125, 4375/4374
: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}


POTE generator: 315.181
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0128{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9895{{c}}
: [[error map]]: {{val| +0.013 +0.024 -0.049 -0.019 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9793{{c}}
: error map: {{val| 0.000 +0.004 -0.080 -0.050 }}


Map: [&lt;2 10 12 24 19|, &lt;0 -13 -14 -35 -23|]
{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd }}
EDOs: 80, 118, 198, 316, 514c, 830c
Badness: 0.0342


=Quincy=
[[Badness]] (Sintel): 1.18
Commas: 4375/4374, 823543/819200


POTE generator: ~1728/1715 = 16.613
=== Monzism ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 2 3|, &lt;0 -30 -49 -14|]
Comma list: 4375/4374, 41503/41472, 184549376/184528125
EDOs: 72, 217, 289
Badness: 0.0797


==11-limit==
Mapping: {{mapping| 1 0 -27 109 -159 | 0 2 37 -134 205 }}
Commas: 441/440, 4000/3993, 41503/41472


POTE generator: ~100/99 = 16.613
Optimal tunings:  
* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}}


Map: [&lt;1 2 2 3 4|, &lt;0 -30 -49 -14 -39|]
{{Optimal ET sequence|legend=0| 53, 559, 612, 3619de, 4231de, …, 6067ddee }}
EDOs: 72, 217, 289
Badness: 0.0309


==13-limit==
Badness (Sintel): 1.89
Commas: 364/363, 441/440, 676/675, 4375/4374


POTE generator: ~100/99 = 16.602
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 2 3 4 5|, &lt;0 -30 -49 -14 -39 -94|]
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
EDOs: 72, 145, 217, 289
Badness: 0.0239


==17-limit==
Mapping: {{mapping| 1 0 -27 109 -159 -70 | 0 2 37 -134 205 93 }}
Commas: 364/363, 441/440, 595/594, 1001/1000, 1156/1155


POTE generator: ~100/99 = 16.602
Optimal tunings:  
* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}}


Map: [&lt;1 2 2 3 4 5 5|, &lt;0 -30 -49 -14 -39 -94 -66|]
{{Optimal ET sequence|legend=0| 53, 559, 612 }}
EDOs: 72, 145, 217, 289
Badness: 0.0147


==19-limit==
Badness (Sintel): 2.22
Commas: 343/342, 364/363, 441/440, 595/594, 676/675, 2601/2600


POTE generator: ~100/99 = 16.594
== Semidimfourth ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimfourth]].''


Map: [&lt;1 2 2 3 4 5 5 4|, &lt;0 -30 -49 -14 -39 -94 -66 18|]
The semidimfourth temperament is featured by a semidiminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, [[235298/234375]].
EDOs: 72, 145, 217
 
Badness: 0.0152</pre></div>
[[Subgroup]]: 2.3.5.7
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Ragismic microtemperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:152:&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:152 --&gt;&lt;!-- ws:start:WikiTextTocRule:153: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Ennealimmal"&gt;Ennealimmal&lt;/a&gt;&lt;/div&gt;
[[Comma list]]: 4375/4374, 235298/234375
&lt;!-- ws:end:WikiTextTocRule:153 --&gt;&lt;!-- ws:start:WikiTextTocRule:154: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Ennealimmal-11 limit hemiennealimmal"&gt;11 limit hemiennealimmal&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:154 --&gt;&lt;!-- ws:start:WikiTextTocRule:155: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Ennealimmal-13 limit hemiennealimmal"&gt;13 limit hemiennealimmal&lt;/a&gt;&lt;/div&gt;
{{Mapping|legend=1| 1 -10 -13 -17 | 0 31 41 53 }}
&lt;!-- ws:end:WikiTextTocRule:155 --&gt;&lt;!-- ws:start:WikiTextTocRule:156: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Ennealimmal-Semiennealimmal"&gt;Semiennealimmal&lt;/a&gt;&lt;/div&gt;
: mapping generators: ~2, ~35/27
&lt;!-- ws:end:WikiTextTocRule:156 --&gt;&lt;!-- ws:start:WikiTextTocRule:157: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Ennealimmal-Semiennealimmal-13 limit semiennealimmal"&gt;13 limit semiennealimmal
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9936{{c}}, ~35/27 = 448.4533{{c}}
: [[error map]]: {{val| -0.007 +0.160 +0.353 -0.694 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 448.4555{{c}}
: error map: {{val| 0.000 +0.165 +0.361 -0.685 }}
 
{{Optimal ET sequence|legend=1| 8d, …, 91, 99, 289, 388, 875 }}
 
[[Badness]] (Sintel): 1.40
 
=== Neusec ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 235298/234375
 
Mapping: {{mapping| 2 -20 -26 -34 -17 | 0 31 41 53 32 }}
: mapping generators: ~99/70, ~35/27
 
Optimal tunings:
* WE: ~99/70 = 600.0381{{c}}, ~35/27 = 448.4812{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4546{{c}}
 
{{Optimal ET sequence|legend=0| 8d, …, 190, 388 }}
 
Badness (Sintel): 1.95
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
 
Mapping: {{mapping| 2 -20 -26 -34 -17 -21 | 0 31 41 53 32 38 }}
 
Optimal tunings:
* WE: ~99/70 = 600.0034{{c}}, ~35/27 = 448.4573{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4549{{c}}
 
{{Optimal ET sequence|legend=0| 8d, …, 190, 198, 388 }}
 
Badness (Sintel): 1.28
 
== Acrokleismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 2202927104/2197265625
 
{{Mapping|legend=1| 1 -22 -22 -65 | 0 32 33 92 }}
: mapping generators: ~2, ~5/3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9305{{c}}, ~5/3 = 884.3923{{c}}
: [[error map]]: {{val| -0.070 +0.126 +0.160 -0.221 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.4423{{c}}
: error map: {{val| 0.000 +0.198 +0.282 -0.136 }}
 
{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}
 
[[Badness]] (Sintel): 1.42
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 41503/41472, 172032/171875
 
Mapping: {{mapping| 1 -22 -22 -65 58 | 0 32 33 92 -74 }}
 
Optimal tunings:
* WE: ~2 = 1199.9698{{c}}, ~5/3 = 884.4193{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4414{{c}}
 
{{Optimal ET sequence|legend=0| 19, 251, 270, 829, 1099, 1369, 1639 }}
 
Badness (Sintel): 1.22
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
 
Mapping: {{mapping| 1 -22 -22 -65 58 -56 | 0 32 33 92 -74 81 }}
 
Optimal tunings:
* WE: ~2 = 1199.9939{{c}}, ~5/3 = 884.4384{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4429{{c}}
 
{{Optimal ET sequence|legend=0| 19, 251, 270 }}
 
Badness (Sintel): 1.11
 
=== Counteracro ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 5632/5625, 117649/117612
 
Mapping: {{mapping| 1 -22 -22 -65 -141 | 0

Latest revision as of 02:29, 16 June 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 is a collection of rank-2 temperaments tempering out the ragisma, 4375/4374 ([-1 -7 4 1⟩). The ragisma is the smallest 7-limit superparticular ratio.

Since (10/9)4 = (4375/4374)⋅(32/21), the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = (4375/4374)⋅(27/25)2, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.

Microtemperaments considered below, sorted by badness, are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, ragitritonic, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:

Supermajor

The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. Note that in the data that follow, the generator is given as its octave complement. 37 of these give 3/222, 46 give 5/227, and 75 give 7/245. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106edo or 1277edo can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80-note generator chain is presumably the place to start, and if that is not enough notes for you, there is always the 171-note generator chain.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 52734375/52706752

Mapping: [⟨1 -22 -27 -45], ⟨0 37 46 75]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1200.0067 ¢, ~14/9 = 764.9222 ¢
error map: ⟨+0.007 +0.019 -0.074 +0.037]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 764.9181 ¢
error map: ⟨0.000 +0.013 -0.083 +0.029]

Optimal ET sequence: 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214

Badness (Sintel): 0.274

Semisupermajor

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 35156250/35153041

Mapping: [⟨2 -7 -8 -15 -6], ⟨0 37 46 75 47]]

mapping generators: ~99/70, ~11/10

Optimal tunings:

  • WE: ~99/70 = 600.0103 ¢, ~11/10 = 164.9205 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~11/10 = 164.9180 ¢

Optimal ET sequence: 80, 262d, 342, 764, 1106, 1448, 2554, 4002e, 6556cee

Badness (Sintel): 0.422

Enneadecal

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Enneadecal (5-limit).

Enneadecal tempers out the enneadeca, [-14 -19 19⟩, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)1/3. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of 19edo up to just ones.

171edo is a good tuning for either the 5- or 7-limit, and 494edo shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use 665edo for a tuning.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 703125/702464

Mapping: [⟨19 0 14 -37], ⟨0 1 1 3]]

mapping generators: ~28/27, ~3

Optimal tunings:

  • WE: ~28/27 = 63.1599 ¢, ~3/2 = 701.9027 ¢ (~225/224 = 7.1437 ¢)
error map: ⟨+0.038 -0.014 -0.134 +0.080]
  • CWE: ~28/27 = 63.1579 ¢, ~3/2 = 701.9002 ¢ (~225/224 = 7.1634 ¢)
error map: ⟨0.000 -0.055 -0.203 +0.033]

Optimal ET sequence: 19, …, 152, 171, 665, 836, 1007, 2185, 3192c

Badness (Sintel): 0.277

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 16384/16335

Mapping: [⟨19 0 14 -37 126], ⟨0 1 1 3 -2]]

Optimal tunings:

  • WE: ~28/27 = 63.1431 ¢, ~3/2 = 702.1956 ¢ (~225/224 = 7.6216 ¢)
  • CWE: ~28/27 = 63.1579 ¢, ~3/2 = 702.3164 ¢ (~225/224 = 7.5795 ¢)

Optimal ET sequence: 19, 133d, 152, 323e, 475de, 627de

Badness (Sintel): 1.45

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 2205/2197

Mapping: [⟨19 0 14 -37 126 -20], ⟨0 1 1 3 -2 3]]

Optimal tunings:

  • WE: ~28/27 = 63.1406 ¢, ~3/2 = 702.0192 ¢ (~225/224 = 7.4730 ¢)
  • CWE: ~28/27 = 63.1579 ¢, ~3/2 = 702.1539 ¢ (~225/224 = 7.4171 ¢)

Optimal ET sequence: 19, 133df, 152f, 323ef

Badness (Sintel): 1.39

Hemienneadecal

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 234375/234256

Mapping: [⟨38 0 28 -74 11], ⟨0 1 1 3 2]]

mapping generators: ~55/54, ~3

Optimal tunings:

  • WE: ~55/54 = 31.5800 ¢, ~3/2 = 701.9053 ¢ (~243/242 = 7.1448 ¢)
  • CWE: ~55/54 = 31.5789 ¢, ~3/2 = 701.9034 ¢ (~243/242 = 7.1666 ¢)

Optimal ET sequence: 152, 342, 836, 1178, 2014, 3192ce, 5206ce

Badness (Sintel): 0.330

Hemienneadecalis

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256

Mapping: [⟨38 0 28 -74 11 -281], ⟨0 1 1 3 2 7]]

Optimal tunings:

  • WE: ~55/54 = 31.5785 ¢, ~3/2 = 701.9995 ¢ (~243/242 = 7.2727 ¢)
  • CWE: ~55/54 = 31.5789 ¢, ~3/2 = 702.0053 ¢ (~243/242 = 7.2685 ¢)

Optimal ET sequence: 152f, 342f, 494

Badness (Sintel): 0.859

Hemienneadec

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213

Mapping: [⟨38 0 28 -74 11 502], ⟨0 1 1 3 2 -6]]

Optimal tunings:

  • WE: ~55/54 = 31.5784 ¢, ~3/2 = 701.9736 ¢ (~243/242 = 7.2493 ¢)
  • CWE: ~55/54 = 31.5789 ¢, ~3/2 = 701.9855 ¢ (~243/242 = 7.2487 ¢)

Optimal ET sequence: 152, 342, 494, 1330, 1824, 2318d

Badness (Sintel): 1.26

Semihemienneadecal

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078

Mapping: [⟨38 1 29 -71 13 111], ⟨0 2 2 6 4 1]]

mapping generators: ~55/54, ~429/250

Optimal tunings:

  • WE: ~55/54 = 31.5799 ¢, ~429/250 = 935.1824 ¢ (~144/143 = 12.2152 ¢)
  • CWE: ~55/54 = 31.5789 ¢, ~429/250 = 935.1617 ¢ (~144/143 = 12.2067 ¢)

Optimal ET sequence: 190, 304d, 494, 684, 1178, 2850, 4028ce

Badness (Sintel): 0.607

Kalium

Named after the 19th element, potassium, and after an archaic variant of the element's name to resolve a name conflict. 19/16 can be used as a generator. Since it is enfactored in the 17-limit and lower, it makes no sense to name it for the lower subgroups.

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344

Mapping: [⟨19 3 17 -28 82 92 159 78], ⟨0 10 10 30 -6 -8 -30 1]]

Optimal tunings:

  • WE: ~28/27 = 63.1582 ¢, ~6545/5928 = 171.2448 ¢
  • CWE: ~28/27 = 63.1579 ¢, ~6545/5928 = 171.2439 ¢

Optimal ET sequence: 855, 988, 1843

Badness (Sintel): 3.15

Semidimi

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

The generator of semidimi is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit [-12 -73 55⟩ and 7-limit 3955078125/3954653486, as well as 4375/4374.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 3955078125/3954653486

Mapping: [⟨1 -19 -25 -32], ⟨0 55 73 93]]

mapping generators: ~2, ~35/27

Optimal tunings:

  • WE: ~2 = 1200.0018 ¢, ~35/27 = 449.1277 ¢
error map: ⟨+0.002 +0.031 -0.040 -0.012]
  • CWE: ~2 = 1200.0000 ¢, ~35/27 = 449.1270 ¢
error map: ⟨0.000 +0.030 -0.043 -0.015]

Optimal ET sequence: 8d, …, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419

Badness (Sintel): 0.382

Brahmagupta

The brahmagupta temperament has a period of 1/7 octave, tempering out the akjaysma ([47 -7 -7 -7⟩), and may be described as the 217 & 224 temperament.

Early in the design of the Sagittal notation system, Secor and Keenan found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4 ¢ many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of brahmagupta that has pure octaves and pure fifths, which can also be described as a 17-limit extension having a 1/7-octave period (171.4286 ¢) and 1/21-apotome generator (5.4136 ¢).

Subgroup: 2.3.5.7

Comma list: 4375/4374, [46 -14 -3 -6⟩

Mapping: [⟨7 2 -8 53], ⟨0 3 8 -11]]

mapping generators: ~1157625/1048576, ~27/20

Optimal tunings:

  • WE: ~1157625/1048576 = 171.4275 ¢, ~27/20 = 519.7125 ¢
error map: ⟨-0.007 +0.037 -0.034 -0.004]
  • CWE: ~1157625/1048576 = 171.4286 ¢, ~27/20 = 519.7156 ¢
error map: ⟨0.000 +0.049 -0.018 +0.017]

Optimal ET sequence: 7, …, 217, 224, 441, 1106, 1547

Badness (Sintel): 0.737

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4000/3993, 4375/4374, 131072/130977

Mapping: [⟨7 2 -8 53 3], ⟨0 3 8 -11 7]]

Optimal tunings:

  • WE: ~243/220 = 171.4208 ¢, ~27/20 = 519.6807 ¢
  • CWE: ~243/220 = 171.4286 ¢, ~27/20 = 519.7034 ¢

Optimal ET sequence: 7, 217, 224, 441, 665

Badness (Sintel): 1.73

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374

Mapping: [⟨7 2 -8 53 3 35], ⟨0 3 8 -11 7 -3]]

Optimal tunings:

  • WE: ~243/220 = 171.4197 ¢, ~27/20 = 519.6789 ¢
  • CWE: ~243/220 = 171.4286 ¢, ~27/20 = 519.7052 ¢

Optimal ET sequence: 7, 217, 224, 441, 665, 1106e

Badness (Sintel): 0.956

Abigail

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

Abigail tempers out the pessoalisma in addition to the ragisma in the 7-limit, and may be described as the 46 & 224 temperament, with a ploidacot signature of diploid wau-hendecacot. It extends into a very strong 11- and 13-limit temperament. 494edo, 764edo and 1258edo are among the possible tunings.

Abigail was named by Gene Ward Smith in 2010 after the birthday of First Lady Abigail Fillmore.[1]

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2147483648/2144153025

Mapping: [⟨2 -4 -11 18], ⟨0 11 24 -19]]

mapping generators: ~46305/32768, ~1536/1225

Optimal tunings:

  • WE: ~46305/32768 = 599.9699 ¢, ~1536/1225 = 391.0818 ¢
error map: ⟨-0.060 +0.065 -0.021 +0.079]
  • CWE: ~46305/32768 = 600.0000 ¢, ~1536/1225 = 391.1007 ¢
error map: ⟨0.000 +0.152 +0.102 +0.262]

Optimal ET sequence: 46, 132, 178, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd

Badness (Sintel): 0.936

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 131072/130977

Mapping: [⟨2 -4 -11 18 18], ⟨0 11 24 -19 -17]]

Optimal tunings:

  • WE: ~99/70 = 599.9782 ¢, ~1536/1225 = 391.0852 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~1536/1225 = 391.0992 ¢

Optimal ET sequence: 46, 132, 178, 224, 270, 494, 764

Badness (Sintel): 0.425

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095

Mapping: [⟨2 -4 -11 18 18 25], ⟨0 11 24 -19 -17 -27]]

Optimal tunings:

  • WE: ~99/70 = 599.9862 ¢, ~351/280 = 391.0879 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~351/280 = 391.0969 ¢

Optimal ET sequence: 46, 178, 224, 270, 494, 764, 1258

Badness (Sintel): 0.366

Gamera

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 589824/588245

Mapping: [⟨1 -17 -30 2], ⟨0 23 40 1]]

mapping generators: ~2, ~7/4

Optimal tunings:

  • WE: ~2 = 1199.8483 ¢, ~7/4 = 969.5415 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 969.6608 ¢

Optimal ET sequence: 26, 73, 99, 224, 323, 422, 745d

Badness (Sintel): 0.953

Hemigamera

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 589824/588245

Mapping: [⟨2 -11 -20 5 10], ⟨0 23 40 1 -5]]

mapping generators: ~99/70, ~99/80

Optimal tunings:

  • WE: ~99/70 = 599.9323 ¢, ~99/80 = 369.6212 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~99/80 = 369.6610 ¢

Optimal ET sequence: 26, 172c, 198, 224, 422, 646, 1068d

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024

Mapping: [⟨2 -11 -20 5 10 -8], ⟨0 23 40 1 -5 25]]

Optimal tunings:

  • WE: ~99/70 = 599.9207 ¢, ~26/21 = 369.6139 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~26/21 = 369.6603 ¢

Optimal ET sequence: 26, 172cf, 198, 224, 422, 646f, 1068df

Badness (Sintel): 0.844

Semigamera

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 14641/14580, 15488/15435

Mapping: [⟨1 -40 -70 1 -77], ⟨0 46 80 2 89]]

mapping generators: ~2, ~144/77

Optimal tunings:

  • WE: ~2 = 1199.8845 ¢, ~144/77 = 1084.7314 ¢
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.8345 ¢

Optimal ET sequence: 73, 125, 198, 323, 521

Badness (Sintel): 2.59

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580

Mapping: [⟨1 -40 -70 1 -77 -131], ⟨0 46 80 2 89 149]]

Optimal tunings:

  • WE: ~2 = 1199.8726 ¢, ~144/77 = 1084.7220 ¢
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.8359 ¢

Optimal ET sequence: 73f, 125f, 198, 323, 521

Badness (Sintel): 1.82

Crazy

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

Crazy tempers out the kwazy comma in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the 118 & 494 temperament, with a ploidacot of diploid alpha-octacot. 1106edo gives a strong tuning.

Crazy was named by Flora Canou in 2025 by removing the mutation from kwazy, the name for the 5-limit microtemperament.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-53 10 16⟩

Mapping: [⟨2 1 6 -15], ⟨0 8 -5 76]]

mapping generators: ~332150625/234881024, ~1125/1024

Optimal tunings:

  • WE: ~332150625/234881024 = 600.0019 ¢, ~1125/1024 = 162.7479 ¢
error map: ⟨+0.004 +0.030 -0.042 -0.014]
  • CWE: ~332150625/234881024 = 600.0000 ¢, ~1125/1024 = 162.7474 ¢
error map: ⟨0.000 +0.024 -0.051 -0.022]

Optimal ET sequence: 118, 376, 494, 612, 1106, 1718

Badness (Sintel): 0.998

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 2791309312/2790703125

Mapping: [⟨2 1 6 -15 -8], ⟨0 8 -5 76 55]]

Optimal tunings:

  • WE: ~99/70 = 600.0047 ¢, ~1125/1024 = 162.7493 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~1125/1024 = 162.7481 ¢

Optimal ET sequence: 118, 376, 494, 612, 1106, 2824, 3930e

Badness (Sintel): 0.562

Orga

Orga may be described as the 26 & 270 temperament, and 1106edo gives a strong tuning.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [41 -4 2 -14⟩

Mapping: [⟨2 -8 -15 6], ⟨0 29 51 -1]]

mapping generators: ~7411887/5242880, ~8/7

Optimal tunings:

  • WE: ~7411887/5242880 = 599.9927 ¢, ~8/7 = 231.1012 ¢
error map: ⟨-0.015 +0.037 -0.045 +0.029]
  • CWE: ~7411887/5242880 = 600.0000 ¢, ~8/7 = 231.1037 ¢
error map: ⟨0.000 +0.053 -0.023 +0.070]

Optimal ET sequence: 26, …, 244, 270, 836, 1106, 1376, 2482

Badness (Sintel): 1.02

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 5767168/5764801

Mapping: [⟨2 -8 -15 6 10], ⟨0 29 51 -1 -8]]

Optimal tunings:

  • WE: ~99/70 = 600.0025 ¢, ~8/7 = 231.1039 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~8/7 = 231.1030 ¢

Optimal ET sequence: 26, 244, 270, 566, 836, 1106

Badness (Sintel): 0.535

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360

Mapping: [⟨2 -8 -15 6 10 -3], ⟨0 29 51 -1 -8 27]]

Optimal tunings:

  • WE: ~99/70 = 600.0192 ¢, ~8/7 = 231.1102 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~8/7 = 231.1033 ¢

Optimal ET sequence: 26, 244, 270, 566, 836f, 1106f

Badness (Sintel): 0.899

Seniority

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

Aside from the ragisma, the seniority temperament tempers out the wadisma, 201768035/201326592, and may be described as 26 & 145. It is so named because the senior comma ([-17 62 -35⟩) is tempered out.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 201768035/201326592

Mapping: [⟨1 -24 -43 5], ⟨0 35 62 -3]]

mapping generators: ~2, ~5120/3087

Optimal tunings:

  • WE: ~2 = 1200.0745 ¢, ~5120/3087 = 877.2500 ¢
error map: ⟨+0.075 +0.008 -0.016 -0.203]
  • CWE: ~2 = 1200.0000 ¢, ~5120/3087 = 877.1965 ¢
error map: ⟨0.000 -0.077 -0.130 -0.415]

Optimal ET sequence: 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d

Badness (Sintel): 1.14

Senator

Senator (26 & 145) extends seniority by tempering out 441/440 and 65536/65219, and can be extended to the 13- and 17-limit immediately by adding 364/363 and 595/594 to the comma list in this order.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4374, 65536/65219

Mapping: [⟨1 -24 -43 5 2], ⟨0 35 62 -3 2]]

Optimal tunings:

  • WE: ~2 = 1199.7665 ¢, ~128/77 = 877.0367 ¢
  • CWE: ~2 = 1200.0000 ¢, ~128/77 = 877.2051 ¢

Optimal ET sequence: 26, 119c, 145, 171, 316e

Badness (Sintel): 3.05

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 2200/2197, 4375/4374

Mapping: [⟨1 -24 -43 5 2 -27], ⟨0 35 62 -3 2 42]]

Optimal tunings:

  • WE: ~2 = 1199.7136 ¢, ~108/65 = 877.9974 ¢
  • CWE: ~2 = 1200.0000 ¢, ~108/65 = 877.2038 ¢

Optimal ET sequence: 26, 119cf, 145, 171, 316ef

Badness (Sintel): 1.85

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197

Mapping: [⟨1 -24 -43 5 2 -27 -31], ⟨0 35 62 -3 2 42 48]]

Optimal tunings:

  • WE: ~2 = 1199.7195 ¢, ~108/65 = 877.0018 ¢
  • CWE: ~2 = 1200.0000 ¢, ~108/65 = 877.2039 ¢

Optimal ET sequence: 26, 119cfg, 145, 171, 316ef

Badness (Sintel): 1.35

Monzismic

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

Monzismic tempers out the monzisma, [54 -37 2⟩, and in the 7-limit, the nanisma, [109 -67 0 -1⟩, as well as the ragisma, 4375/4374. It may be described as the 53 & 612 temperament, with a ploidacot signature of alpha-dicot. A notable tuning not appearing on the optimal ET sequence is 665edo, which is nearly equivalent to the pure-3's tuning.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-55 30 2 1⟩

Mapping: [⟨1 0 -27 109], ⟨0 2 37 -134]]

mapping generators: ~2, ~[28 -11 -3 -1⟩

Optimal tunings:

  • WE: ~2 = 1200.0128 ¢, ~[28 -11 -3 -1⟩ = 950.9895 ¢
error map: ⟨+0.013 +0.024 -0.049 -0.019]
  • CWE: ~2 = 1200.0000 ¢, ~[28 -11 -3 -1⟩ = 950.9793 ¢
error map: ⟨0.000 +0.004 -0.080 -0.050]

Optimal ET sequence: 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd

Badness (Sintel): 1.18

Monzism

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 41503/41472, 184549376/184528125

Mapping: [⟨1 0 -27 109 -159], ⟨0 2 37 -134 205]]

Optimal tunings:

  • WE: ~2 = 1200.0347 ¢, ~400/231 = 951.0082 ¢
  • CWE: ~2 = 1200.0000 ¢, ~400/231 = 950.9807 ¢

Optimal ET sequence: 53, 559, 612, 3619de, 4231de, …, 6067ddee

Badness (Sintel): 1.89

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625

Mapping: [⟨1 0 -27 109 -159 -70], ⟨0 2 37 -134 205 93]]

Optimal tunings:

  • WE: ~2 = 1200.0036 ¢, ~400/231 = 950.9829 ¢
  • CWE: ~2 = 1200.0000 ¢, ~400/231 = 950.9801 ¢

Optimal ET sequence: 53, 559, 612

Badness (Sintel): 2.22

Semidimfourth

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

The semidimfourth temperament is featured by a semidiminished fourth inverval which is 128/125 above the pythagorean major third 81/64. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, 235298/234375.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 235298/234375

Mapping: [⟨1 -10 -13 -17], ⟨0 31 41 53]]

mapping generators: ~2, ~35/27

Optimal tunings:

  • WE: ~2 = 1199.9936 ¢, ~35/27 = 448.4533 ¢
error map: ⟨-0.007 +0.160 +0.353 -0.694]
  • CWE: ~2 = 1200.0000 ¢, ~35/27 = 448.4555 ¢
error map: ⟨0.000 +0.165 +0.361 -0.685]

Optimal ET sequence: 8d, …, 91, 99, 289, 388, 875

Badness (Sintel): 1.40

Neusec

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 235298/234375

Mapping: [⟨2 -20 -26 -34 -17], ⟨0 31 41 53 32]]

mapping generators: ~99/70, ~35/27

Optimal tunings:

  • WE: ~99/70 = 600.0381 ¢, ~35/27 = 448.4812 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~35/27 = 448.4546 ¢

Optimal ET sequence: 8d, …, 190, 388

Badness (Sintel): 1.95

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374

Mapping: [⟨2 -20 -26 -34 -17 -21], ⟨0 31 41 53 32 38]]

Optimal tunings:

  • WE: ~99/70 = 600.0034 ¢, ~35/27 = 448.4573 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~35/27 = 448.4549 ¢

Optimal ET sequence: 8d, …, 190, 198, 388

Badness (Sintel): 1.28

Acrokleismic

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2202927104/2197265625

Mapping: [⟨1 -22 -22 -65], ⟨0 32 33 92]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1199.9305 ¢, ~5/3 = 884.3923 ¢
error map: ⟨-0.070 +0.126 +0.160 -0.221]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4423 ¢
error map: ⟨0.000 +0.198 +0.282 -0.136]

Optimal ET sequence: 19, …, 251, 270, 2449c, 2719c, 2989bc

Badness (Sintel): 1.42

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 41503/41472, 172032/171875

Mapping: [⟨1 -22 -22 -65 58], ⟨0 32 33 92 -74]]

Optimal tunings:

  • WE: ~2 = 1199.9698 ¢, ~5/3 = 884.4193 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4414 ¢

Optimal ET sequence: 19, 251, 270, 829, 1099, 1369, 1639

Badness (Sintel): 1.22

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976

Mapping: [⟨1 -22 -22 -65 58 -56], ⟨0 32 33 92 -74 81]]

Optimal tunings:

  • WE: ~2 = 1199.9939 ¢, ~5/3 = 884.4384 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4429 ¢

Optimal ET sequence: 19, 251, 270

Badness (Sintel): 1.11

Counteracro

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 117649/117612

Mapping: [⟨1 -22 -22 -65 -141], ⟨0 32 33 92 196]]

Optimal tunings:

  • WE: ~2 = 1199.8877 ¢, ~5/3 = 884.3639 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4457 ¢

Optimal ET sequence: 19e, …, 251e, 270, 1061e, 1331c, 1601c, 1871bc

Badness (Sintel): 1.41

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374

Mapping: [⟨1 -22 -22 -65 -141 -56], ⟨0 32 33 92 196 81]]

Optimal tunings:

  • WE: ~2 = 1199.9285 ¢, ~5/3 = 884.3937 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4458 ¢

Optimal ET sequence: 19e, …, 251e, 270, 1331c

Badness (Sintel): 1.08

Quasithird

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

Quasithird may be described as the 224 & 388 temperament, featured by a major third interval which is 1600000/1594323 (amity comma) or 5120/5103 (hemifamity comma) below the just major third 5/4 as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows it to temper out the ragisma and [-60 29 0 5⟩. Its ploidacot is tetraploid delta-pentacot.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-60 29 0 5⟩

Mapping: [⟨4 0 -11 48], ⟨0 5 16 -29]]

mapping generators: ~65536/55125, ~5103/4096

Optimal tunings:

  • WE: ~65536/55125 = 300.0052 ¢, ~5103/4096 = 380.3949 ¢
error map: ⟨+0.021 +0.020 -0.052 -0.031]
  • CWE: ~65536/55125 = 300.0000 ¢, ~5103/4096 = 380.3884 ¢
error map: ⟨0.000 -0.013 -0.100 -0.089]

Optimal ET sequence: 60d, 164, 224, 388, 612, 1448, 2060

Badness (Sintel): 1.56

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 4296700485/4294967296

Mapping: [⟨4 0 -11 48 43], ⟨0 5 16 -29 -23]]

Optimal tunings:

  • WE: ~65536/51125 = 300.0073 ¢, ~5103/4096 = 380.3963 ¢ (or ~22/21 = 80.3890 ¢)
  • CWE: ~65536/51125 = 300.0000 ¢, ~5103/4096 = 380.3868 ¢ (or ~22/21 = 80.3868 ¢)

Optimal ET sequence: 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee

Badness (Sintel): 0.698

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374

Mapping: [⟨4 0 -11 48 43 11], ⟨0 5 16 -29 -23 3]]

Optimal tunings:

  • WE: ~65536/51125 = 299.9985 ¢, ~81/65 = 380.3833 ¢ (or ~22/21 = 80.3848 ¢)
  • CWE: ~65536/51125 = 300.0000 ¢, ~81/65 = 380.3852 ¢ (or ~22/21 = 80.3852 ¢)

Optimal ET sequence: 60d, 164, 224, 388, 612, 836

Badness (Sintel): 1.22

Deca

For 5-limit version, see 10th-octave temperaments #Neon.

Deca has a period of 1/10 octave and tempers out the neon comma [21 60 -50⟩ in the 5-limit, the linus comma[11 -10 -10 10⟩ and [12 -3 -14 9⟩ (165288374272/164794921875) in the 7-limit. It may be described as the 80 & 190 temperament, and has a ploidacot of decaploid wau-pentacot.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 165288374272/164794921875

Mapping: [⟨10 4 9 2], ⟨0 5 6 11]]

mapping generators: ~15/14, ~460992/390625

Optimal tunings:

  • WE: ~15/14 = 119.9966 ¢, ~460992/390625 = 284.4150 ¢ (5625/5488 = 44.4219 ¢)
error map: ⟨-0.034 +0.106 +0.145 -0.268]
  • CWE: ~15/14 = 120.0000 ¢, ~460992/390625 = 284.4182 ¢ (5625/5488 = 44.4182 ¢)
error map: ⟨0.000 +0.136 +0.195 -0.226]

Optimal ET sequence: 80, 190, 270, 1270, 1540, 1810, 2080

Badness (Sintel): 2.04

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 391314/390625

Mapping: [⟨10 4 9 2 18], ⟨0 5 6 11 7]]

Optimal tunings:

  • WE: ~15/14 = 120.0004 ¢, ~33/28 = 284.4193 ¢ (77/75 = 44.4185 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~33/28 = 284.4189 ¢ (77/75 = 44.4189 ¢)

Optimal ET sequence: 80, 190, 270, 1000, 1270, 1540e, 1810e

Badness (Sintel): 0.804

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374

Mapping: [⟨10 4 9 2 18 37], ⟨0 5 6 11 7 0]]

Optimal tunings:

  • WE: ~15/14 = 120.0067 ¢, ~33/28 = 284.4139 ¢ (~40/39 = 44.4006 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~33/28 = 284.4048 ¢ (~40/39 = 44.4048 ¢)

Optimal ET sequence: 80, 190, 270, 730, 1000

Badness (Sintel): 0.695

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1001/1000, 1521/1520, 3025/3024, 4225/4224, 4375/4374

Mapping: [⟨10 4 9 2 18 37 33], ⟨0 5 6 11 7 0 4]]

Optimal tunings:

  • WE: ~15/14 = 120.0045 ¢, ~33/28 = 284.4140 ¢ (~39/38 = 44.4050 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~33/28 = 284.4075 ¢ (~39/38 = 44.4075 ¢)

Optimal ET sequence: 80, 190, 270, 730, 1000

Badness (Sintel): 0.556

Keenanose

Keenanose, the 270 & 1889 temperament, was named by Eliora in 2022 for the fact that it uses 385/384, the keenanisma, as the generator.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-56 1 -8 26⟩

Mapping: [⟨1 2 3 3], ⟨0 -112 -183 -52]]

mapping generators: ~2, ~[21 3 1 -10⟩

Optimal tunings:

  • WE: ~2 = 1200.0068 ¢, ~[21 3 1 -10⟩ = 4.4467 ¢
error map: ⟨+0.007 +0.031 -0.035 -0.032]
  • CWE: ~2 = 1200.0000 ¢, ~[21 3 1 -10⟩ = 4.4466 ¢
error map: ⟨0.000 +0.025 -0.043 -0.050]

Optimal ET sequence: 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd

Badness (Sintel): 2.17

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 117649/117612, 67110351/67108864

Mapping: [⟨1 2 3 3 3], ⟨0 -112 -183 -52 124]]

Optimal tunings:

  • WE: ~2 = 1199.9970 ¢, ~385/384 = 4.4465 ¢
  • CWE: ~2 = 1200.0000 ¢, ~385/384 = 4.4465 ¢

Optimal ET sequence: 270, 1349, 1619, 1889, 2159, 11065, 13224

Badness (Sintel): 1.02

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612

Mapping: [⟨1 2 3 3 3 3], ⟨0 -112 -183 -52 124 189]]

Optimal tunings:

  • WE: ~2 = 1200.0065 ¢, ~385/384 = 4.4467 ¢
  • CWE: ~2 = 1200.0000 ¢, ~385/384 = 4.4467 ¢

Optimal ET sequence: 270, 1079, 1349, 1619, 1889, 4048

Badness (Sintel): 0.879

Aluminium

For the 5-limit version, see 13th-octave temperaments #Aluminium.

Aluminium was named by Eliora in 2023 after the 13th element. It tempers out the [92 -39 -13⟩ comma, which sets 135/128 interval to be equal to 1/13th of the octave.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [92 -39 -13⟩

Mapping: [⟨13 0 92 -355], ⟨0 1 -3 19]]

Mapping generators: ~135/128, ~3

Optimal tunings:

  • WE: ~135/128 = 92.3072 ¢, ~3/2 = 701.9995 ¢
error map: ⟨-0.006 +0.038 -0.030 -0.013]
  • CWE: ~135/128 = 92.3077 ¢, ~3/2 = 702.0030 ¢
error map: ⟨0.000 +0.048 -0.015 +0.001]

Optimal ET sequence: 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b

Badness (Sintel): 3.20

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 234375/234256, 2097152/2096325

Mapping: [⟨13 0 92 -355 148], ⟨0 1 -3 19 -5]]

Optimal tunings:

  • WE: ~135/128 = 92.3062 ¢, ~3/2 = 701.9946 ¢
  • CWE: ~135/128 = 92.3077 ¢, ~3/2 = 702.0056 ¢

Optimal ET sequence: 494, 1053, 1547, 3588e, 5135e

Badness (Sintel): 1.39

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078

Mapping: [⟨13 0 92 -355 148 419], ⟨0 1 -3 19 -5 -18]]

Optimal tunings:

  • WE: ~135/128 = 92.3055 ¢, ~3/2 = 701.9928 ¢
  • CWE: ~135/128 = 92.3077 ¢, ~3/2 = 702.0098 ¢

Optimal ET sequence: 494, 1547, 2041, 4576def

Badness (Sintel): 1.18

Ragitritonic

For the 5-limit version, see Schismic–Mercator equivalence continuum #Countritonic.

Ragitritonic may be described as the 53 & 369 temperament, splitting the 24th harmonic into nine tritone generators; its ploidacot is thus delta-enneacot. 422edo makes for a strong tuning.

Ragitritonic was named by Flora Canou in 2026 as a contraction of ragismic and tritonic.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 68719476736/68356598625

Mapping: [⟨1 -3 -15 40], ⟨0 9 34 -73]]

mapping generators: ~2, ~65536/45927

Optimal tunings:

  • WE: ~2 = 1199.8189 ¢, ~65536/45927 = 611.2850 ¢
error map: ⟨-0.181 +0.153 +0.094 +0.123]
  • CWE: ~2 = 1200.0000 ¢, ~65536/45927 = 611.3775 ¢
error map: ⟨0.000 +0.443 +0.522 +0.615]

Optimal ET sequence: 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd

Badness (Sintel): 3.37

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 2621440/2614689

Mapping: [⟨1 -3 -15 40 -75], ⟨0 9 34 -73 154]]

Optimal tunings:

  • WE: ~2 = 1199.8147 ¢, ~768/539 = 611.2822 ¢
  • CWE: ~2 = 1200.0000 ¢, ~768/539 = 611.3762 ¢

Optimal ET sequence: 53, 316e, 369, 422, 791e, 1213cde

Badness (Sintel): 2.34

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625

Mapping: [⟨1 -3 -15 40 -75 -34], ⟨0 9 34 -73 154 74]]

Optimal tunings:

  • WE: ~2 = 1199.7916 ¢, ~91/64 = 611.2698 ¢
  • CWE: ~2 = 1200.0000 ¢, ~91/64 = 611.3754 ¢

Optimal ET sequence: 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff

Badness (Sintel): 1.51

Quatracot

See also: Stratosphere

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-32 5 14 -3⟩

Mapping: [⟨2 -6 -1 -36], ⟨0 13 8 59]]

mapping generators: ~2278125/1605632, ~7168/5625

Optimal tunings:

  • WE: ~2278125/1605632 = 600.0888 ¢, ~7168/5625 = 423.2574 ¢
error map: ⟨+0.178 -0.141 -0.343 +0.165]
  • CWE: ~2278125/1605632 = 600.0000 ¢, ~7168/5625 = 423.1986 ¢
error map: ⟨0.000 -0.374 -0.725 -0.111]

Optimal ET sequence: 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc

Badness (Sintel): 4.45

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 1265625/1261568

Mapping: [⟨2 -6 -1 -36 -22], ⟨0 13 8 59 41]]

Optimal tunings:

  • WE: ~99/70 = 600.0847 ¢, ~225/176 = 423.2536 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~225/176 = 423.1977 ¢

Optimal ET sequence: 34d, 156de, 190, 224, 414, 638, 1052c

Badness (Sintel): 1.36

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1575/1573, 2200/2197

Mapping: [⟨2 -6 -1 -36 -22 -6], ⟨0 13 8 59 41 19]]

Optimal tunings:

  • WE: ~99/70 = 600.0571 ¢, ~143/112 = 423.2366 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~143/112 = 423.1987 ¢

Optimal ET sequence: 34d, 156de, 190, 224, 414, 638

Badness (Sintel): 0.936

Moulin

Moulin can be described as the 494 & 1619 temperament. It has a generator of ~22/13, and it was named by Eliora in 2022 after the Law & Order: Special Victims Unit episode Season 22, Episode 13. "Trick-Rolled At The Moulin". However, the functional generator is ~13/11, and 73 of them octave reduced reach the perfect fifth. Since 11/8 is within 23 generators, the 25-tone generator chain (4L 21s) of this temperament contains the 8:11:13 triad.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-88 2 45 -7⟩

Mapping: [⟨1 -16 -9 -75], ⟨0 73 47 323]]

mapping generators: ~2, ~3796875/3211264

Optimal tunings:

  • WE: ~2 = 1200.0272 ¢, ~3796875/3211264 = 289.0675 ¢
error map: ⟨+0.027 +0.007 -0.084 +0.013]
  • CWE: ~2 = 1200.0000 ¢, ~3796875/3211264 = 289.0675 ¢
error map: ⟨0.000 -0.029 -0.142 -0.029]

Optimal ET sequence: 494, 1125, 1619, 8589cc, 10208cc

Badness (Sintel): 5.93

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 759375/758912, 100663296/100656875

Mapping: [⟨1 -16 -9 -75 9], ⟨0 73 47 323 -23]]

Optimal tunings:

  • WE: ~2 = 1200.0043 ¢, ~605/512 = 289.0687 ¢
  • CWE: ~2 = 1200.0000 ¢, ~605/512 = 289.0677 ¢

Optimal ET sequence: 494, 1125, 1619, 2113

Badness (Sintel): 2.24

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078

Mapping: [⟨1 -16 -9 -75 9 9], ⟨0 73 47 323 -23 -22]]

Optimal tunings:

  • WE: ~2 = 1200.0043 ¢, ~13/11 = 289.0687 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/11 = 289.0677 ¢

Optimal ET sequence: 494, 1125, 1619, 2113

Badness (Sintel): 1.12

Palladium

For the 5-limit version, see 46th-octave temperaments #Palladium.

The name of the palladium temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, [-39 92 -46⟩, by which 46 minor whole tones (10/9) fall short of seven octaves. This temperament can be described as 46 & 414 temperament, which tempers out [-51 8 2 12⟩ as well as the ragisma.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-51 8 2 12⟩

Mapping: [⟨46 0 -39 202], ⟨0 1 2 -1]]

mapping generators: ~83349/81920, ~3

Optimal tunings:

  • WE: ~83349/81920 = 26.0910 ¢, ~3/2 = 701.7155 ¢
error map: ⟨+0.185 -0.055 -0.061 +0.349]
  • CWE: ~83349/81920 = 26.0870 ¢, ~3/2 = 701.6491 ¢
error map: ⟨0.000 -0.306 -0.407 -0.910]

Optimal ET sequence: 46, …, 368, 414, 460, 874d

Badness (Sintel): 7.81

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 134775333/134217728

Mapping: [⟨46 0 -39 202 232], ⟨0 1 2 -1 -1]]

Optimal tunings:

  • WE: ~8192/8085 = 26.0912 ¢, ~3/2 = 701.7082 ¢
  • CWE: ~8192/8085 = 26.0870 ¢, ~3/2 = 701.6173 ¢

Optimal ET sequence: 46, …, 368, 414, 460, 874de

Badness (Sintel): 2.44

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364

Mapping: [⟨46 0 -39 202 232 316], ⟨0 1 2 -1 -1 -2]]

Optimal tunings:

  • WE: ~65/64 = 26.0906 ¢, ~3/2 = 701.7411 ¢
  • CWE: ~65/64 = 26.0870 ¢, ~3/2 = 701.6465 ¢

Optimal ET sequence: 46, 368, 414, 460, 874de, 1334dde

Badness (Sintel): 1.68

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224

Mapping: [⟨46 0 -39 202 232 316 188], ⟨0 1 2 -1 -1 -2 0]]

Optimal tunings:

  • WE: ~65/64 = 26.0906 ¢, ~3/2 = 701.7399 ¢
  • CWE: ~65/64 = 26.0870 ¢, ~3/2 = 701.6464 ¢

Optimal ET sequence: 46, 368, 414, 460, 874de, 1334ddeg

Badness (Sintel): 1.14

Counterorson

Counterorson tempers out the [147 -103 7⟩ comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the semicomma family, 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on 53edo.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [154 -54 -21 -7⟩

Mapping: [⟨1 0 -21 85], ⟨0 7 103 -363]]

mapping generators: ~2, ~[66 -23 -9 -3⟩

Optimal tunings:

  • WE: ~2 = 1200.0040 ¢, ~[66 -23 -9 -3⟩ = 271.7122 ¢
error map: ⟨+0.004 -0.303 -0.041 -0.015]
  • CWE: ~2 = 1200.0000 ¢, ~[66 -23 -9 -3⟩ = 271.7113 ¢
error map: ⟨0.000 +0.024 -0.051 -0.025]

Optimal ET sequence: 53, …, 1612, 1665, 1718

Badness (Sintel): 7.92

Octoid

Main article: Octoid
For the 5-limit version, see 8th-octave temperaments #Octoid.

The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 (ragisma) and 16875/16807 (mirkwai comma). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives ~12/11, two give ~25/21, three give ~35/27, and four give 99/70~140/99.

The 11-limit is the last place where all the extensions of octoid shown here agree in the mappings of primes. 80edo is an alternative tuning for octoid in the 11-limit; though 72edo does better for minimizing the average damage on the 11-odd-limit, 80edo damages prime 7 in favor of practically-just 17/16's, 11/10's and 9/7's. In higher limits, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 16875/16807

Mapping: [⟨8 1 3 3], ⟨0 3 4 5]]

mapping generators: ~49/45, ~7/5

Optimal tunings:

  • WE: ~49/45 = 150.0003 ¢, ~7/5 = 583.9416 ¢
error map: ⟨+0.002 -0.130 -0.547 +0.883]
  • CWE: ~49/45 = 150.0000 ¢, ~7/5 = 583.9411 ¢
error map: ⟨0.000 -0.132 -0.549 +0.880]

Tuning ranges:

  • 7-odd-limit diamond monotone: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
  • 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
  • 7-odd-limit diamond tradeoff: ~7/5 = [582.512, 584.359]
  • 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

Optimal ET sequence: 8d, …, 72, 152, 224

Badness (Sintel): 1.08

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 4000/3993

Mapping: [⟨8 1 3 3 16], ⟨0 3 4 5 3]]

Optimal tunings:

  • WE: ~12/11 = 149.9932 ¢, ~7/5 = 583.9356 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9477 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
  • 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

Optimal ET sequence: 8d, …, 72, 152, 224, 824d

Badness (Sintel): 0.466

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 1375/1372

Mapping: [⟨8 1 3 3 16 -21], ⟨0 3 4 5 3 13]]

Optimal tunings:

  • WE: ~12/11 = 150.0005 ¢, ~7/5 = 583.9066 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9052 ¢

Optimal ET sequence: 72, 152f, 224

Badness (Sintel): 0.631

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 715/714, 729/728

Mapping: [⟨8 1 3 3 16 -21 -14], ⟨0 3 4 5 3 13 12]]

Optimal tunings:

  • WE: ~12/11 = 150.0064 ¢, ~7/5 = 583.8666 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.8489 ¢

Optimal ET sequence: 72, 152fg, 224, 296, 520g

Badness (Sintel): 0.729

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714

Mapping: [⟨8 1 3 3 16 -21 -14 34], ⟨0 3 4 5 3 13 12 0]]

Optimal tunings:

  • WE: ~12/11 = 149.9785 ¢, ~7/5 = 583.8482 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9138 ¢

Optimal ET sequence: 72, 152fg, 224

Badness (Sintel): 0.975

Octopus

A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is 80edo, which has a strong sharp tendency that can be thought of as matching the sharpness of mapping 19/16 to 1\4 = 300 ¢.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 540/539

Mapping: [⟨8 1 3 3 16 14], ⟨0 3 4 5 3 4]]

Optimal tunings:

  • WE: ~12/11 = 150.0313 ¢, ~7/5 = 584.0134 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9583 ¢

Optimal ET sequence: 8d, …, 72, 152, 224f

Badness (Sintel): 0.896

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 289/288, 325/324, 540/539

Mapping: [⟨8 1 3 3 16 14 21], ⟨0 3 4 5 3 4 3]]

Optimal tunings:

  • WE: ~12/11 = 150.0528 ¢, ~7/5 = 584.0161 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9166 ¢

Optimal ET sequence: 8d, …, 72, 152, 224fg, 296ffg

Badness (Sintel): 0.795

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399

Mapping: [⟨8 1 3 3 16 14 21 34], ⟨0 3 4 5 3 4 3 0]]

Optimal tunings:

  • WE: ~12/11 = 150.0049 ¢, ~7/5 = 584.0833 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 584.0712 ¢

Optimal ET sequence: 8d, 72, 152

Badness (Sintel): 0.993

Scales: Octoid72, Octoid80

Hexadecoid

See also: 16th-octave temperaments

Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224

Mapping: [⟨16 2 6 6 32 67], ⟨0 3 4 5 3 -1]]

mapping generators: ~448/429, ~7/5

Optimal tunings:

  • WE: ~448/429 = 74.9943 ¢, ~7/5 = 583.9408 ¢
  • CWE: ~448/429 = 75.0000 ¢, ~7/5 = 583.9709 ¢

Optimal ET sequence: 80, 144, 224

Badness (Sintel): 1.27

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224

Mapping: [⟨16 2 6 6 32 67 81], ⟨0 3 4 5 3 -1 -2]]

Optimal tunings:

  • WE: ~117/112 = 74.9865 ¢, ~7/5 = 583.9626 ¢
  • CWE: ~117/112 = 75.0000 ¢, ~7/5 = 584.0463 ¢

Optimal ET sequence: 80, 144, 224, 528dg

Badness (Sintel): 1.46

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444

Mapping: [⟨16 2 6 6 32 67 81 68], ⟨0 3 4 5 3 -1 -2 0]]

Optimal tunings:

  • WE: ~117/112 = 74.9865 ¢, ~7/5 = 583.9642 ¢
  • CWE: ~117/112 = 75.0000 ¢, ~7/5 = 584.0803 ¢

Optimal ET sequence: 80, 144, 224, 304dh, 528dghh

Badness (Sintel): 1.44

Parakleismic

Main article: Parakleismic
For the 5-limit version, see Syntonic–kleismic equivalence continuum #Parakleismic (5-limit).

In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, [8 14 -13⟩, with the 118edo tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. While 118 no longer has better than a cent of accuracy in the 7-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, for which 99edo, 118edo, and especially 217edo are accurate tunings.

Parakleismic does not extend easily to the 11- or 13-limit. Possible 11-limit extensions include undecimal parakleismic (99 & 118), paralytic (99e & 118), parkleismic (80 & 99), and paradigmic (80 & 99e).

Subgroup: 2.3.5.7

Comma list: 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23], ⟨0 13 14 35]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1199.7820 ¢, ~5/3 = 884.6581 ¢
error map: ⟨-0.218 +0.344 +0.644 -0.779]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.8088 ¢
error map: ⟨0.000 +0.560 +1.010 -0.516]

Optimal ET sequence: 19, 61d, 80, 99, 217, 316, 415

Badness (Sintel): 0.694

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23 30], ⟨0 13 14 35 -36]]

Optimal tunings:

  • WE: ~2 = 1200.3296 ¢, ~5/3 = 884.9921 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7519 ¢

Optimal ET sequence: 19, 99, 118

Badness (Sintel): 1.64

Paralytic

Paralytic (99e & 118) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23 -57], ⟨0 13 14 35 82]]

Optimal tunings:

  • WE: ~2 = 1199.9940 ¢, ~5/3 = 884.7757 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7800 ¢

Optimal ET sequence: 19e, …, 99e, 118, 217, 335

Badness (Sintel): 1.19

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23 -57 59], ⟨0 13 14 35 82 -75]]

Optimal tunings:

  • WE: ~2 = 1199.9218 ¢, ~5/3 = 884.7285 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7858 ¢

Optimal ET sequence: 99e, 118, 217

Badness (Sintel): 1.85

Paraklein

Paraklein (19e & 118) is another 13-limit extension of paralytic, which equates 13/11 with 32/27, 14/13 with 15/14, 25/24 with 26/25, and 27/26 with 28/27.

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 625/624, 729/728

Mapping: [⟨1 -8 -8 -23 -57 -28], ⟨0 13 14 35 82 43]]

Optimal tunings:

  • WE: ~2 = 1199.8239 ¢, ~5/3 = 884.6449 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7709 ¢

Optimal ET sequence: 19e, …, 99ef, 118

Badness (Sintel): 1.55

Parkleismic

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1375/1372, 2200/2187

Mapping: [⟨1 -8 -8 -23 -43], ⟨0 13 14 35 63]]

Optimal tunings:

  • WE: ~2 = 1199.1848 ¢, ~5/3 = 884.3386 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.9158 ¢

Optimal ET sequence: 19e, 61de, 80, 179, 259cd

Badness (Sintel): 1.85

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 1375/1372

Mapping: [⟨1 -8 -8 -23 -43 -14], ⟨0 13 14 35 63 24]]

Optimal tunings:

  • WE: ~2 = 1199.5318 ¢, ~5/3 = 884.5800 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.9118 ¢

Optimal ET sequence: 19e, 61de, 80, 179

Badness (Sintel): 1.51

Paradigmic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 3136/3125

Mapping: [⟨1 -8 -8 -23 16], ⟨0 13 14 35 -17]]

Optimal tunings:

  • WE: ~2 = 1199.0616 ¢, ~5/3 = 884.2124 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.8877 ¢

Optimal ET sequence: 19, 61d, 80, 99e, 179e, 457bcddeeee

Badness (Sintel): 1.38

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 540/539, 832/825

Mapping: [⟨1 -8 -8 -23 16 -14], ⟨0 13 14 35 -17 24]]

Optimal tunings:

  • WE: ~2 = 1199.2683 ¢, ~5/3 = 884.3805 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.9061 ¢

Optimal ET sequence: 19, 61d, 80, 99e

Badness (Sintel): 1.48

Semiparakleismic

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 4375/4374

Mapping: [⟨2 -3 -2 -11 -4], ⟨0 13 14 35 23]]

mapping generators: ~99/70, ~33/28

Optimal tunings:

  • WE: ~99/70 = 599.9270 ¢, ~33/28 = 284.7841 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~33/28 = 284.8119 ¢

Optimal ET sequence: 80, 118, 198, 316, 514c

Badness (Sintel): 1.13

Semiparamint

This extension was named semiparakleismic in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374

Mapping: [⟨2 -3 -2 -11 -4 15], ⟨0 13 14 35 23 -16]]

Optimal tunings:

  • WE: ~99/70 = 599.8253 ¢, ~33/28 = 284.7608 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~33/28 = 284.8366 ¢

Optimal ET sequence: 80, 118, 198

Badness (Sintel): 1.40

Semiparawolf

This extension was named gentsemiparakleismic in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 3136/3125

Mapping: [⟨2 -3 -2 -11 -4 -4], ⟨0 13 14 35 23 24]]

Optimal tunings:

  • WE: ~99/70 = 600.0569 ¢, ~13/11 = 284.8431 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~13/11 = 284.8216 ¢

Optimal ET sequence: 80, 118f, 198f

Badness (Sintel): 1.67

Counterkleismic

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Counterhanson.

In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, [-20 -24 25⟩, the amount by which six major dieses ((648/625)6) fall short of the classic major third (5/4). It can be described as 19 & 224 temperament, tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). It was named by analogy to catakleismic and parakleismic)

Subgroup: 2.3.5.7

Comma list: 4375/4374, 158203125/157351936

Mapping: [⟨1 -5 -4 -18], ⟨0 25 24 79]]

mapping generators: ~2, ~6/5

Optimal tunings:

  • WE: ~2 = 1200.1778 ¢, ~6/5 = 316.1065 ¢
error map: ⟨+0.178 -0.181 -0.469 +0.388]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0631 ¢
error map: ⟨0.000 -0.377 -0.799 +0.161]

Optimal ET sequence: 19, …, 205, 224, 243, 467

Badness (Sintel): 2.29

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 2097152/2096325

Mapping: [⟨1 -5 -4 -18 19], ⟨0 25 24 79 -59]]

Optimal tunings:

  • WE: ~2 = 1199.9944 ¢, ~6/5 = 316.0690 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0705 ¢

Optimal ET sequence: 19, 205, 224

Badness (Sintel): 2.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 10985/10976

Mapping: [⟨1 -5 -4 -18 19 -15], ⟨0 25 24 79 -59 71]]

Optimal tunings:

  • WE: ~2 = 1199.9827 ¢, ~6/5 = 316.0650 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0695 ¢

Optimal ET sequence: 19, 205, 224

Badness (Sintel): 1.40

Counterlytic

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4375/4374, 496125/495616

Mapping: [⟨1 -5 -4 -18 -40], ⟨0 25 24 79 165]]

Optimal tunings:

  • WE: ~2 = 1200.1247 ¢, ~6/5 = 316.0976 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0660 ¢

Optimal ET sequence: 19e, 205e, 224, 467e, 691, 915c

Badness (Sintel): 2.16

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1375/1372, 10985/10976

Mapping: [⟨1 -5 -4 -18 -40 -15], ⟨0 25 24 79 165 71]]

Optimal tunings:

  • WE: ~2 = 1200.0987 ¢, ~6/5 = 316.0908 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0658 ¢

Optimal ET sequence: 19e, 205e, 224, 467e, 691, 915c

Badness (Sintel): 1.23

Quincy

Subgroup: 2.3.5.7

Comma list: 4375/4374, 823543/819200

Mapping: [⟨1 2 3 3], ⟨0 -30 -49 -14]]

mapping generators: ~2, ~1728/1715

Optimal tunings:

  • WE: ~2 = 1200.2169 ¢, ~1728/1715 = 16.6160 ¢
error map: ⟨+0.217 +0.000 +0.155 -0.799]
  • CWE: ~2 = 1200.0000 ¢, ~1728/1715 = 16.6083 ¢
error map: ⟨0.000 -0.205 -0.122 -1.343]

Optimal ET sequence: 72, 217, 289, 650d, 939dd

Badness (Sintel): 2.02

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4000/3993, 4375/4374

Mapping: [⟨1 2 3 3 4], ⟨0 -30 -49 -14 -39]]

Optimal tunings:

  • WE: ~2 = 1200.1286 ¢, ~100/99 = 16.6147 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 16.6101 ¢

Optimal ET sequence: 72, 217, 289

Badness (Sintel): 1.02

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675, 4375/4374

Mapping: [⟨1 2 3 3 4 5], ⟨0 -30 -49 -14 -39 -94]]

Optimal tunings:

  • WE: ~2 = 1200.0554 ¢, ~100/99 = 16.6028 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 16.6011 ¢

Optimal ET sequence: 72, 145, 217, 289

Badness (Sintel): 0.986

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155

Mapping: [⟨1 2 3 3 4 5 5], ⟨0 -30 -49 -14 -39 -94 -66]]

Optimal tunings:

  • WE: ~2 = 1200.0647 ¢, ~100/99 = 16.6025 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 16.6004 ¢

Optimal ET sequence: 72, 145, 217, 289

Badness (Sintel): 0.751

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675

Mapping: [⟨1 2 3 3 4 5 5 4], ⟨0 -30 -49 -14 -39 -94 -66 18]]

Optimal tunings:

  • WE: ~2 = 1199.9287 ¢, ~100/99 = 16.5930 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 16.5948 ¢

Optimal ET sequence: 72, 145, 217

Badness (Sintel): 0.924

Sfourth

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 64827/64000

Mapping: [⟨1 2 3 3], ⟨0 -19 -31 -9]]

mapping generators: ~2, ~49/48

Optimal tunings:

  • WE: ~2 = 1200.8332 ¢, ~49/48 = 26.3053 ¢
error map: ⟨+0.833 -0.090 +0.721 -3.074]
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2590 ¢
error map: ⟨0.000 -0.876 -0.343 -5.157]

Optimal ET sequence: 45, 46, 91, 137d

Badness (Sintel): 3.12

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 4375/4374

Mapping: [⟨1 2 3 3 4], ⟨0 -19 -31 -9 -25]]

Optimal tunings:

  • WE: ~2 = 1201.1486 ¢, ~49/48 = 26.3112 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2461 ¢

Optimal ET sequence: 45e, 46, 91e, 137de

Badness (Sintel): 1.78

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 325/324, 441/440

Mapping: [⟨1 2 3 3 4 4], ⟨0 -19 -31 -9 -25 -14]]

Optimal tunings:

  • WE: ~2 = 1201.4956 ¢, ~49/48 = 26.3423 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2614 ¢

Optimal ET sequence: 45ef, 46, 91ef, 137def, 228ddeeefff

Badness (Sintel): 1.37

Sfour

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2401/2376, 4375/4374

Mapping: [⟨1 2 3 3 3], ⟨0 -19 -31 -9 21]]

Optimal tunings:

  • WE: ~2 = 1200.4402 ¢, ~49/48 = 26.2557 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2403 ¢

Optimal ET sequence: 45, 46, 91, 137d, 183d

Badness (Sintel): 2.53

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 364/363, 385/384, 4375/4374

Mapping: [⟨1 2 3 3 3 3], ⟨0 -19 -31 -9 21 32]]

Optimal tunings:

  • WE: ~2 = 1200.3796 ¢, ~49/48 = 26.2473 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2372 ¢

Optimal ET sequence: 45, 46, 91, 137d, 183d

Badness (Sintel): 2.14

Trideci

For the 5-limit version, see 13th-octave temperaments #Tridecatonic.

The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the tridecatonic temperament, but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name trideci comes from tridecim (Latin for "thirteen").

Subgroup: 2.3.5.7

Comma list: 4375/4374, 83349/81920

Mapping: [⟨13 0 -11 57], ⟨0 1 2 -1]]

mapping generators: ~256/245, ~3

Optimal tunings:

  • WE: ~256/245 = 92.4141 ¢, ~3/2 = 699.9466 ¢
error map: ⟨+1.383 -0.626 -0.210 -2.554]
  • CWE: ~256/245 = 92.3077 ¢, ~3/2 = 699.4521 ¢
error map: ⟨0.000 -2.503 -2.794 -6.740]

Optimal ET sequence: 26, 65, 91

Badness (Sintel): 4.67

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 385/384, 4375/4374

Mapping: [⟨13 0 -11 57 45], ⟨0 1 2 -1 0]]

Optimal tunings:

  • WE: ~22/21 = 92.3729 ¢, ~3/2 = 700.1118 ¢
  • CWE: ~22/21 = 92.3077 ¢, ~3/2 = 699.7703 ¢

Optimal ET sequence: 26, 65, 91

Badness (Sintel): 2.80

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 245/242, 325/324, 385/384

Mapping: [⟨13 0 -11 57 45 48], ⟨0 1 2 -1 0 0]]

Optimal tunings:

  • WE: ~22/21 = 92.4003 ¢, ~3/2 = 699.9983 ¢
  • CWE: ~22/21 = 92.3077 ¢, ~3/2 = 699.4772 ¢

Optimal ET sequence: 26, 65f, 91f

Badness (Sintel): 2.16

References

  1. ↑ Yahoo! Tuning Group | 11-limit rank 2 using only wedgies "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things." —Gene Ward Smith
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