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Ragismic microtemperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This 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>2014-04-17 23:12:50 UTC</tt>.<br>
: The original revision id was <tt>503137770</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.
Temperaments discussed elsewhere are:
* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]
* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]]
* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]]
* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]
* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]]
* [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]]
* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]
* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Zarvo]]'' (+33075/32768) → [[Gravity family #Zarvo|Gravity family]]
* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]
* [[Parakleismic]] (+3136/3125) → [[Parakleismic family #Septimal parakleismic|Parakleismic family]]
* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnu family #Septimal vishnu|Vishnu family]]
* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Nuwell temperaments #Maja|Nuwell temperaments]]
* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortone family #Mitonic|Minortone family]]
* ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]]
* [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* [[Octoid]] (+16875/16807) → [[Stearnsmic clan #Octoid|Stearnsmic clan]]
* ''[[Ludizemlic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments #Ludizemlic|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.
Considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, chlorine, seniority, monzismic, semidimfourth, acrokleismic, quasithird, quincy, deca, keenanose, counterkleismic, sfourth, aluminium, ragitritonic, quatracot, trideci, moulin, and palladium.  


Commas: 2401/2400, 4375/4374
== 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.


POTE generators: 36/35: 49.0205; 10/9: 182.354; 6/5: 315.687; 49/40: 350.980
[[Subgroup]]: 2.3.5.7


Map: [&lt;9 1 1 2|, &lt;0 2 3 2|]
[[Comma list]]: 4375/4374, 52734375/52706752
Wedgie: &lt;&lt;18 27 18 1 -22 -34||
EDOs: [[27edo|27]], [[45edo|45]], [[72edo|72]], [[99edo|99]], [[171edo|171]], [[270edo|270]], [[441edo|441]], [[612edo|612]], [[3600edo|3600]]
Badness: 0.00361


==11 limit hemiennealimmal==
{{Mapping|legend=1| 1 -22 -27 -45 | 0 37 46 75 }}
Commas: 2401/2400, 4375/4374, 3025/3024
: mapping generators: ~2, ~14/9


POTE generator: 99/98: 17.6219 or 6/5: 315.7114
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0067{{c}}, ~14/9 = 764.9222{{c}}
: [[error map]]: {{val| +0.007 +0.019 -0.074 +0.037 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 764.9181{{c}}
: error map: {{val| 0.000 +0.013 -0.083 +0.029 }}


Map: [&lt;18 0 -1 22 48|, &lt;0 2 3 2 1|]
{{Optimal ET sequence|legend=1| 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214 }}
EDOs: 72, 198, 270, 342, 612, 954, 1566
Badness: 0.00628


==13 limit hemiennealimmal==
[[Badness]] (Sintel): 0.274
Commas: 676/675, 1001/1000, 1716/1715, 3025/3024


POTE generator ~99/98 = 17.7504
=== Semisupermajor ===
Subgroup: 2.3.5.7.11


Map: [&lt;18 0 -1 22 48 -19|, &lt;0 2 3 2 1 6|]
Comma list: 3025/3024, 4375/4374, 35156250/35153041
EDOs: 72, 198, 270
Badness: 0.0125


==Semiennealimmal==
{{Mapping|legend=0| 2 -7 -8 -15 -6 | 0 37 46 75 47 }}
Commas: 2401/2400, 4375/4374, 4000/3993
: mapping generators: ~99/70, ~11/10


POTE generator: ~140/121 = 250.3367
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;9 3 4 14 18|, &lt;0 6 9 6 7|]
{{Optimal ET sequence|legend=0| 80, 262d, 342, 764, 1106, 1448, 2554, 4002e, 6556cee }}
EDOs: 72, 369, 441
Badness: 0.0342


===13 limit semiennealimmal===
Badness (Sintel): 0.422
Commas: 1575/1573, 2080/2079, 2401/2400, 4375/4374


POTE generator: ~140/121 = 250.3375
== Enneadecal ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Enneadecal (5-limit)]].''


Map: [&lt;9 3 4 14 18 -8|, &lt;0 6 9 6 7 22|]
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, 441
Badness: 0.0261


==Quadraennealimmal==
[[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: 2401/2400 4375/4374 234375/234256


POTE generator: ~77/75 = 45.595
[[Subgroup]]: 2.3.5.7


Map: [&lt;9 1 1 12 -7|, [0 8 12 8 23]]
[[Comma list]]: 4375/4374, 703125/702464
EDOs: 342, 1053, 1395, 1737, 4869d, 6606cd
Badness: 0.0213


==Ennealimnic==
{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}
Commas: 243/242, 441/440, 4375/4356
: mapping generators: ~28/27, ~3


POTE generator: ~36/35 = 49.395
[[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 -2|, &lt;0 2 3 2 5|]
{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}
EDOs: 72, 171, 243
Badness: 0.0203


===13 limit ennealimnic===
[[Badness]] (Sintel): 0.277
Commas: 243/242, 364/363, 441/440, 625/624


POTE generator: ~36/35 = 49.341
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;9 1 1 12 -2 -33|, &lt;0 2 3 2 5 10|]
Comma list: 540/539, 4375/4374, 16384/16335
EDOs: 72, 171, 243
Badness: 0.0233


====17 limit ennealimnic====
{{Mapping|legend=0| 19 0 14 -37 126 | 0 1 1 3 -2 }}
Commas: 243/242, 364/363, 375/374, 441/440, 595/594


POTE generator: ~36/35 = 49.335
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 -3|, &lt;0 2 3 2 5 10 6|]
{{Optimal ET sequence|legend=0| 19, 133d, 152, 323e, 475de, 627de }}
EDOs: 72, 171, 243
Badness: 0.0146


===Ennealim===
Badness (Sintel): 1.45
Commas: 169/168, 243/242, 325/324, 441/440


POTE generator: ~36/35 = 49.708
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;9 1 1 12 -2 20|, &lt;0 2 3 2 5 2|]
Comma list: 540/539, 625/624, 729/728, 2205/2197
EDOs: 27e, 45f, 72, 315ff, 387cff, 459cdfff
Badness: 0.0207


==Ennealiminal==
{{Mapping|legend=0| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}
Commas: 385/384, 1375/1372, 4375/4374


POTE generator: ~36/35 = 49.504
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 51|, &lt;0 2 3 2 -3|]
{{Optimal ET sequence|legend=0| 19, 133df, 152f, 323ef }}
EDOs: 27, 45, 72, 171e, 243e, 315e
Badness: 0.0311


===13-limit===
Badness (Sintel): 1.39
Commas: 169/168, 325/324, 385/384, 1375/1372


POTE generator: ~36/35 = 49.486
=== Hemienneadecal ===
Subgroup: 2.3.5.7.11


Map: [&lt;9 1 1 12 51 20|, &lt;0 2 3 2 -3 2|]
Comma list: 3025/3024, 4375/4374, 234375/234256
EDOs: 27, 45f, 72, 171ef, 243ef
Badness: 0.0303


==Trinealimmal==
{{Mapping|legend=0| 38 0 28 -74 11 | 0 1 1 3 2 }}
Commas: 2401/2400, 4375/4374, 2097152/2096325
: mapping generators: ~55/54, ~3


POTE generator: ~6/5 = 315.644
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;27 1 0 34 177|, &lt;0 2 3 2 -4|]
{{Optimal ET sequence|legend=0| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}
EDOs: 27, 243, 270, 783, 1053, 1323, 10854bcde
Badness: 0.0298


==Semihemiennealimmal==
Badness (Sintel): 0.330
Commas: 2401/2400, 4375/4374, 3025/3024, 4225/4224


POTE generator: ~39/32 = 342.139
==== Hemienneadecalis ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;18 0 -1 22 48 88|, &lt;0 4 6 4 2 -3|]
Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256
EDOs: 126, 144, 270, 684, 954
Badness: 0.0131


=Gamera=
{{Mapping|legend=0| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}
Commas: 4375/4374, 589824/588245


POTE generator ~8/7 = 230.336
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;1 6 10 3|, &lt;0 -23 -40 -1|]
{{Optimal ET sequence|legend=0| 152f, 342f, 494 }}
EDOs: 26, 73, 99, 224, 323, 422, 735
Badness: 0.0376


==Hemigamera==
Badness (Sintel): 0.859
Commas: 3025/3024, 4375/4374, 202397184/201768035


POTE generator: ~8/7 = 230.337
==== Hemienneadec ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 12 20 6 5|, &lt;0 -23 -40 -1 5|]
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
EDOs: 26, 198, 224, 422, 646, 1068d
Badness: 0.0410


===13-limit===
{{Mapping|legend=0| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}
Commas: 1716/1715 2080/2079 2200/2197 3025/3024


Map: [&lt;2 12 20 6 5 17|, &lt;0 -23 -40 -1 5 -25|]
Optimal tunings:  
EDOs: 26, 198, 224, 422, 646f, 1068df
* WE: ~55/54 = 31.5784{{c}}, ~3/2 = 701.9736{{c}} (~243/242 = 7.2493{{c}})
Badness: 0.0204
* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9855{{c}} (~243/242 = 7.2487{{c}})


=Supermajor=
{{Optimal ET sequence|legend=0| 152, 342, 494, 1330, 1824, 2318d }}
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
Badness (Sintel): 1.26


POTE generator: ~9/7 = 435.082
==== Semihemienneadecal ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 15 19 30|, &lt;0 -37 -46 -75|]
Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078
EDOs: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214
Badness: 0.0108


==Semisupermajor==
{{Mapping|legend=0| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}
Commas: 3025/3024, 4375/4374, 35156250/35153041
: 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;2 30 38 60 41|, &lt;0 -37 -46 -75 -47|]
{{Optimal ET sequence|legend=0| 190, 304d, 494, 684, 1178, 2850, 4028ce }}
EDOs: 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf
Badness: 0.0128


=Enneadecal=
Badness (Sintel): 0.607
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
=== 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.


POTE generator: ~3/2 = 701.880
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;19 0 14 -37|, &lt;0 1 1 3|]
Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344
Generators: 28/27, 3
EDOs: 19, 152, 171, 665, 836, 1007, 2185
Badness: 0.0110


=Deca=
{{Mapping|legend=0| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}
Commas: 4375/4374, 165288374272/164794921875


POTE generator: ~460992/390625 = 284.423
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;10 4 2 9|, &lt;0 5 6 11|]
{{Optimal ET sequence|legend=0| 855, 988, 1843 }}
EDOs: 80, 190, 270, 1270, 1540, 1810, 2080
Badness: 0.0806


==11-limit==
Badness (Sintel): 3.15
Commas: 3025/3024, 4375/4374, 422576/421875


POTE generator: ~33/28 = 284.418
== Semidimi ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimi]].''


Map: [&lt;10 4 2 9 18|, &lt;0 5 6 11 7|]
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, 1000, 1270
Badness: 0.0243


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 1001/1000, 3025/3024, 4225/4224, 4375/4374


POTE generator: ~33/28 = 284.398
[[Comma list]]: 4375/4374, 3955078125/3954653486


Map: [&lt;10 4 2 9 18 37|, &lt;0 5 6 11 7 0|]
{{Mapping|legend=1| 1 -19 -25 -32 | 0 55 73 93 }}
EDOs: 80, 190, 270, 730, 1000
: mapping generators: ~2, ~35/27
Badness: 0.0168


=Mitonic=  
[[Optimal tuning]]s:
Commas: 4375/4374, 2100875/2097152
* [[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: ~10/9 = 182.458
{{Optimal ET sequence|legend=1| 8d, …, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}


Map: [&lt;1 16 32 -15|, &lt;0 -17 -35 21|]
[[Badness]] (Sintel): 0.382
EDOs: 46, 125, 171
Badness: 0.0252


=Abigail=  
== Brahmagupta ==
Commas: 4375/4374, 2147483648/2144153025
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 tuning|POTE generator]]: 208.899
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;2 7 13 -1|, &lt;0 -11 -24 19|]
[[Subgroup]]: 2.3.5.7
Wedgie: &lt;&lt;22 48 -38 25 -122 -223||
EDOs: 46, 132, 178, 224, 270, 494, 764, 1034, 1798
Badness: 0.0370


==11-limit==
[[Comma list]]: 4375/4374, {{monzo| 46 -14 -3 -6 }}
Comma: 3025/3024, 4375/4374, 20614528/20588575


[[POTE tuning|POTE generator]]: 208.901
{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}
: mapping generators: ~1157625/1048576, ~27/20


Map: [&lt;2 7 13 -1 1|, &lt;0 -11 -24 19 17|]
[[Optimal tuning]]s:
EDOs: 46, 132, 178, 224, 270, 494, 764
* [[WE]]: ~1157625/1048576 = 171.4275{{c}}, ~27/20 = 519.7125{{c}}
Badness: 0.0129
: [[error map]]: {{val| -0.007 +0.037 -0.034 -0.004 }}
* [[CWE]]: ~1157625/1048576 = 171.4286{{c}}, ~27/20 = 519.7156{{c}}
: error map: {{val| 0.000 +0.049 -0.018 +0.017 }}


==13-limit==
{{Optimal ET sequence|legend=1| 7, …, 217, 224, 441, 1106, 1547 }}
Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095


[[POTE tuning|POTE generator]]: 208.903
[[Badness]] (Sintel): 0.737


Map: [&lt;2 7 13 -1 1 -2|, &lt;0 -11 -24 19 17 27|]
=== 11-limit ===
EDOs: 46, 178, 224, 270, 494, 764, 1258
Subgroup: 2.3.5.7.11
Badness: 0.00886


=Semidimi=
Comma list: 4000/3993, 4375/4374, 131072/130977
Commas: 4375/4374, 3955078125/3954653486


POTE generator: ~35/27 = 449.127
{{Mapping|legend=0| 7 2 -8 53 3 | 0 3 8 -11 7 }}


Map: [&lt;1 36 48 61|, &lt;0 -55 -73 -93|]
Optimal tunings:  
Wedgie: &lt;&lt;55 73 93 -12 -7 11||
* WE: ~243/220 = 171.4208{{c}}, ~27/20 = 519.6807{{c}}
EDOs: 171, 863, 8419, 1205, 1376, 1547, 1718, 4983, 6701, 8419
* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7034{{c}}
Badness: 0.0151


=Brahmagupta=
{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665 }}
Commas: 4375/4374, 70368744177664/70338939985125


POTE generator: ~27/20 = 519.716
Badness (Sintel): 1.73


Map: [&lt;7 2 -8 53|, &lt;0 3 8 -11|]
=== 13-limit ===
Wedgie: &lt;&lt;21 56 -77 40 -181 -336||
Subgroup: 2.3.5.7.11.13
EDOs: 217, 224, 441, 1106, 1547
Badness: 0.0291


==11-limit==
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374
Commas: 4000/3993, 4375/4374, 131072/130977


POTE generator: ~27/20 = 519.704
{{Mapping|legend=0| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}


Map: [&lt;7 2 -8 53 3|, &lt;0 3 8 -11 7|]
Optimal tunings:  
EDOs: 217, 224, 441, 665, 1771e
* WE: ~243/220 = 171.4197{{c}}, ~27/20 = 519.6789{{c}}
Badness: 0.0522
* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7052{{c}}


=Neusec=
{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1106e }}
Commas: 3025/3024, 4375/4374, 235298/234375


POTE generator: ~12/11 = 151.547
Badness (Sintel): 0.956


Map: [&lt;2 11 15 19 15|, &lt;0 -31 -41 -53 -32|]
== Abigail ==
EDOs: 190, 388
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Abigail]].''
Badness: 0.0591


==13-limit==
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: 847/845, 1001/1000, 3025/3024, 4375/4374


POTE generator: ~12/11 = 151.545
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 17|, &lt;0 -31 -41 -53 -32 -38|]
[[Subgroup]]: 2.3.5.7
EDOs: 190, 198, 388
Badness: 0.0309


=Quasithird=
[[Comma list]]: 4375/4374, 2147483648/2144153025
Commas: 4375/4374, 1153470752371588581/1152921504606846976


POTE generator: ~5103/4096 = 380.388
{{Mapping|legend=1| 2 -4 -11 18 | 0 11 24 -19 }}
: mapping generators: ~46305/32768, ~1536/1225


Map: [&lt;4 0 -11 48|, &lt;0 5 16 -29|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;20 64 -116 55 -240 -449||
* [[WE]]: ~46305/32768 = 599.9699{{c}}, ~1536/1225 = 391.0818{{c}}
EDOs: 164, 224, 388, 612, 1448, 2060
: [[error map]]: {{val| -0.060 +0.065 -0.021 +0.079 }}
Badness: 0.0618
* [[CWE]]: ~46305/32768 = 600.0000{{c}}, ~1536/1225 = 391.1007{{c}}
: error map: {{val| 0.000 +0.152 +0.102 +0.262 }}


==11-limit==
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd }}
Commas: 3025/3024, 4375/4374, 4296700485/4294967296


POTE generator: ~5103/4096 = 380.387
[[Badness]] (Sintel): 0.936


Map: [&lt;4 0 -11 48 43|, &lt;0 5 16 -29 -23|]
=== 11-limit ===
EDOs: 164, 224, 388, 612, 836, 1448
Subgroup: 2.3.5.7.11
Badness: 0.0211


==13-limit==
Comma list: 3025/3024, 4375/4374, 131072/130977
Commas: 2200/2197, 3025/3024, 4375/4374, 468512/468195


POTE generator: ~5103/4096 = 380.385
{{Mapping|legend=0| 2 -4 -11 18 18 | 0 11 24 -19 -17 }}


Map: [&lt;4 0 -11 48 43 11|, &lt;0 5 16 -29 -23 3|]
Optimal tunings:  
EDOs: 164, 224, 388, 612, 836, 1448f, 2284f
* WE: ~99/70 = 599.9782{{c}}, ~1536/1225 = 391.0852{{c}}
Badness: 0.0295
* CWE: ~99/70 = 600.0000{{c}}, ~1536/1225 = 391.0992{{c}}


=Semidimfourth=
{{Optimal ET sequence|legend=0| 46, 132, 178, 224, 270, 494, 764 }}
Commas: 4375/4374, 235298/234375


POTE generator: ~35/27 = 448.457
Badness (Sintel): 0.425


Map: [&lt;1 21 28 36|, &lt;0 -31 -41 -53|]
=== 13-limit ===
Wedgie: &lt;&lt;31 41 53 -7 -3 8||
Subgroup: 2.3.5.7.11.13
EDOs: 91, 99, 289, 388, 875, 1263d, 1651d
Badness: 0.0552


=Acrokleismic=
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
Commas: 4375/4374, 2202927104/2197265625


POTE generator: ~6/5 = 315.557
{{Mapping|legend=0| 2 -4 -11 18 18 25 | 0 11 24 -19 -17 -27 }}


Map: [&lt;1 10 11 27|, &lt;0 -32 -33 -92|]
Optimal tunings:  
Wedgie: &lt;&lt;32 33 92 -22 56 121||
* WE: ~99/70 = 599.9862{{c}}, ~351/280 = 391.0879{{c}}
EDOs: 19, 251, 270
* CWE: ~99/70 = 600.0000{{c}}, ~351/280 = 391.0969{{c}}
Badness: 0.0562


==11-limit==
{{Optimal ET sequence|legend=0| 46, 178, 224, 270, 494, 764, 1258 }}
Commas: 4375/4374, 41503/41472, 172032/171875


POTE generator: ~6/5 = 315.558
Badness (Sintel): 0.366


Map: [&lt;1 10 11 27 -16|, &lt;0 -32 -33 -92 74|]
== Gamera ==
EDOs: 19, 251, 270, 829, 1099, 1369, 1639
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Gamera]].''
Badness: 0.0369


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 676/675, 1001/1000, 4375/4374, 10985/10976


POTE generator: ~6/5 = 315.557
[[Comma list]]: 4375/4374, 589824/588245


Map: [&lt;1 10 11 27 -16 25|, &lt;0 -32 -33 -92 74 -81|]
{{Mapping|legend=1| 1 -17 -30 2 | 0 23 40 1 }}
EDOs: 19, 251, 270
: mapping generators: ~2, ~7/4
Badness: 0.0268


==Counteracro==
[[Optimal tuning]]s:
Commas: 4375/4374, 5632/5625, 117649/117612
* [[WE]]: ~2 = 1199.8483{{c}}, ~7/4 = 969.5415{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 969.6608{{c}}


POTE generator: ~6/5 = 315.553
{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}


Map: [&lt;1 10 11 27 55|, &lt;0 -32 -33 -92 -196|]
[[Badness]] (Sintel): 0.953
EDOs: 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde
Badness: 0.0426


===13-limit===  
=== Hemigamera ===
Commas: 676/675, 1716/1715, 4225/4224, 4375/4374
Subgroup: 2.3.5.7.11


POTE generator: ~6/5 = 315.554
Comma list: 3025/3024, 4375/4374, 589824/588245


Map: [&lt;1 10 11 27 55 25|, &lt;0 -32 -33 -92 -196 -81|]
{{Mapping|legend=0| 2 -11 -20 5 10 | 0 23 40 1 -5 }}
EDOs: 270, 1331c, 1601c, 1871bcf, 2141bcf
: mapping generators: ~99/70, ~99/80
Badness: 0.0260


=Seniority=
Optimal tunings:
Commas: 4375/4374 201768035/201326592
* WE: ~99/70 = 599.9323{{c}}, ~99/80 = 369.6212{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~99/80 = 369.6610{{c}}


POTE generator: ~3087/2560 = 322.804
{{Optimal ET sequence|legend=0| 26, 172c, 198, 224, 422, 646, 1068d }}


Map: [&lt;1 11 19 2|, &lt;0 -35 -62 3|]
Badness (Sintel): 1.35
Wedgie: &lt;&lt;35 62 -3 17 -103 -181||
EDOs: 26, 145, 171, 2710d
Badness: 0.0449


=Orga=
==== 13-limit ====
Commas: 4375/4374 54975581388800/54936068900769
Subgroup: 2.3.5.7.11.13


POTE generator: ~8/7 = 231.104
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024


Map: [&lt;2 21 36 5|, &lt;0 -29 -51 1|]
{{Mapping|legend=0| 2 -11 -20 5 10 -8 | 0 23 40 1 -5 25 }}
Wedgie: &lt;&lt;58 102 -2 27 -166 -291||
EDOs: 26, 244, 270, 836, 1106, 1376, 2482, 19856bd, 23714bd
Badness: 0.0402


==11-limit==
Optimal tunings:
Commas: 3025/3024 4375/4374 5767168/5764801
* 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 = 231.103
{{Optimal ET sequence|legend=0| 26, 172cf, 198, 224, 422, 646f, 1068df }}


Map: [&lt;2 21 36 5 2|, &lt;0 -29 -51 1 8|]
Badness (Sintel): 0.844
EDOs: 26, 244, 270, 566, 836, 1106, 7472e, 8578de, 9684cde, 10790cde, 11896cde
Badness: 0.0162


=Quatracot=
=== Semigamera ===
Commas: 4375/4374, 1483154296875/1473173782528
Subgroup: 2.3.5.7.11


POTE generator: ~448/405 = 176.805
Comma list: 4375/4374, 14641/14580, 15488/15435


Map: [&lt;2 7 7 23|, &lt;0 -13 -8 -59|]
{{Mapping|legend=0| 1 -40 -70 1 -77 | 0 46 80 2 89 }}
Wedgie: &lt;&lt;26 16 118 -35 114 229||
: mapping generators: ~2, ~144/77
EDOs: 190, 224, 414, 638, 1052c, 1690bc
Badness: 0.1760


==11-limit==
Optimal tunings:
Commas: 3025/3024, 4375/4374, 1265625/1261568
* WE: ~2 = 1199.8845{{c}}, ~144/77 = 1084.7314{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8345{{c}}


POTE generator: ~448/405 = 176.806
{{Optimal ET sequence|legend=0| 73, 125, 198, 323, 521 }}


Map: [&lt;2 7 7 23 19|, &lt;0 -13 -8 -59 -41|]
Badness (Sintel): 2.59
EDOs: 190, 224, 414, 638, 1052c
Badness: 0.0410


==13-limit==
==== 13-limit ====
Commas: 625/624, 729/728, 1575/1573, 2200/2197
Subgroup: 2.3.5.7.11.13


POTE generator: ~448/405 = 176.804
Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580


Map: [&lt;2 7 7 23 19 13|, &lt;0 -13 -8 -59 -41 -19|]
{{Mapping|legend=0| 1 -40 -70 1 -77 -131 | 0 46 80 2 89 149 }}
EDOs: 190, 224, 414, 638, 1690bc, 2328bcde
Badness: 0.0226


=Nearly Micro=  
Optimal tunings:
* WE: ~2 = 1199.8726{{c}}, ~144/77 = 1084.7220{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8359{{c}}


=Octoid=
{{Optimal ET sequence|legend=0| 73f, 125f, 198, 323, 521 }}
Commas: 4375/4374, 16875/16807


POTE generator: ~7/5 = 583.940
Badness (Sintel): 1.82


Map: [&lt;8 1 3 3|, &lt;0 3 4 5|]
== Crazy ==
Generators: 49/45, 7/5
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
EDOs: 72, 152, 224
Badness: 0.0427


==11-limit==
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: 540/539, 1375/1372, 4000/3993


POTE generator: ~7/5 = 583.692
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 16|, &lt;0 3 4 5 3|]
[[Subgroup]]: 2.3.5.7
EDOs: 72, 152, 224
Badness: 0.0141


==13-limit==
[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}
Commas: 540/539, 1375/1372, 4000/3993, 625/624


POTE generator: ~7/5 = 583.905
{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}
: mapping generators: ~332150625/234881024, ~1125/1024


Map: [&lt;8 1 3 3 16 -21|, &lt;0 3 4 5 3 13|]
[[Optimal tuning]]s:
EDOs: 72, 224
* [[WE]]: ~332150625/234881024 = 600.0019{{c}}, ~1125/1024 = 162.7479{{c}}
Badness: 0.0153
: [[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 }}


==Music==
{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}
http://www.archive.org/details/Dreyfus
[[http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3|play]]


==Octopus==
[[Badness]] (Sintel): 0.998
Commas: 169/168, 325/324, 364/363, 540/539


POTE generator: ~7/5 = 583.892
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;8 1 3 3 16 14|, &lt;0 3 4 5 3 4|]
Comma list: 3025/3024, 4375/4374, 2791309312/2790703125
EDOs: 72, 152, 224f
Badness: 0.0217


=Amity=
{{Mapping|legend=0| 2 1 6 -15 -8 | 0 8 -5 76 55 }}
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.
Optimal tunings:
* WE: ~99/70 = 600.0047{{c}}, ~1125/1024 = 162.7493{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~1125/1024 = 162.7481{{c}}


==5-limit==
{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}
Comma: 1600000/1594323


POTE generator: ~243/200 = 339.519
Badness (Sintel): 0.562


Map: [&lt;1 3 6|, &lt;0 -5 -13|]
== Orga ==
EDOs: 7, 39, 46, 53, 152, 205, 463, 668, 873
Orga may be described as the {{nowrap| 26 & 270 }} temperament, and [[1106edo]] gives a strong tuning.  
Badness: 0.0220


==7-limit==
[[Subgroup]]: 2.3.5.7
Commas: 4375/4374, 5120/5103


POTE generator: ~243/200 = 339.432
[[Comma list]]: 4375/4374, {{monzo| 41 -4 2 -14 }}


Map: [&lt;1 3 6 -2|, &lt;0 -5 -13 17|]
{{Mapping|legend=1| 2 -8 -15 6 | 0 29 51 -1 }}
Wedgie: &lt;&lt;5 13 -17 9 -41 -76||
: mapping generators: ~7411887/5242880, ~8/7
EDOs: 7, 39, 46, 53, 99, 251, 350
Badness: 0.0236


==11-limit==
[[Optimal tuning]]s:
Commas: 540/539, 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 = 339.464
{{Optimal ET sequence|legend=1| 26, …, 244, 270, 836, 1106, 1376, 2482 }}


Map: [&lt;1 3 6 -2 21|, &lt;0 -5 -13 17 -62|]
[[Badness]] (Sintel): 1.02
EDOs: 53, 99e, 152, 555de, 707de, 859bde
Badness: 0.0315


==13-limit==
=== 11-limit ===
Commas: 352/351, 540/539, 625/624, 847/845
Subgroup: 2.3.5.7.11


POTE generator: ~243/200 = 339.481
Comma list: 3025/3024, 4375/4374, 5767168/5764801


Map: [&lt;1 3 6 -2 21 17|, &lt;0 -5 -13 17 -62 -47|]
{{Mapping|legend=0| 2 -8 -15 6 10 | 0 29 51 -1 -8 }}
EDOS: 53, 99ef, 152f, 205
Badness: 0.0280


==Accord==
Optimal tunings:
Commas: 126/125, 100352/98415
* 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 = 338.993
{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836, 1106 }}


Map: [&lt;1 3 6 11|, &lt;0 -5 -13 -29|]
Badness (Sintel): 0.535
Wedgie: &lt;&lt;5 13 29 9 32 31||
EDOs: 46, 131c, 177c
Badness: 0.0956


===11-limit===
=== 13-limit ===
Commas: 121/120, 126/125, 896/891
Subgroup: 2.3.5.7.11.13


POTE generator: ~11/9 = 339.047
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360


Map: [&lt;1 3 6 11 6|, &lt;0 -5 -13 -29 -9|]
{{Mapping|legend=0| 2 -8 -15 6 10 -3 | 0 29 51 -1 -8 27 }}
EDOs: 46, 177c, 223bc, 269bce
Badness: 0.0425


==Hitchcock==
Optimal tunings:
Commas: 121/120, 176/175, 2200/2187
* 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 = 339.340
{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836f, 1106f }}


Map: [&lt;1 3 6 -2 6|, &lt;0 -5 -13 17 -9|]
Badness (Sintel): 0.899
EDOs: 7, 39, 46, 53, 99
Badness: 0.0352


===13-limit===
== Chlorine ==
Commas: 121/120, 169/168, 176/175, 325/324
: ''For the 5-limit version, see [[17th-octave temperaments #Chlorine]].''


POTE generator: ~11/9 = 339.419
Chlorine (named after the 17th element) tempers out the [[septendecima]] in the 5-limit, and {{monzo| -49 4 22 -3 }} as well as the ragisma in the 7-limit. It has a 1/17-octave period, and can be described as {{nowrap| 289 & 323 }} temperament. Not only the semitwelfth, but also the ~5/4 can be used as a generator.  


Map: [&lt;1 3 6 -2 6 2|, &lt;0 -5 -13 17 -9 6|]
[[Subgroup]]: 2.3.5.7
EDOs: 7, 39, 46, 53, 99
Badness: 0.0224


==Hemiamity==
[[Comma list]]: 4375/4374, {{monzo| -49 4 22 -3 }}
Commas: 4375/4374, 5120/5103, 3025/3024


POTE generator: ~ 243/200 = 339.493
{{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }}


Map: [&lt;2 1 -1 13 13|, &lt;0 5 13 -17 -14|]
[[Optimal tuning]]s:  
EDOs: 14, 46, 106, 152, 350
* [[WE]]: ~25/24 = 70.5880{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9962{{c}}
: [[error map]]: {{val| -0.004 +0.037 -0.030 -0.019 }}
* [[CWE]]: ~25/24 = 70.5882{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9990{{c}}
: error map: {{val| 0.000 +0.043 -0.021 -0.013 }}


=Parakleismic=
{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547, 3706, 5253 }}
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
[[Badness]] (Sintel): 1.05


POTE generator: ~6/5 = 315.240
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 5 6|, &lt;0 -13 -14|]
Comma list: 4375/4374, 41503/41472, 1879453125/1879048192
EDOs: 19, 61, 80, 99, 118, 453, 571, 689, 1496
Badness: 0.0433


==7-limit==
{{Mapping|legend=0| 17 0 26 -87 207 | 0 2 1 10 -11 }}
Commas: 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.181
Optimal tunings:  
* WE: ~25/24 = 70.5905{{c}}, ~693/400 = 951.0054{{c}}
* CWE: ~25/24 = 70.5882{{c}}, ~693/400 = 950.9754{{c}}


Map: [&lt;1 5 6 12|, &lt;0 -13 -14 -35|]
{{Optimal ET sequence|legend=0| 289, 323, 612, 3349de, 3961de, …, 5797ddee }}
EDOs: 19, 80, 99, 217, 316, 415
Badness: 0.0274


==11-limit==
Badness (Sintel): 2.11
Commas: 385/384, 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.251
== Seniority ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]].  


Map: [&lt;1 5 6 12 -6|, &lt;0 -13 -14 -35 36|]
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.
EDOs: 19, 99, 118
Badness: 0.0497


==Parkleismic==
[[Subgroup]]: 2.3.5.7
Commas: 176/175, 1375/1372, 2200/2187


POTE generator: ~6/5 = 315.060
[[Comma list]]: 4375/4374, 201768035/201326592


Map: [&lt;1 5 6 12 20|, &lt;0 -13 -14 -35 -63|]
{{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }}
EDOs: 80, 179, 259cd
: mapping generators: ~2, ~5120/3087
Badness: 0.0559


===13-limit===  
[[Optimal tuning]]s:
Commas: 169/168, 176/175, 325/324, 1375/1372
* [[WE]]: ~2 = 1200.0745{{c}}, ~5120/3087 = 877.2500{{c}}
: [[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 }}


POTE generator: ~6/5 = 315.075
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d }}


Map: [&lt;1 5 6 12 20 10|, &lt;0 -13 -14 -35 -63 -24|]
[[Badness]] (Sintel): 1.14
EDOs: 15, 19, 80, 179
Badness: 0.0366


==Paradigmic==  
=== Senator ===
Commas: 540/539, 896/891, 3136/3125
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.


POTE generator: ~6/5 = 315.096
Subgroup: 2.3.5.7.11


Map: [&lt;1 5 6 12 -1|, &lt;0 -13 -14 -35 17|]
Comma list: 441/440, 4375/4374, 65536/65219
EDOs: 19, 80, 99e, 179e
Badness: 0.0417


===13-limit===
{{Mapping|legend=0| 1 -24 -43 5 2 | 0 35 62 -3 2 }}
Commas: 169/168, 325/324, 540/539, 832/825


POTE generator: ~6/5 = 315.080
Optimal tunings:  
* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}}


Map: [&lt;1 5 6 12 -1 10|, &lt;0 -13 -14 -35 17 -24|]
{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e }}
EDOs: 19, 80, 99e, 179e
Badness: 0.0358


=Semiparakleismic=
Badness (Sintel): 3.05
Commas: 3025/3024, 3136/3125, 4375/4374


POTE generator: 315.181
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 10 12 24 19|, &lt;0 -13 -14 -35 -23|]
Comma list: 364/363, 441/440, 2200/2197, 4375/4374
EDOs: 80, 118, 198, 316, 514c, 830c
Badness: 0.0342


=Quincy=
{{Mapping|legend=0| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}
Commas: 4375/4374, 823543/819200


POTE generator: ~1728/1715 = 16.613
Optimal tunings:  
* WE: ~2 = 1199.7136{{c}}, ~108/65 = 877.9974{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2038{{c}}


Map: [&lt;1 2 2 3|, &lt;0 -30 -49 -14|]
{{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }}
EDOs: 72, 217, 289
Badness: 0.0797


==11-limit==
Badness (Sintel): 1.85
Commas: 441/440, 4000/3993, 41503/41472


POTE generator: ~100/99 = 16.613
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 2 2 3 4|, &lt;0 -30 -49 -14 -39|]
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197
EDOs: 72, 217, 289
Badness: 0.0309


==13-limit==
{{Mapping|legend=0| 1 -24 -43 5 2 -27 -31 | 0 35 62 -3 2 42 48 }}
Commas: 364/363, 441/440, 676/675, 4375/4374


POTE generator: ~100/99 = 16.602
Optimal tunings:  
* WE: ~2 = 1199.7195{{c}}, ~108/65 = 877.0018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}}


Map: [&lt;1 2 2 3 4 5|, &lt;0 -30 -49 -14 -39 -94|]
{{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }}
EDOs: 72, 145, 217, 289
Badness: 0.0239


==17-limit==
Badness (Sintel): 1.35
Commas: 364/363, 441/440, 595/594, 1001/1000, 1156/1155


POTE generator: ~100/99 = 16.602
== Monzismic ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]].  


Map: [&lt;1 2 2 3 4 5 5|, &lt;0 -30 -49 -14 -39 -94 -66|]
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.
EDOs: 72, 145, 217, 289
Badness: 0.0147


==19-limit==
[[Subgroup]]: 2.3.5.7
Commas: 343/342, 364/363, 441/440, 595/594, 676/675, 2601/2600


POTE generator: ~100/99 = 16.594
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}


Map: [&lt;1 2 2 3 4 5 5 4|, &lt;0 -30 -49 -14 -39 -94 -66 18|]
{{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }}
EDOs: 72, 145, 217
: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}
Badness: 0.0152</pre></div>
 
<h4>Original HTML content:</h4>
[[Optimal tuning]]s:
<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;
* [[WE]]: ~2 = 1200.0128{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9895{{c}}
&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;
: [[error map]]: {{val| +0.013 +0.024 -0.049 -0.019 }}
&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;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9793{{c}}
&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;
: error map: {{val| 0.000 +0.004 -0.080 -0.050 }}
&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
 
{{Optimal ET sequence|legend=1| 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|legend=0| 1 0 -27 109 -159 | 0 2 37 -134 205 }}
 
Optimal tunings:
* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=0| 1 0 -27 109 -159 -70 | 0 2 37 -134 205 93 }}
 
Optimal tunings:
* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=1| 1 -10 -13 -17 | 0 31 41 53 }}
: mapping generators: ~2, ~35/27
 
[[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 =
Retrieved from "https://en.xen.wiki/w/Ragismic_microtemperaments"