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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>2011-06-27 02:52:36 UTC</tt>.<br>
: The original revision id was <tt>238903247</tt>.<br>
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]


The ragisma is 4375/4374, 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 cmplexity, with the same caveat about "relatively"; however 27/25 is the period for 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=
Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
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||.
* ''[[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]]


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.  
== Supermajor ==
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2<sup>15</sup>)/3, 46 give (2<sup>19</sup>)/5, and 75 give (2<sup>30</sup>)/7. 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 is not enough notes for you, there is always the 171-note mos.


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.
[[Subgroup]]: 2.3.5.7


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 15 19 30 | 0 -37 -46 -75 }}


Map: [&lt;9 1 1 2|, &lt;0 2 3 2|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082
Wedgie: &lt;&lt;18 27 18 1 -22 -34||
EDOs: 27, 45, 72, 99, 171, 270, 441, 612
Badness: 0.00361


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


POTE generator: 99/98: 17.6219 or 6/5: 315.7114
[[Badness]]: 0.010836


Map: [&lt;18 0 -1 22 48|, &lt;0 2 3 2 1|]
=== Semisupermajor ===
EDOs: 72, 198, 270, 342, 612, 954, 1566
Subgroup: 2.3.5.7.11
Badness: 0.00628


==13 limit hemiennealimmal==
Comma list: 3025/3024, 4375/4374, 35156250/35153041
Commas: 676/675, 1001/1000, 1716/1715, 3025/3024


POTE generator ~99/98 = 17.7504
Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}


Map: [&lt;18 0 -1 22 48 -19|, &lt;0 2 3 2 1 6|]
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082
EDOs: 72, 198, 270
Badness: 0.0125


=Gamera=
{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}
Commas: 4375/4374, 589824/588245


POTE generator ~8/7 = 230.336
Badness: 0.012773


Map: [&lt;1 6 10 3|, &lt;0 -23 -40 -1|]
== Enneadecal ==
EDOs: 26, 73, 99, 224, 323, 422, 735
Enneadecal temperament 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 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. [[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.
Badness: 0.0376


=Supermajor=
''For the 5-limit temperament, see [[19th-octave temperaments#(5-limit) enneadecal]].''
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
[[Subgroup]]: 2.3.5.7


POTE generator: ~9/7 = 435.082
[[Comma list]]: 4375/4374, 703125/702464


Map: [&lt;1 15 19 30|, &lt;0 -37 -46 -75|]
{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}
EDOs: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214
Badness: 0.0108


=Enneadecal=
: mapping generators: ~28/27, ~3
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
[[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)


POTE generator: ~3/2 = 701.880
{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}


Map: [&lt;19 0 14 -37|, &lt;0 1 1 3|]
[[Badness]]: 0.010954
Generators: 28/27, 3
EDOs: 19, 152, 171, 665, 836, 1007, 2185
Badness: 0.0110


=Abigail=
=== 11-limit ===
Commas: 4375/4374, 2147483648/2144153025
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE generator]]: 208.899
Comma list: 540/539, 4375/4374, 16384/16335


Map: [&lt;2 7 13 -1|, &lt;0 -11 -24 19|]
Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}
Wedgie: &lt;&lt;22 48 -38 25 -122 -223||
EDOs: 46, 132, 178, 224, 270, 494, 764, 1034, 1798
Badness: 0.0370


==11-limit==
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)
Comma: 3025/3024, 4375/4374, 20614528/20588575


[[POTE tuning|POTE generator]]: 208.901
{{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }}


Map: [&lt;2 7 13 -1 1|, &lt;0 -11 -24 19 17|]
Badness: 0.043734
EDOs: 46, 132, 178, 224, 270, 494, 764
Badness: 0.0129


==13-limit==
==== 13-limit ====
Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095
Subgroup: 2.3.5.7.11.13


[[POTE tuning|POTE generator]]: 208.903
Comma list: 540/539, 625/624, 729/728, 2205/2197


Map: [&lt;2 7 13 -1 1 -2|, &lt;0 -11 -24 19 17 27|]
Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}
EDOs: 46, 178, 224, 270, 494, 764, 1258
Badness: 0.00886


=Nearly Micro=
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)


=Octoid=
{{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }}
Commas: 4375/4374, 16875/16807


POTE generator: ~7/5 = 583.940
Badness: 0.033545


Map: [&lt;8 1 3 3|, &lt;0 3 4 5|]
=== Hemienneadecal ===
Generators: 49/45, 7/5
Subgroup: 2.3.5.7.11
EDOs: 72, 152, 224
Badness: 0.0427


==11-limit==
Comma list: 3025/3024, 4375/4374, 234375/234256
Commas: 540/539, 1375/1372, 4000/3993


POTE generator: ~7/5 = 583.692
Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}


Map: Map: [&lt;8 1 3 3 16|, &lt;0 3 4 5 3|]
: mapping generators: ~55/54, ~3
EDOs: 72, 152, 224
Badness: 0.0141


==13-limit==
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)
Commas: 540/539, 1375/1372, 4000/3993, 625/624


POTE generator: ~7/5 = 583.905
{{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}


Map: Map: [&lt;8 1 3 3 16 -21|, &lt;0 3 4 5 3 13|]
Badness: 0.009985
EDOs: 72, 224
Badness: 0.0153


=Amity=
==== Hemienneadecalis ====
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.
Subgroup: 2.3.5.7.11.13


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


==5-limit==
Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}
Comma: 1600000/1594323


POTE generator: ~243/200 = 339.519
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)


Map: [&lt;1 3 6|, &lt;0 -5 -13|]
{{Optimal ET sequence|legend=1| 152f, 342f, 494 }}
EDOs: 7, 39, 46, 53, 152, 205, 463, 668, 873
Badness: 0.0220


==7-limit==
Badness: 0.020782
Commas: 4375/4374, 5120/5103


POTE generator: ~243/200 = 339.432
==== Hemienneadec ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 3 6 -2|, &lt;0 -5 -13 17|]
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
EDOs: 7, 39, 46, 53, 99, 251, 350
Badness: 0.0236


==Hitchcock==
Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}
Commas: 121/120, 176/175, 2200/2187


POTE generator: ~11/9 = 339.340
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444)


Map: [&lt;1 3 6 -2 6|, &lt;0 -5 -13 17 -9|]
{{Optimal ET sequence|legend=1| 152, 342, 494, 1330, 1824, 2318d }}
EDOs: 7, 39, 46, 53, 99
Badness: 0.0352


==Hemiamity==
Badness: 0.030391
Commas: 4375/4374, 5120/5103, 3025/3024


POTE generator: ~ 243/200 = 339.493
==== Semihemienneadecal ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 1 -1 13 13|, &lt;0 5 13 -17 -14|]
Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078
EDOs: 14, 46, 106, 152, 350


=Parakleismic=
Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}
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
: mapping generators: ~55/54 = 1\38, ~55/54, ~429/250


POTE generator: ~6/5 = 315.240
Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)


Map: [&lt;1 5 6|, &lt;0 -13 -14|]
{{Optimal ET sequence|legend=1| 190, 304d, 494, 684, 1178, 2850, 4028ce }}
EDOs: 19, 61, 80, 99, 118, 453, 571, 689, 1496
Badness: 0.0433


==7-limit==
Badness: 0.014694
Commas: 3136/3125, 5475/4374


POTE generator: ~6/5 = 315.181
=== 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;1 5 6 12|, &lt;0 -13 -14 -35|]
Subgroup: 2.3.5.7.11.13.17.19
EDOs: 19, 80, 99, 217, 316, 415
Badness: 0.0274


==11-limit==
Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344
Commas: 385/384, 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.251
Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}


Map: [&lt;1 5 6 12 -6|, &lt;0 -13 -14 -35 36|]
Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244
EDOs: 19, 99, 118
 
Badness: 0.0497</pre></div>
{{Optimal ET sequence|legend=1| 855, 988, 1843 }}
<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;
== Semidimi ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].''
 
The generator of semidimi temperament 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 3955078125/3954653486
 
{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270
 
{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
 
[[Badness]]: 0.015075
 
== Brahmagupta ==
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.
 
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 temperament that has pure octaves and pure fifths, which can also be described as a 17-limit extension having 1/7th octave period (171.4286 ¢) and 1/21st apotome generator (5.4136 ¢).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 70368744177664/70338939985125
 
{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}
 
: mapping generators: ~1157625/1048576, ~27/20
 
[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716
 
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }}
 
[[Badness]]: 0.029122
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4000/3993, 4375/4374, 131072/130977
 
Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}
 
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704
 
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }}
 
Badness: 0.052190
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374
 
Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}
 
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.706
 
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771eef }}
 
Badness: 0.023132
 
== Abigail ==
Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930]: "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."</ref>
 
''For the 5-limit temperament, see [[Very high accuracy temperaments#Abigail]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 2147483648/2144153025
 
{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }}
 
: mapping generators: ~46305/32768, ~27/20
 
[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899
 
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}
 
[[Badness]]: 0.037000
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 131072/130977
 
Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901
 
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }}
 
Badness: 0.012860
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
 
Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903
 
{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }}
 
Badness: 0.008856
 
== Gamera ==
''For the 5-limit temperament, see [[High badness temperaments#Gamera]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 589824/588245
 
{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }}
 
: mapping generators: ~2, ~8/7
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336
 
{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323
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