Ragismic microtemperaments: Difference between revisions

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The ragisma is [[4375/4374]] with a [[monzo]] of {{monzo| -1 -7 4 1 }}, the smallest 7-limit [[superparticular]] ratio. Since (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 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.
{{Technical data page}}
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]].  


Temperaments discussed elsewhere include:
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.
* ''[[Hystrix]]'', {36/35, 160/147} → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Rhinoceros]]'', {49/48, 4375/4374} → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Crepuscular]]'', {50/49, 4375/4374} → [[Jubilismic clan #Crepuscular|Jubilismic clan]] and [[Fifive family #Crepuscular|Fifive family]]
* ''[[Modus]]'', {64/63, 4375/4374} → [[Tetracot family #Modus|Tetracot family]]
* ''[[Flattone]]'', {81/80, 525/512} → [[Meantone family #Flattone|Meantone family]]
* [[Sensi]], {126/125, 245/243} → [[Sensipent family #Sensi|Sensipent family]] and [[Sensamagic clan #Sensi|Sensamagic clan]]
* [[Catakleismic]], {225/224, 4375/4374} → [[Kleismic family #Catakleismic|Kleismic family]]
* [[Unidec]], {1029/1024, 4375/4374} → [[Gamelismic clan #Unidec|Gamelismic clan]]
* ''[[Quartonic]]'', {1728/1715, 4000/3969} → [[Orwellismic temperaments #Quartonic|Orwellismic temperaments]]
* ''[[Maja]]'', {2430/2401, 3125/3087} → [[Maja family #Septimal maja|Maja family]]
* [[Pontiac]], {4375/4374, 32805/32768} → [[Schismatic family #Pontiac|Schismatic family]]
* ''[[Zarvo]]'', {4375/4374, 33075/32768} → [[Gravity family #Zarvo|Gravity family]]
* ''[[Whirrschmidt]]'', {4375/4374, 393216/390625} → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
* ''[[Mitonic]]'', {4375/4374, 2100875/2097152} → [[Minortonic family #Mitonic|Minortonic family]]
* ''[[Vishnu]]'', {4375/4374, 29360128/29296875} → [[Vishnuzmic family #Septimal vishnu|Vishnuzmic family]]
* ''[[Vulture]]'', {4375/4374, 33554432/33480783} → [[Vulture family #Septimal vulture|Vulture family]]
* ''[[Trillium]]'', {4375/4374, {{monzo| 40 -22 -1 -1 }}} → [[Tricot family #Trillium|Tricot family]]
* ''[[Unlit]]'', {4375/4374, {{monzo| 41 -20 -4 }}} → [[Undim family #Unlit|Undim family]]
* ''[[Quindro]]'', {4375/4374, {{monzo| 56 -28 -5 }}} → [[Quindromeda family #Quindro|Quindromeda family]]


Considered below are ennealimmal, gamera, supermajor, enneadeca, decal, sfourth, abigail, semidimi, brahmagupta, quasithird, semidimfourth, acrokleismic, seniority, orga, quatracot, octoid, amity, parakleismic, counterkleismic, quincy, chlorine, palladium, and monzism.  
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:
* ''[[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]]


== Ennealimmal ==
== Supermajor ==
{{Main| Ennealimmal }}
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.


[[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|ennealimmal comma]], {{monzo|1 -27 18}}, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. Its [[pergen]] is (P8/9, P5/2). While 27/25 is a 5-limit interval, two period 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 {{multival|18 27 18 1 -22 -34}}.
[[Subgroup]]: 2.3.5.7


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 EDOs, though its hardly likely anyone could tell the difference.
[[Comma list]]: 4375/4374, 52734375/52706752


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.
{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }}


Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]], [[Tritrizo clan #Undecentic|undecentic]], [[Tritrizo clan #Schisennealimmal|schisennealimmal]], and [[Tritrizo clan #Lunennealimmal|lunennealimmal]].
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082


{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}


Subgroup: 2.3.5.7
[[Badness]]: 0.010836


[[Comma list]]: 2401/2400, 4375/4374
=== Semisupermajor ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val|9 1 1 12}}, {{val|0 2 3 2}}]
Comma list: 3025/3024, 4375/4374, 35156250/35153041


{{Multival|legend=1|18 27 18 1 -22 -34}}
Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}


Mapping generators: ~27/25, ~5/3
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082


[[POTE generator]]s: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980
{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}


[[Tuning ranges]]:  
Badness: 0.012773
* 7-odd-limit [[diamond monotone]]: ~36/35 = [26.667, 66.667] (1\45 to 1\18)
* 9-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~36/35 = [48.920, 49.179]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 49.179]


{{Val list|legend=1| 27, 45, 72, 99, 171, 441, 612 }}
== Enneadecal ==
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]]: 0.003610
''For the 5-limit temperament, see [[19th-octave temperaments#(5-limit) enneadecal]].''


=== 11-limit ===
[[Subgroup]]: 2.3.5.7
The ennealimmal temperament can be described as 99e&amp;270 temperament, which tempers out 5632/5625 (vishdel comma) and 19712/19683 (symbiotic comma).


Subgroup: 2.3.5.7.11
[[Comma list]]: 4375/4374, 703125/702464


Comma list: 2401/2400, 4375/4374, 5632/5625
{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}


Mapping: [{{val|9 1 1 12 -75}}, {{val|0 2 3 2 16}}]
: mapping generators: ~28/27, ~3


POTE generator: ~36/35 = 48.8654
[[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)


Vals: {{Val list| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}


Badness: 0.027332
[[Badness]]: 0.010954
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374
 
Mapping: [{{val|9 1 1 12 -75 93}}, {{val|0 2 3 2 16 -9}}]
 
POTE generator: ~36/35 = 48.9030
 
Vals: {{Val list| 99e, 171e, 270 }}
 
Badness: 0.029404
 
=== Ennealimmia ===
Ennealimmal temperament has various extensions to the 11-limit. Tempering out 131072/130977 (salururu comma) leads to the ''ennealimmia'' temperament (171&amp;270).


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 4375/4374, 131072/130977
Comma list: 540/539, 4375/4374, 16384/16335


Mapping: [{{val|9 1 1 12 124}}, {{val|0 2 3 2 -14}}]
Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}


POTE generator: ~36/35 = 48.9244
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)


Vals: {{Val list| 99, 171, 270, 711, 981, 1251, 2232e }}
{{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }}


Badness: 0.026463
Badness: 0.043734


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374
Comma list: 540/539, 625/624, 729/728, 2205/2197


Mapping: [{{val|9 1 1 12 124 93}}, {{val|0 2 3 2 -14 -9}}]
Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}


POTE generator: ~36/35 = 48.9336
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)


Vals: {{Val list| 99, 171, 270, 711, 981, 1692e, 2673e }}
{{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }}


Badness: 0.016607
Badness: 0.033545
 
=== Ennealimnic ===
Ennealimnic temperament (72&amp;171) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.


=== Hemienneadecal ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 243/242, 441/440, 4375/4356
Comma list: 3025/3024, 4375/4374, 234375/234256


Mapping: [{{val|9 1 1 12 -2}}, {{val|0 2 3 2 5}}]
Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}


POTE generator: ~36/35 = 49.395
: mapping generators: ~55/54, ~3


Tuning ranges:
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)
* 11-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
* 11-odd-limit diamond tradeoff: ~36/35 = [48.920, 52.592]
* 11-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 52.592]


Vals: {{Val list| 72, 171, 243 }}
{{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}


Badness: 0.020347
Badness: 0.009985


==== 13-limit ====
==== Hemienneadecalis ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 243/242, 364/363, 441/440, 625/624
Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256


Mapping: [{{val|9 1 1 12 -2 -33}}, {{val|0 2 3 2 5 10}}]
Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}


POTE generator: ~36/35 = 49.341
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)


Tuning ranges:
{{Optimal ET sequence|legend=1| 152f, 342f, 494 }}
* 13- and 15-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)
* 13- and 15-odd-limit diamond tradeoff: ~36/35 = [48.825, 52.592]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~36/35 = [48.825, 50.000]


Vals: {{Val list| 72, 171, 243 }}
Badness: 0.020782


Badness: 0.023250
==== Hemienneadec ====
Subgroup: 2.3.5.7.11.13


===== 17-limit =====
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
Subgroup: 2.3.5.7.11.13.17


Comma list: 243/242, 364/363, 375/374, 441/440, 595/594
Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}


Mapping: [{{val|9 1 1 12 -2 -33 -3}}, {{val|0 2 3 2 5 10 6}}]
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444)


POTE generator: ~36/35 = 49.335
{{Optimal ET sequence|legend=1| 152, 342, 494, 1330, 1824, 2318d }}


Tuning ranges:  
Badness: 0.030391
* 17-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)
* 17-odd-limit diamond tradeoff: ~36/35 = [46.363, 52.592]
* 17-odd-limit diamond monotone and tradeoff: ~36/35 = [48.485, 50.000]


Vals: {{Val list| 72, 171, 243 }}
==== Semihemienneadecal ====
Subgroup: 2.3.5.7.11.13


Badness: 0.014602
Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078


==== Ennealim ====
Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}
Subgroup: 2.3.5.7.13


Comma list: 169/168, 243/242, 325/324, 441/440
: mapping generators: ~55/54 = 1\38, ~55/54, ~429/250


Mapping: [{{val|9 1 1 12 -2 20}}, {{val|0 2 3 2 5 2}}]
Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)


POTE generator: ~36/35 = 49.708
{{Optimal ET sequence|legend=1| 190, 304d, 494, 684, 1178, 2850, 4028ce }}


Vals: {{Val list| 27e, 45ef, 72 }}
Badness: 0.014694


Badness: 0.020697
=== 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.


=== Ennealiminal ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11


Comma list: 385/384, 1375/1372, 4375/4374
Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344


Mapping: [{{val|9 1 1 12 51}}, {{val|0 2 3 2 -3}}]
Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}


POTE generator: ~36/35 = 49.504
Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244


Vals: {{Val list| 27, 45, 72, 171e, 243e, 315e }}
{{Optimal ET sequence|legend=1| 855, 988, 1843 }}


Badness: 0.031123
== Semidimi ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].''


==== 13-limit ====
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: 2.3.5.7.11.13


Comma list: 169/168, 325/324, 385/384, 1375/1372
[[Subgroup]]: 2.3.5.7


Mapping: [{{val|9 1 1 12 51 20}}, {{val|0 2 3 2 -3 2}}]
[[Comma list]]: 4375/4374, 3955078125/3954653486


POTE generator: ~36/35 = 49.486
{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}


Vals: {{Val list| 27, 45f, 72, 171ef, 243ef }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270


Badness: 0.030325
{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}


=== Hemiennealimmal ===
[[Badness]]: 0.015075
Hemiennealimmal (72&amp;198) has a period of 1/18 octave and tempers out the four smallest superparticular commas of the 11-limit JI, 2401/2400, 3025/3024, 4375/4374, and 9801/9800. Tempering out [[9801/9800]] leads an octave split into two equal parts.


Subgroup: 2.3.5.7.11
== Brahmagupta ==
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.  


Comma list: 2401/2400, 3025/3024, 4375/4374
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 ¢).


Mapping: [{{val| 18 0 -1 22 48 }}, {{val| 0 2 3 2 1 }}]
[[Subgroup]]: 2.3.5.7


Mapping generators: ~80/77, ~400/231
[[Comma list]]: 4375/4374, 70368744177664/70338939985125


POTE generator: ~99/98 = 17.6219
{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}


Tuning ranges:  
: mapping generators: ~1157625/1048576, ~27/20
* 11-odd-limit diamond monotone: ~99/98 = [13.333, 22.222] (1\90 to 1\54)
* 11-odd-limit diamond tradeoff: ~99/98 = [17.304, 17.985]
* 11-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 17.985]


Vals: {{Val list| 72, 198, 270, 342, 612, 954, 1566 }}
[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716


Badness: 0.006283
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }}


==== 13-limit ====
[[Badness]]: 0.029122
Subgroup: 2.3.5.7.11.13


Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 18 0 -1 22 48 -19 }}, {{val| 0 2 3 2 1 6 }}]
Comma list: 4000/3993, 4375/4374, 131072/130977


POTE generator ~99/98 = 17.7504
Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}


Tuning ranges:
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704
* 13-odd-limit diamond monotone: ~99/98 = [16.667, 22.222] (1\72 to 1\54)
* 15-odd-limit diamond monotone: ~99/98 = [16.667, 19.048] (1\72 to 2\126)
* 13-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.309]
* 15-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.926]
* 13-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.309]
* 15-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.926]


Vals: {{Val list| 72, 198, 270 }}
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }}


Badness: 0.012505
Badness: 0.052190


==== Semihemiennealimmal ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374


Mapping: [{{val| 18 0 -1 22 48 88 }}, {{val| 0 4 6 4 2 -3 }}]
Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}


Mapping generators: ~80/77, ~1053/800
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.706


POTE generator: ~39/32 = 342.139
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771eef }}


Vals: {{Val list| 126, 144, 270, 684, 954 }}
Badness: 0.023132


Badness: 0.013104
== 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>


=== Semiennealimmal ===
''For the 5-limit temperament, see [[Very high accuracy temperaments#Abigail]].''
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 4000/3993, 4375/4374
[[Subgroup]]: 2.3.5.7


Mapping: [{{val|9 3 4 14 18}}, {{val|0 6 9 6 7}}]
[[Comma list]]: 4375/4374, 2147483648/2144153025


POTE generator: ~140/121 = 250.3367
{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }}


Vals: {{Val list| 72, 369, 441 }}
: mapping generators: ~46305/32768, ~27/20


Badness: 0.034196
[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899


==== 13-limit ====
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}
Subgroup: 2.3.5.7.11.13


Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374
[[Badness]]: 0.037000


Mapping: [{{val|9 3 4 14 18 -8}}, {{val|0 6 9 6 7 22}}]
=== 11-limit ===
 
POTE generator: ~140/121 = 250.3375
 
Vals: {{Val list| 72, 297ef, 369f, 441 }}
 
Badness: 0.026122
 
=== Quadraennealimmal ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 4375/4374, 234375/234256
Comma list: 3025/3024, 4375/4374, 131072/130977


Mapping: [{{val|9 1 1 12 -7}}, {{val|0 8 12 8 23}}]
Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }}


POTE generator: ~77/75 = 45.595
Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901


Vals: {{Val list| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }}


Badness: 0.021320
Badness: 0.012860


=== Trinealimmal ===
=== 13-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 2401/2400, 4375/4374, 2097152/2096325
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095


Mapping: [{{val| 27 1 0 34 177 }}, {{val| 0 2 3 2 -4 }}]
Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}


Mapping generators: ~2744/2673, ~2352/1375
Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903


POTE generator: ~6/5 = 315.644
{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }}


Vals: {{Val list| 27, 243, 270, 783, 1053, 1323 }}
Badness: 0.008856


Badness: 0.029812
== Gamera ==
''For the 5-limit temperament, see [[High badness temperaments#Gamera]].


== Gamera ==
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 589824/588245
[[Comma list]]: 4375/4374, 589824/588245


[[Mapping]]: [{{val|1 6 10 3}}, {{val|0 -23 -40 -1}}]
{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }}


{{Multival|legend=1|23 40 1 10 -63 -110}}
: mapping generators: ~2, ~8/7


[[POTE generator]] ~8/7 = 230.336
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336


{{Val list|legend=1| 26, 73, 99, 224, 323, 422, 745d }}
{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}


[[Badness]]: 0.037648
[[Badness]]: 0.037648
Line 341: Line 302:
Comma list: 3025/3024, 4375/4374, 589824/588245
Comma list: 3025/3024, 4375/4374, 589824/588245


Mapping: [{{val|2 12 20 6 5}}, {{val|0 -23 -40 -1 5}}]
Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}
 
: mapping generators: ~99/70, ~8/7


POTE generator: ~8/7 = 230.3370
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3370


Vals: {{Val list| 26, 198, 224, 422, 646, 1068d }}
{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646, 1068d }}


Badness: 0.040955
Badness: 0.040955
Line 354: Line 317:
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024


Mapping: [{{val|2 12 20 6 5 17}}, {{val|0 -23 -40 -1 5 -25}}]
Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}


POTE generator: ~8/7 = 230.3373
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373


Vals: {{Val list| 26, 198, 224, 422, 646f, 1068df }}
{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646f, 1068df }}


Badness: 0.020416
Badness: 0.020416


== Supermajor ==
=== Semigamera ===
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 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 {{multival|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.
Subgroup: 2.3.5.7.11


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


Subgroup: 2.3.5.7
Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }}


[[Comma list]]: 4375/4374, 52734375/52706752
: mapping generators: ~2, ~77/72


[[Mapping]]: [{{val|1 15 19 30}}, {{val|0 -37 -46 -75}}]
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1642


{{Multival|legend=1|37 46 75 -13 15 45}}
{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }}


[[POTE generator]]: ~9/7 = 435.082
Badness: 0.078


{{Val list|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.010836
Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580


=== Semisupermajor ===
Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }}
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4375/4374, 35156250/35153041
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1628
 
Mapping: [{{val|2 30 38 60 41}}, {{val|0 -37 -46 -75 -47}}]
 
POTE generator: ~9/7 = 435.082
 
EDOs: {{Val list| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}


Badness: 0.012773
{{Optimal ET sequence|legend=1| 73f, 125f, 198, 323, 521 }}


== Enneadecal ==
Badness: 0.044
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)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo|19EDO]] up to just ones. [[171edo|171EDO]] is a good tuning for either the 5 or 7 limits, and [[494edo|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|665EDO]] for a tuning.


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


Subgroup: 2.3.5.7
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. [[1106edo]] is an strong tuning.  


[[Comma list]]: 4375/4374, 703125/702464
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|19 0 14 -37}}, {{val|0 1 1 3}}]
[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}


{{Multival|legend=1|19 19 57 -14 37 79}}
{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}


Mapping generators: ~28/27, ~3
: mapping generators: ~332150625/234881024, ~1125/1024


[[POTE generator]]: ~3/2 = 701.880
[[Optimal tuning]]s:
* [[CTE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7475
* [[error map]]: {{val| 0.0000 +0.0253 -0.0514 -0.0133 }}
* [[CWE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7474
* error map: {{val| 0.0000 +0.0244 -0.0508 -0.0218 }}


{{Val list|legend=1| 19, 152, 171, 665, 836, 1007, 2185 }}
{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}


[[Badness]]: 0.010954
[[Badness]] (Smith): 0.0394


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 540/539, 4375/4374, 16384/16335
Comma list: 3025/3024, 4375/4374, 2791309312/2790703125
 
Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}


Mapping: [{{val|19 0 14 -37 126}}, {{val|0 1 1 3 -2}}]
Optimal tunings:
* CTE: ~99/70 = 162.7485, ~1125/1024 = 162.7485
* CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481


POTE generator: ~3/2 = 702.360
{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}


Vals: {{Val list| 19, 152, 323e, 475de, 627de }}
Badness (Smith): 0.0170


Badness: 0.043734
== Orga ==
[[Subgroup]]: 2.3.5.7


==== 13-limit ====
[[Comma list]]: 4375/4374, 54975581388800/54936068900769
Subgroup: 2.3.5.7.11.13


Comma list: 540/539, 625/624, 729/728, 2205/2197
{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }}


Mapping: [{{val|19 0 14 -37 126 -20}}, {{val|0 1 1 3 -2 3}}]
: mapping generators: ~7411887/5242880, ~1310720/1058841


POTE generator: ~3/2 = 702.212
[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104


Vals: {{Val list| 19, 152f, 323e }}
{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}


Badness: 0.033545
[[Badness]]: 0.040236


=== Hemienneadecal ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4375/4374, 234375/234256
Comma list: 3025/3024, 4375/4374, 5767168/5764801


Mapping: [{{val|38 0 28 -74 11}}, {{val|0 1 1 3 2}}]
Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }}


POTE generator: ~3/2 = 701.881
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103


Vals: {{Val list| 152, 342, 494, 836, 1178, 2014 }}
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836, 1106 }}


Badness: 0.009985
Badness: 0.016188


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360


Mapping: [{{val|38 0 28 -74 11 502}}, {{val|0 1 1 3 2 -6}}]
Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }}


POTE generator: ~3/2 = 701.986
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103


Vals: {{Val list| 152, 342, 494, 836 }}
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836f, 1106f }}


Badness: 0.030391
Badness: 0.021762
 
== Seniority ==
{{See also| Very high accuracy temperaments #Senior }}


== Deca ==
Aside from the ragisma, the seniority temperament (26 &amp; 145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ({{monzo| -17 62 -35 }}, quadla-sepquingu) is tempered out.
Deca temperament has a period of 1/10 octave and tempers out the [[15/14ths equal temperament #Linus temperaments|linus comma]], {{monzo|11 -10 -10 10}} and {{monzo|12 -3 -14 9}} = 165288374272/164794921875 (satritrizo-asepbigu).


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 165288374272/164794921875
[[Comma list]]: 4375/4374, 201768035/201326592


[[Mapping]]: [{{val|10 4 9 2}}, {{val|0 5 6 11}}]
{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}


{{Multival|legend=1|50 60 110 -21 34 87}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3087/2560 = 322.804


[[POTE generator]]: ~6/5 = 315.577
{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}


{{Val list|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}
[[Badness]]: 0.044877


[[Badness]]: 0.080637
=== Senator ===
The senator temperament (26 &amp; 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It can be extended to the 13- and 17-limit immediately, by adding 364/363 and 595/594 to the comma list in this order.


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4375/4374, 422576/421875
Comma list: 441/440, 4375/4374, 65536/65219


Mapping: [{{val|10 4 9 2 18}}, {{val|0 5 6 11 7}}]
Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }}


POTE generator: ~6/5 = 315.582
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793


Vals: {{Val list| 80, 190, 270, 1000, 1270 }}
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316e, 487ee }}


Badness: 0.024329
Badness: 0.092238


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
Comma list: 364/363, 441/440, 2200/2197, 4375/4374


Mapping: [{{val|10 4 9 2 18 37}}, {{val|0 5 6 11 7 0}}]
Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }}


POTE generator: ~6/5 = 315.602
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793


Vals: {{Val list| 80, 190, 270, 730, 1000 }}
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}


Badness: 0.016810
Badness: 0.044662


== Sfourth ==
==== 17-limit ====
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 4375/4374, 64827/64000
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197


[[Mapping]]: [{{val|1 2 3 3}}, {{val|0 -19 -31 -9}}]
Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }}


{{Multival|legend=1|19 31 9 5 -39 -66}}
Optimal tuning (POTE): ~77/64 = 322.793


[[POTE generator]]: ~49/48 = 26.287
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}


{{Val list|legend=1| 45, 46, 91, 137d }}
Badness: 0.026562
 
[[Badness]]: 0.123291


=== 11-limit ===
== Monzismic ==
Subgroup: 2.3.5.7.11
: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Monzismic]].  


Comma list: 121/120, 441/440, 4375/4374
The monzismic temperament (53 &amp; 612) 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]].


Mapping: [{{val|1 2 3 3 4}}, {{val|0 -19 -31 -9 -25}}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~49/48 = 26.286
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}


Vals: {{Val list| 45e, 46, 91e, 137de }}
{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}


Badness: 0.054098
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207


==== 13-limit ====
{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 169/168, 325/324, 441/440
[[Badness]]: 0.046569


Mapping: [{{val|1 2 3 3 4 4}}, {{val|0 -19 -31 -9 -25 -14}}]
=== Monzism ===
 
POTE generator: ~49/48 = 26.310
 
Vals: {{Val list| 45ef, 46, 91ef, 137def }}
 
Badness: 0.033067
 
=== Sfour ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 2401/2376, 4375/4374
Comma list: 4375/4374, 41503/41472, 184549376/184528125


Mapping: [{{val|1 2 3 3 3}}, {{val|0 -19 -31 -9 21}}]
Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}


POTE generator: ~49/48 = 26.246
Optimal tuning (POTE): ~231/200 = 249.0193


Vals: {{Val list| 45, 46, 91, 137d }}
{{Optimal ET sequence|legend=1| 53, 559, 612 }}


Badness: 0.076567
Badness: 0.057083


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 196/195, 364/363, 385/384, 4375/4374
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
 
Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}


Mapping: [{{val|1 2 3 3 3 3}}, {{val|0 -19 -31 -9 21 32}}]
Optimal tuning (POTE): ~231/200 = 249.0199


POTE generator: ~49/48 = 26.239
{{Optimal ET sequence|legend=1| 53, 559, 612 }}


Vals: {{Val list| 45, 46, 91, 137d }}
Badness: 0.053780


Badness: 0.051893
== Semidimfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].''


== Abigail ==
The semidimfourth temperament is featured by a semi-diminished 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, 2147483648/2144153025
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|2 7 13 -1}}, {{val|0 -11 -24 19}}]
[[Comma list]]: 4375/4374, 235298/234375


{{Multival|legend=1|22 48 -38 25 -122 -223}}
[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}


[[POTE generator]]: ~6912/6125 = 208.899
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456


{{Val list|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}
{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}


[[Badness]]: 0.037000
[[Badness]]: 0.055249


=== 11-limit ===
=== Neusec ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4375/4374, 131072/130977
Comma list: 3025/3024, 4375/4374, 235298/234375


Mapping: [{{val|2 7 13 -1 1}}, {{val|0 -11 -24 19 17}}]
Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }}


POTE generator: ~1155/1024 = 208.901
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547


Vals: {{Val list| 46, 132, 178, 224, 270, 494, 764 }}
{{Optimal ET sequence|legend=1| 8d, 190, 388 }}


Badness: 0.012860
Badness: 0.059127


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374


Mapping: [{{val|2 7 13 -1 1 -2}}, {{val|0 -11 -24 19 17 27}}]
Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }}


POTE generator: ~44/39 = 208.903
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545


Vals: {{Val list| 46, 178, 224, 270, 494, 764, 1258 }}
{{Optimal ET sequence|legend=1| 8d, 190, 198, 388 }}


Badness: 0.008856
Badness: 0.030941


== Semidimi ==
== Acrokleismic ==
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


Subgroup: 2.3.5
[[Comma list]]: 4375/4374, 2202927104/2197265625


[[Comma]]: {{monzo|-12 -73 55}}
{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }}


[[Mapping]]: [{{val|1 36 48}}, {{val|0 -55 -73}}]
: mapping generators: ~2, ~6/5


[[POTE generator]]: ~162/125 = 449.1269
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557


{{Val list|legend=1| 8, 163, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}


[[Badness]]: 0.754866
[[Badness]]: 0.056184


=== 7-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 4375/4374, 3955078125/3954653486
Comma list: 4375/4374, 41503/41472, 172032/171875


[[Mapping]]: [{{val|1 36 48 61}}, {{val|0 -55 -73 -93}}]
Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }}


{{Multival|legend=1|55 73 93 -12 -7 11}}
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558


[[POTE generator]]: ~35/27 = 449.1270
{{Optimal ET sequence|legend=1| 19, 251, 270, 829, 1099, 1369, 1639 }}


{{Val list|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
Badness: 0.036878


[[Badness]]: 0.015075
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


== Brahmagupta ==
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo|47 -7 -7 -7}} = 140737488355328 / 140710042265625.


Subgroup: 2.3.5.7
Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }}


[[Comma list]]: 4375/4374, 70368744177664/70338939985125
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557


[[Mapping]]: [{{val|7 2 -8 53}}, {{val|0 3 8 -11}}]
{{Optimal ET sequence|legend=1| 19, 251, 270 }}


{{Multival|legend=1|21 56 -77 40 -181 -336}}
Badness: 0.026818


[[POTE generator]]: ~27/20 = 519.716
=== Counteracro ===
 
{{Val list|legend=1| 7, 217, 224, 441, 1106, 1547 }}
 
[[Badness]]: 0.029122
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 4000/3993, 4375/4374, 131072/130977
Comma list: 4375/4374, 5632/5625, 117649/117612


Mapping: [{{val|7 2 -8 53 3}}, {{val|0 3 8 -11 7}}]
Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }}


POTE generator: ~27/20 = 519.704
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.553


Vals: {{Val list| 7, 217, 224, 441, 665, 1771ee }}
{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}


Badness: 0.052190
Badness: 0.042572


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374


Mapping: [{{val|7 2 -8 53 3 35}}, {{val|0 3 8 -11 7 -3}}]
Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }}


POTE generator: ~27/20 = 519.706
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.554


Vals: {{Val list| 7, 217, 224, 441, 665, 1771eef }}
{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}


Badness: 0.023132
Badness: 0.026028


== Quasithird ==
== Quasithird ==
The '''quasithird''' temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[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 to temper out the [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}.
The quasithird temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[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 to temper out the [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}.


[[Subgroup]]: 2.3.5


Subgroup: 2.3.5
[[Comma list]]: {{monzo| 55 -64 20 }}


[[Comma]]: {{monzo|55 -64 20}}
{{Mapping|legend=1| 4 0 -11 | 0 5 16 }}


[[Mapping]]: [{{val|4 0 -11}}, {{val|0 5 16}}]
: mapping generators: ~51200000/43046721, ~1594323/1280000


[[POTE generator]]: ~1594323/1280000 = 380.395
[[Optimal tuning]] ([[POTE]]): ~51200000/43046721, ~1594323/1280000 = 380.395


{{Val list|legend=1| 60, 164, 224, 388, 612, 836, 1000, 1448, 1612, 2224, 2836 }}
{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }}


[[Badness]]: 0.099519
[[Badness]]: 0.099519


=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 1153470752371588581/1152921504606846976


[[Mapping]]: [{{val|4 0 -11 48}}, {{val|0 5 16 -29}}]
[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}


[[Wedgie]]: {{multival|20 64 -116 55 -240 -449}}
{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}


[[POTE generator]]: ~5103/4096 = 380.388  
[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 1\4, ~5103/4096 = 380.388  


{{Val list|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}


[[Badness]]: 0.061813
[[Badness]]: 0.061813
Line 725: Line 678:
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296


Mapping: [{{val|4 0 -11 48 43}}, {{val|0 5 16 -29 -23}}]
Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}


POTE generator: ~22/21 = 80.387 (or ~5103/4096 = 380.387)
Optimal tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387)


Vals: {{Val list| 60d, 164, 224, 388, 612, 836, 1448 }}
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448 }}


Badness: 0.021125
Badness: 0.021125
Line 736: Line 689:
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 2200/2197, 3025/3024, 4375/4374, 468512/468195
Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374


Mapping: [{{val|4 0 -11 48 43 11}}, {{val|0 5 16 -29 -23 3}}]
Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}


POTE generator: ~22/21 = 80.385 (or ~5103/4096 = 380.385)
Optimal tuning (POTE): ~81/65 = 380.385 (or ~22/21 = 80.385)


Vals: {{Val list| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}


Badness: 0.029501
Badness: 0.029501


== Semidimfourth ==
== Deca ==
The '''semidimifourth''' temperament is featured by a semi-diminished 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.
: ''For 5-limit version of this temperament, see [[10th-octave temperaments #Neon]].''


Subgroup: 2.3.5
Deca temperament has a period of 1/10 octave and tempers out the [[linus comma]], {{monzo| 11 -10 -10 10 }}, neon comma {{monzo| 21 60 -50 }} and {{monzo| 12 -3 -14 9 }} = 165288374272/164794921875 (satritrizo-asepbigu).


[[Comma]]: {{monzo|7 41 -31}}
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 21 28}}, {{val|0 -31 -41}}]
[[Comma list]]: 4375/4374, 165288374272/164794921875


[[POTE generator]]: ~162/125 = 448.449
{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}


{{Val list|legend=1| 8, 91, 99, 190, 289, 388, 677, 3674, 4351, 5028, 5705, 6382, 13441c, 19823bcc }}
: mapping generators: ~15/14, ~6/5


[[Badness]]: 0.233376
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577


=== 7-limit ===
{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 235298/234375
[[Badness]]: 0.080637


[[Mapping]]: [{{val|1 21 28 36}}, {{val|0 -31 -41 -53}}]
Badness (Sintel): 2.041


[[Wedgie]]: {{multival|31 41 53 -7 -3 8}}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~35/27 = 448.456
Comma list: 3025/3024, 4375/4374, 391314/390625


{{Val list|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}
Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}


[[Badness]]: 0.055249
Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.582


=== Neusec ===
{{Optimal ET sequence|legend=1| 80, 190, 270, 1000, 1270, 1540e, 1810e }}
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4375/4374, 235298/234375
Badness: 0.024329


Mapping: [{{val|2 11 15 19 15}}, {{val|0 -31 -41 -53 -32}}]
Badness (Sintel): 0.804


POTE generator: ~12/11 = 151.547
=== 13-limit ===
 
Vals: {{Val list| 8d, 190, 388 }}
 
Badness: 0.059127
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374


Mapping: [{{val|2 11 15 19 15 17}}, {{val|0 -31 -41 -53 -32 -38}}]
Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}


POTE generator: ~12/11 = 151.545
Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.602 (~40/39 = 44.398)


Vals: {{Val list| 8d, 190, 198, 388 }}
{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}


Badness: 0.030941
Badness: 0.016810


== Acrokleismic ==
Badness (Sintel): 0.695
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 2202927104/2197265625
=== no-17's 19-limit ===
Subgroup: 2.3.5.7.11.13.19


[[Mapping]]: [{{val|1 10 11 27}}, {{val|0 -32 -33 -92}}]
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374, 1521/1520


[[Wedgie]]: {{multival|32 33 92 -22 56 121}}
Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}


[[POTE generator]]: ~6/5 = 315.557
Optimal tuning (CTE): ~15/14 = 1\10, ~6/5 = 315.581 (~39/38 = 44.419)


{{Val list|legend=1| 19, 251, 270 }}
{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}


[[Badness]]: 0.056184
Badness (Sintel): 0.556


=== 11-limit ===
== Keenanose ==
Subgroup: 2.3.5.7.11
Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.


Comma list: 4375/4374, 41503/41472, 172032/171875
[[Subgroup]]: 2.3.5.7


Mapping: [{{val|1 10 11 27 -16}}, {{val|0 -32 -33 -92 74}}]
[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}


POTE generator: ~6/5 = 315.558
{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}


Vals: {{Val list| 19, 251, 270, 829, 1099, 1369, 1639 }}
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}


Badness: 0.036878
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 21 3 1 -10 }} = 4.4465


==== 13-limit ====
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}
Subgroup: 2.3.5.7.11.13


Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
[[Badness]]: 0.0858


Mapping: [{{val|1 10 11 27 -16 25}}, {{val|0 -32 -33 -92 74 -81}}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~6/5 = 315.557
Comma list: 4375/4374, 117649/117612, 67110351/67108864


Vals: {{Val list| 19, 251, 270 }}
Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}


Badness: 0.026818
Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4465


=== Counteracro ===
{{Optimal ET sequence|legend=1| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}
Subgroup: 2.3.5.7.11


Comma list: 4375/4374, 5632/5625, 117649/117612
Badness: 0.0308


Mapping: [{{val|1 10 11 27 55}}, {{val|0 -32 -33 -92 -196}}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


POTE generator: ~6/5 = 315.553
Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612


Vals: {{Val list| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}


Badness: 0.042572
Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4466


==== 13-limit ====
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 4048 }}
Subgroup: 2.3.5.7.11.13


Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
Badness: 0.0213


Mapping: [{{val|1 10 11 27 55 25}}, {{val|0 -32 -33 -92 -196 -81}}]
== Aluminium ==
Aluminium is named after the 13th element, and tempers out the {{monzo| 92 -39 -13 }} comma which sets [[135/128]] interval to be equal to 1/13th of the octave.


POTE generator: ~6/5 = 315.554
[[Subgroup]]: 2.3.5


Vals: {{Val list| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}
[[Comma list]]: {{monzo| 92 -39 -13 }}


Badness: 0.026028
[[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}


== Seniority ==
: mapping generators: ~135/128, ~3
{{see also|Very high accuracy temperaments #Senior}}


Aside from the ragisma, the seniority temperament (26&amp;145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ({{monzo|-17 62 -35}}, quadla-sepquingu) is tempered out.
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}


[[Comma list]]: 4375/4374, 201768035/201326592
[[Badness]]: 0.123


[[Mapping]]: [{{val|1 11 19 2}}, {{val|0 -35 -62 3}}]
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


[[Wedgie]]: {{multival|35 62 -3 17 -103 -181}}
[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}


[[POTE generator]]: ~3087/2560 = 322.804
[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}


{{Val list|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 702.0024


[[Badness]]: 0.044877
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}


=== Senator ===
[[Badness]]: 0.126
The senator temperament (26&amp;145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It can be extended to the 13- and 17-limit immediately, by adding 364/363 and 595/594 to the comma list in this order.


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 4375/4374, 65536/65219
Comma list: 4375/4374, 234375/234256, 2097152/2096325


Mapping: [{{val|1 11 19 2 4}}, {{val|0 -35 -62 3 -2}}]
Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}


POTE generator: ~77/64 = 322.793
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042


Vals: {{Val list| 26, 119c, 145, 171, 316e, 487ee }}
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 3588e, 5135e }}


Badness: 0.092238
Badness: 0.0421


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 364/363, 441/440, 2200/2197, 4375/4374
Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078


Mapping: [{{val|1 11 19 2 4 15}}, {{val|0 -35 -62 3 -2 -42}}]
Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}


POTE generator: ~77/64 = 322.793
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099


Vals: {{Val list| 26, 119c, 145, 171, 316ef, 487eef }}
{{Optimal ET sequence|legend=1| 494, 1547, 2041, 4576def }}


Badness: 0.044662
Badness: 0.0286


==== 17-limit ====
== Countritonic ==
Subgroup: 2.3.5.7.11.13.17
: ''For the 5-limit version of this temperament, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197
Countritonic (''co-un-tritonic'') can be described as the 53 & 422 temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit.


Mapping: [{{val|1 11 19 2 4 15 17}}, {{val|0 -35 -62 3 -2 -42 -48}}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~77/64 = 322.793
[[Comma list]]: 4375/4374, 68719476736/68356598625


Vals: {{Val list| 26, 119c, 145, 171, 316ef, 487eef }}
{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }}


Badness: 0.026562
: mapping generators: ~2, ~45927/32768


== Orga ==
[[Optimal tuning]] (CTE): ~2 = 1\1, ~45927/32768 = 588.6216
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 54975581388800/54936068900769
{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}
 
[[Mapping]]: [{{val|2 21 36 5}}, {{val|0 -29 -51 1}}]
 
[[Wedgie]]: {{multival|58 102 -2 27 -166 -291}}
 
[[POTE generator]]: ~8/7 = 231.104
 
{{Val list|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}


[[Badness]]: 0.040236
[[Badness]]: 0.133


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4375/4374, 5767168/5764801
Comma list: 4375/4374, 5632/5625, 2621440/2614689


Mapping: [{{val|2 21 36 5 2}}, {{val|0 -29 -51 1 8}}]
Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 }}


POTE generator: ~8/7 = 231.103
Optimal tuning (CTE): ~2 = 1\1, ~539/384 = 588.6258


Vals: {{Val list| 26, 244, 270, 566, 836, 1106 }}
{{Optimal ET sequence|legend=1| 53, 316e, 369, 422, 791e, 1213cde }}


Badness: 0.016188
Badness: 0.0707


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360
Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625


Mapping: [{{val|2 21 36 5 2 24}}, {{val|0 -29 -51 1 8 -27}}]
Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 -74 }}


POTE generator: ~8/7 = 231.103
Optimal tuning (CTE): ~2 = 1\1, ~128/91 = 588.6277


Vals: {{Val list| 26, 244, 270, 566, 836f, 1106f }}
{{Optimal ET sequence|legend=1| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}


Badness: 0.021762
Badness: 0.0366


== Quatracot ==
== Quatracot ==
{{See also| Stratosphere }}
{{See also| Stratosphere }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 1483154296875/1473173782528
[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}


[[Mapping]]: [{{val| 2 7 7 23 }}, {{val| 0 -13 -8 -59 }}]
{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }}


{{Multival|legend=1| 26 16 118 -35 114 229 }}
: mapping generators: ~2278125/1605632, ~448/405


[[POTE generator]]: ~448/405 = 176.805
[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 1\2, ~448/405 = 176.805


{{Val list|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}


[[Badness]]: 0.175982
[[Badness]]: 0.175982
Line 992: Line 929:
Comma list: 3025/3024, 4375/4374, 1265625/1261568
Comma list: 3025/3024, 4375/4374, 1265625/1261568


Mapping: [{{val| 2 7 7 23 19 }}, {{val| 0 -13 -8 -59 -41 }}]
Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }}


POTE generator: ~448/405 = 176.806
Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 176.806


Vals: {{Val list| 190, 224, 414, 638, 1052c }}
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c }}


Badness: 0.041043
Badness: 0.041043
Line 1,005: Line 942:
Comma list: 625/624, 729/728, 1575/1573, 2200/2197
Comma list: 625/624, 729/728, 1575/1573, 2200/2197


Mapping: [{{val| 2 7 7 23 19 13 }}, {{val| 0 -13 -8 -59 -41 -19 }}]
Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }}


POTE generator: ~195/176 = 176.804
Optimal tuning (POTE): ~99/70 = 1\2, ~195/176 = 176.804


Vals: {{Val list| 190, 224, 414, 638, 1690bcc, 2328bccde }}
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1690bcc, 2328bccde }}


Badness: 0.022643
Badness: 0.022643


== Octoid ==
== Moulin ==
The '''octoid''' temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.
Moulin has a generator of 22/13, and it is named after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". It can be described as the 494 & 1619 temperament.


[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}


[[Comma list]]: 4375/4374, 16875/16807
{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }}


[[Mapping]]: [{{val|8 1 3 3}}, {{val|0 3 4 5}}]
: mapping generators: ~2, ~6422528/3796875


[[Wedgie]]: {{multival|24 32 40 -5 -4 3}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6422528/3796875 = 910.9323


Mapping generators: ~49/45, ~7/5
{{Optimal ET sequence|legend=1| 494, 1125, 1619 }}


[[POTE generator]]: ~7/5 = 583.940
[[Badness]]: 0.234


[[Tuning ranges]]:
=== 11-limit ===
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
Subgroup: 2.3.5.7.11
* 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]
Comma list: 4375/4374, 759375/758912, 100663296/100656875
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
* 7-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 585.084]


{{Val list|legend=1| 8d, 72, 152, 224 }}
Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }}


[[Badness]]: 0.042670
Optimal tuning (CTE): ~2 = 1\1, ~1024/605 = 910.9323


Scales: [[Octoid72]], [[Octoid80]]
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}


=== 11-limit ===
Badness: 0.0678
Subgroup: 2.3.5.7.11


Comma list: 540/539, 1375/1372, 4000/3993
=== 13-limit ===
Since 11/8 is within 23 generators, the 25 tone MOS (4L 21s) of this temperament contains the 8:11:13 triad.


Mapping: [{{val|8 1 3 3 16}}, {{val|0 3 4 5 3}}]
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 583.962
Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078


Tuning ranges:  
Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }}
* 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]
* 11-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 585.084]


Vals: {{Val list| 72, 152, 224 }}
Optimal tuning (CTE): ~2 = 1\1, ~22/13 = 910.9323


Badness: 0.014097
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}


Scales: [[Octoid72]], [[Octoid80]]
Badness: 0.0271


=== 13-limit ===
== Palladium ==
Subgroup: 2.3.5.7.11.13
: ''For the 5-limit version of this temperament, see [[46th-octave temperaments]]''.


Comma list: 540/539, 625/624, 729/728, 1375/1372
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, {{monzo| -39 92 -46 }}, by which 46 minortones (10/9) fall short of seven octaves. This temperament can be described as 46 &amp; 414 temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.


Mapping: [{{val|8 1 3 3 16 -21}}, {{val|0 3 4 5 3 13}}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~7/5 = 583.905
[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}


Vals: {{Val list| 72, 152f, 224 }}
{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}


Badness: 0.015274
: mapping generators: ~83349/81920, ~3


Scales: [[Octoid72]], [[Octoid80]]
[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 1\46, ~3/2 = 701.6074


; Music
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }}
* [http://www.archive.org/details/Dreyfus http://www.archive.org/details/Dreyfus] [http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play]


==== 17-limit ====
[[Badness]]: 0.308505
Subgroup: 2.3.5.7.11.13.17


Comma list: 375/374, 540/539, 625/624, 715/714, 729/728
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val|8 1 3 3 16 -21 -14}}, {{val|0 3 4 5 3 13 12}}]
Comma list: 3025/3024, 4375/4374, 134775333/134217728


POTE generator: ~7/5 = 583.842
Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}


Vals: {{Val list| 72, 152fg, 224, 296, 520g }}
Optimal tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951


Badness: 0.014304
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de }}


Scales: [[Octoid72]], [[Octoid80]]
Badness: 0.073783


==== 19-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13


Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364


Mapping: [{{val|8 1 3 3 16 -21 -14 34}}, {{val|0 3 4 5 3 13 12 0}}]
Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}


POTE generator: ~7/5 = 583.932
Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6419


Vals: {{Val list| 72, 152fg, 224 }}
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334de }}


Badness: 0.016036
Badness: 0.040751


Scales: [[Octoid72]], [[Octoid80]]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


=== Octopus ===
Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 325/324, 364/363, 540/539
Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}


Mapping: [{{val|8 1 3 3 16 14}}, {{val|0 3 4 5 3 4}}]
Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425


POTE generator: ~7/5 = 583.892
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334deg }}


Vals: {{Val list| 72, 152, 224f }}
Badness: 0.022441


Badness: 0.021679
== Oviminor ==
{{See also| Syntonic–kleismic equivalence continuum }}


Scales: [[Octoid72]], [[Octoid80]]
Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.


==== 17-limit ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17


Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}


Mapping: [{{val|8 1 3 3 16 14 21}}, {{val|0 3 4 5 3 4 3}}]
{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }}


POTE generator: ~7/5 = 583.811
: mapping generators: ~2, ~6/5


Vals: {{Val list| 72, 152, 224fg, 296ffg }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 315.7501


Badness: 0.015614
{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}


Scales: [[Octoid72]], [[Octoid80]]
[[Badness]]: 0.582


==== 19-limit ====
== Octoid ==
Subgroup: 2.3.5.7.11.13.17.19
''For the 5-limit temperament, see [[8th-octave temperaments#Octoid (5-limit)]].''


Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.


Mapping: [{{val|8 1 3 3 16 14 21 34}}, {{val|0 3 4 5 3 4 3 0}}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~7/5 = 584.064
[[Comma list]]: 4375/4374, 16875/16807


Vals: {{Val list| 72, 152, 224fg, 376ffgh }}
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}


Badness: 0.016321
: mapping generators: ~49/45, ~7/5


Scales: [[Octoid72]], [[Octoid80]]
[[Optimal tuning]] ([[POTE]]): ~49/45 = 1\8, ~7/5 = 583.940


=== Hexadecoid ===
[[Tuning ranges]]:
Hexadecoid (80&amp;144) has a period of 1/16 octave and tempers out 4225/4224.
* 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]


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}


Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
[[Badness]]: 0.042670


Mapping: [{{val|16 26 38 46 56 59}}, {{val|0 -3 -4 -5 -3 1}}]
Scales: [[octoid72]], [[octoid80]]


POTE generator: ~13/8 = 841.015
=== 11-limit ===
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 minimaxing the 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, if one wants to use 80edo as the tuning, one must use octopus — not octoid — as 80edo doesn't temper 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.


Vals: {{Val list| 80, 144, 224 }}
Subgroup: 2.3.5.7.11


Badness: 0.030818
Comma list: 540/539, 1375/1372, 4000/3993


==== 17-limit ====
Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.962


Mapping: [{{val|16 26 38 46 56 59 65}}, {{val|0 -3 -4 -5 -3 1 2}}]
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]


POTE generator: ~13/8 = 840.932
{{Optimal ET sequence|legend=1| 72, 152, 224 }}


Vals: {{Val list| 80, 144, 224, 528dg }}
Badness: 0.014097


Badness: 0.028611
Scales: [[octoid72]], [[octoid80]]


==== 19-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13


Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
Comma list: 540/539, 625/624, 729/728, 1375/1372


Mapping: [{{val|16 26 38 46 56 59 65 68}}, {{val|0 -3 -4 -5 -3 1 2 0}}]
Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}


POTE generator: ~13/8 = 840.896
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.905


Vals: {{Val list| 80, 144, 224, 304dh, 528dghh }}
{{Optimal ET sequence|legend=1| 72, 152f, 224 }}


Badness: 0.023731
Badness: 0.015274


== Amity ==
Scales: [[octoid72]], [[octoid80]]
{{main| Amity }}
{{see also| Amity family #Amity }}


The generator for amity temperament is the acute minor third, which means the 6/5 just 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. [[99edo|99EDO]] is a good tuning for amity, with generator 28\99, and MOS of 11, 18, 25, 32, 39, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.
; Music
* ''Dreyfus'' (archived 2010) by [[Gene Ward Smith]] – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – octoid[72] in 224edo tuning


In the 5-limit amity is a genuine microtemperament, with 58\205 being a possible tuning. Another good choice is (64/5)<sup>1/13</sup>, which gives pure major thirds.
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


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


Subgroup: 2.3.5.7
Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}


[[Comma list]]: 4375/4374, 5120/5103
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.842


[[Mapping]]: [{{val| 1 3 6 -2 }}, {{val| 0 -5 -13 17 }}]
{{Optimal ET sequence|legend=1| 72, 152fg, 224, 296, 520g }}


{{Multival|legend=1| 5 13 -17 9 -41 -76 }}
Badness: 0.014304


[[POTE generator]]: ~128/105 = 339.432
Scales: [[octoid72]], [[octoid80]]


{{Val list|legend=1| 7, 32c, 39, 46, 53, 99, 251, 350, 601cd, 951bcdd }}
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


[[Badness]]: 0.023649
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714


=== 11-limit ===
Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}
Subgroup: 2.3.5.7.11


Comma list: 540/539, 4375/4374, 5120/5103
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.932


Mapping: [{{val| 1 3 6 -2 21 }}, {{val| 0 -5 -13 17 -62 }}]
{{Optimal ET sequence|legend=1| 72, 152fg, 224 }}


POTE generator: ~128/105 = 339.464
Badness: 0.016036


Vals: {{Val list| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}
Scales: [[octoid72]], [[octoid80]]


Badness: 0.031506
==== 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{{cent}}.


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 540/539, 625/624, 847/845
Comma list: 169/168, 325/324, 364/363, 540/539
 
Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
 
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.892
 
{{Optimal ET sequence|legend=1| 72, 152, 224f }}
 
Badness: 0.021679
 
Scales: [[octoid72]], [[octoid80]]
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 221/220, 289/288, 325/324, 540/539


Mapping: [{{val| 1 3 6 -2 21 17 }}, {{val| 0 -5 -13 17 -62 -47 }}]
Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}


POTE generator: ~128/105 = 339.481
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.811


Vals: {{Val list| 46ef, 53, 99ef, 152f }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=1| 72, 152, 224fg, 296ffg }}


<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
Badness: 0.015614


Badness: 0.028008
Scales: [[Octoid72]], [[Octoid80]]


=== Hitchcock ===
===== 19-limit =====
{{see also|Amity family #Hitchcock}}
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}
 
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 584.064


Comma list: 121/120, 176/175, 2200/2187
{{Optimal ET sequence|legend=1| 72, 152, 224fg, 376ffgh }}


Mapping: [{{val| 1 3 6 -2 6 }}, {{val| 0 -5 -13 17 -9 }}]
Badness: 0.016321


POTE generator: ~11/9 = 339.390
Scales: [[Octoid72]], [[Octoid80]]


Vals: {{Val list| 7, 39, 46, 53, 99 }}
==== Hexadecoid ====
{{ See also | 16th-octave temperaments }}


Badness: 0.035187
Hexadecoid (80 &amp; 144) has a period of 1/16 octave and tempers out 4225/4224.


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 169/168, 176/175, 325/324
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
 
Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}


Mapping: [{{val| 1 3 6 -2 6 2 }}, {{val| 0 -5 -13 17 -9 6 }}]
: mapping generators: ~448/429, ~7/5


POTE generator: ~11/9 = 339.419
Optimal tuning (POTE): ~448/429 = 1\16, ~13/8 = 841.015


Vals: {{Val list| 7, 39, 46, 53, 99 }}
{{Optimal ET sequence|legend=1| 80, 144, 224 }}


Badness: 0.022448
Badness: 0.030818


==== 17-limit ====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 121/120, 154/153, 169/168, 176/175, 273/272
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224


Mapping: [{{val| 1 3 6 -2 6 2 -1 }}, {{val| 0 -5 -13 17 -9 6 18 }}]
Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}


POTE generator: ~11/9 = 339.366
Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932


Vals: {{Val list| 7, 39, 46, 53, 99 }}
{{Optimal ET sequence|legend=1| 80, 144, 224, 528dg }}


Badness: 0.019395
Badness: 0.028611


=== Hemiamity ===
===== 19-limit =====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 3025/3024, 4375/4374, 5120/5103
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444


Mapping: [{{val| 2 1 -1 13 13 }}, {{val| 0 5 13 -17 -14 }}]
Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }}


POTE generator: ~64/55 = 339.439
Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896


Vals: {{Val list| 14cde, 46, 106, 152, 350, 502d }}
{{Optimal ET sequence|legend=1| 80, 144, 224, 304dh, 528dghh }}


Badness: 0.031307
Badness: 0.023731


== Parakleismic ==
== Parakleismic ==
{{main| Parakleismic }}
{{Main| Parakleismic }}


In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo|8 14 -13}}, with the [[118edo|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. 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 {{multival|13 14 35 -8 19 42}} and adding 3136/3125 and 4375/4374, and the 11-limit wedgie {{multival|13 14 35 -36 -8 19 -102 42 -132 -222}} adding 385/384. For the 7-limit [[99edo|99EDO]] may be preferred, but in the 11-limit it is best to stick with 118.
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 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. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, and 11-limit adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.


[[Subgroup]]: 2.3.5


Subgroup: 2.3.5
[[Comma list]]: 1224440064/1220703125


[[Comma list]]: 1224440064/1220703125
{{Mapping|legend=1| 1 5 6 | 0 -13 -14 }}


[[Mapping]]: [{{val|1 5 6}}, {{val|0 -13 -14}}]
: mapping generators: ~2, ~6/5


[[POTE generator]]: ~6/5 = 315.240
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.240


{{Val list|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}
{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}


[[Badness]]: 0.043279
[[Badness]]: 0.043279


=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 4375/4374
[[Comma list]]: 3136/3125, 4375/4374


[[Mapping]]: [{{val|1 5 6 12}}, {{val|0 -13 -14 -35}}]
{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }}


[[Wedgie]]: {{multival|13 14 35 -8 19 42}}


[[POTE generator]]: ~6/5 = 315.181
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.181


{{Val list|legend=1| 19, 80, 99, 217, 316, 415 }}
{{Optimal ET sequence|legend=1| 19, 80, 99, 217, 316, 415 }}


[[Badness]]: 0.027431
[[Badness]]: 0.027431
Line 1,340: Line 1,298:
Comma list: 385/384, 3136/3125, 4375/4374
Comma list: 385/384, 3136/3125, 4375/4374


Mapping: [{{val|1 5 6 12 -6}}, {{val|0 -13 -14 -35 36}}]
Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }}


POTE generator: ~6/5 = 315.251
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.251


Vals: {{Val list| 19, 99, 118 }}
{{Optimal ET sequence|legend=1| 19, 99, 118 }}


Badness: 0.049711
Badness: 0.049711


=== Paralytic ===
=== Paralytic ===
The ''paralytic'' temperament (118&amp;217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118&amp;217 tempers out 1001/1000, 1575/1573, and 3584/3575.
The ''paralytic'' temperament (118&amp;217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 &amp; 217 tempers out 1001/1000, 1575/1573, and 3584/3575.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,355: Line 1,313:
Comma list: 441/440, 3136/3125, 4375/4374
Comma list: 441/440, 3136/3125, 4375/4374


Mapping: [{{val|1 5 6 12 25}}, {{val|0 -13 -14 -35 -82}}]
Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }}


POTE generator: ~6/5 = 315.220
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.220


Vals: {{Val list| 19e, 99e, 118, 217, 335, 552d, 887dd }}
{{Optimal ET sequence|legend=1| 19e, 99e, 118, 217, 335, 552d, 887dd }}


Badness: 0.036027
Badness: 0.036027
Line 1,368: Line 1,326:
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374


Mapping: [{{val|1 5 6 12 25 -16}}, {{val|0 -13 -14 -35 -82 75}}]
Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }}


POTE generator: ~6/5 = 315.214
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.214


Vals: {{Val list| 99e, 118, 217, 552d, 769de }}
{{Optimal ET sequence|legend=1| 99e, 118, 217, 552d, 769de }}


Badness: 0.044710
Badness: 0.044710


==== Paraklein ====
==== Paraklein ====
The ''paraklein'' temperament (19e&amp;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]].
The ''paraklein'' temperament (19e &amp; 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
Subgroup: 2.3.5.7.11.13
Line 1,383: Line 1,341:
Comma list: 196/195, 352/351, 625/624, 729/728
Comma list: 196/195, 352/351, 625/624, 729/728


Mapping: [{{val|1 5 6 12 25 15}}, {{val|0 -13 -14 -35 -82 -43}}]
Mapping: {{mapping| 1 5 6 12 25 15 | 0 -13 -14 -35 -82 -43 }}


POTE generator: ~6/5 = 315.225
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.225


Vals: {{Val list| 19e, 99ef, 118, 217ff, 335ff }}
{{Optimal ET sequence|legend=1| 19e, 99ef, 118, 217ff, 335ff }}


Badness: 0.037618
Badness: 0.037618
Line 1,396: Line 1,354:
Comma list: 176/175, 1375/1372, 2200/2187
Comma list: 176/175, 1375/1372, 2200/2187


Mapping: [{{val|1 5 6 12 20}}, {{val|0 -13 -14 -35 -63}}]
Mapping: {{mapping| 1 5 6 12 20 | 0 -13 -14 -35 -63 }}


POTE generator: ~6/5 = 315.060
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.060


Vals: {{Val list| 19e, 80, 179, 259cd }}
{{Optimal ET sequence|legend=1| 19e, 80, 179, 259cd }}


Badness: 0.055884
Badness: 0.055884
Line 1,409: Line 1,367:
Comma list: 169/168, 176/175, 325/324, 1375/1372
Comma list: 169/168, 176/175, 325/324, 1375/1372


Mapping: [{{val|1 5 6 12 20 10}}, {{val|0 -13 -14 -35 -63 -24}}]
Mapping: {{mapping| 1 5 6 12 20 10 | 0 -13 -14 -35 -63 -24 }}


POTE generator: ~6/5 = 315.075
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.075


Vals: {{Val list| 19e, 80, 179 }}
{{Optimal ET sequence|legend=1| 19e, 80, 179 }}


Badness: 0.036559
Badness: 0.036559
Line 1,422: Line 1,380:
Comma list: 540/539, 896/891, 3136/3125
Comma list: 540/539, 896/891, 3136/3125


Mapping: [{{val|1 5 6 12 -1}}, {{val|0 -13 -14 -35 17}}]
Mapping: {{mapping| 1 5 6 12 -1 | 0 -13 -14 -35 17 }}


POTE generator: ~6/5 = 315.096
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.096


Vals: {{Val list| 19, 61d, 80, 99e, 179e }}
{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}


Badness: 0.041720
Badness: 0.041720
Line 1,435: Line 1,393:
Comma list: 169/168, 325/324, 540/539, 832/825
Comma list: 169/168, 325/324, 540/539, 832/825


Mapping: [{{val|1 5 6 12 -1 10}}, {{val|0 -13 -14 -35 17 -24}}]
Mapping: {{mapping| 1 5 6 12 -1 10 | 0 -13 -14 -35 17 -24 }}


POTE generator: ~6/5 = 315.080
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.080


Vals: {{Val list| 19, 61d, 80, 99e, 179e }}
{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}


Badness: 0.035781
Badness: 0.035781
Line 1,448: Line 1,406:
Comma list: 3025/3024, 3136/3125, 4375/4374
Comma list: 3025/3024, 3136/3125, 4375/4374


Mapping: [{{val|2 10 12 24 19}}, {{val|0 -13 -14 -35 -23}}]
Mapping: {{mapping| 2 10 12 24 19 | 0 -13 -14 -35 -23 }}


POTE generator: ~6/5 = 315.181
Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.181


Vals: {{Val list| 80, 118, 198, 316, 514c, 830c }}
{{Optimal ET sequence|legend=1| 80, 118, 198, 316, 514c, 830c }}


Badness: 0.034208
Badness: 0.034208
Line 1,463: Line 1,421:
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374


Mapping: [{{val|2 10 12 24 19 -1}}, {{val|0 -13 -14 -35 -23 16}}]
Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }}


POTE generator: ~6/5 = 315.156
Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.156


Vals: {{Val list| 80, 118, 198 }}
{{Optimal ET sequence|legend=1| 80, 118, 198 }}


Badness: 0.033775
Badness: 0.033775
Line 1,478: Line 1,436:
Comma list: 169/168, 325/324, 364/363, 3136/3125
Comma list: 169/168, 325/324, 364/363, 3136/3125


Mapping: [{{val|2 10 12 24 19 20}}, {{val|0 -13 -14 -35 -23 -24}}]
Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }}


POTE generator: ~6/5 = 315.184
Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 315.184


Vals: {{Val list| 80, 118f, 198f }}
{{Optimal ET sequence|legend=1| 80, 118f, 198f }}


Badness: 0.040467
Badness: 0.040467


== Counterkleismic ==
== Counterkleismic ==
{{see also|Syntonic-enneadecal equivalence continuum #Counterhanson}}
{{See also| High badness temperaments #Counterhanson}}
 
In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo|-20 -24 25}}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as 19&amp;224 temperament (''counterkleismic'', named by analogy to [[catakleismic]] and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 158203125/157351936
[[Comma list]]: 4375/4374, 158203125/157351936


[[Mapping]]: [{{val|1 -5 -4 -18}}, {{val|0 25 24 79}}]
{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }}


[[Wedgie]]: {{multival|25 24 79 -20 55 116}}
: mapping generators: ~2, ~5/3


[[POTE generator]]: ~6/5 = 316.060
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.060


{{Val list|legend=1| 19, 205, 224, 243, 467 }}
{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }}


[[Badness]]: 0.090553
[[Badness]]: 0.090553
Line 1,511: Line 1,468:
Comma list: 540/539, 4375/4374, 2097152/2096325
Comma list: 540/539, 4375/4374, 2097152/2096325


Mapping: [{{val|1 -5 -4 -18 19}}, {{val|0 25 24 79 -59}}]
Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }}


POTE generator: ~6/5 = 316.071
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.071


Vals: {{Val list| 19, 205, 224 }}
{{Optimal ET sequence|legend=1| 19, 205, 224 }}


Badness: 0.070952
Badness: 0.070952
Line 1,524: Line 1,481:
Comma list: 540/539, 625/624, 729/728, 10985/10976
Comma list: 540/539, 625/624, 729/728, 10985/10976


Mapping: [{{val|1 -5 -4 -18 19 -15}}, {{val|0 25 24 79 -59 71}}]
Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }}


POTE generator: ~6/5 = 316.070
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.070


Vals: {{Val list| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}
{{Optimal ET sequence|legend=1| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}


Badness: 0.033874
Badness: 0.033874
Line 1,537: Line 1,494:
Comma list: 1375/1372, 4375/4374, 496125/495616
Comma list: 1375/1372, 4375/4374, 496125/495616


Mapping: [{{val|1 -5 -4 -18 -40}}, {{val|0 25 24 79 165}}]
Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }}


POTE generator: ~6/5 = 316.065
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065


Vals: {{Val list| 19e, 205e, 224 }}
{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}


Badness: 0.065400
Badness: 0.065400
Line 1,550: Line 1,507:
Comma list: 625/624, 729/728, 1375/1372, 10985/10976
Comma list: 625/624, 729/728, 1375/1372, 10985/10976


Mapping: [{{val|1 -5 -4 -18 -40 -15}}, {{val|0 25 24 79 165 71}}]
Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }}


POTE generator: ~6/5 = 316.065
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065


Vals: {{Val list| 19e, 205e, 224 }}
{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}


Badness: 0.029782
Badness: 0.029782


== Quincy ==
== Quincy ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 823543/819200
[[Comma list]]: 4375/4374, 823543/819200


[[Mapping]]: [{{val|1 2 3 3}}, {{val|0 -30 -49 -14}}]
{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
 
[[Wedgie]]: {{multival|30 49 14 8 -62 -105}}


[[POTE generator]]: ~1728/1715 = 16.613
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1715 = 16.613


{{Val list|legend=1| 72, 217, 289 }}
{{Optimal ET sequence|legend=1| 72, 217, 289 }}


[[Badness]]: 0.079657
[[Badness]]: 0.079657
Line 1,578: Line 1,533:
Comma list: 441/440, 4000/3993, 4375/4374
Comma list: 441/440, 4000/3993, 4375/4374


Mapping: [{{val|1 2 3 3 4}}, {{val|0 -30 -49 -14 -39}}]
Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}


POTE generator: ~100/99 = 16.613
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.613


Vals: {{Val list| 72, 217, 289 }}
{{Optimal ET sequence|legend=1| 72, 217, 289 }}


Badness: 0.030875
Badness: 0.030875
Line 1,591: Line 1,546:
Comma list: 364/363, 441/440, 676/675, 4375/4374
Comma list: 364/363, 441/440, 676/675, 4375/4374


Mapping: [{{val|1 2 3 3 4 5}}, {{val|0 -30 -49 -14 -39 -94}}]
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}


POTE generator: ~100/99 = 16.602
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602


Vals: {{Val list| 72, 145, 217, 289 }}
{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}


Badness: 0.023862
Badness: 0.023862
Line 1,604: Line 1,559:
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155


Mapping: [{{val|1 2 3 3 4 5 5}}, {{val|0 -30 -49 -14 -39 -94 -66}}]
Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}


POTE generator: ~100/99 = 16.602
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602


Vals: {{Val list| 72, 145, 217, 289 }}
{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}


Badness: 0.014741
Badness: 0.014741
Line 1,617: Line 1,572:
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675


Mapping: [{{val|1 2 3 3 4 5 5 4}}, {{val|0 -30 -49 -14 -39 -94 -66 18}}]
Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}


POTE generator: ~100/99 = 16.594
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.594


Vals: {{Val list| 72, 145, 217 }}
{{Optimal ET sequence|legend=1| 72, 145, 217 }}


Badness: 0.015197
Badness: 0.015197


== Trideci ==
== Sfourth ==
{{see also| High badness temperaments #Tridecatonic }}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''


The ''trideci'' temperament (26&amp;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 [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from "tridecim" (Latin for "[[wikipedia:13|thirteen]]").
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, 64827/64000


[[Comma list]]: 4375/4374, 83349/81920
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}


[[Mapping]]: [{{val|13 21 31 36}}, {{val|0 -1 -2 1}}]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 26.287


[[POTE generator]]: ~3/2 = 699.1410
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


{{Val list|legend=1| 26, 65, 91, 156d, 247cdd }}
[[Badness]]: 0.123291
 
[[Badness]]: 0.184585


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 245/242, 385/384, 4375/4374
Comma list: 121/120, 441/440, 4375/4374


Mapping: [{{val|13 21 31 36 45}}, {{val|0 -1 -2 1 0}}]
Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}


POTE generator: ~3/2 = 699.6179
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.286


Vals: {{Val list| 26, 65, 91, 156d, 247cdde }}
{{Optimal ET sequence|legend=1| 45e, 46, 91e, 137de }}


Badness: 0.084590
Badness: 0.054098


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 245/242, 325/324, 385/384
Comma list: 121/120, 169/168, 325/324, 441/440


Mapping: [{{val|13 21 31 36 45 48}}, {{val|0 -1 -2 1 0 0}}]
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}


POTE generator: ~3/2 = 699.2969
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.310


Vals: {{Val list| 26, 65f, 91f, 156dff }}
{{Optimal ET sequence|legend=1| 45ef, 46, 91ef, 137def }}


Badness: 0.052366
Badness: 0.033067


== Chlorine ==
=== Sfour ===
The name of chlorine temperament comes from Chlorine, the 17th element.
Subgroup: 2.3.5.7.11


Chlorine temperament has a period of 1/17 octave. It tempers out the septendecima, {{monzo|-52 -17 34}}, by which 17 chromatic semitones (25/24) exceed an octave. This temperament can be described as 289&amp;323 temperament, which tempers out {{monzo|-49 4 22 -3}} as well as the ragisma.
Comma list: 385/384, 2401/2376, 4375/4374


Subgroup: 2.3.5
Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}


[[Comma]]: {{monzo|-52 -17 34}}
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.246


[[Mapping]]: [{{val|17 26 39}}, {{val|0 2 1}}]
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


[[POTE tuning|POTE generators]]: ~25/24 = 70.5882, ~5/4 = 386.2687
Badness: 0.076567


{{Val list|legend=1| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.077072
Comma list: 196/195, 364/363, 385/384, 4375/4374


=== 7-limit ===
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 193119049072265625/193091834023510016
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.239


[[Mapping]]: [{{val|17 26 39 43}}, {{val|0 2 1 10}}]
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


[[Wedgie]]: {{multival|34 17 170 -52 174 347}}
Badness: 0.051893


[[POTE tuning|POTE generators]]: ~25/24 = 70.5882, ~5/4 = 386.2936
== Trideci ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tridecatonic]].''


{{Val list|legend=1| 289, 323, 612, 935, 1547 }}
The trideci temperament (26 &amp; 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 [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from "tridecim" (Latin for "[[wikipedia:13|thirteen]]").


[[Badness]]: 0.041658
[[Subgroup]]: 2.3.5.7


=== 11-limit ===
[[Comma list]]: 4375/4374, 83349/81920
Subgroup: 2.3.5.7.11


Comma list: 4375/4374, 41503/41472, 1879453125/1879048192
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}


Mapping: [{{val|17 26 39 43 64}}, {{val|0 2 1 10 -11}}]
[[Optimal tuning]] ([[POTE]]): ~256/245 = 1\13, ~3/2 = 699.1410


POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2690
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }}


Vals: {{Val list| 289, 323, 612 }}
[[Badness]]: 0.184585
 
Badness: 0.063706
 
== Palladium ==
The name of ''palladium temperament'' comes from Palladium, the 46th element.
 
Palladium temperament has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo|-39 92 -46}}, by which 46 minortones (10/9) fall short of seven octaves. This temperament can be described as 46&amp;414 temperament, which tempers out {{monzo|-51 8 2 12}} as well as the ragisma.
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 4375/4374, 2270317133144025/2251799813685248
 
[[Mapping]]: [{{val|46 73 107 129}}, {{val|0 -1 -2 1}}]
 
[[Wedgie]]: {{multival|46 92 -46 39 -202 -365}}
 
[[POTE generator]]: ~3/2 = 701.6074
 
{{Val list|legend=1| 46, 368, 414, 460, 874d }}
 
[[Badness]]: 0.308505


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 9801/9800, 134775333/134217728
Comma list: 245/242, 385/384, 4375/4374


Mapping: [{{val|46 73 107 129 159}}, {{val|0 -1 -2 1 1}}]
Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}


POTE generator: ~3/2 = 701.5951
Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.6179


Vals: {{Val list| 46, 368, 414, 460, 874de }}
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdde }}


Badness: 0.073783
Badness: 0.084590


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
Comma list: 169/168, 245/242, 325/324, 385/384


Mapping: [{{val|46 73 107 129 159 170}}, {{val|0 -1 -2 1 1 2}}]
Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}


POTE generator: ~3/2 = 701.6419
Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.2969


Vals: {{Val list| 46, 368, 414, 460, 874de, 1334de }}
{{Optimal ET sequence|legend=1| 26, 65f, 91f, 156dff }}


Badness: 0.040751
Badness: 0.052366
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
== Counterorson ==
 
Counterorson tempers out the {{monzo| 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]].  
Mapping: [{{val|46 73 107 129 159 170 188}}, {{val|0 -1 -2 1 1 2 0}}]
 
POTE generator: ~3/2 = 701.6425
 
Vals: {{Val list| 46, 368, 414, 460, 874de, 1334deg }}
 
Badness: 0.022441
 
== Monzism ==
The ''monzism'' temperament (53&amp;612) is a rank-two temperament which tempers out the [[monzisma]], {{monzo|54 -37 2}} and the [[nanisma]], {{monzo|109 -67 0 -1}}, as well as the ragisma, [[4375/4374]].  


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 36030948116563575/36028797018963968
Comma list: 4375/4374, {{monzo| 154 -54 -21 -7 }}


[[Mapping]]: [{{val|1 2 10 -25}}, {{val|0 -2 -37 134}}]
Mapping: {{mapping| 1 0 -21 85 | 0 7 103 -363 }}


[[Wedgie]]: {{multival|2 37 -134 54 -218 -415}}
Optimal tuning (CTE): ~2 = 1\1, ~{{monzo| 66 -23 -9 -3 }} = 271.7113


[[POTE generator]]: ~310078125/268435456 = 249.0207
{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}


{{Val list|legend=1| 53, 559, 612, 1277, 1889 }}
Badness: 0.312806


[[Badness]]: 0.046569
== Notes ==
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 41503/41472, 184549376/184528125
 
Mapping: [{{val|1 2 10 -25 46}}, {{val|0 -2 -37 134 -205}}]
 
POTE generator: ~231/200 = 249.0193
 
Vals: {{Val list| 53, 559, 612 }}
 
Badness: 0.057083
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
 
Mapping: [{{val|1 2 10 -25 46 23}}, {{val|0 -2 -37 134 -205 -93}}]
 
POTE generator: ~231/200 = 249.0199
 
Vals: {{Val list| 53, 559, 612 }}
 
Badness: 0.053780


[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
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[[Category:Ragismic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Microtemperaments]]
[[Category:Abigail]]
[[Category:Abigail]]
[[Category:Amity]]
[[Category:Deca]]
[[Category:Deca]]
[[Category:Enneadecal]]
[[Category:Enneadecal]]
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[[Category:Quincy]]
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[[Category:Supermajor]]
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[[Category:Microtemperament]]
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[[Category:Rank 2]]
[[Category:Temperament collection]]
[[Category:Ragismic microtemperaments| ]] <!-- main article -->

Latest revision as of 00:36, 24 June 2025

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, 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. 37 of these give (215)/3, 46 give (219)/5, and 75 give (230)/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.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 52734375/52706752

Mapping: [⟨1 15 19 30], ⟨0 -37 -46 -75]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 435.082

Optimal ET sequence: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214

Badness: 0.010836

Semisupermajor

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 30 38 60 41], ⟨0 -37 -46 -75 -47]]

Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082

Optimal ET sequence: 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf

Badness: 0.012773

Enneadecal

Enneadecal temperament 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.

For the 5-limit temperament, see 19th-octave temperaments#(5-limit) enneadecal.

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 tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)

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

Badness: 0.010954

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 tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)

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

Badness: 0.043734

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

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

Badness: 0.033545

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 tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)

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

Badness: 0.009985

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 tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)

Optimal ET sequence: 152f, 342f, 494

Badness: 0.020782

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 tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444)

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

Badness: 0.030391

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 = 1\38, ~55/54, ~429/250

Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)

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

Badness: 0.014694

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 tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244

Optimal ET sequence: 855, 988, 1843

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 [-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 36 48 61], ⟨0 -55 -73 -93]]

Optimal tuning (POTE): ~2 = 1\1, ~35/27 = 449.1270

Optimal ET sequence: 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, [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: [⟨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: 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: [⟨7 2 -8 53 3], ⟨0 3 8 -11 7]]

Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704

Optimal ET sequence: 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: [⟨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: 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.[1]

For the 5-limit temperament, see Very high accuracy temperaments#Abigail.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2147483648/2144153025

Mapping: [⟨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: 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: [⟨2 7 13 -1 1], ⟨0 -11 -24 19 17]]

Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901

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

Badness: 0.037648

Hemigamera

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 12 20 6 5], ⟨0 -23 -40 -1 5]]

mapping generators: ~99/70, ~8/7

Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3370

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

Badness: 0.040955

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 12 20 6 5 17], ⟨0 -23 -40 -1 5 -25]]

Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373

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

Badness: 0.020416

Semigamera

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 6 10 3 12], ⟨0 -46 -80 -2 -89]]

mapping generators: ~2, ~77/72

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1642

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

Badness: 0.078

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 6 10 3 12 18], ⟨0 -46 -80 -2 -89 -149]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1628

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

Badness: 0.044

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. 1106edo is an strong tuning.

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:

  • CTE: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7475
  • error map: ⟨0.0000 +0.0253 -0.0514 -0.0133]
  • CWE: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7474
  • error map: ⟨0.0000 +0.0244 -0.0508 -0.0218]

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

Badness (Smith): 0.0394

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:

  • CTE: ~99/70 = 162.7485, ~1125/1024 = 162.7485
  • CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481

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

Badness (Smith): 0.0170

Orga

Subgroup: 2.3.5.7

Comma list: 4375/4374, 54975581388800/54936068900769

Mapping: [⟨2 21 36 5], ⟨0 -29 -51 1]]

mapping generators: ~7411887/5242880, ~1310720/1058841

Optimal tuning (POTE): ~7411887/5242880 = 1\2, ~8/7 = 231.104

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

Badness: 0.040236

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 21 36 5 2], ⟨0 -29 -51 1 8]]

Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103

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

Badness: 0.016188

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 21 36 5 2 24], ⟨0 -29 -51 1 8 -27]]

Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103

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

Badness: 0.021762

Seniority

See also: Very high accuracy temperaments § Senior

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 201768035/201326592

Mapping: [⟨1 11 19 2], ⟨0 -35 -62 3]]

Optimal tuning (POTE): ~2 = 1\1, ~3087/2560 = 322.804

Optimal ET sequence: 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d

Badness: 0.044877

Senator

The senator temperament (26 & 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It 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 11 19 2 4], ⟨0 -35 -62 3 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793

Optimal ET sequence: 26, 119c, 145, 171, 316e, 487ee

Badness: 0.092238

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 11 19 2 4 15], ⟨0 -35 -62 3 -2 -42]]

Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793

Optimal ET sequence: 26, 119c, 145, 171, 316ef, 487eef

Badness: 0.044662

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 11 19 2 4 15 17], ⟨0 -35 -62 3 -2 -42 -48]]

Optimal tuning (POTE): ~77/64 = 322.793

Optimal ET sequence: 26, 119c, 145, 171, 316ef, 487eef

Badness: 0.026562

Monzismic

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

The monzismic temperament (53 & 612) 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.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-55 30 2 1⟩

Mapping: [⟨1 2 10 -25], ⟨0 -2 -37 134]]

Optimal tuning (POTE): ~2 = 1\1, ~[-27 11 3 1⟩ = 249.0207

Optimal ET sequence: 53, …, 559, 612, 1277, 1889

Badness: 0.046569

Monzism

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 10 -25 46], ⟨0 -2 -37 134 -205]]

Optimal tuning (POTE): ~231/200 = 249.0193

Optimal ET sequence: 53, 559, 612

Badness: 0.057083

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 10 -25 46 23], ⟨0 -2 -37 134 -205 -93]]

Optimal tuning (POTE): ~231/200 = 249.0199

Optimal ET sequence: 53, 559, 612

Badness: 0.053780

Semidimfourth

For the 5-limit version of this temperament, see High badness temperaments #Semidimfourth.

The semidimfourth temperament is featured by a semi-diminished 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 21 28 36], ⟨0 -31 -41 -53]]

Optimal tuning (POTE): ~2 = 1\1, ~35/27 = 448.456

Optimal ET sequence: 8d, 91, 99, 289, 388, 875, 1263d, 1651d

Badness: 0.055249

Neusec

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 11 15 19 15], ⟨0 -31 -41 -53 -32]]

Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547

Optimal ET sequence: 8d, 190, 388

Badness: 0.059127

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 11 15 19 15 17], ⟨0 -31 -41 -53 -32 -38]]

Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545

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

Badness: 0.030941

Acrokleismic

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2202927104/2197265625

Mapping: [⟨1 10 11 27], ⟨0 -32 -33 -92]]

mapping generators: ~2, ~6/5

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557

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

Badness: 0.056184

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 10 11 27 -16], ⟨0 -32 -33 -92 74]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558

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

Badness: 0.036878

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 10 11 27 -16 25], ⟨0 -32 -33 -92 74 -81]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557

Optimal ET sequence: 19, 251, 270

Badness: 0.026818

Counteracro

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 10 11 27 55], ⟨0 -32 -33 -92 -196]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.553

Optimal ET sequence: 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde

Badness: 0.042572

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 10 11 27 55 25], ⟨0 -32 -33 -92 -196 -81]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.554

Optimal ET sequence: 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf

Badness: 0.026028

Quasithird

The quasithird temperament is 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 to temper out the ragisma and [-60 29 0 5⟩.

Subgroup: 2.3.5

Comma list: [55 -64 20⟩

Mapping: [⟨4 0 -11], ⟨0 5 16]]

mapping generators: ~51200000/43046721, ~1594323/1280000

Optimal tuning (POTE): ~51200000/43046721, ~1594323/1280000 = 380.395

Optimal ET sequence: 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404

Badness: 0.099519

7-limit

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-60 29 0 5⟩

Mapping: [⟨4 0 -11 48], ⟨0 5 16 -29]]

Optimal tuning (POTE): ~65536/55125 = 1\4, ~5103/4096 = 380.388

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

Badness: 0.061813

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 tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387)

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

Badness: 0.021125

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 tuning (POTE): ~81/65 = 380.385 (or ~22/21 = 80.385)

Optimal ET sequence: 60d, 164, 224, 388, 612, 836, 1448f, 2284f

Badness: 0.029501

Deca

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

Deca temperament has a period of 1/10 octave and tempers out the linus comma, [11 -10 -10 10⟩, neon comma [21 60 -50⟩ and [12 -3 -14 9⟩ = 165288374272/164794921875 (satritrizo-asepbigu).

Subgroup: 2.3.5.7

Comma list: 4375/4374, 165288374272/164794921875

Mapping: [⟨10 4 9 2], ⟨0 5 6 11]]

mapping generators: ~15/14, ~6/5

Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.577

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

Badness: 0.080637

Badness (Sintel): 2.041

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 tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.582

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

Badness: 0.024329

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 tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.602 (~40/39 = 44.398)

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

Badness: 0.016810

Badness (Sintel): 0.695

no-17's 19-limit

Subgroup: 2.3.5.7.11.13.19

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

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

Optimal tuning (CTE): ~15/14 = 1\10, ~6/5 = 315.581 (~39/38 = 44.419)

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

Badness (Sintel): 0.556

Keenanose

Keenanose is named 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 tuning (CTE): ~2 = 1\1, ~[21 3 1 -10⟩ = 4.4465

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

Badness: 0.0858

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 tuning (CTE): ~2 = 1\1, ~385/384 = 4.4465

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

Badness: 0.0308

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 tuning (CTE): ~2 = 1\1, ~385/384 = 4.4466

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

Badness: 0.0213

Aluminium

Aluminium is named after the 13th element, and 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

Comma list: [92 -39 -13⟩

Mapping: [⟨13 0 92], ⟨0 1 -3]]

mapping generators: ~135/128, ~3

Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 701.9897

Optimal ET sequence: 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc

Badness: 0.123

7-limit

Subgroup: 2.3.5.7

Comma list: 4375/4374, [92 -39 -13⟩

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

Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0024

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

Badness: 0.126

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 tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042

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

Badness: 0.0421

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 tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099

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

Badness: 0.0286

Countritonic

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

Countritonic (co-un-tritonic) can be described as the 53 & 422 temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 68719476736/68356598625

Mapping: [⟨1 6 19 -33], ⟨0 -9 -34 73]]

mapping generators: ~2, ~45927/32768

Optimal tuning (CTE): ~2 = 1\1, ~45927/32768 = 588.6216

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

Badness: 0.133

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 6 19 -13 79], ⟨0 -9 -34 73 154]]

Optimal tuning (CTE): ~2 = 1\1, ~539/384 = 588.6258

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

Badness: 0.0707

13-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 6 19 -13 79], ⟨0 -9 -34 73 154 -74]]

Optimal tuning (CTE): ~2 = 1\1, ~128/91 = 588.6277

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

Badness: 0.0366

Quatracot

See also: Stratosphere

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-32 5 14 -3⟩

Mapping: [⟨2 7 7 23], ⟨0 -13 -8 -59]]

mapping generators: ~2278125/1605632, ~448/405

Optimal tuning (POTE): ~2278125/1605632 = 1\2, ~448/405 = 176.805

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

Badness: 0.175982

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 7 7 23 19], ⟨0 -13 -8 -59 -41]]

Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 176.806

Optimal ET sequence: 190, 224, 414, 638, 1052c

Badness: 0.041043

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 7 7 23 19 13], ⟨0 -13 -8 -59 -41 -19]]

Optimal tuning (POTE): ~99/70 = 1\2, ~195/176 = 176.804

Optimal ET sequence: 190, 224, 414, 638, 1690bcc, 2328bccde

Badness: 0.022643

Moulin

Moulin has a generator of 22/13, and it is named after the Law & Order: Special Victims Unit episode Season 22, Episode 13. "Trick-Rolled At The Moulin". It can be described as the 494 & 1619 temperament.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-88 2 45 -7⟩

Mapping: [⟨1 57 38 248], ⟨0 -73 -47 -323]]

mapping generators: ~2, ~6422528/3796875

Optimal tuning (CTE): ~2 = 1\1, ~6422528/3796875 = 910.9323

Optimal ET sequence: 494, 1125, 1619

Badness: 0.234

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 57 38 248 -14], ⟨0 -73 -47 -323 23]]

Optimal tuning (CTE): ~2 = 1\1, ~1024/605 = 910.9323

Optimal ET sequence: 494, 1125, 1619, 2113

Badness: 0.0678

13-limit

Since 11/8 is within 23 generators, the 25 tone MOS (4L 21s) of this temperament contains the 8:11:13 triad.

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 57 38 248 -14 -13], ⟨0 -73 -47 -323 23 22]]

Optimal tuning (CTE): ~2 = 1\1, ~22/13 = 910.9323

Optimal ET sequence: 494, 1125, 1619, 2113

Badness: 0.0271

Palladium

For the 5-limit version of this temperament, see 46th-octave temperaments.

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 minortones (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 tuning (POTE): ~83349/81920 = 1\46, ~3/2 = 701.6074

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

Badness: 0.308505

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 tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951

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

Badness: 0.073783

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 tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6419

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

Badness: 0.040751

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 tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425

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

Badness: 0.022441

Oviminor

See also: Syntonic–kleismic equivalence continuum

Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past egads, though it is less accurate.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-100 53 48 -34⟩

Mapping: [⟨1 50 51 147], ⟨0 -184 -185 -548]]

mapping generators: ~2, ~6/5

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 315.7501

Optimal ET sequence: 19, …, 1600, 1619, 4838, 6457c

Badness: 0.582

Octoid

For the 5-limit temperament, see 8th-octave temperaments#Octoid (5-limit).

The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 (ragisma) and 16875/16807 (mirkwai). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.

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 tuning (POTE): ~49/45 = 1\8, ~7/5 = 583.940

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: 0.042670

Scales: octoid72, octoid80

11-limit

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 minimaxing the 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, if one wants to use 80edo as the tuning, one must use octopus — not octoid — as 80edo doesn't temper 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.

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 tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.962

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: 72, 152, 224

Badness: 0.014097

Scales: octoid72, octoid80

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 tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.905

Optimal ET sequence: 72, 152f, 224

Badness: 0.015274

Scales: octoid72, octoid80

Music
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 tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.842

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

Badness: 0.014304

Scales: octoid72, octoid80

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 tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.932

Optimal ET sequence: 72, 152fg, 224

Badness: 0.016036

Scales: octoid72, octoid80

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 tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.892

Optimal ET sequence: 72, 152, 224f

Badness: 0.021679

Scales: octoid72, octoid80

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 tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.811

Optimal ET sequence: 72, 152, 224fg, 296ffg

Badness: 0.015614

Scales: Octoid72, Octoid80

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 tuning (POTE): ~12/11 = 1\8, ~7/5 = 584.064

Optimal ET sequence: 72, 152, 224fg, 376ffgh

Badness: 0.016321

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 tuning (POTE): ~448/429 = 1\16, ~13/8 = 841.015

Optimal ET sequence: 80, 144, 224

Badness: 0.030818

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 tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932

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

Badness: 0.028611

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 tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896

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

Badness: 0.023731

Parakleismic

Main article: Parakleismic

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. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, and 11-limit adding 385/384. For the 7-limit 99edo may be preferred, but in the 11-limit it is best to stick with 118.

Subgroup: 2.3.5

Comma list: 1224440064/1220703125

Mapping: [⟨1 5 6], ⟨0 -13 -14]]

mapping generators: ~2, ~6/5

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.240

Optimal ET sequence: 19, 61, 80, 99, 118, 453, 571, 689, 1496

Badness: 0.043279

7-limit

Subgroup: 2.3.5.7

Comma list: 3136/3125, 4375/4374

Mapping: [⟨1 5 6 12], ⟨0 -13 -14 -35]]


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.181

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

Badness: 0.027431

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 5 6 12 -6], ⟨0 -13 -14 -35 36]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.251

Optimal ET sequence: 19, 99, 118

Badness: 0.049711

Paralytic

The paralytic temperament (118&217) 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 5 6 12 25], ⟨0 -13 -14 -35 -82]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.220

Optimal ET sequence: 19e, 99e, 118, 217, 335, 552d, 887dd

Badness: 0.036027

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 5 6 12 25 -16], ⟨0 -13 -14 -35 -82 75]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.214

Optimal ET sequence: 99e, 118, 217, 552d, 769de

Badness: 0.044710

Paraklein

The paraklein temperament (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 5 6 12 25 15], ⟨0 -13 -14 -35 -82 -43]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.225

Optimal ET sequence: 19e, 99ef, 118, 217ff, 335ff

Badness: 0.037618

Parkleismic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 5 6 12 20], ⟨0 -13 -14 -35 -63]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.060

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

Badness: 0.055884

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 5 6 12 20 10], ⟨0 -13 -14 -35 -63 -24]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.075

Optimal ET sequence: 19e, 80, 179

Badness: 0.036559

Paradigmic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 5 6 12 -1], ⟨0 -13 -14 -35 17]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.096

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

Badness: 0.041720

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 5 6 12 -1 10], ⟨0 -13 -14 -35 17 -24]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.080

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

Badness: 0.035781

Semiparakleismic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 10 12 24 19], ⟨0 -13 -14 -35 -23]]

Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.181

Optimal ET sequence: 80, 118, 198, 316, 514c, 830c

Badness: 0.034208

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 10 12 24 19 -1], ⟨0 -13 -14 -35 -23 16]]

Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.156

Optimal ET sequence: 80, 118, 198

Badness: 0.033775

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 10 12 24 19 20], ⟨0 -13 -14 -35 -23 -24]]

Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 315.184

Optimal ET sequence: 80, 118f, 198f

Badness: 0.040467

Counterkleismic

See also: High badness temperaments § 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) fall short of the classic major third (5/4). It can be described as 19 & 224 temperament (counterkleismic, named by analogy to catakleismic and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).

Subgroup: 2.3.5.7

Comma list: 4375/4374, 158203125/157351936

Mapping: [⟨1 20 20 61], ⟨0 -25 -24 -79]]

mapping generators: ~2, ~5/3

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.060

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

Badness: 0.090553

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 20 20 61 -40], ⟨0 -25 -24 -79 59]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.071

Optimal ET sequence: 19, 205, 224

Badness: 0.070952

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 20 20 61 -40 56], ⟨0 -25 -24 -79 59 -71]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.070

Optimal ET sequence: 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef

Badness: 0.033874

Counterlytic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 20 20 61 125], ⟨0 -25 -24 -79 -165]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065

Optimal ET sequence: 19e, 205e, 224

Badness: 0.065400

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 20 20 61 125 56], ⟨0 -25 -24 -79 -165 -71]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065

Optimal ET sequence: 19e, 205e, 224

Badness: 0.029782

Quincy

Subgroup: 2.3.5.7

Comma list: 4375/4374, 823543/819200

Mapping: [⟨1 2 3 3], ⟨0 -30 -49 -14]]

Optimal tuning (POTE): ~2 = 1\1, ~1728/1715 = 16.613

Optimal ET sequence: 72, 217, 289

Badness: 0.079657

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 tuning (POTE): ~2 = 1\1, ~100/99 = 16.613

Optimal ET sequence: 72, 217, 289

Badness: 0.030875

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 tuning (POTE): ~2 = 1\1, ~100/99 = 16.602

Optimal ET sequence: 72, 145, 217, 289

Badness: 0.023862

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 tuning (POTE): ~2 = 1\1, ~100/99 = 16.602

Optimal ET sequence: 72, 145, 217, 289

Badness: 0.014741

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 tuning (POTE): ~2 = 1\1, ~100/99 = 16.594

Optimal ET sequence: 72, 145, 217

Badness: 0.015197

Sfourth

For the 5-limit version of this temperament, see High badness temperaments #Sfourth.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 64827/64000

Mapping: [⟨1 2 3 3], ⟨0 -19 -31 -9]]

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.287

Optimal ET sequence: 45, 46, 91, 137d

Badness: 0.123291

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 tuning (POTE): ~2 = 1\1, ~49/48 = 26.286

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

Badness: 0.054098

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 tuning (POTE): ~2 = 1\1, ~49/48 = 26.310

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

Badness: 0.033067

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 tuning (POTE): ~2 = 1\1, ~49/48 = 26.246

Optimal ET sequence: 45, 46, 91, 137d

Badness: 0.076567

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 tuning (POTE): ~2 = 1\1, ~49/48 = 26.239

Optimal ET sequence: 45, 46, 91, 137d

Badness: 0.051893

Trideci

For the 5-limit version of this temperament, see High badness 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]]

Optimal tuning (POTE): ~256/245 = 1\13, ~3/2 = 699.1410

Optimal ET sequence: 26, 65, 91, 156d, 247cdd

Badness: 0.184585

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 tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.6179

Optimal ET sequence: 26, 65, 91, 156d, 247cdde

Badness: 0.084590

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 tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.2969

Optimal ET sequence: 26, 65f, 91f, 156dff

Badness: 0.052366

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]]

Optimal tuning (CTE): ~2 = 1\1, ~[66 -23 -9 -3⟩ = 271.7113

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

Badness: 0.312806

Notes

  1. ↑ [1]: "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."
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