Xenharmonic Wiki

Ragismic microtemperaments: Difference between revisions

← Older edit
VisualWikitext
Gud2228 (talk | contribs)
autopatrolled, emailconfirmed
227 edits
Tags: Mobile edit Mobile web edit
 
(119 intermediate revisions by 16 users not shown)
Line 1: Line 1:
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]]
* ''[[Srutal]]'', {2048/2025, 4375/4374} → [[Diaschismic family #Srutal|Diaschismic family]]
* ''[[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, enneadecal, decal, sfourth, abigail, semidimi, brahmagupta, quasithird, semidimfourth, acrokleismic, seniority, orga, quatracot, octoid, amity, parakleismic, counterkleismic, quincy, trideci, 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]] 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]], {{monzo|1 -27 18}}, which leads to the identification of (27/25)<sup>9</sup> 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.
[[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 49/40~60/49, all of which have their own interesting advantages. Possible tunings are 441-, 612-, or 3600edo, 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


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}


[[Comma list]]: 2401/2400, 4375/4374
[[Badness]]: 0.010836


[[Mapping]]: [{{val| 9 1 1 12 }}, {{val| 0 2 3 2 }}]
=== Semisupermajor ===
Subgroup: 2.3.5.7.11


{{Multival|legend=1| 18 27 18 1 -22 -34 }}
Comma list: 3025/3024, 4375/4374, 35156250/35153041


Mapping generators: ~27/25, ~5/3
Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}


[[POTE generator]]s: ~5/3 = 884.3129 or ~36/35 = 49.0205
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082


[[Tuning ranges]]:
{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}
* 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 }}
Badness: 0.012773


[[Badness]]: 0.003610
== 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.


=== 11-limit ===
''For the 5-limit temperament, see [[19th-octave temperaments#(5-limit) enneadecal]].''
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
[[Subgroup]]: 2.3.5.7


Comma list: 2401/2400, 4375/4374, 5632/5625
[[Comma list]]: 4375/4374, 703125/702464


Mapping: [{{val| 9 1 1 12 -75 }}, {{val| 0 2 3 2 16 }}]
{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}


POTE generator: ~5/3 = 884.4679 or ~36/35 = 48.8654
: mapping generators: ~28/27, ~3


Optimal GPV sequence: {{Val list| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
[[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)


Badness: 0.027332
{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}


==== 13-limit ====
[[Badness]]: 0.010954
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: ~5/3 = 884.4304 or ~36/35 = 48.9030
 
Optimal GPV sequence: {{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: ~5/3 = 884.4089 or ~36/35 = 48.9244
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)


Optimal GPV sequence: {{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: ~5/3 = 884.3997 or ~36/35 = 48.9336
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)


Optimal GPV sequence: {{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: ~5/3 = 883.9386 or ~36/35 = 49.3948
: 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]


Optimal GPV sequence: {{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: ~5/3 = 883.9920 or ~36/35 = 49.3414
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]


Optimal GPV sequence: {{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: ~5/3 = 883.9981 or ~36/35 = 49.3353
{{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]
 
Optimal GPV sequence: {{Val list| 72, 171, 243 }}
 
Badness: 0.014602


==== Ennealim ====
==== Semihemienneadecal ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 243/242, 325/324, 441/440
Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078


Mapping: [{{val| 9 1 1 12 -2 20 }}, {{val| 0 2 3 2 5 2 }}]
Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}


POTE generator: ~5/3 = 883.6257 or ~36/35 = 49.7076
: mapping generators: ~55/54 = 1\38, ~55/54, ~429/250


Optimal GPV sequence: {{Val list| 27e, 45ef, 72 }}
Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)


Badness: 0.020697
{{Optimal ET sequence|legend=1| 190, 304d, 494, 684, 1178, 2850, 4028ce }}


=== Ennealiminal ===
Badness: 0.014694
Subgroup: 2.3.5.7.11


Comma list: 385/384, 1375/1372, 4375/4374
=== 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.


Mapping: [{{val| 9 1 1 12 51 }}, {{val| 0 2 3 2 -3 }}]
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~5/3 = 883.8298 or ~36/35 = 49.5036
Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344


Optimal GPV sequence: {{Val list| 27, 45, 72, 171e, 243e, 315e }}
Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}


Badness: 0.031123
Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244


==== 13-limit ====
{{Optimal ET sequence|legend=1| 855, 988, 1843 }}
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 325/324, 385/384, 1375/1372
== Semidimi ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].''


Mapping: [{{val| 9 1 1 12 51 20 }}, {{val| 0 2 3 2 -3 2 }}]
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.


POTE generator: ~5/3 = 883.8476 or ~36/35 = 49.4857
[[Subgroup]]: 2.3.5.7


Optimal GPV sequence: {{Val list| 27, 45f, 72, 171ef, 243ef }}
[[Comma list]]: 4375/4374, 3955078125/3954653486


Badness: 0.030325
{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}


=== Hemiennealimmal ===
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270
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
{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}


Comma list: 2401/2400, 3025/3024, 4375/4374
[[Badness]]: 0.015075


Mapping: [{{val| 18 0 -1 22 48 }}, {{val| 0 2 3 2 1 }}]
== Brahmagupta ==
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.


Mapping generators: ~80/77, ~400/231
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 ¢).


POTE generator: ~400/231 = 950.9553
[[Subgroup]]: 2.3.5.7


Tuning ranges:
[[Comma list]]: 4375/4374, 70368744177664/70338939985125
* 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]


Optimal GPV sequence: {{Val list| 72, 198, 270, 342, 612, 954, 1566 }}
{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}


Badness: 0.006283
: mapping generators: ~1157625/1048576, ~27/20


==== 13-limit ====
[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716
Subgroup: 2.3.5.7.11.13


Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }}


Mapping: [{{val| 18 0 -1 22 48 -19 }}, {{val| 0 2 3 2 1 6 }}]
[[Badness]]: 0.029122


POTE generator ~26/15 = 951.0837
=== 11-limit ===
Subgroup: 2.3.5.7.11


Tuning ranges:  
Comma list: 4000/3993, 4375/4374, 131072/130977
* 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]


Optimal GPV sequence: {{Val list| 72, 198, 270 }}
Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}


Badness: 0.012505
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704


==== 17-limit ====
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 676/675, 715/714, 1001/1000, 1716/1715, 3025/3024
Badness: 0.052190


Mapping: [{{val| 18 0 -1 22 48 -19 -6 }}, {{val| 0 2 3 2 1 6 6 }}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


POTE generator ~26/15 = 951.0837
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374


Optimal GPV sequence: {{Val list| 72, 198g, 270 }}
Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}


==== 19-limit ====
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.706
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 676/675, 1001/1000, 1331/1330, 1716/1715, 3025/3024
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771eef }}


Mapping: [{{val| 18 0 -1 22 48 -19 -6 103 }}, {{val| 0 2 3 2 1 6 6 -2 }}]
Badness: 0.023132


POTE generator ~26/15 = 951.0837
== 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>


Optimal GPV sequence: {{Val list| 72, 198g, 270 }}
''For the 5-limit temperament, see [[Very high accuracy temperaments#Abigail]].''


==== Semihemiennealimmal ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13


Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374
[[Comma list]]: 4375/4374, 2147483648/2144153025


Mapping: [{{val| 18 0 -1 22 48 88 }}, {{val| 0 4 6 4 2 -3 }}]
{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }}


Mapping generators: ~80/77, ~1053/800
: mapping generators: ~46305/32768, ~27/20


POTE generator: ~1053/800 = 475.4727
[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899


Optimal GPV sequence: {{Val list| 126, 144, 270, 684, 954 }}
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}


Badness: 0.013104
[[Badness]]: 0.037000
 
=== Semiennealimmal ===
Semiennealimmal tempers out [[4000/3993]], and uses a ~140/121 semifourth generator. Notably, however, two generator steps do not reach ~4/3, despite that the name may suggest so. In fact, it splits the generator of ennealimmal into three.  


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


Comma list: 2401/2400, 4000/3993, 4375/4374
Comma list: 3025/3024, 4375/4374, 131072/130977


Mapping: [{{val| 9 3 4 14 18 }}, {{val| 0 6 9 6 7 }}]
Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }}


Mapping generators: ~27/25, ~140/121
Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901


POTE generator: ~140/121 = 250.3367
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }}


Optimal GPV sequence: {{Val list| 72, 369, 441 }}
Badness: 0.012860


Badness: 0.034196
=== 13-limit ===
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095


Mapping: [{{val| 9 3 4 14 18 -8 }}, {{val| 0 6 9 6 7 22 }}]
Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}


POTE generator: ~140/121 = 250.3375
Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903


Optimal GPV sequence: {{Val list| 72, 297ef, 369f, 441 }}
{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }}


Badness: 0.026122
Badness: 0.008856


=== Quadraennealimmal ===
== Gamera ==
Subgroup: 2.3.5.7.11
''For the 5-limit temperament, see [[High badness temperaments#Gamera]].


Comma list: 2401/2400, 4375/4374, 234375/234256
[[Subgroup]]: 2.3.5.7
 
Mapping: [{{val| 9 1 1 12 -7 }}, {{val| 0 8 12 8 23 }}]
 
Mapping generators: ~27/25, ~25/22
 
POTE generator: ~25/22 = 221.0717
 
Optimal GPV sequence: {{Val list| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
 
Badness: 0.021320
 
=== Trinealimmal ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 4375/4374, 2097152/2096325
 
Mapping: [{{val| 27 1 0 34 177 }}, {{val| 0 2 3 2 -4 }}]
 
Mapping generators: ~2744/2673, ~2352/1375
 
POTE generator: ~2352/1375 = 928.8000
 
Optimal GPV sequence: {{Val list| 27, 243, 270, 783, 1053, 1323 }}
 
Badness: 0.029812
 
== Gamera ==
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 }}
 
Mapping generators: ~2, ~8/7


{{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 371: 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
: mapping generators: ~99/70, ~8/7


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


Optimal GPV sequence: {{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 386: 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


Optimal GPV sequence: {{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
Line 399: Line 330:
Comma list: 4375/4374, 14641/14580, 15488/15435
Comma list: 4375/4374, 14641/14580, 15488/15435


Mapping: [{{val| 1 6 10 3 12 }}, {{val| 0 -46 -80 -2 -89 }}]
Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }}


Mapping generators: ~2, ~77/72
: mapping generators: ~2, ~77/72


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


Optimal GPV sequence: {{Val list| 73, 125, 198, 323, 521 }}
{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }}


Badness: 0.078
Badness: 0.078
Line 414: Line 345:
Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580
Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580


Mapping: [{{val| 1 6 10 3 12 18 }}, {{val| 0 -46 -80 -2 -89 -149 }}]
Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }}


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


Optimal GPV sequence: {{Val list| 73f, 125f, 198, 323, 521 }}
{{Optimal ET sequence|legend=1| 73f, 125f, 198, 323, 521 }}


Badness: 0.044
Badness: 0.044


== Supermajor ==
== Crazy ==
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.
: ''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 {{nowrap| 118 & 494 }} temperament. [[1106edo]] is an strong tuning.  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 52734375/52706752
[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}


[[Mapping]]: [{{val|1 15 19 30}}, {{val|0 -37 -46 -75}}]
{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}


{{Multival|legend=1|37 46 75 -13 15 45}}
: mapping generators: ~332150625/234881024, ~1125/1024


[[POTE generator]]: ~9/7 = 435.082
[[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| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}
{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}


[[Badness]]: 0.010836
[[Badness]] (Smith): 0.0394


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


Comma list: 3025/3024, 4375/4374, 35156250/35153041
Comma list: 3025/3024, 4375/4374, 2791309312/2790703125


Mapping: [{{val|2 30 38 60 41}}, {{val|0 -37 -46 -75 -47}}]
Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}


POTE generator: ~9/7 = 435.082
Optimal tunings:
* CTE: ~99/70 = 162.7485, ~1125/1024 = 162.7485
* CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481


Optimal GPV sequence: {{Val list| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}
{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}


Badness: 0.012773
Badness (Smith): 0.0170


== Enneadecal ==
== Orga ==
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.
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, 54975581388800/54936068900769


[[Comma list]]: 4375/4374, 703125/702464
{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }}


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


{{Multival|legend=1|19 19 57 -14 37 79}}
[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104


Mapping generators: ~28/27, ~3
{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}


[[POTE generator]]: ~3/2 = 701.880
[[Badness]]: 0.040236
 
{{Val list|legend=1| 19, 152, 171, 665, 836, 1007, 2185 }}
 
[[Badness]]: 0.010954


=== 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, 5767168/5764801


Mapping: [{{val|19 0 14 -37 126}}, {{val|0 1 1 3 -2}}]
Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }}


POTE generator: ~3/2 = 702.360
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103


Optimal GPV sequence: {{Val list| 19, 152, 323e, 475de, 627de }}
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836, 1106 }}


Badness: 0.043734
Badness: 0.016188


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


Comma list: 540/539, 625/624, 729/728, 2205/2197
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360
 
Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }}


Mapping: [{{val|19 0 14 -37 126 -20}}, {{val|0 1 1 3 -2 3}}]
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103


POTE generator: ~3/2 = 702.212
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836f, 1106f }}


Optimal GPV sequence: {{Val list| 19, 152f, 323e }}
Badness: 0.021762


Badness: 0.033545
== Seniority ==
{{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.


=== Hemienneadecal ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4375/4374, 234375/234256
[[Comma list]]: 4375/4374, 201768035/201326592


Mapping: [{{val|38 0 28 -74 11}}, {{val|0 1 1 3 2}}]
{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}


Mapping generators: ~55/54, ~3
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3087/2560 = 322.804


POTE generator: ~3/2 = 701.881
{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}


Optimal GPV sequence: {{Val list| 152, 342, 494, 836, 1178, 2014 }}
[[Badness]]: 0.044877


Badness: 0.009985
=== 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.


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


Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
Comma list: 441/440, 4375/4374, 65536/65219


Mapping: [{{val|38 0 28 -74 11 502}}, {{val|0 1 1 3 2 -6}}]
Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }}


POTE generator: ~3/2 = 701.986
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793


Optimal GPV sequence: {{Val list| 152, 342, 494, 836 }}
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316e, 487ee }}


Badness: 0.030391
Badness: 0.092238


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


Subgroup: 2.3.5.7
Comma list: 364/363, 441/440, 2200/2197, 4375/4374


[[Comma list]]: 4375/4374, 165288374272/164794921875
Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }}


[[Mapping]]: [{{val|10 4 9 2}}, {{val|0 5 6 11}}]
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793


{{Multival|legend=1|50 60 110 -21 34 87}}
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}


[[POTE generator]]: ~6/5 = 315.577
Badness: 0.044662


{{Val list|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


[[Badness]]: 0.080637
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197


=== 11-limit ===
Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }}
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4375/4374, 422576/421875
Optimal tuning (POTE): ~77/64 = 322.793


Mapping: [{{val|10 4 9 2 18}}, {{val|0 5 6 11 7}}]
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}


POTE generator: ~6/5 = 315.582
Badness: 0.026562


Optimal GPV sequence: {{Val list| 80, 190, 270, 1000, 1270 }}
== Monzismic ==
: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Monzismic]].


Badness: 0.024329
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]].  


=== 13-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13


Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}


Mapping: [{{val|10 4 9 2 18 37}}, {{val|0 5 6 11 7 0}}]
{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}


POTE generator: ~6/5 = 315.602
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207


Optimal GPV sequence: {{Val list| 80, 190, 270, 730, 1000 }}
{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}


Badness: 0.016810
[[Badness]]: 0.046569


== Sfourth ==
=== Monzism ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''
Subgroup: 2.3.5.7.11


Subgroup: 2.3.5.7
Comma list: 4375/4374, 41503/41472, 184549376/184528125


[[Comma list]]: 4375/4374, 64827/64000
Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}


[[Mapping]]: [{{val|1 2 3 3}}, {{val|0 -19 -31 -9}}]
Optimal tuning (POTE): ~231/200 = 249.0193


{{Multival|legend=1|19 31 9 5 -39 -66}}
{{Optimal ET sequence|legend=1| 53, 559, 612 }}


[[POTE generator]]: ~49/48 = 26.287
Badness: 0.057083


{{Val list|legend=1| 45, 46, 91, 137d }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.123291
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625


=== 11-limit ===
Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}
Subgroup: 2.3.5.7.11


Comma list: 121/120, 441/440, 4375/4374
Optimal tuning (POTE): ~231/200 = 249.0199


Mapping: [{{val|1 2 3 3 4}}, {{val|0 -19 -31 -9 -25}}]
{{Optimal ET sequence|legend=1| 53, 559, 612 }}


POTE generator: ~49/48 = 26.286
Badness: 0.053780


Optimal GPV sequence: {{Val list| 45e, 46, 91e, 137de }}
== Semidimfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].''


Badness: 0.054098
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.


==== 13-limit ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 169/168, 325/324, 441/440
[[Comma list]]: 4375/4374, 235298/234375


Mapping: [{{val|1 2 3 3 4 4}}, {{val|0 -19 -31 -9 -25 -14}}]
[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}


POTE generator: ~49/48 = 26.310
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456


Optimal GPV sequence: {{Val list| 45ef, 46, 91ef, 137def }}
{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}


Badness: 0.033067
[[Badness]]: 0.055249


=== Sfour ===
=== Neusec ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 2401/2376, 4375/4374
Comma list: 3025/3024, 4375/4374, 235298/234375


Mapping: [{{val|1 2 3 3 3}}, {{val|0 -19 -31 -9 21}}]
Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }}


POTE generator: ~49/48 = 26.246
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547


Optimal GPV sequence: {{Val list| 45, 46, 91, 137d }}
{{Optimal ET sequence|legend=1| 8d, 190, 388 }}


Badness: 0.076567
Badness: 0.059127


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


Mapping: [{{val|1 2 3 3 3 3}}, {{val|0 -19 -31 -9 21 32}}]
Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }}


POTE generator: ~49/48 = 26.239
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545


Optimal GPV sequence: {{Val list| 45, 46, 91, 137d }}
{{Optimal ET sequence|legend=1| 8d, 190, 198, 388 }}


Badness: 0.051893
Badness: 0.030941


== Abigail ==
== Acrokleismic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 2147483648/2144153025
[[Comma list]]: 4375/4374, 2202927104/2197265625


[[Mapping]]: [{{val|2 7 13 -1}}, {{val|0 -11 -24 19}}]
{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }}


{{Multival|legend=1|22 48 -38 25 -122 -223}}
: mapping generators: ~2, ~6/5


[[POTE generator]]: ~6912/6125 = 208.899
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557


{{Val list|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}
{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}


[[Badness]]: 0.037000
[[Badness]]: 0.056184


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


Comma list: 3025/3024, 4375/4374, 131072/130977
Comma list: 4375/4374, 41503/41472, 172032/171875


Mapping: [{{val|2 7 13 -1 1}}, {{val|0 -11 -24 19 17}}]
Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }}


POTE generator: ~1155/1024 = 208.901
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558


Optimal GPV sequence: {{Val list| 46, 132, 178, 224, 270, 494, 764 }}
{{Optimal ET sequence|legend=1| 19, 251, 270, 829, 1099, 1369, 1639 }}


Badness: 0.012860
Badness: 0.036878


=== 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: 676/675, 1001/1000, 4375/4374, 10985/10976


Mapping: [{{val|2 7 13 -1 1 -2}}, {{val|0 -11 -24 19 17 27}}]
Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }}


POTE generator: ~44/39 = 208.903
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557


Optimal GPV sequence: {{Val list| 46, 178, 224, 270, 494, 764, 1258 }}
{{Optimal ET sequence|legend=1| 19, 251, 270 }}


Badness: 0.008856
Badness: 0.026818


== Semidimi ==
=== Counteracro ===
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].''
 
The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo|-12 -73 55}} and 7-limit 3955078125/3954653486, as well as 4375/4374.
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 4375/4374, 3955078125/3954653486
 
[[Mapping]]: [{{val|1 36 48 61}}, {{val|0 -55 -73 -93}}]
 
{{Multival|legend=1|55 73 93 -12 -7 11}}
 
[[POTE generator]]: ~35/27 = 449.1270
 
{{Val list|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
 
[[Badness]]: 0.015075
 
== Brahmagupta ==
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo|47 -7 -7 -7}} = 140737488355328 / 140710042265625.
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 4375/4374, 70368744177664/70338939985125
 
[[Mapping]]: [{{val|7 2 -8 53}}, {{val|0 3 8 -11}}]
 
{{Multival|legend=1|21 56 -77 40 -181 -336}}
 
[[POTE generator]]: ~27/20 = 519.716
 
{{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


Optimal GPV sequence: {{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


Optimal GPV sequence: {{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]]: {{monzo| 55 -64 20 }}
[[Comma list]]: {{monzo| 55 -64 20 }}


[[Mapping]]: [{{val| 4 0 -11 }}, {{val| 0 5 16 }}]
{{Mapping|legend=1| 4 0 -11 | 0 5 16 }}


Mapping generators: ~51200000/43046721, ~1594323/1280000
: 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, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }}
{{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 }}


{{Multival|legend=1| 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 777: 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: ~5103/4096 = 380.387 (or ~22/21 = 80.387)
Optimal tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387)


Optimal GPV sequence: {{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 790: Line 691:
Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374
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: ~81/65 = 380.385 (or ~22/21 = 80.385)
Optimal tuning (POTE): ~81/65 = 380.385 (or ~22/21 = 80.385)


Optimal GPV sequence: {{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 ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].''
: ''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]], {{monzo| 11 -10 -10 10 }}, neon comma {{monzo| 21 60 -50 }} and {{monzo| 12 -3 -14 9 }} = 165288374272/164794921875 (satritrizo-asepbigu).


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


Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, 165288374272/164794921875


[[Comma list]]: 4375/4374, 235298/234375
{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}


[[Mapping]]: [{{val|1 21 28 36}}, {{val|0 -31 -41 -53}}]
: mapping generators: ~15/14, ~6/5


[[Wedgie]]: {{multival|31 41 53 -7 -3 8}}
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577


[[POTE generator]]: ~35/27 = 448.456
{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}


{{Val list|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}
[[Badness]]: 0.080637


[[Badness]]: 0.055249
Badness (Sintel): 2.041


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


Comma list: 3025/3024, 4375/4374, 235298/234375
Comma list: 3025/3024, 4375/4374, 391314/390625
 
Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}


Mapping: [{{val|2 11 15 19 15}}, {{val|0 -31 -41 -53 -32}}]
Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.582


POTE generator: ~12/11 = 151.547
{{Optimal ET sequence|legend=1| 80, 190, 270, 1000, 1270, 1540e, 1810e }}


Optimal GPV sequence: {{Val list| 8d, 190, 388 }}
Badness: 0.024329


Badness: 0.059127
Badness (Sintel): 0.804


==== 13-limit ====
=== 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)


Optimal GPV sequence: {{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 }}


Optimal GPV sequence: {{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


Optimal GPV sequence: {{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


Optimal GPV sequence: {{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


Optimal GPV sequence: {{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


Optimal GPV sequence: {{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


Optimal GPV sequence: {{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
 
Optimal GPV sequence: {{Val list| 26, 119c, 145, 171, 316ef, 487eef }}
 
Badness: 0.026562
 
== Orga ==
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 54975581388800/54936068900769
[[Comma list]]: 4375/4374, 68719476736/68356598625


[[Mapping]]: [{{val|2 21 36 5}}, {{val|0 -29 -51 1}}]
{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }}


[[Wedgie]]: {{multival|58 102 -2 27 -166 -291}}
: mapping generators: ~2, ~45927/32768


[[POTE generator]]: ~8/7 = 231.104
[[Optimal tuning]] (CTE): ~2 = 1\1, ~45927/32768 = 588.6216


{{Val list|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}
{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}


[[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


Optimal GPV sequence: {{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


Optimal GPV sequence: {{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 1,033: 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


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


Badness: 0.041043
Badness: 0.041043
Line 1,046: 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


Optimal GPV sequence: {{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, 16875/16807


[[Mapping]]: [{{val|8 1 3 3}}, {{val|0 3 4 5}}]
[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}


[[Wedgie]]: {{multival|24 32 40 -5 -4 3}}
{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }}


Mapping generators: ~49/45, ~7/5
: mapping generators: ~2, ~6422528/3796875


[[POTE generator]]: ~7/5 = 583.940
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6422528/3796875 = 910.9323


[[Tuning ranges]]:
{{Optimal ET sequence|legend=1| 494, 1125, 1619 }}
* 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]
* 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 }}
[[Badness]]: 0.234
 
[[Badness]]: 0.042670
 
Scales: [[Octoid72]], [[Octoid80]]


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


Comma list: 540/539, 1375/1372, 4000/3993
Comma list: 4375/4374, 759375/758912, 100663296/100656875


Mapping: [{{val|8 1 3 3 16}}, {{val|0 3 4 5 3}}]
Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }}


POTE generator: ~7/5 = 583.962
Optimal tuning (CTE): ~2 = 1\1, ~1024/605 = 910.9323


Tuning ranges:
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}
* 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]


Optimal GPV sequence: {{Val list| 72, 152, 224 }}
Badness: 0.0678


Badness: 0.014097
=== 13-limit ===
Since 11/8 is within 23 generators, the 25 tone MOS (4L 21s) of this temperament contains the 8:11:13 triad.


Scales: [[Octoid72]], [[Octoid80]]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 540/539, 625/624, 729/728, 1375/1372
Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078


Mapping: [{{val|8 1 3 3 16 -21}}, {{val|0 3 4 5 3 13}}]
Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }}


POTE generator: ~7/5 = 583.905
Optimal tuning (CTE): ~2 = 1\1, ~22/13 = 910.9323


Optimal GPV sequence: {{Val list| 72, 152f, 224 }}
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}


Badness: 0.015274
Badness: 0.0271


Scales: [[Octoid72]], [[Octoid80]]
== Palladium ==
: ''For the 5-limit version of this temperament, see [[46th-octave temperaments]]''.


; Music
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.
* [http://www.archive.org/details/Dreyfus http://www.archive.org/details/Dreyfus] [http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play]


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


Comma list: 375/374, 540/539, 625/624, 715/714, 729/728
[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}


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


POTE generator: ~7/5 = 583.842
: mapping generators: ~83349/81920, ~3


Optimal GPV sequence: {{Val list| 72, 152fg, 224, 296, 520g }}
[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 1\46, ~3/2 = 701.6074


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


Scales: [[Octoid72]], [[Octoid80]]
[[Badness]]: 0.308505


===== 19-limit =====
=== 11-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11


Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
Comma list: 3025/3024, 4375/4374, 134775333/134217728


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 | 0 1 2 -1 -1 }}


POTE generator: ~7/5 = 583.932
Optimal tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951


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


Badness: 0.016036
Badness: 0.073783


Scales: [[Octoid72]], [[Octoid80]]
=== 13-limit ===
 
==== Octopus ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 325/324, 364/363, 540/539
Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364


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


POTE generator: ~7/5 = 583.892
Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6419


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


Badness: 0.021679
Badness: 0.040751


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


Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224


Mapping: [{{val|8 1 3 3 16 14 21}}, {{val|0 3 4 5 3 4 3}}]
Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}


POTE generator: ~7/5 = 583.811
Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425


Optimal GPV sequence: {{Val list| 72, 152, 224fg, 296ffg }}
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334deg }}


Badness: 0.015614
Badness: 0.022441


Scales: [[Octoid72]], [[Octoid80]]
== Oviminor ==
{{See also| Syntonic–kleismic equivalence continuum }}


===== 19-limit =====
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: 2.3.5.7.11.13.17.19


Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
[[Subgroup]]: 2.3.5.7


Mapping: [{{val|8 1 3 3 16 14 21 34}}, {{val|0 3 4 5 3 4 3 0}}]
[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}


POTE generator: ~7/5 = 584.064
{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }}


Optimal GPV sequence: {{Val list| 72, 152, 224fg, 376ffgh }}
: mapping generators: ~2, ~6/5


Badness: 0.016321
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 315.7501


Scales: [[Octoid72]], [[Octoid80]]
{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}


==== Hexadecoid ====
[[Badness]]: 0.582
Hexadecoid (80&amp;144) has a period of 1/16 octave and tempers out 4225/4224.


Subgroup: 2.3.5.7.11.13
== Octoid ==
''For the 5-limit temperament, see [[8th-octave temperaments#Octoid (5-limit)]].''


Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
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|16 26 38 46 56 59}}, {{val|0 -3 -4 -5 -3 1}}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~13/8 = 841.015
[[Comma list]]: 4375/4374, 16875/16807


Optimal GPV sequence: {{Val list| 80, 144, 224 }}
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}


Badness: 0.030818
: mapping generators: ~49/45, ~7/5


===== 17-limit =====
[[Optimal tuning]] ([[POTE]]): ~49/45 = 1\8, ~7/5 = 583.940
Subgroup: 2.3.5.7.11.13.17


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


Mapping: [{{val|16 26 38 46 56 59 65}}, {{val|0 -3 -4 -5 -3 1 2}}]
{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}


POTE generator: ~13/8 = 840.932
[[Badness]]: 0.042670


Optimal GPV sequence: {{Val list| 80, 144, 224, 528dg }}
Scales: [[octoid72]], [[octoid80]]


Badness: 0.028611
=== 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.


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


Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
Comma list: 540/539, 1375/1372, 4000/3993


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 | 0 3 4 5 3 }}


POTE generator: ~13/8 = 840.896
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.962


Optimal GPV sequence: {{Val list| 80, 144, 224, 304dh, 528dghh }}
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]


Badness: 0.023731
{{Optimal ET sequence|legend=1| 72, 152, 224 }}


== Amity ==
Badness: 0.014097
{{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.
Scales: [[octoid72]], [[octoid80]]


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


Subgroup: 2.3.5.7
Comma list: 540/539, 625/624, 729/728, 1375/1372


[[Comma list]]: 4375/4374, 5120/5103
Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}


[[Mapping]]: [{{val| 1 3 6 -2 }}, {{val| 0 -5 -13 17 }}]
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.905


{{Multival|legend=1| 5 13 -17 9 -41 -76 }}
{{Optimal ET sequence|legend=1| 72, 152f, 224 }}


[[POTE generator]]: ~128/105 = 339.432
Badness: 0.015274


{{Val list|legend=1| 7, 32c, 39, 46, 53, 99, 251, 350, 601cd, 951bcdd }}
Scales: [[octoid72]], [[octoid80]]


[[Badness]]: 0.023649
; 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


=== 11-limit ===
===== 17-limit =====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17


Comma list: 540/539, 4375/4374, 5120/5103
Comma list: 375/374, 540/539, 625/624, 715/714, 729/728


Mapping: [{{val| 1 3 6 -2 21 }}, {{val| 0 -5 -13 17 -62 }}]
Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}


POTE generator: ~128/105 = 339.464
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.842


Optimal GPV sequence: {{Val list| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}
{{Optimal ET sequence|legend=1| 72, 152fg, 224, 296, 520g }}


Badness: 0.031506
Badness: 0.014304


==== 13-limit ====
Scales: [[octoid72]], [[octoid80]]
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 540/539, 625/624, 847/845
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 3 6 -2 21 17 }}, {{val| 0 -5 -13 17 -62 -47 }}]
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714


POTE generator: ~128/105 = 339.481
Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}


Optimal GPV sequence: {{Val list| 46ef, 53, 99ef, 152f }} <nowiki>*</nowiki>
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.932


<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
{{Optimal ET sequence|legend=1| 72, 152fg, 224 }}


Badness: 0.028008
Badness: 0.016036


=== Hitchcock ===
Scales: [[octoid72]], [[octoid80]]
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 2200/2187
==== 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}}.


Mapping: [{{val| 1 3 6 -2 6 }}, {{val| 0 -5 -13 17 -9 }}]
Subgroup: 2.3.5.7.11.13


POTE generator: ~11/9 = 339.390
Comma list: 169/168, 325/324, 364/363, 540/539


Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}


Badness: 0.035187
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.892


==== 13-limit ====
{{Optimal ET sequence|legend=1| 72, 152, 224f }}
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 169/168, 176/175, 325/324
Badness: 0.021679


Mapping: [{{val| 1 3 6 -2 6 2 }}, {{val| 0 -5 -13 17 -9 6 }}]
Scales: [[octoid72]], [[octoid80]]


POTE generator: ~11/9 = 339.419
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
Comma list: 169/168, 221/220, 289/288, 325/324, 540/539


Badness: 0.022448
Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}


==== 17-limit ====
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.811
Subgroup: 2.3.5.7.11.13.17


Comma list: 121/120, 154/153, 169/168, 176/175, 273/272
{{Optimal ET sequence|legend=1| 72, 152, 224fg, 296ffg }}


Mapping: [{{val| 1 3 6 -2 6 2 -1 }}, {{val| 0 -5 -13 17 -9 6 18 }}]
Badness: 0.015614


POTE generator: ~11/9 = 339.366
Scales: [[Octoid72]], [[Octoid80]]


Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
===== 19-limit =====
 
Badness: 0.019395
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189
Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399


Mapping: [{{val| 1 3 6 -2 6 2 -1 0 }}, {{val| 0 -5 -13 17 -9 6 18 15 }}]
Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}


POTE generator: ~11/9 = 339.407
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 584.064


Optimal GPV sequence: {{Val list| 7, 39h, 46, 53, 99h }}
{{Optimal ET sequence|legend=1| 72, 152, 224fg, 376ffgh }}


Badness: 0.017513
Badness: 0.016321


=== Catamite ===
Scales: [[Octoid72]], [[Octoid80]]
Subgroup: 2.3.5.7.11


Comma list: 441/440, 896/891, 4375/4374
==== Hexadecoid ====
{{ See also | 16th-octave temperaments }}


Mapping: [{{val|1 3 6 -2 -7}}, {{val|0 -5 -13 17 37}}]
Hexadecoid (80 &amp; 144) has a period of 1/16 octave and tempers out 4225/4224.


POTE generator: ~128/105 = 339.340
Subgroup: 2.3.5.7.11.13


Optimal GPV sequence: {{Val list| 46, 99e, 145, 244e }}
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224


Badness: 0.040976
Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}


==== 13-limit ====
: mapping generators: ~448/429, ~7/5
Subgroup: 2.3.5.7.11.13


Comma list: 196/195, 352/351, 364/363, 4375/4374
Optimal tuning (POTE): ~448/429 = 1\16, ~13/8 = 841.015


Mapping: [{{val|1 3 6 -2 -7 -11}}, {{val|0 -5 -13 17 37 52}}]
{{Optimal ET sequence|legend=1| 80, 144, 224 }}


POTE generator: ~128/105 = 339.313
Badness: 0.030818


Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
===== 17-limit =====
 
Badness: 0.034215
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224


Mapping: [{{val|1 3 6 -2 -7 -11 -1}}, {{val|0 -5 -13 17 37 52 18}}]
Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}


POTE generator: ~17/14 = 339.313
Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932


Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
{{Optimal ET sequence|legend=1| 80, 144, 224, 528dg }}


Badness: 0.021193
Badness: 0.028611


==== 19-limit ====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444


Mapping: [{{val|1 3 6 -2 -7 -11 -1 -13}}, {{val|0 -5 -13 17 37 52 18 61}}]
Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }}


POTE generator: ~17/14 = 339.325
Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896


Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
{{Optimal ET sequence|legend=1| 80, 144, 224, 304dh, 528dghh }}


Badness: 0.018864
Badness: 0.023731
 
=== Hemiamity ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 5120/5103
 
Mapping: [{{val| 2 1 -1 13 13 }}, {{val| 0 5 13 -17 -14 }}]
 
Mapping generators: ~99/70, ~64/55
 
POTE generator: ~64/55 = 260.561
 
Optimal GPV sequence: {{Val list| 14cde, 46, 106, 152, 350, 502d }}
 
Badness: 0.031307
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 847/845, 1716/1715, 3025/3024
 
Mapping: [{{val| 2 1 -1 13 13 20 }}, {{val| 0 5 13 -17 -14 -29 }}]
 
POTE generator: ~64/55 = 260.583
 
Optimal GPV sequence: {{Val list| 46, 106f, 152f, 198, 350f, 548cdff }}
 
Badness: 0.025784


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


[[POTE generator]]: ~6/5 = 315.240
: mapping generators: ~2, ~6/5


{{Val list|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.240
 
{{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 1,456: 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


Optimal GPV sequence: {{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 1,471: 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


Optimal GPV sequence: {{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 1,484: 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


Optimal GPV sequence: {{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 1,499: 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


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


Badness: 0.037618
Badness: 0.037618
Line 1,512: 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


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


Badness: 0.055884
Badness: 0.055884
Line 1,525: 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


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


Badness: 0.036559
Badness: 0.036559
Line 1,538: 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


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


Badness: 0.041720
Badness: 0.041720
Line 1,551: 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


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


Badness: 0.035781
Badness: 0.035781
Line 1,564: 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


Optimal GPV sequence: {{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 1,579: 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


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


Badness: 0.033775
Badness: 0.033775
Line 1,594: 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


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


Badness: 0.040467
Badness: 0.040467


== Counterkleismic ==
== Counterkleismic ==
{{see also| High badness temperaments #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 1,626: 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


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


Badness: 0.070952
Badness: 0.070952
Line 1,639: 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


Optimal GPV sequence: {{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 1,652: 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


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


Badness: 0.065400
Badness: 0.065400
Line 1,665: 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


Optimal GPV sequence: {{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 1,693: 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


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


Badness: 0.030875
Badness: 0.030875
Line 1,706: 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


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


Badness: 0.023862
Badness: 0.023862
Line 1,719: 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


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


Badness: 0.014741
Badness: 0.014741
Line 1,732: 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


Optimal GPV sequence: {{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


Optimal GPV sequence: {{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


Optimal GPV sequence: {{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. Not only the semitwelfth, but also the ~5/4 can be used as a generator.
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 0 26 }}, {{val| 0 2 1 }}]
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


Mapping generators: ~25/24, ~{{monzo| 26 9 -17 }}
Badness: 0.076567


[[POTE generator]]: ~{{monzo| 26 9 -17 }} = 950.9746
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}
Comma list: 196/195, 364/363, 385/384, 4375/4374


[[Badness]]: 0.077072
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}


=== 7-limit ===
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.239
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 193119049072265625/193091834023510016
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


[[Mapping]]: [{{val| 17 0 26 -87 }}, {{val| 0 2 1 10 }}]
Badness: 0.051893


{{Multival|legend=1| 34 17 170 -52 174 347 }}
== Trideci ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tridecatonic]].''


[[POTE generator]]: ~822083584/474609375 = 950.9995
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]]").


{{Val list|legend=1| 289, 323, 612, 935, 1547 }}
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.041658
[[Comma list]]: 4375/4374, 83349/81920


=== 11-limit ===
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
Subgroup: 2.3.5.7.11


Comma list: 4375/4374, 41503/41472, 1879453125/1879048192
[[Optimal tuning]] ([[POTE]]): ~256/245 = 1\13, ~3/2 = 699.1410


Mapping: [{{val| 17 0 26 -87 207 }}, {{val| 0 2 1 10 -11 }}]
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }}


POTE generators: ~822083584/474609375 = 950.9749
[[Badness]]: 0.184585
 
Optimal GPV sequence: {{Val list| 289, 323, 612 }}
 
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


Optimal GPV sequence: {{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


Optimal GPV sequence: {{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
 
Mapping: [{{val|46 73 107 129 159 170 188}}, {{val|0 -1 -2 1 1 2 0}}]
 
POTE generator: ~3/2 = 701.6425
 
Optimal GPV sequence: {{Val list| 46, 368, 414, 460, 874de, 1334deg }}


Badness: 0.022441
== 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]].  
== 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
 
Optimal GPV sequence: {{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
 
Optimal GPV sequence: {{Val list| 53, 559, 612 }}
 
Badness: 0.053780


[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Ragismic microtemperaments| ]] <!-- main article -->
[[Category:Ragismic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Microtemperaments]]
[[Category:Abigail]]
[[Category:Abigail]]
[[Category:Amity]]
[[Category:Deca]]
[[Category:Deca]]
[[Category:Enneadecal]]
[[Category:Enneadecal]]
Line 1,942: Line 1,723:
[[Category:Quincy]]
[[Category:Quincy]]
[[Category:Supermajor]]
[[Category:Supermajor]]
[[Category:Microtemperaments]]
[[Category:Ragismic]]
[[Category:Rank 2]]
[[Category:Temperament collections]]
[[Category:Ragismic microtemperaments| ]] <!-- main article -->
Retrieved from "https://en.xen.wiki/w/Ragismic_microtemperaments"