Ragismic microtemperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
m +whirrschmidt to the list
 
(214 intermediate revisions by 20 users not shown)
Line 1: Line 1:
The ragisma is [[4375/4374]] with a [[monzo]] of |-1 -7 4 1>, the smallest 7-limit [[superparticular]] ratio. Since (10/9)^4=4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[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)^2, 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 not discussed here include [[Meantone family #Flattone|flattone]], [[Porcupine family #Hystrix|hystrix]], [[Starling temperaments #Sensi|sensi]], [[Gamelismic clan #Unidec|unidec]], [[Orwellismic temperaments #Quartonic|quartonic]], [[Kleismic family #Catakleismic|catakleismic]], [[Tetracot family #Modus|modus]], [[Schismatic family #Pontiac|pontiac]], [[Würschmidt family #Whirrschmidt|whirrschmidt]], [[Vishnuzmic family #Vishnu|vishnu]], and [[Vulture family #Vulture|vulture]].  
Since {{nowrap|(10/9)<sup>4</sup> {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)<sup>2</sup> }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.


=Ennealimmal=
Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
Ennealimmal temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the ennealimmal comma, |1 -27 18&gt;, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is &lt;&lt;18 27 18 1 -22 -34||.
* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
* ''[[Modus]]'' (+64/63) → [[Tetracot family #Modus|Tetracot family]]
* ''[[Flattone]]'' (+81/80) → [[Meantone family #Flattone|Meantone family]]
* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]
* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]]
* [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]
* [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]]
* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]
* ''[[Zarvo]]'' (+33075/32768) → [[Gravity family #Zarvo|Gravity family]]
* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortonic family #Mitonic|Minortonic family]]
* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnuzmic family #Septimal vishnu|Vishnuzmic family]]
* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]
* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]]
* ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]]
* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]]
* ''[[Chlorine]]'' (+{{monzo| -52 -17 34}}) → [[17th-octave temperaments #Chlorine|17th-octave temperaments]]
* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]
* ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments#Dzelic|37th-octave temperaments]]


Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 EDOs, though its hardly likely anyone could tell the difference.
== Supermajor ==
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2<sup>15</sup>)/3, 46 give (2<sup>19</sup>)/5, and 75 give (2<sup>30</sup>)/7. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80-note mos is presumably the place to start, and if that is not enough notes for you, there is always the 171-note mos.


If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example.) In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS.
[[Subgroup]]: 2.3.5.7


[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [26.667, 66.667] (45bcd to 18bcd)
[[Comma list]]: 4375/4374, 52734375/52706752


nice range: [48.920, 49.179]
{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }}


strict range: [48.920, 49.179]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082


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


POTE generators: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980
[[Badness]]: 0.010836


Map: [&lt;9 1 1 2|, &lt;0 2 3 2|]
=== Semisupermajor ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;18 27 18 1 -22 -34||
Comma list: 3025/3024, 4375/4374, 35156250/35153041


EDOs: [[27edo|27]], [[45edo|45]], [[72edo|72]], [[99edo|99]], [[171edo|171]], [[270edo|270]], [[441edo|441]], [[612edo|612]], [[3600edo|3600]]
Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}


Badness: 0.00361
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082


==Hemiennealimmal==
{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}
Commas: 2401/2400, 4375/4374, 3025/3024


valid range: [13.333, 22.222] (90bcd, 54c)
Badness: 0.012773


nice range: [17.304, 17.985]
== 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.


strict range:  [17.304, 17.985]
''For the 5-limit temperament, see [[19th-octave temperaments#(5-limit) enneadecal]].''


POTE generator: ~99/98 = 17.6219
[[Subgroup]]: 2.3.5.7


Map: [&lt;18 0 -1 22 48|, &lt;0 2 3 2 1|]
[[Comma list]]: 4375/4374, 703125/702464


EDOs: 72, 198, 270, 342, 612, 954, 1566
{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}


Badness: 0.00628
: mapping generators: ~28/27, ~3


===13-limit===
[[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)
Commas: 676/675, 1001/1000, 1716/1715, 3025/3024


valid range: [16.667, 22.222] (72 to 54cf)
{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}


nice range: [17.304, 18.309]
[[Badness]]: 0.010954


strict range: [17.304, 18.309]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator ~99/98 = 17.7504
Comma list: 540/539, 4375/4374, 16384/16335


Map: [&lt;18 0 -1 22 48 -19|, &lt;0 2 3 2 1 6|]
Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}


EDOs: 72, 198, 270
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)


Badness: 0.0125
{{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }}


=== Semihemiennealimmal ===
Badness: 0.043734
Commas: 2401/2400, 4375/4374, 3025/3024, 4225/4224


POTE generator: ~39/32 = 342.139
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;18 0 -1 22 48 88|, &lt;0 4 6 4 2 -3|]
Comma list: 540/539, 625/624, 729/728, 2205/2197


EDOs: 126, 144, 270, 684, 954
Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}


Badness: 0.0131
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)


==Semiennealimmal==
{{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }}
Commas: 2401/2400, 4375/4374, 4000/3993


POTE generator: ~140/121 = 250.3367
Badness: 0.033545


Map: [&lt;9 3 4 14 18|, &lt;0 6 9 6 7|]
=== Hemienneadecal ===
Subgroup: 2.3.5.7.11


EDOs: 72, 369, 441
Comma list: 3025/3024, 4375/4374, 234375/234256


Badness: 0.0342
Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}


===13-limit===
: mapping generators: ~55/54, ~3
Commas: 1575/1573, 2080/2079, 2401/2400, 4375/4374


POTE generator: ~140/121 = 250.3375
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)


Map: [&lt;9 3 4 14 18 -8|, &lt;0 6 9 6 7 22|]
{{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}


EDOs: 72, 441
Badness: 0.009985


Badness: 0.0261
==== Hemienneadecalis ====
Subgroup: 2.3.5.7.11.13


==Quadraennealimmal==
Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256
Commas: 2401/2400, 4375/4374, 234375/234256


POTE generator: ~77/75 = 45.595
Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}


Map: [&lt;9 1 1 12 -7|, &lt;0 8 12 8 23|]
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)


EDOs: 342, 1053, 1395, 1737, 4869d, 6606cd
{{Optimal ET sequence|legend=1| 152f, 342f, 494 }}


Badness: 0.0213
Badness: 0.020782
 
==== Hemienneadec ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
 
Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}
 
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444)
 
{{Optimal ET sequence|legend=1| 152, 342, 494, 1330, 1824, 2318d }}
 
Badness: 0.030391
 
==== Semihemienneadecal ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078
 
Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}
 
: mapping generators: ~55/54 = 1\38, ~55/54, ~429/250
 
Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)
 
{{Optimal ET sequence|legend=1| 190, 304d, 494, 684, 1178, 2850, 4028ce }}
 
Badness: 0.014694
 
=== Kalium ===
Named after the 19th element, potassium, and after an archaic variant of the element's name to resolve a name conflict. [[19/16]] can be used as a generator. Since it is enfactored in the 17-limit and lower, it makes no sense to name it for the lower subgroups.
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344
 
Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}
 
Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244
 
{{Optimal ET sequence|legend=1| 855, 988, 1843 }}
 
== Semidimi ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].''
 
The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 3955078125/3954653486
 
{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270
 
{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
 
[[Badness]]: 0.015075
 
== Brahmagupta ==
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.
 
Early in the design of the [[Sagittal]] notation system, Secor and Keenan found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4 ¢ many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of Brahmagupta temperament that has pure octaves and pure fifths, which can also be described as a 17-limit extension having 1/7th octave period (171.4286 ¢) and 1/21st apotome generator (5.4136 ¢).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 70368744177664/70338939985125
 
{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}
 
: mapping generators: ~1157625/1048576, ~27/20
 
[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716
 
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }}
 
[[Badness]]: 0.029122
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4000/3993, 4375/4374, 131072/130977
 
Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}
 
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704
 
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }}
 
Badness: 0.052190
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374
 
Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}
 
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.706
 
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771eef }}
 
Badness: 0.023132
 
== Abigail ==
Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930]: "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things."</ref>
 
''For the 5-limit temperament, see [[Very high accuracy temperaments#Abigail]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 2147483648/2144153025
 
{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }}
 
: mapping generators: ~46305/32768, ~27/20
 
[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899
 
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}
 
[[Badness]]: 0.037000
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 131072/130977
 
Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901
 
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }}
 
Badness: 0.012860
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==Ennealimnic==
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
Commas: 243/242, 441/440, 4375/4356


valid range: [44.444, 53.333] (27e to 45e)
Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}


nice range: [48.920, 52.592]
Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903


strict range: [48.920, 52.592]
{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }}


POTE generator: ~36/35 = 49.395
Badness: 0.008856


Map: [&lt;9 1 1 12 -2|, &lt;0 2 3 2 5|]
== Gamera ==
''For the 5-limit temperament, see [[High badness temperaments#Gamera]].


EDOs: 72, 171, 243
[[Subgroup]]: 2.3.5.7


Badness: 0.0203
[[Comma list]]: 4375/4374, 589824/588245


===13-limit===
{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }}
Commas: 243/242, 364/363, 441/440, 625/624


valid range: [48.485, 50.000] (99ef to 72)
: mapping generators: ~2, ~8/7


nice range: [48.825, 52.592]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336


strict range: [48.825, 50.000]
{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}


POTE generator: ~36/35 = 49.341
[[Badness]]: 0.037648


Map: [&lt;9 1 1 12 -2 -33|, &lt;0 2 3 2 5 10|]
=== Hemigamera ===
Subgroup: 2.3.5.7.11


EDOs: 72, 171, 243
Comma list: 3025/3024, 4375/4374, 589824/588245


Badness: 0.0233
Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}


==== 17-limit ====
: mapping generators: ~99/70, ~8/7
Commas: 243/242, 364/363, 375/374, 441/440, 595/594


valid range: [48.485, 50.000] (99ef to 72)
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3370


nice range: [46.363, 52.592]
{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646, 1068d }}


strict range: [48.485, 50.000]
Badness: 0.040955


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


Map: [&lt;9 1 1 12 -2 -33 -3|, &lt;0 2 3 2 5 10 6|]
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024


EDOs: 72, 171, 243
Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}


Badness: 0.0146
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373


=== Ennealim ===
{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646f, 1068df }}
Commas: 169/168, 243/242, 325/324, 441/440


POTE generator: ~36/35 = 49.708
Badness: 0.020416


Map: [&lt;9 1 1 12 -2 20|, &lt;0 2 3 2 5 2|]
=== Semigamera ===
Subgroup: 2.3.5.7.11


EDOs: 27e, 45ef, 72, 315ff, 387cff, 459cdfff
Comma list: 4375/4374, 14641/14580, 15488/15435


Badness: 0.0207
Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }}


==Ennealiminal==
: mapping generators: ~2, ~77/72
Commas: 385/384, 1375/1372, 4375/4374


POTE generator: ~36/35 = 49.504
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1642


Map: [&lt;9 1 1 12 51|, &lt;0 2 3 2 -3|]
{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }}


EDOs: 27, 45, 72, 171e, 243e, 315e
Badness: 0.078


Badness: 0.0311
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


===13-limit===
Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580
Commas: 169/168, 325/324, 385/384, 1375/1372


POTE generator: ~36/35 = 49.486
Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }}


Map: [&lt;9 1 1 12 51 20|, &lt;0 2 3 2 -3 2|]
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1628


EDOs: 27, 45f, 72, 171ef, 243ef
{{Optimal ET sequence|legend=1| 73f, 125f, 198, 323, 521 }}


Badness: 0.0303
Badness: 0.044


==Trinealimmal==
== Crazy ==
Commas: 2401/2400, 4375/4374, 2097152/2096325
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


POTE generator: ~6/5 = 315.644
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.  


Map: [&lt;27 1 0 34 177|, &lt;0 2 3 2 -4|]
[[Subgroup]]: 2.3.5.7


EDOs: 27, 243, 270, 783, 1053, 1323, 10854bcde
[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}


Badness: 0.0298
{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}


=Gamera=
: mapping generators: ~332150625/234881024, ~1125/1024
Commas: 4375/4374, 589824/588245


POTE generator ~8/7 = 230.336
[[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 }}


Map: [&lt;1 6 10 3|, &lt;0 -23 -40 -1|]
{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}


EDOs: 26, 73, 99, 224, 323, 422, 735
[[Badness]] (Smith): 0.0394


Badness: 0.0376
=== 11-limit ===
Subgroup: 2.3.5.7.11


==Hemigamera==
Comma list: 3025/3024, 4375/4374, 2791309312/2790703125
Commas: 3025/3024, 4375/4374, 202397184/201768035


POTE generator: ~8/7 = 230.337
Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}


Map: [&lt;2 12 20 6 5|, &lt;0 -23 -40 -1 5|]
Optimal tunings:
* CTE: ~99/70 = 162.7485, ~1125/1024 = 162.7485
* CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481


EDOs: 26, 198, 224, 422, 646, 1068d
{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}


Badness: 0.0410
Badness (Smith): 0.0170


===13-limit===
== Orga ==
Commas: 1716/1715 2080/2079 2200/2197 3025/3024
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 12 20 6 5 17|, &lt;0 -23 -40 -1 5 -25|]
[[Comma list]]: 4375/4374, 54975581388800/54936068900769


EDOs: 26, 198, 224, 422, 646f, 1068df
{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }}


Badness: 0.0204
: mapping generators: ~7411887/5242880, ~1310720/1058841


=Supermajor=
[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.0002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of &lt;&lt;37 46 75 -13 15 45||. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.


Commas: 4375/4374, 52734375/52706752
{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}


POTE generator: ~9/7 = 435.082
[[Badness]]: 0.040236


Map: [&lt;1 15 19 30|, &lt;0 -37 -46 -75|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214
Comma list: 3025/3024, 4375/4374, 5767168/5764801


Badness: 0.0108
Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }}


==Semisupermajor==
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
Commas: 3025/3024, 4375/4374, 35156250/35153041


POTE generator: ~9/7 = 435.082
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836, 1106 }}


Map: [&lt;2 30 38 60 41|, &lt;0 -37 -46 -75 -47|]
Badness: 0.016188


EDOs: 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0128
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360


=Enneadecal=
Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }}
Enndedecal temperament tempers out the enneadeca, |-14 -19 19&gt;, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5 or 7 limits, and [[494edo]] shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.


Commas: 4375/4374, 703125/702464
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103


POTE generator: ~3/2 = 701.880
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836f, 1106f }}


Map: [&lt;19 0 14 -37|, &lt;0 1 1 3|]
Badness: 0.021762


Generators: 28/27, 3
== Seniority ==
{{See also| Very high accuracy temperaments #Senior }}


EDOs: 19, 152, 171, 665, 836, 1007, 2185
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.


Badness: 0.0110
[[Subgroup]]: 2.3.5.7


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


POTE generator: ~3/2 = 701.881
{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}


Map: [&lt;38 0 28 -74 11|, &lt;0 1 1 3 2|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3087/2560 = 322.804


EDOs: 152, 342, 494, 836, 1178, 2014
{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}


Badness: 0.00999
[[Badness]]: 0.044877


===13-limit===
=== Senator ===
Commas: 3025/3024, 4096/4095, 4375/4374, 31250/31213
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.


POTE generator: ~3/2 = 701.986
Subgroup: 2.3.5.7.11


Map: [&lt;38 0 28 -74 11 502|, &lt;0 1 1 3 2 -6|]
Comma list: 441/440, 4375/4374, 65536/65219


EDOs: 152, 342, 494, 836
Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }}


Badness: 0.0304
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793


=Deca=
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316e, 487ee }}
Commas: 4375/4374, 165288374272/164794921875


POTE generator: ~460992/390625 = 284.423
Badness: 0.092238


Map: [&lt;10 4 2 9|, &lt;0 5 6 11|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 80, 190, 270, 1270, 1540, 1810, 2080
Comma list: 364/363, 441/440, 2200/2197, 4375/4374


Badness: 0.0806
Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }}


==11-limit==
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
Commas: 3025/3024, 4375/4374, 422576/421875


POTE generator: ~33/28 = 284.418
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}


Map: [&lt;10 4 2 9 18|, &lt;0 5 6 11 7|]
Badness: 0.044662


EDOs: 80, 190, 270, 1000, 1270
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0243
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197


==13-limit==
Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }}
Commas: 1001/1000, 3025/3024, 4225/4224, 4375/4374


POTE generator: ~33/28 = 284.398
Optimal tuning (POTE): ~77/64 = 322.793


Map: [&lt;10 4 2 9 18 37|, &lt;0 5 6 11 7 0|]
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}


EDOs: 80, 190, 270, 730, 1000
Badness: 0.026562


Badness: 0.0168
== Monzismic ==
: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Monzismic]].  


= Mitonic =
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]].
{{see also|Minortonic family #Mitonic}}


Commas: 4375/4374, 2100875/2097152
[[Subgroup]]: 2.3.5.7


POTE generator: ~10/9 = 182.458
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}


Map: [&lt;1 16 32 -15|, &lt;0 -17 -35 21|]
{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}


EDOs: 46, 125, 171
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207


Badness: 0.0252
{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}


=Abigail=
[[Badness]]: 0.046569
Commas: 4375/4374, 2147483648/2144153025


[[POTE_tuning|POTE generator]]: 208.899
=== Monzism ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 7 13 -1|, &lt;0 -11 -24 19|]
Comma list: 4375/4374, 41503/41472, 184549376/184528125


Wedgie: &lt;&lt;22 48 -38 25 -122 -223||
Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}


EDOs: 46, 132, 178, 224, 270, 494, 764, 1034, 1798
Optimal tuning (POTE): ~231/200 = 249.0193


Badness: 0.0370
{{Optimal ET sequence|legend=1| 53, 559, 612 }}


==11-limit==
Badness: 0.057083
Comma: 3025/3024, 4375/4374, 20614528/20588575


[[POTE_tuning|POTE generator]]: 208.901
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 7 13 -1 1|, &lt;0 -11 -24 19 17|]
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625


EDOs: 46, 132, 178, 224, 270, 494, 764
Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}


Badness: 0.0129
Optimal tuning (POTE): ~231/200 = 249.0199


==13-limit==
{{Optimal ET sequence|legend=1| 53, 559, 612 }}
Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095


[[POTE_tuning|POTE generator]]: 208.903
Badness: 0.053780


Map: [&lt;2 7 13 -1 1 -2|, &lt;0 -11 -24 19 17 27|]
== Semidimfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].''


EDOs: 46, 178, 224, 270, 494, 764, 1258
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.


Badness: 0.00886
[[Subgroup]]: 2.3.5.7


=Semidimi=
[[Comma list]]: 4375/4374, 235298/234375
The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit |-12 -73 55&gt; and 7-limit 3955078125/3954653486, as well as 4375/4374.


Comma: |-12 -73 55&gt;
[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}


POTE generator: ~162/125 = 449.127
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456


Map: [&lt;1 36 48|, &lt;0 -55 -73|]
{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}


Wedgie: &lt;&lt;55 73 -12||
[[Badness]]: 0.055249


EDOs: 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419
=== Neusec ===
Subgroup: 2.3.5.7.11


Badness: 0.7549
Comma list: 3025/3024, 4375/4374, 235298/234375


==7-limit==
Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }}
Commas: 4375/4374, 3955078125/3954653486


POTE generator: ~35/27 = 449.127
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547


Map: [&lt;1 36 48 61|, &lt;0 -55 -73 -93|]
{{Optimal ET sequence|legend=1| 8d, 190, 388 }}


Wedgie: &lt;&lt;55 73 93 -12 -7 11||
Badness: 0.059127


EDOs: 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0151
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374


=Brahmagupta=
Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }}
Commas: 4375/4374, 70368744177664/70338939985125


POTE generator: ~27/20 = 519.716
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545


Map: [&lt;7 2 -8 53|, &lt;0 3 8 -11|]
{{Optimal ET sequence|legend=1| 8d, 190, 198, 388 }}


Wedgie: &lt;&lt;21 56 -77 40 -181 -336||
Badness: 0.030941


EDOs: 217, 224, 441, 1106, 1547
== Acrokleismic ==
[[Subgroup]]: 2.3.5.7


Badness: 0.0291
[[Comma list]]: 4375/4374, 2202927104/2197265625


==11-limit==
{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }}
Commas: 4000/3993, 4375/4374, 131072/130977


POTE generator: ~27/20 = 519.704
: mapping generators: ~2, ~6/5


Map: [&lt;7 2 -8 53 3|, &lt;0 3 8 -11 7|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557


EDOs: 217, 224, 441, 665, 1771ee
{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}


Badness: 0.0522
[[Badness]]: 0.056184


==13-limit==
=== 11-limit ===
Commas: 1575/1573, 2080/2079, 4096/4095, 4375/4374
Subgroup: 2.3.5.7.11


POTE generator: ~27/20 = 519.706
Comma list: 4375/4374, 41503/41472, 172032/171875


Map: [&lt;7 2 -8 53 3 35|, &lt;0 3 8 -11 7 -3|]
Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }}


EDOs: 217, 224, 441, 665, 1771eef
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558


Badness: 0.0231
{{Optimal ET sequence|legend=1| 19, 251, 270, 829, 1099, 1369, 1639 }}


=Quasithird=
Badness: 0.036878
Comma: |55 -64 20&gt;


POTE generator: ~1594323/1280000 = 380.395
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;4 0 -11|, &lt;0 5 16|]
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976


Wedgie: &lt;&lt;20 64 55||
Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }}


EDOs: 164, 224, 388, 612, 836, 1000, 1448, 1612, 2224, 2836
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557


Badness: 0.0995
{{Optimal ET sequence|legend=1| 19, 251, 270 }}


==7-limit==
Badness: 0.026818
Commas: 4375/4374, 1153470752371588581/1152921504606846976


POTE generator: ~5103/4096 = 380.388
=== Counteracro ===
Subgroup: 2.3.5.7.11


Map: [&lt;4 0 -11 48|, &lt;0 5 16 -29|]
Comma list: 4375/4374, 5632/5625, 117649/117612


Wedgie: &lt;&lt;20 64 -116 55 -240 -449||
Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }}


EDOs: 164, 224, 388, 612, 1448, 2060
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.553


Badness: 0.0618
{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}


==11-limit==
Badness: 0.042572
Commas: 3025/3024, 4375/4374, 4296700485/4294967296


POTE generator: ~5103/4096 = 380.387
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;4 0 -11 48 43|, &lt;0 5 16 -29 -23|]
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374


EDOs: 164, 224, 388, 612, 836, 1448
Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }}


Badness: 0.0211
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.554


==13-limit==
{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}
Commas: 2200/2197, 3025/3024, 4375/4374, 468512/468195


POTE generator: ~5103/4096 = 380.385
Badness: 0.026028


Map: [&lt;4 0 -11 48 43 11|, &lt;0 5 16 -29 -23 3|]
== 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}}.


EDOs: 164, 224, 388, 612, 836, 1448f, 2284f
[[Subgroup]]: 2.3.5


Badness: 0.0295
[[Comma list]]: {{monzo| 55 -64 20 }}


=Semidimfourth=
{{Mapping|legend=1| 4 0 -11 | 0 5 16 }}
Comma: |7 41 -31&gt;


POTE generator: ~162/125 = 448.449
: mapping generators: ~51200000/43046721, ~1594323/1280000


Map: [&lt;1 21 28|, &lt;0 -31 -41|]
[[Optimal tuning]] ([[POTE]]): ~51200000/43046721, ~1594323/1280000 = 380.395


Wedgie: &lt;&lt;31 41 -7||
{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }}


EDOs: 91, 99, 190, 289, 388, 487, 677, 875, 966
[[Badness]]: 0.099519


Badness: 0.1930
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


==7-limit==
[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}
Commas: 4375/4374, 235298/234375


POTE generator: ~35/27 = 448.457
{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}


Map: [&lt;1 21 28 36|, &lt;0 -31 -41 -53|]
[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 1\4, ~5103/4096 = 380.388


Wedgie: &lt;&lt;31 41 53 -7 -3 8||
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}


EDOs: 91, 99, 289, 388, 875, 1263d, 1651d
[[Badness]]: 0.061813


Badness: 0.0552
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Neusec ==
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
Commas: 3025/3024, 4375/4374, 235298/234375


POTE generator: ~12/11 = 151.547
Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}


Map: [&lt;2 11 15 19 15|, &lt;0 -31 -41 -53 -32|]
Optimal tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387)


EDOs: 190, 388
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448 }}


Badness: 0.0591
Badness: 0.021125


=== 13-limit ===
=== 13-limit ===
Commas: 847/845, 1001/1000, 3025/3024, 4375/4374
Subgroup: 2.3.5.7.11.13
 
Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374
 
Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}
 
Optimal tuning (POTE): ~81/65 = 380.385 (or ~22/21 = 80.385)
 
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}
 
Badness: 0.029501
 
== Deca ==
: ''For 5-limit version of this temperament, see [[10th-octave temperaments #Neon]].''
 
Deca temperament has a period of 1/10 octave and tempers out the [[linus comma]], {{monzo| 11 -10 -10 10 }}, neon comma {{monzo| 21 60 -50 }} and {{monzo| 12 -3 -14 9 }} = 165288374272/164794921875 (satritrizo-asepbigu).


POTE generator: ~12/11 = 151.545
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 11 15 19 15 17|, &lt;0 -31 -41 -53 -32 -38|]
[[Comma list]]: 4375/4374, 165288374272/164794921875


EDOs: 190, 198, 388
{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}


Badness: 0.0309
: mapping generators: ~15/14, ~6/5


=Acrokleismic=
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577
Commas: 4375/4374, 2202927104/2197265625


POTE generator: ~6/5 = 315.557
{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}


Map: [&lt;1 10 11 27|, &lt;0 -32 -33 -92|]
[[Badness]]: 0.080637


Wedgie: &lt;&lt;32 33 92 -22 56 121||
Badness (Sintel): 2.041


EDOs: 19, 251, 270
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0562
Comma list: 3025/3024, 4375/4374, 391314/390625


==11-limit==
Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}
Commas: 4375/4374, 41503/41472, 172032/171875


POTE generator: ~6/5 = 315.558
Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.582


Map: [&lt;1 10 11 27 -16|, &lt;0 -32 -33 -92 74|]
{{Optimal ET sequence|legend=1| 80, 190, 270, 1000, 1270, 1540e, 1810e }}


EDOs: 19, 251, 270, 829, 1099, 1369, 1639
Badness: 0.024329


Badness: 0.0369
Badness (Sintel): 0.804


=== 13-limit ===
=== 13-limit ===
Commas: 676/675, 1001/1000, 4375/4374, 10985/10976
Subgroup: 2.3.5.7.11.13


POTE generator: ~6/5 = 315.557
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374


Map: [&lt;1 10 11 27 -16 25|, &lt;0 -32 -33 -92 74 -81|]
Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}


EDOs: 19, 251, 270
Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.602 (~40/39 = 44.398)


Badness: 0.0268
{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}


==Counteracro==
Badness: 0.016810
Commas: 4375/4374, 5632/5625, 117649/117612


POTE generator: ~6/5 = 315.553
Badness (Sintel): 0.695


Map: [&lt;1 10 11 27 55|, &lt;0 -32 -33 -92 -196|]
=== no-17's 19-limit ===
Subgroup: 2.3.5.7.11.13.19


EDOs: 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374, 1521/1520


Badness: 0.0426
Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}


===13-limit===
Optimal tuning (CTE): ~15/14 = 1\10, ~6/5 = 315.581 (~39/38 = 44.419)
Commas: 676/675, 1716/1715, 4225/4224, 4375/4374


POTE generator: ~6/5 = 315.554
{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}


Map: [&lt;1 10 11 27 55 25|, &lt;0 -32 -33 -92 -196 -81|]
Badness (Sintel): 0.556


EDOs: 270, 1331c, 1601c, 1871bcf, 2141bcf
== Keenanose ==
Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.


Badness: 0.0260
[[Subgroup]]: 2.3.5.7


=Seniority=
[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}
Commas: 4375/4374, 201768035/201326592


POTE generator: ~3087/2560 = 322.804
{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}


Map: [&lt;1 11 19 2|, &lt;0 -35 -62 3|]
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}


Wedgie: &lt;&lt;35 62 -3 17 -103 -181||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 21 3 1 -10 }} = 4.4465


EDOs: 26, 145, 171, 2710d
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}


Badness: 0.0449
[[Badness]]: 0.0858


=Orga=
=== 11-limit ===
Commas: 4375/4374, 54975581388800/54936068900769
Subgroup: 2.3.5.7.11


POTE generator: ~8/7 = 231.104
Comma list: 4375/4374, 117649/117612, 67110351/67108864


Map: [&lt;2 21 36 5|, &lt;0 -29 -51 1|]
Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}


Wedgie: &lt;&lt;58 102 -2 27 -166 -291||
Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4465


EDOs: 26, 244, 270, 836, 1106, 1376, 2482
{{Optimal ET sequence|legend=1| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}


Badness: 0.0402
Badness: 0.0308


==11-limit==
=== 13-limit ===
Commas: 3025/3024, 4375/4374, 5767168/5764801
Subgroup: 2.3.5.7.11.13


POTE generator: ~8/7 = 231.103
Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612


Map: [&lt;2 21 36 5 2|, &lt;0 -29 -51 1 8|]
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}


EDOs: 26, 244, 270, 566, 836, 1106
Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4466


Badness: 0.0162
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 4048 }}


==13-limit==
Badness: 0.0213
Commas: 1716/1715, 2080/2079, 3025/3024, 15379/15360


POTE generator: ~8/7 = 231.103
== 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.


Map: [&lt;2 21 36 5 2 24|, &lt;0 -29 -51 1 8 -27|]
[[Subgroup]]: 2.3.5


EDOs: 26, 244, 270, 566, 836f, 1106f
[[Comma list]]: {{monzo| 92 -39 -13 }}


Badness: 0.0218
[[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}


=Quatracot=
: mapping generators: ~135/128, ~3
Commas: 4375/4374, 1483154296875/1473173782528


POTE generator: ~448/405 = 176.805
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897


Map: [&lt;2 7 7 23|, &lt;0 -13 -8 -59|]
{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}


Wedgie: &lt;&lt;26 16 118 -35 114 229||
[[Badness]]: 0.123


EDOs: 190, 224, 414, 638, 1052c, 1690bc
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


Badness: 0.1760
[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}


==11-limit==
[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}
Commas: 3025/3024, 4375/4374, 1265625/1261568


POTE generator: ~448/405 = 176.806
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 702.0024


Map: [&lt;2 7 7 23 19|, &lt;0 -13 -8 -59 -41|]
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}


EDOs: 190, 224, 414, 638, 1052c
[[Badness]]: 0.126


Badness: 0.0410
=== 11-limit ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 4375/4374, 234375/234256, 2097152/2096325
Commas: 625/624, 729/728, 1575/1573, 2200/2197


POTE generator: ~448/405 = 176.804
Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}


Map: [&lt;2 7 7 23 19 13|, &lt;0 -13 -8 -59 -41 -19|]
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042


EDOs: 190, 224, 414, 638, 1690bc, 2328bcde
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 3588e, 5135e }}


Badness: 0.0226
Badness: 0.0421


=Octoid=
=== 13-limit ===
Commas: 4375/4374, 16875/16807
Subgroup: 2.3.5.7.11.13


valid range: [578.571, 600.000] (56bcd to 8d)
Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078


nice range: [582.512, 584.359]
Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}


strict range: [582.512, 584.359]
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099


POTE generator: ~7/5 = 583.940
{{Optimal ET sequence|legend=1| 494, 1547, 2041, 4576def }}


Map: [&lt;8 1 3 3|, &lt;0 3 4 5|]
Badness: 0.0286


Generators: 49/45, 7/5
== Countritonic ==
: ''For the 5-limit version of this temperament, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


EDOs: 72, 152, 224
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.


Badness: 0.0427
[[Subgroup]]: 2.3.5.7


==11-limit==
[[Comma list]]: 4375/4374, 68719476736/68356598625
Commas: 540/539, 1375/1372, 4000/3993


valid range: [581.250, 586.364] (64cd, 88bcde)
{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }}


nice range: [582.512, 585.084]
: mapping generators: ~2, ~45927/32768


strict range: [582.512, 585.084]
[[Optimal tuning]] (CTE): ~2 = 1\1, ~45927/32768 = 588.6216


POTE generator: ~7/5 = 583.692
{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}


Map: [&lt;8 1 3 3 16|, &lt;0 3 4 5 3|]
[[Badness]]: 0.133


EDOs: 72, 152, 224
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0141
Comma list: 4375/4374, 5632/5625, 2621440/2614689
 
Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~539/384 = 588.6258
 
{{Optimal ET sequence|legend=1| 53, 316e, 369, 422, 791e, 1213cde }}
 
Badness: 0.0707
 
=== 13-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625
 
Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 -74 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~128/91 = 588.6277
 
{{Optimal ET sequence|legend=1| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}
 
Badness: 0.0366
 
== Quatracot ==
{{See also| Stratosphere }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
 
{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }}
 
: mapping generators: ~2278125/1605632, ~448/405
 
[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 1\2, ~448/405 = 176.805
 
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}
 
[[Badness]]: 0.175982
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 1265625/1261568
 
Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 176.806
 
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c }}
 
Badness: 0.041043


=== 13-limit ===
=== 13-limit ===
Commas: 540/539, 1375/1372, 4000/3993, 625/624
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 729/728, 1575/1573, 2200/2197
 
Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~195/176 = 176.804
 
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1690bcc, 2328bccde }}
 
Badness: 0.022643
 
== Moulin ==
Moulin has a generator of 22/13, and it is named after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". It can be described as the 494 & 1619 temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
 
{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }}
 
: mapping generators: ~2, ~6422528/3796875
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6422528/3796875 = 910.9323


POTE generator: ~7/5 = 583.905
{{Optimal ET sequence|legend=1| 494, 1125, 1619 }}


Map: [&lt;8 1 3 3 16 -21|, &lt;0 3 4 5 3 13|]
[[Badness]]: 0.234


EDOs: 72, 224
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0153
Comma list: 4375/4374, 759375/758912, 100663296/100656875


=== Music ===
Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }}
* [http://www.archive.org/details/Dreyfus http://www.archive.org/details/Dreyfus]
* [http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play]


=== Octopus ===
Optimal tuning (CTE): ~2 = 1\1, ~1024/605 = 910.9323
Commas: 169/168, 325/324, 364/363, 540/539


POTE generator: ~7/5 = 583.892
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}


Map: [&lt;8 1 3 3 16 14|, &lt;0 3 4 5 3 4|]
Badness: 0.0678


EDOs: 72, 152, 224f
=== 13-limit ===
Since 11/8 is within 23 generators, the 25 tone MOS (4L 21s) of this temperament contains the 8:11:13 triad.


Badness: 0.0217
Subgroup: 2.3.5.7.11.13


= Amity =
Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
{{main|Amity}}
{{see also|Amity family #Amity}}


The generator for [[amity]] temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&amp;53 temperament, or by its wedgie, &lt;&lt;5 13 -17 9 -41 -76||. [[99edo]] is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.
Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }}


In the 5-limit amity is a genuine microtemperament, with 58/205 being a possible tuning. Another good choice is (64/5)^(1/13), which gives pure major thirds.
Optimal tuning (CTE): ~2 = 1\1, ~22/13 = 910.9323


Comma: 1600000/1594323
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}


POTE generator: ~243/200 = 339.519
Badness: 0.0271


Map: [&lt;1 3 6|, &lt;0 -5 -13|]
== Palladium ==
: ''For the 5-limit version of this temperament, see [[46th-octave temperaments]]''.


EDOs: 7, 39, 46, 53, 152, 205, 463, 668, 873
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.


Badness: 0.0220
[[Subgroup]]: 2.3.5.7


==7-limit==
[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}
Commas: 4375/4374, 5120/5103


POTE generator: ~128/105 = 339.432
{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}


Map: [&lt;1 3 6 -2|, &lt;0 -5 -13 17|]
: mapping generators: ~83349/81920, ~3


Wedgie: &lt;&lt;5 13 -17 9 -41 -76||
[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 1\46, ~3/2 = 701.6074


EDOs: 7, 39, 46, 53, 99, 251, 350
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }}


Badness: 0.0236
[[Badness]]: 0.308505


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


POTE generator: ~128/105 = 339.464
Comma list: 3025/3024, 4375/4374, 134775333/134217728


Map: [&lt;1 3 6 -2 21|, &lt;0 -5 -13 17 -62|]
Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}


EDOs: 53, 99e, 152, 555de, 707de, 859bde
Optimal tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951


Badness: 0.0315
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de }}
 
Badness: 0.073783


=== 13-limit ===
=== 13-limit ===
Commas: 352/351, 540/539, 625/624, 847/845
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
 
Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}
 
Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6419
 
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334de }}
 
Badness: 0.040751
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
 
Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}
 
Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425
 
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334deg }}
 
Badness: 0.022441
 
== Oviminor ==
{{See also| Syntonic–kleismic equivalence continuum }}
 
Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}
 
{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }}
 
: mapping generators: ~2, ~6/5
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 315.7501
 
{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}
 
[[Badness]]: 0.582
 
== Octoid ==
''For the 5-limit temperament, see [[8th-octave temperaments#Octoid (5-limit)]].''
 
The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 16875/16807
 
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
 
: mapping generators: ~49/45, ~7/5
 
[[Optimal tuning]] ([[POTE]]): ~49/45 = 1\8, ~7/5 = 583.940
 
[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
 
{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}
 
[[Badness]]: 0.042670
 
Scales: [[octoid72]], [[octoid80]]
 
=== 11-limit ===
The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimaxing the damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, if one wants to use 80edo as the tuning, one must use octopus — not octoid — as 80edo doesn't temper 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.
 
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 4000/3993
 
Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}
 
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.962
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
 
{{Optimal ET sequence|legend=1| 72, 152, 224 }}
 
Badness: 0.014097
 
Scales: [[octoid72]], [[octoid80]]
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 625/624, 729/728, 1375/1372
 
Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}
 
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.905
 
{{Optimal ET sequence|legend=1| 72, 152f, 224 }}
 
Badness: 0.015274


POTE generator: ~128/105 = 339.481
Scales: [[octoid72]], [[octoid80]]


Map: [&lt;1 3 6 -2 21 17|, &lt;0 -5 -13 17 -62 -47|]
; 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


EDOS: 53, 99ef, 152f, 205
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0280
Comma list: 375/374, 540/539, 625/624, 715/714, 729/728


==Hitchcock==
Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}
Commas: 121/120, 176/175, 2200/2187


POTE generator: ~11/9 = 339.340
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.842


Map: [&lt;1 3 6 -2 6|, &lt;0 -5 -13 17 -9|]
{{Optimal ET sequence|legend=1| 72, 152fg, 224, 296, 520g }}


EDOs: 7, 39, 46, 53, 99
Badness: 0.014304


Badness: 0.0352
Scales: [[octoid72]], [[octoid80]]


===13-limit===
===== 19-limit =====
Commas: 121/120, 169/168, 176/175, 325/324
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~11/9 = 339.419
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714


Map: [&lt;1 3 6 -2 6 2|, &lt;0 -5 -13 17 -9 6|]
Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}


EDOs: 7, 39, 46, 53, 99
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.932


Badness: 0.0224
{{Optimal ET sequence|legend=1| 72, 152fg, 224 }}


==Hemiamity==
Badness: 0.016036
Commas: 3025/3024, 4375/4374, 5120/5103


POTE generator: ~64/55 = 339.493
Scales: [[octoid72]], [[octoid80]]


Map: [&lt;2 1 -1 13 13|, &lt;0 5 13 -17 -14|]
==== 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}}.


EDOs: 14cde, 46, 106, 152, 198, 350
Subgroup: 2.3.5.7.11.13


Badness: 0.0313
Comma list: 169/168, 325/324, 364/363, 540/539


=Parakleismic=
Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, |8 14 -13&gt;, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 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 &lt;&lt;13 14 35 -8 19 42|| and adding 3136/3125 and 4375/4374, and the 11-limit wedgie &lt;&lt;13 14 35 -36 ...|| adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.


Comma: 124440064/1220703125
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.892


POTE generator: ~6/5 = 315.240
{{Optimal ET sequence|legend=1| 72, 152, 224f }}


Map: [&lt;1 5 6|, &lt;0 -13 -14|]
Badness: 0.021679


EDOs: 19, 61, 80, 99, 118, 453, 571, 689, 1496
Scales: [[octoid72]], [[octoid80]]


Badness: 0.0433
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


==7-limit==
Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
Commas: 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.181
Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}


Map: [&lt;1 5 6 12|, &lt;0 -13 -14 -35|]
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.811


EDOs: 19, 80, 99, 217, 316, 415
{{Optimal ET sequence|legend=1| 72, 152, 224fg, 296ffg }}


Badness: 0.0274
Badness: 0.015614


==11-limit==
Scales: [[Octoid72]], [[Octoid80]]
Commas: 385/384, 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.251
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 5 6 12 -6|, &lt;0 -13 -14 -35 36|]
Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399


EDOs: 19, 99, 118
Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}


Badness: 0.0497
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 584.064


==Parkleismic==
{{Optimal ET sequence|legend=1| 72, 152, 224fg, 376ffgh }}
Commas: 176/175, 1375/1372, 2200/2187


POTE generator: ~6/5 = 315.060
Badness: 0.016321


Map: [&lt;1 5 6 12 20|, &lt;0 -13 -14 -35 -63|]
Scales: [[Octoid72]], [[Octoid80]]


EDOs: 80, 179, 259cd
==== Hexadecoid ====
{{ See also | 16th-octave temperaments }}


Badness: 0.0559
Hexadecoid (80 &amp; 144) has a period of 1/16 octave and tempers out 4225/4224.


===13-limit===
Subgroup: 2.3.5.7.11.13
Commas: 169/168, 176/175, 325/324, 1375/1372


POTE generator: ~6/5 = 315.075
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224


Map: [&lt;1 5 6 12 20 10|, &lt;0 -13 -14 -35 -63 -24|]
Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}


EDOs: 15, 19, 80, 179
: mapping generators: ~448/429, ~7/5


Badness: 0.0366
Optimal tuning (POTE): ~448/429 = 1\16, ~13/8 = 841.015
 
{{Optimal ET sequence|legend=1| 80, 144, 224 }}
 
Badness: 0.030818
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
 
Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
 
Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932
 
{{Optimal ET sequence|legend=1| 80, 144, 224, 528dg }}
 
Badness: 0.028611
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
 
Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }}
 
Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896
 
{{Optimal ET sequence|legend=1| 80, 144, 224, 304dh, 528dghh }}
 
Badness: 0.023731
 
== Parakleismic ==
{{Main| Parakleismic }}
 
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
 
[[Comma list]]: 1224440064/1220703125
 
{{Mapping|legend=1| 1 5 6 | 0 -13 -14 }}
 
: mapping generators: ~2, ~6/5
 
[[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
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3136/3125, 4375/4374
 
{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }}
 
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.181
 
{{Optimal ET sequence|legend=1| 19, 80, 99, 217, 316, 415 }}
 
[[Badness]]: 0.027431
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 3136/3125, 4375/4374
 
Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.251
 
{{Optimal ET sequence|legend=1| 19, 99, 118 }}
 
Badness: 0.049711
 
=== 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.
 
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 3136/3125, 4375/4374
 
Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.220
 
{{Optimal ET sequence|legend=1| 19e, 99e, 118, 217, 335, 552d, 887dd }}
 
Badness: 0.036027
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
 
Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.214
 
{{Optimal ET sequence|legend=1| 99e, 118, 217, 552d, 769de }}
 
Badness: 0.044710
 
==== 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]].
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 625/624, 729/728
 
Mapping: {{mapping| 1 5 6 12 25 15 | 0 -13 -14 -35 -82 -43 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.225
 
{{Optimal ET sequence|legend=1| 19e, 99ef, 118, 217ff, 335ff }}
 
Badness: 0.037618
 
=== Parkleismic ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 1375/1372, 2200/2187
 
Mapping: {{mapping| 1 5 6 12 20 | 0 -13 -14 -35 -63 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.060
 
{{Optimal ET sequence|legend=1| 19e, 80, 179, 259cd }}
 
Badness: 0.055884
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 176/175, 325/324, 1375/1372
 
Mapping: {{mapping| 1 5 6 12 20 10 | 0 -13 -14 -35 -63 -24 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.075
 
{{Optimal ET sequence|legend=1| 19e, 80, 179 }}
 
Badness: 0.036559
 
=== Paradigmic ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 896/891, 3136/3125
 
Mapping: {{mapping| 1 5 6 12 -1 | 0 -13 -14 -35 17 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.096
 
{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}
 
Badness: 0.041720
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 325/324, 540/539, 832/825
 
Mapping: {{mapping| 1 5 6 12 -1 10 | 0 -13 -14 -35 17 -24 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.080
 
{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}
 
Badness: 0.035781
 
=== Semiparakleismic ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 3136/3125, 4375/4374
 
Mapping: {{mapping| 2 10 12 24 19 | 0 -13 -14 -35 -23 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.181
 
{{Optimal ET sequence|legend=1| 80, 118, 198, 316, 514c, 830c }}
 
Badness: 0.034208
 
==== Semiparamint ====
This extension was named ''semiparakleismic'' in the earlier materials.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
 
Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.156
 
{{Optimal ET sequence|legend=1| 80, 118, 198 }}
 
Badness: 0.033775
 
==== Semiparawolf ====
This extension was named ''gentsemiparakleismic'' in the earlier materials.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 325/324, 364/363, 3136/3125
 
Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }}
 
Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 315.184
 
{{Optimal ET sequence|legend=1| 80, 118f, 198f }}
 
Badness: 0.040467
 
== Counterkleismic ==
{{See also| High badness temperaments #Counterhanson}}
 
In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{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
 
[[Comma list]]: 4375/4374, 158203125/157351936
 
{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }}
 
: mapping generators: ~2, ~5/3
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.060
 
{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }}
 
[[Badness]]: 0.090553
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4375/4374, 2097152/2096325
 
Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.071
 
{{Optimal ET sequence|legend=1| 19, 205, 224 }}
 
Badness: 0.070952
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 625/624, 729/728, 10985/10976
 
Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.070
 
{{Optimal ET sequence|legend=1| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}
 
Badness: 0.033874
 
=== Counterlytic ===
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 4375/4374, 496125/495616
 
Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065
 
{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}
 
Badness: 0.065400
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 729/728, 1375/1372, 10985/10976
 
Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }}


==Paradigmic==
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065
Commas: 540/539, 896/891, 3136/3125


POTE generator: ~6/5 = 315.096
{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}


Map: [&lt;1 5 6 12 -1|, &lt;0 -13 -14 -35 17|]
Badness: 0.029782


EDOs: 19, 80, 99e, 179e
== Quincy ==
[[Subgroup]]: 2.3.5.7


Badness: 0.0417
[[Comma list]]: 4375/4374, 823543/819200


===13-limit===
{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
Commas: 169/168, 325/324, 540/539, 832/825


POTE generator: ~6/5 = 315.080
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1715 = 16.613


Map: [&lt;1 5 6 12 -1 10|, &lt;0 -13 -14 -35 17 -24|]
{{Optimal ET sequence|legend=1| 72, 217, 289 }}


EDOs: 19, 80, 99e, 179e
[[Badness]]: 0.079657


Badness: 0.0358
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Semiparakleismic ==
Comma list: 441/440, 4000/3993, 4375/4374
Commas: 3025/3024, 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.181
Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}


Map: [&lt;2 10 12 24 19|, &lt;0 -13 -14 -35 -23|]
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.613


EDOs: 80, 118, 198, 316, 514c, 830c
{{Optimal ET sequence|legend=1| 72, 217, 289 }}


Badness: 0.0342
Badness: 0.030875


=== 13-limit ===
=== 13-limit ===
Commas: 352/351, 1001/1000, 3025/3024, 4375/4374
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 676/675, 4375/4374
 
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602
 
{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}
 
Badness: 0.023862
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155
 
Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602
 
{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}
 
Badness: 0.014741
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675
 
Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.594
 
{{Optimal ET sequence|legend=1| 72, 145, 217 }}
 
Badness: 0.015197
 
== Sfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 64827/64000
 
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 26.287


POTE generator: ~6/5 = 315.1563
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


Map: [<2 10 12 24 19 -1|, <0 -13 -14 -35 -23 16|]
[[Badness]]: 0.123291


EDOs: {{EDOs|80, 118, 198}}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0338
Comma list: 121/120, 441/440, 4375/4374


=== Gentsemiparakleismic ===
Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}
Commas: 169/168, 325/324, 364/363, 3136/3125


POTE generator: ~6/5 = 315.1839
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.286


Map: [<2 10 12 24 19 20|, <0 -13 -14 -35 -23 -24|]
{{Optimal ET sequence|legend=1| 45e, 46, 91e, 137de }}


EDOs: {{EDOs|80, 118f, 198f}}
Badness: 0.054098


Badness: 0.0405
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=Quincy=
Comma list: 121/120, 169/168, 325/324, 441/440
Commas: 4375/4374, 823543/819200


POTE generator: ~1728/1715 = 16.613
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}


Map: [&lt;1 2 2 3|, &lt;0 -30 -49 -14|]
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.310


EDOs: 72, 217, 289
{{Optimal ET sequence|legend=1| 45ef, 46, 91ef, 137def }}


Badness: 0.0797
Badness: 0.033067


==11-limit==
=== Sfour ===
Commas: 441/440, 4000/3993, 41503/41472
Subgroup: 2.3.5.7.11


POTE generator: ~100/99 = 16.613
Comma list: 385/384, 2401/2376, 4375/4374


Map: [&lt;1 2 2 3 4|, &lt;0 -30 -49 -14 -39|]
Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}


EDOs: 72, 217, 289
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.246


Badness: 0.0309
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


==13-limit==
Badness: 0.076567
Commas: 364/363, 441/440, 676/675, 4375/4374


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


Map: [&lt;1 2 2 3 4 5|, &lt;0 -30 -49 -14 -39 -94|]
Comma list: 196/195, 364/363, 385/384, 4375/4374


EDOs: 72, 145, 217, 289
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}


Badness: 0.0239
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.239


==17-limit==
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
Commas: 364/363, 441/440, 595/594, 1001/1000, 1156/1155


POTE generator: ~100/99 = 16.602
Badness: 0.051893


Map: [&lt;1 2 2 3 4 5 5|, &lt;0 -30 -49 -14 -39 -94 -66|]
== Trideci ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tridecatonic]].''


EDOs: 72, 145, 217, 289
The trideci temperament (26 &amp; 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from "tridecim" (Latin for "[[wikipedia:13|thirteen]]").


Badness: 0.0147
[[Subgroup]]: 2.3.5.7


==19-limit==
[[Comma list]]: 4375/4374, 83349/81920
Commas: 343/342, 364/363, 441/440, 595/594, 676/675, 2601/2600


POTE generator: ~100/99 = 16.594
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}


Map: [&lt;1 2 2 3 4 5 5 4|, &lt;0 -30 -49 -14 -39 -94 -66 18|]
[[Optimal tuning]] ([[POTE]]): ~256/245 = 1\13, ~3/2 = 699.1410


EDOs: 72, 145, 217
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }}


Badness: 0.0152
[[Badness]]: 0.184585


=Chlorine=
=== 11-limit ===
The name of chlorine temperament comes from Chlorine, the 17th element.
Subgroup: 2.3.5.7.11


Chlorine microtemperament has a period of 1/17 octave. It tempers out the septendecima, |-52 -17 34&gt;, by which 17 chromatic semitones (25/24) fall short of an octave. Possible tunings for chlorine are [[289edo|289]], [[323edo|323]], and [[612edo|612]] EDOs, though its hardly likely anyone could tell the difference. In the 7-limit, 289&amp;323 temperament tempers out |-49 4 22 -3&gt; as well as the ragisma.
Comma list: 245/242, 385/384, 4375/4374


Comma: |-52 -17 34&gt;
Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}


POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2687
Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.6179
 
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdde }}
 
Badness: 0.084590
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;17 26 39|, &lt;0 2 1|]
Comma list: 169/168, 245/242, 325/324, 385/384


EDOs: 34, 289, 323, 612, 901
Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}


Badness: 0.0771
Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.2969


==7-limit==
{{Optimal ET sequence|legend=1| 26, 65f, 91f, 156dff }}
Commas: 4375/4374, 193119049072265625/193091834023510016


POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2936
Badness: 0.052366


Map: [&lt;17 26 39 43|, &lt;0 2 1 10|]
== 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]].


EDOs: 34d, 289, 323, 612, 935, 1547
Subgroup: 2.3.5.7


Badness: 0.0417
Comma list: 4375/4374, {{monzo| 154 -54 -21 -7 }}


==11-limit==
Mapping: {{mapping| 1 0 -21 85 | 0 7 103 -363 }}
Commas: 4375/4374, 41503/41472, 1879453125/1879048192


POTE generators: ~25/24 = 70.5882, ~5/4 = 386.2690
Optimal tuning (CTE): ~2 = 1\1, ~{{monzo| 66 -23 -9 -3 }} = 271.7113


Map: [&lt;17 26 39 43 64|, &lt;0 2 1 10 -11|]
{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}


EDOs: 34de, 289, 323, 612, 901
Badness: 0.312806


Badness: 0.0637
== Notes ==


[[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 1,034: Line 1,721:
[[Category:Octoid]]
[[Category:Octoid]]
[[Category:Parakleismic]]
[[Category:Parakleismic]]
[[Category:Quincy]]
[[Category:Supermajor]]
[[Category:Supermajor]]
[[Category:Microtemperament]]
[[Category:Ragismic]]

Latest revision as of 00:36, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-2 temperaments tempering out the ragisma, 4375/4374 ([-1 -7 4 1⟩). The ragisma is the smallest 7-limit superparticular ratio.

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

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

Supermajor

The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (215)/3, 46 give (219)/5, and 75 give (230)/7. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80-note mos is presumably the place to start, and if that is not enough notes for you, there is always the 171-note mos.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 52734375/52706752

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

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

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

Badness: 0.010836

Semisupermajor

Subgroup: 2.3.5.7.11

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

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

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

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

Badness: 0.012773

Enneadecal

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

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 703125/702464

Mapping: [⟨19 0 14 -37], ⟨0 1 1 3]]

mapping generators: ~28/27, ~3

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

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

Badness: 0.010954

11-limit

Subgroup: 2.3.5.7.11

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

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

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

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

Badness: 0.043734

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness: 0.033545

Hemienneadecal

Subgroup: 2.3.5.7.11

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

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

mapping generators: ~55/54, ~3

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

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

Badness: 0.009985

Hemienneadecalis

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal ET sequence: 152f, 342f, 494

Badness: 0.020782

Hemienneadec

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness: 0.030391

Semihemienneadecal

Subgroup: 2.3.5.7.11.13

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

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

mapping generators: ~55/54 = 1\38, ~55/54, ~429/250

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

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

Badness: 0.014694

Kalium

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

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

Optimal ET sequence: 855, 988, 1843

Semidimi

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

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 3955078125/3954653486

Mapping: [⟨1 36 48 61], ⟨0 -55 -73 -93]]

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

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

Badness: 0.015075

Brahmagupta

The brahmagupta temperament has a period of 1/7 octave, tempering out the akjaysma, [47 -7 -7 -7⟩ = 140737488355328 / 140710042265625.

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 70368744177664/70338939985125

Mapping: [⟨7 2 -8 53], ⟨0 3 8 -11]]

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

Optimal tuning (POTE): ~1157625/1048576 = 1\7, ~27/20 = 519.716

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

Badness: 0.029122

11-limit

Subgroup: 2.3.5.7.11

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

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

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

Optimal ET sequence: 7, 217, 224, 441, 665, 1771ee

Badness: 0.052190

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.706

Optimal ET sequence: 7, 217, 224, 441, 665, 1771eef

Badness: 0.023132

Abigail

Abigail temperament tempers out the pessoalisma in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.[1]

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2147483648/2144153025

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

mapping generators: ~46305/32768, ~27/20

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

Optimal ET sequence: 46, 132, 178, 224, 270, 494, 764, 1034, 1798

Badness: 0.037000

11-limit

Subgroup: 2.3.5.7.11

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

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

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

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

Badness: 0.012860

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 7 13 -1 1 -2], ⟨0 -11 -24 19 17 27]]

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

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

Badness: 0.008856

Gamera

For the 5-limit temperament, see High badness temperaments#Gamera.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 589824/588245

Mapping: [⟨1 6 10 3], ⟨0 -23 -40 -1]]

mapping generators: ~2, ~8/7

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

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

Badness: 0.037648

Hemigamera

Subgroup: 2.3.5.7.11

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

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

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

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

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

Badness: 0.040955

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness: 0.020416

Semigamera

Subgroup: 2.3.5.7.11

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

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

mapping generators: ~2, ~77/72

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

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

Badness: 0.078

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness: 0.044

Crazy

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

Crazy tempers out the kwazy comma in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the 118 & 494 temperament. 1106edo is an strong tuning.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-53 10 16⟩

Mapping: [⟨2 1 6 -15], ⟨0 8 -5 76]]

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

Optimal tunings:

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

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

Badness (Smith): 0.0394

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Smith): 0.0170

Orga

Subgroup: 2.3.5.7

Comma list: 4375/4374, 54975581388800/54936068900769

Mapping: [⟨2 21 36 5], ⟨0 -29 -51 1]]

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

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

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

Badness: 0.040236

11-limit

Subgroup: 2.3.5.7.11

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

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

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

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

Badness: 0.016188

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness: 0.021762

Seniority

See also: Very high accuracy temperaments § Senior

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 201768035/201326592

Mapping: [⟨1 11 19 2], ⟨0 -35 -62 3]]

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

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

Badness: 0.044877

Senator

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

Subgroup: 2.3.5.7.11

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

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

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

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

Badness: 0.092238

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness: 0.044662

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

Badness: 0.026562

Monzismic

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

The monzismic temperament (53 & 612) tempers out the monzisma, [54 -37 2⟩, and in the 7-limit, the nanisma, [109 -67 0 -1⟩, as well as the ragisma, 4375/4374.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-55 30 2 1⟩

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

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

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

Badness: 0.046569

Monzism

Subgroup: 2.3.5.7.11

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

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

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

Optimal ET sequence: 53, 559, 612

Badness: 0.057083

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal ET sequence: 53, 559, 612

Badness: 0.053780

Semidimfourth

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

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 235298/234375

Mapping: [⟨1 21 28 36], ⟨0 -31 -41 -53]]

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

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

Badness: 0.055249

Neusec

Subgroup: 2.3.5.7.11

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

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

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

Optimal ET sequence: 8d, 190, 388

Badness: 0.059127

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness: 0.030941

Acrokleismic

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2202927104/2197265625

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

mapping generators: ~2, ~6/5

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

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

Badness: 0.056184

11-limit

Subgroup: 2.3.5.7.11

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

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

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

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

Badness: 0.036878

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal ET sequence: 19, 251, 270

Badness: 0.026818

Counteracro

Subgroup: 2.3.5.7.11

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

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

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

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

Badness: 0.042572

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness: 0.026028

Quasithird

The quasithird temperament is featured by a major third interval which is 1600000/1594323 (amity comma) or 5120/5103 (hemifamity comma) below the just major third 5/4 as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the ragisma and [-60 29 0 5⟩.

Subgroup: 2.3.5

Comma list: [55 -64 20⟩

Mapping: [⟨4 0 -11], ⟨0 5 16]]

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

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

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

Badness: 0.099519

7-limit

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-60 29 0 5⟩

Mapping: [⟨4 0 -11 48], ⟨0 5 16 -29]]

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

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

Badness: 0.061813

11-limit

Subgroup: 2.3.5.7.11

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

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

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

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

Badness: 0.021125

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness: 0.029501

Deca

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

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 165288374272/164794921875

Mapping: [⟨10 4 9 2], ⟨0 5 6 11]]

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

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

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

Badness: 0.080637

Badness (Sintel): 2.041

11-limit

Subgroup: 2.3.5.7.11

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

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

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

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

Badness: 0.024329

Badness (Sintel): 0.804

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness: 0.016810

Badness (Sintel): 0.695

no-17's 19-limit

Subgroup: 2.3.5.7.11.13.19

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

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

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

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

Badness (Sintel): 0.556

Keenanose

Keenanose is named for the fact that it uses 385/384, the keenanisma, as the generator.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-56 1 -8 26⟩

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

mapping generators: ~2, ~[21 3 1 -10⟩

Optimal tuning (CTE): ~2 = 1\1, ~[21 3 1 -10⟩ = 4.4465

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

Badness: 0.0858

11-limit

Subgroup: 2.3.5.7.11

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

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

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

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

Badness: 0.0308

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness: 0.0213

Aluminium

Aluminium is named after the 13th element, and tempers out the [92 -39 -13⟩ comma which sets 135/128 interval to be equal to 1/13th of the octave.

Subgroup: 2.3.5

Comma list: [92 -39 -13⟩

Mapping: [⟨13 0 92], ⟨0 1 -3]]

mapping generators: ~135/128, ~3

Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 701.9897

Optimal ET sequence: 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc

Badness: 0.123

7-limit

Subgroup: 2.3.5.7

Comma list: 4375/4374, [92 -39 -13⟩

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

Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0024

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

Badness: 0.126

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042

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

Badness: 0.0421

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099

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

Badness: 0.0286

Countritonic

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

Countritonic (co-un-tritonic) can be described as the 53 & 422 temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 68719476736/68356598625

Mapping: [⟨1 6 19 -33], ⟨0 -9 -34 73]]

mapping generators: ~2, ~45927/32768

Optimal tuning (CTE): ~2 = 1\1, ~45927/32768 = 588.6216

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

Badness: 0.133

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 6 19 -13 79], ⟨0 -9 -34 73 154]]

Optimal tuning (CTE): ~2 = 1\1, ~539/384 = 588.6258

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

Badness: 0.0707

13-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 6 19 -13 79], ⟨0 -9 -34 73 154 -74]]

Optimal tuning (CTE): ~2 = 1\1, ~128/91 = 588.6277

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

Badness: 0.0366

Quatracot

See also: Stratosphere

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-32 5 14 -3⟩

Mapping: [⟨2 7 7 23], ⟨0 -13 -8 -59]]

mapping generators: ~2278125/1605632, ~448/405

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

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

Badness: 0.175982

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 7 7 23 19], ⟨0 -13 -8 -59 -41]]

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

Optimal ET sequence: 190, 224, 414, 638, 1052c

Badness: 0.041043

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 7 7 23 19 13], ⟨0 -13 -8 -59 -41 -19]]

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

Optimal ET sequence: 190, 224, 414, 638, 1690bcc, 2328bccde

Badness: 0.022643

Moulin

Moulin has a generator of 22/13, and it is named after the Law & Order: Special Victims Unit episode Season 22, Episode 13. "Trick-Rolled At The Moulin". It can be described as the 494 & 1619 temperament.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-88 2 45 -7⟩

Mapping: [⟨1 57 38 248], ⟨0 -73 -47 -323]]

mapping generators: ~2, ~6422528/3796875

Optimal tuning (CTE): ~2 = 1\1, ~6422528/3796875 = 910.9323

Optimal ET sequence: 494, 1125, 1619

Badness: 0.234

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 57 38 248 -14], ⟨0 -73 -47 -323 23]]

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

Optimal ET sequence: 494, 1125, 1619, 2113

Badness: 0.0678

13-limit

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

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 57 38 248 -14 -13], ⟨0 -73 -47 -323 23 22]]

Optimal tuning (CTE): ~2 = 1\1, ~22/13 = 910.9323

Optimal ET sequence: 494, 1125, 1619, 2113

Badness: 0.0271

Palladium

For the 5-limit version of this temperament, see 46th-octave temperaments.

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-51 8 2 12⟩

Mapping: [⟨46 0 -39 202], ⟨0 1 2 -1]]

mapping generators: ~83349/81920, ~3

Optimal tuning (POTE): ~83349/81920 = 1\46, ~3/2 = 701.6074

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

Badness: 0.308505

11-limit

Subgroup: 2.3.5.7.11

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

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

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

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

Badness: 0.073783

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness: 0.040751

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

Badness: 0.022441

Oviminor

See also: Syntonic–kleismic equivalence continuum

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-100 53 48 -34⟩

Mapping: [⟨1 50 51 147], ⟨0 -184 -185 -548]]

mapping generators: ~2, ~6/5

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

Optimal ET sequence: 19, …, 1600, 1619, 4838, 6457c

Badness: 0.582

Octoid

For the 5-limit temperament, see 8th-octave temperaments#Octoid (5-limit).

The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 (ragisma) and 16875/16807 (mirkwai). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 16875/16807

Mapping: [⟨8 1 3 3], ⟨0 3 4 5]]

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

Optimal tuning (POTE): ~49/45 = 1\8, ~7/5 = 583.940

Tuning ranges:

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

Optimal ET sequence: 8d, 72, 152, 224

Badness: 0.042670

Scales: octoid72, octoid80

11-limit

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

Subgroup: 2.3.5.7.11

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

Mapping: [⟨8 1 3 3 16], ⟨0 3 4 5 3]]

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

Tuning ranges:

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

Optimal ET sequence: 72, 152, 224

Badness: 0.014097

Scales: octoid72, octoid80

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal ET sequence: 72, 152f, 224

Badness: 0.015274

Scales: octoid72, octoid80

Music
17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

Badness: 0.014304

Scales: octoid72, octoid80

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.932

Optimal ET sequence: 72, 152fg, 224

Badness: 0.016036

Scales: octoid72, octoid80

Octopus

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

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal ET sequence: 72, 152, 224f

Badness: 0.021679

Scales: octoid72, octoid80

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

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

Optimal ET sequence: 72, 152, 224fg, 296ffg

Badness: 0.015614

Scales: Octoid72, Octoid80

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

Optimal ET sequence: 72, 152, 224fg, 376ffgh

Badness: 0.016321

Scales: Octoid72, Octoid80

Hexadecoid

See also: 16th-octave temperaments

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

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal ET sequence: 80, 144, 224

Badness: 0.030818

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

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

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

Badness: 0.028611

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

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

Badness: 0.023731

Parakleismic

Main article: Parakleismic

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

Subgroup: 2.3.5

Comma list: 1224440064/1220703125

Mapping: [⟨1 5 6], ⟨0 -13 -14]]

mapping generators: ~2, ~6/5

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

Optimal ET sequence: 19, 61, 80, 99, 118, 453, 571, 689, 1496

Badness: 0.043279

7-limit

Subgroup: 2.3.5.7

Comma list: 3136/3125, 4375/4374

Mapping: [⟨1 5 6 12], ⟨0 -13 -14 -35]]


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

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

Badness: 0.027431

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 5 6 12 -6], ⟨0 -13 -14 -35 36]]

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

Optimal ET sequence: 19, 99, 118

Badness: 0.049711

Paralytic

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

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 5 6 12 25], ⟨0 -13 -14 -35 -82]]

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

Optimal ET sequence: 19e, 99e, 118, 217, 335, 552d, 887dd

Badness: 0.036027

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 5 6 12 25 -16], ⟨0 -13 -14 -35 -82 75]]

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

Optimal ET sequence: 99e, 118, 217, 552d, 769de

Badness: 0.044710

Paraklein

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

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 5 6 12 25 15], ⟨0 -13 -14 -35 -82 -43]]

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

Optimal ET sequence: 19e, 99ef, 118, 217ff, 335ff

Badness: 0.037618

Parkleismic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 5 6 12 20], ⟨0 -13 -14 -35 -63]]

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

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

Badness: 0.055884

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 5 6 12 20 10], ⟨0 -13 -14 -35 -63 -24]]

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

Optimal ET sequence: 19e, 80, 179

Badness: 0.036559

Paradigmic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 5 6 12 -1], ⟨0 -13 -14 -35 17]]

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

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

Badness: 0.041720

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 5 6 12 -1 10], ⟨0 -13 -14 -35 17 -24]]

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

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

Badness: 0.035781

Semiparakleismic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 10 12 24 19], ⟨0 -13 -14 -35 -23]]

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

Optimal ET sequence: 80, 118, 198, 316, 514c, 830c

Badness: 0.034208

Semiparamint

This extension was named semiparakleismic in the earlier materials.

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 10 12 24 19 -1], ⟨0 -13 -14 -35 -23 16]]

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

Optimal ET sequence: 80, 118, 198

Badness: 0.033775

Semiparawolf

This extension was named gentsemiparakleismic in the earlier materials.

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 10 12 24 19 20], ⟨0 -13 -14 -35 -23 -24]]

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

Optimal ET sequence: 80, 118f, 198f

Badness: 0.040467

Counterkleismic

See also: High badness temperaments § Counterhanson

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 158203125/157351936

Mapping: [⟨1 20 20 61], ⟨0 -25 -24 -79]]

mapping generators: ~2, ~5/3

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

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

Badness: 0.090553

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 20 20 61 -40], ⟨0 -25 -24 -79 59]]

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

Optimal ET sequence: 19, 205, 224

Badness: 0.070952

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 20 20 61 -40 56], ⟨0 -25 -24 -79 59 -71]]

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

Optimal ET sequence: 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef

Badness: 0.033874

Counterlytic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 20 20 61 125], ⟨0 -25 -24 -79 -165]]

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

Optimal ET sequence: 19e, 205e, 224

Badness: 0.065400

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 20 20 61 125 56], ⟨0 -25 -24 -79 -165 -71]]

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

Optimal ET sequence: 19e, 205e, 224

Badness: 0.029782

Quincy

Subgroup: 2.3.5.7

Comma list: 4375/4374, 823543/819200

Mapping: [⟨1 2 3 3], ⟨0 -30 -49 -14]]

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

Optimal ET sequence: 72, 217, 289

Badness: 0.079657

11-limit

Subgroup: 2.3.5.7.11

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

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

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

Optimal ET sequence: 72, 217, 289

Badness: 0.030875

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal ET sequence: 72, 145, 217, 289

Badness: 0.023862

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

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

Optimal ET sequence: 72, 145, 217, 289

Badness: 0.014741

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

Optimal ET sequence: 72, 145, 217

Badness: 0.015197

Sfourth

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 64827/64000

Mapping: [⟨1 2 3 3], ⟨0 -19 -31 -9]]

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.287

Optimal ET sequence: 45, 46, 91, 137d

Badness: 0.123291

11-limit

Subgroup: 2.3.5.7.11

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

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

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

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

Badness: 0.054098

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness: 0.033067

Sfour

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.246

Optimal ET sequence: 45, 46, 91, 137d

Badness: 0.076567

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.239

Optimal ET sequence: 45, 46, 91, 137d

Badness: 0.051893

Trideci

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

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 83349/81920

Mapping: [⟨13 0 -11 57], ⟨0 1 2 -1]]

Optimal tuning (POTE): ~256/245 = 1\13, ~3/2 = 699.1410

Optimal ET sequence: 26, 65, 91, 156d, 247cdd

Badness: 0.184585

11-limit

Subgroup: 2.3.5.7.11

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

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

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

Optimal ET sequence: 26, 65, 91, 156d, 247cdde

Badness: 0.084590

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal ET sequence: 26, 65f, 91f, 156dff

Badness: 0.052366

Counterorson

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

Subgroup: 2.3.5.7

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

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

Optimal tuning (CTE): ~2 = 1\1, ~[66 -23 -9 -3⟩ = 271.7113

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

Badness: 0.312806

Notes

  1. ↑ [1]: "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things."
Retrieved from "https://en.xen.wiki/index.php?title=Ragismic_microtemperaments&oldid=201066"