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Ragismic microtemperaments: Difference between revisions

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* ''[[Quindro]]'', {4375/4374, {{monzo| 56 -28 -5 }}} → [[Quindromeda family #Quindro|Quindromeda family]]
* ''[[Quindro]]'', {4375/4374, {{monzo| 56 -28 -5 }}} → [[Quindromeda family #Quindro|Quindromeda family]]


Microtemperaments considered below are ennealimmal, supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, orga, chlorine, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, quatracot, and palladium.  
Microtemperaments considered below are ennealimmal, supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, orga, chlorine, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, quatracot, moulin, and palladium.  


Some near-microtemperaments are appended as octoid, amity, parakleismic, counterkleismic, quincy, sfourth, and trideci.  
Some near-microtemperaments are appended as octoid, amity, parakleismic, counterkleismic, quincy, sfourth, and trideci.  
Line 1,258: Line 1,258:
Badness: 0.022643
Badness: 0.022643


== Palladium ==
== Moulin ==
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&414 temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.
Moulin has a generator of 22/13, and it is named after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". It can be described as the 494 & 1619 temperament.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 2270317133144025/2251799813685248
[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}


[[Mapping]]: [{{val| 46 73 107 129 }}, {{val| 0 -1 -2 1 }}]
[[Mapping]]: [{{val| 1 57 38 248 }}, {{val| 0 -73 -47 -323 }}]


{{Multival|legend=1| 46 92 -46 39 -202 -365 }}
[[Optimal tuning]] ([[CTE]]): ~22/13 = 910.9323


[[Optimal tuning]] ([[POTE]]): ~3/2 = 701.6074
{{Val list|legend=1| 494, 1125, 1619 }}


{{Val list|legend=1| 46, 368, 414, 460, 874d }}
[[Badness]]: 0.234
 
[[Badness]]: 0.308505


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


Comma list: 3025/3024, 4375/4374, 134775333/134217728
Comma list: 4375/4374, 759375/758912, 100663296/100656875


Mapping: [{{val| 46 73 107 129 159 }}, {{val| 0 -1 -2 1 1 }}]
Mapping: [{{val| 1 57 38 248 -14 }}, {{val| 0 -73 -47 -323 23 }}]


Optimal tuning (POTE): ~3/2 = 701.5951
Optimal tuning (CTE): ~22/13 = 910.9323


Optimal GPV sequence: {{Val list| 46, 368, 414, 460, 874de }}
Optimal GPV sequence: {{val list| 494, 1125, 1619, 2113 }}


Badness: 0.073783
Badness: 0.0678


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


Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078


Mapping: [{{val| 46 73 107 129 159 170 }}, {{val| 0 -1 -2 1 1 2 }}]
Mapping: [{{val| 1 57 38 248 -14 -13 }}, {{val| 0 -73 -47 -323 23 22 }}]


Optimal tuning (POTE): ~3/2 = 701.6419
Optimal tuning (CTE): ~22/13 = 910.9323


Optimal GPV sequence: {{Val list| 46, 368, 414, 460, 874de, 1334de }}
Optimal GPV sequence: {{val list| 494, 1125, 1619, 2113 }}


Badness: 0.040751
Badness: 0.0271


=== 17-limit ===
== Palladium ==
Subgroup: 2.3.5.7.11.13.17
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&414 temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.


Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 46 73 107 129 159 170 188 }}, {{val| 0 -1 -2 1 1 2 0 }}]
[[Comma list]]: 4375/4374, 2270317133144025/2251799813685248


Optimal tuning (POTE): ~3/2 = 701.6425
[[Mapping]]: [{{val| 46 73 107 129 }}, {{val| 0 -1 -2 1 }}]


Optimal GPV sequence: {{Val list| 46, 368, 414, 460, 874de, 1334deg }}
{{Multival|legend=1| 46 92 -46 39 -202 -365 }}


Badness: 0.022441
[[Optimal tuning]] ([[POTE]]): ~3/2 = 701.6074


== Octoid ==
{{Val list|legend=1| 46, 368, 414, 460, 874d }}
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
[[Badness]]: 0.308505


[[Comma list]]: 4375/4374, 16875/16807
=== 11-limit ===
Subgroup: 2.3.5.7.11


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


[[Wedgie]]: {{multival|24 32 40 -5 -4 3}}
Mapping: [{{val| 46 73 107 129 159 }}, {{val| 0 -1 -2 1 1 }}]


Mapping generators: ~49/45, ~7/5
Optimal tuning (POTE): ~3/2 = 701.5951


[[POTE generator]]: ~7/5 = 583.940
Optimal GPV sequence: {{Val list| 46, 368, 414, 460, 874de }}


[[Tuning ranges]]:  
Badness: 0.073783
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
* 7-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 585.084]


{{Val list|legend=1| 8d, 72, 152, 224 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.042670
Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364


Scales: [[Octoid72]], [[Octoid80]]
Mapping: [{{val| 46 73 107 129 159 170 }}, {{val| 0 -1 -2 1 1 2 }}]


=== 11-limit ===
Optimal tuning (POTE): ~3/2 = 701.6419
Subgroup: 2.3.5.7.11


Comma list: 540/539, 1375/1372, 4000/3993
Optimal GPV sequence: {{Val list| 46, 368, 414, 460, 874de, 1334de }}


Mapping: [{{val|8 1 3 3 16}}, {{val|0 3 4 5 3}}]
Badness: 0.040751


POTE generator: ~7/5 = 583.962
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Tuning ranges:  
Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
* 11-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 585.084]


Optimal GPV sequence: {{Val list| 72, 152, 224 }}
Mapping: [{{val| 46 73 107 129 159 170 188 }}, {{val| 0 -1 -2 1 1 2 0 }}]


Badness: 0.014097
Optimal tuning (POTE): ~3/2 = 701.6425


Scales: [[Octoid72]], [[Octoid80]]
Optimal GPV sequence: {{Val list| 46, 368, 414, 460, 874de, 1334deg }}


==== 13-limit ====
Badness: 0.022441
Subgroup: 2.3.5.7.11.13


Comma list: 540/539, 625/624, 729/728, 1375/1372
== Octoid ==
The '''octoid''' temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.


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


POTE generator: ~7/5 = 583.905
[[Comma list]]: 4375/4374, 16875/16807


Optimal GPV sequence: {{Val list| 72, 152f, 224 }}
[[Mapping]]: [{{val|8 1 3 3}}, {{val|0 3 4 5}}]


Badness: 0.015274
[[Wedgie]]: {{multival|24 32 40 -5 -4 3}}


Scales: [[Octoid72]], [[Octoid80]]
Mapping generators: ~49/45, ~7/5


; Music
[[POTE generator]]: ~7/5 = 583.940
* [https://www.archive.org/details/Dreyfus http://www.archive.org/details/Dreyfus] [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play]


===== 17-limit =====
[[Tuning ranges]]:
Subgroup: 2.3.5.7.11.13.17
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
* 7-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 585.084]


Comma list: 375/374, 540/539, 625/624, 715/714, 729/728
{{Val list|legend=1| 8d, 72, 152, 224 }}


Mapping: [{{val|8 1 3 3 16 -21 -14}}, {{val|0 3 4 5 3 13 12}}]
[[Badness]]: 0.042670


POTE generator: ~7/5 = 583.842
Scales: [[Octoid72]], [[Octoid80]]


Optimal GPV sequence: {{Val list| 72, 152fg, 224, 296, 520g }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.014304
Comma list: 540/539, 1375/1372, 4000/3993


Scales: [[Octoid72]], [[Octoid80]]
Mapping: [{{val|8 1 3 3 16}}, {{val|0 3 4 5 3}}]


===== 19-limit =====
POTE generator: ~7/5 = 583.962
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
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]
* 11-odd-limit diamond monotone and tradeoff: ~7/5 = [582.512, 585.084]


Mapping: [{{val|8 1 3 3 16 -21 -14 34}}, {{val|0 3 4 5 3 13 12 0}}]
Optimal GPV sequence: {{Val list| 72, 152, 224 }}


POTE generator: ~7/5 = 583.932
Badness: 0.014097
 
Optimal GPV sequence: {{Val list| 72, 152fg, 224 }}
 
Badness: 0.016036


Scales: [[Octoid72]], [[Octoid80]]
Scales: [[Octoid72]], [[Octoid80]]


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


Comma list: 169/168, 325/324, 364/363, 540/539
Comma list: 540/539, 625/624, 729/728, 1375/1372


Mapping: [{{val|8 1 3 3 16 14}}, {{val|0 3 4 5 3 4}}]
Mapping: [{{val|8 1 3 3 16 -21}}, {{val|0 3 4 5 3 13}}]


POTE generator: ~7/5 = 583.892
POTE generator: ~7/5 = 583.905


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


Badness: 0.021679
Badness: 0.015274


Scales: [[Octoid72]], [[Octoid80]]
Scales: [[Octoid72]], [[Octoid80]]
; Music
* [https://www.archive.org/details/Dreyfus http://www.archive.org/details/Dreyfus] [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play]


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


Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
Comma list: 375/374, 540/539, 625/624, 715/714, 729/728


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


POTE generator: ~7/5 = 583.811
POTE generator: ~7/5 = 583.842


Optimal GPV sequence: {{Val list| 72, 152, 224fg, 296ffg }}
Optimal GPV sequence: {{Val list| 72, 152fg, 224, 296, 520g }}


Badness: 0.015614
Badness: 0.014304


Scales: [[Octoid72]], [[Octoid80]]
Scales: [[Octoid72]], [[Octoid80]]
Line 1,444: Line 1,440:
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714


Mapping: [{{val|8 1 3 3 16 14 21 34}}, {{val|0 3 4 5 3 4 3 0}}]
Mapping: [{{val|8 1 3 3 16 -21 -14 34}}, {{val|0 3 4 5 3 13 12 0}}]


POTE generator: ~7/5 = 584.064
POTE generator: ~7/5 = 583.932


Optimal GPV sequence: {{Val list| 72, 152, 224fg, 376ffgh }}
Optimal GPV sequence: {{Val list| 72, 152fg, 224 }}


Badness: 0.016321
Badness: 0.016036


Scales: [[Octoid72]], [[Octoid80]]
Scales: [[Octoid72]], [[Octoid80]]


==== Hexadecoid ====
==== Octopus ====
Hexadecoid (80&144) has a period of 1/16 octave and tempers out 4225/4224.
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 325/324, 364/363, 540/539


Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
Mapping: [{{val|8 1 3 3 16 14}}, {{val|0 3 4 5 3 4}}]


Mapping: [{{val|16 26 38 46 56 59}}, {{val|0 -3 -4 -5 -3 1}}]
POTE generator: ~7/5 = 583.892


POTE generator: ~13/8 = 841.015
Optimal GPV sequence: {{Val list| 72, 152, 224f }}


Optimal GPV sequence: {{Val list| 80, 144, 224 }}
Badness: 0.021679


Badness: 0.030818
Scales: [[Octoid72]], [[Octoid80]]


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


Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
Comma list: 169/168, 221/220, 289/288, 325/324, 540/539


Mapping: [{{val|16 26 38 46 56 59 65}}, {{val|0 -3 -4 -5 -3 1 2}}]
Mapping: [{{val|8 1 3 3 16 14 21}}, {{val|0 3 4 5 3 4 3}}]


POTE generator: ~13/8 = 840.932
POTE generator: ~7/5 = 583.811


Optimal GPV sequence: {{Val list| 80, 144, 224, 528dg }}
Optimal GPV sequence: {{Val list| 72, 152, 224fg, 296ffg }}
 
Badness: 0.015614


Badness: 0.028611
Scales: [[Octoid72]], [[Octoid80]]


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


Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399


Mapping: [{{val|16 26 38 46 56 59 65 68}}, {{val|0 -3 -4 -5 -3 1 2 0}}]
Mapping: [{{val|8 1 3 3 16 14 21 34}}, {{val|0 3 4 5 3 4 3 0}}]


POTE generator: ~13/8 = 840.896
POTE generator: ~7/5 = 584.064


Optimal GPV sequence: {{Val list| 80, 144, 224, 304dh, 528dghh }}
Optimal GPV sequence: {{Val list| 72, 152, 224fg, 376ffgh }}


Badness: 0.023731
Badness: 0.016321


== Amity ==
Scales: [[Octoid72]], [[Octoid80]]
{{main| Amity }}
{{see also| Amity family #Amity }}


The generator for amity temperament is the acute minor third, which means the 6/5 just minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit [[amity comma]], 1600000/1594323, [[5120/5103]] and [[6144/6125]]. It can also be described as the 46&53 temperament. [[99edo|99EDO]] is a good tuning for amity, with generator 28\99, and MOS of 11, 18, 25, 32, 39, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.
==== Hexadecoid ====
Hexadecoid (80&144) has a period of 1/16 octave and tempers out 4225/4224.


In the 5-limit amity is a genuine microtemperament, with 58\205 being a possible tuning. Another good choice is (64/5)<sup>1/13</sup>, which gives pure major thirds.
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5.7
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224


[[Comma list]]: 4375/4374, 5120/5103
Mapping: [{{val|16 26 38 46 56 59}}, {{val|0 -3 -4 -5 -3 1}}]


[[Mapping]]: [{{val| 1 3 6 -2 }}, {{val| 0 -5 -13 17 }}]
POTE generator: ~13/8 = 841.015


{{Multival|legend=1| 5 13 -17 9 -41 -76 }}
Optimal GPV sequence: {{Val list| 80, 144, 224 }}


[[POTE generator]]: ~128/105 = 339.432
Badness: 0.030818


{{Val list|legend=1| 7, 32c, 39, 46, 53, 99, 251, 350, 601cd, 951bcdd }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Badness]]: 0.023649
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224


=== 11-limit ===
Mapping: [{{val|16 26 38 46 56 59 65}}, {{val|0 -3 -4 -5 -3 1 2}}]
Subgroup: 2.3.5.7.11


Comma list: 540/539, 4375/4374, 5120/5103
POTE generator: ~13/8 = 840.932


Mapping: [{{val| 1 3 6 -2 21 }}, {{val| 0 -5 -13 17 -62 }}]
Optimal GPV sequence: {{Val list| 80, 144, 224, 528dg }}


POTE generator: ~128/105 = 339.464
Badness: 0.028611


Optimal GPV sequence: {{Val list| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.031506
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444


==== 13-limit ====
Mapping: [{{val|16 26 38 46 56 59 65 68}}, {{val|0 -3 -4 -5 -3 1 2 0}}]
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 540/539, 625/624, 847/845
POTE generator: ~13/8 = 840.896


Mapping: [{{val| 1 3 6 -2 21 17 }}, {{val| 0 -5 -13 17 -62 -47 }}]
Optimal GPV sequence: {{Val list| 80, 144, 224, 304dh, 528dghh }}


POTE generator: ~128/105 = 339.481
Badness: 0.023731


Optimal GPV sequence: {{Val list| 46ef, 53, 99ef, 152f }} <nowiki>*</nowiki>
== Amity ==
{{main| Amity }}
{{see also| Amity family #Amity }}


<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
The generator for amity temperament is the acute minor third, which means the 6/5 just minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit [[amity comma]], 1600000/1594323, [[5120/5103]] and [[6144/6125]]. It can also be described as the 46&amp;53 temperament. [[99edo|99EDO]] is a good tuning for amity, with generator 28\99, and MOS of 11, 18, 25, 32, 39, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.


Badness: 0.028008
In the 5-limit amity is a genuine microtemperament, with 58\205 being a possible tuning. Another good choice is (64/5)<sup>1/13</sup>, which gives pure major thirds.


=== Hitchcock ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 2200/2187
[[Comma list]]: 4375/4374, 5120/5103


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


POTE generator: ~11/9 = 339.390
{{Multival|legend=1| 5 13 -17 9 -41 -76 }}


Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
[[POTE generator]]: ~128/105 = 339.432


Badness: 0.035187
{{Val list|legend=1| 7, 32c, 39, 46, 53, 99, 251, 350, 601cd, 951bcdd }}


==== 13-limit ====
[[Badness]]: 0.023649
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 169/168, 176/175, 325/324
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 3 6 -2 6 2 }}, {{val| 0 -5 -13 17 -9 6 }}]
Comma list: 540/539, 4375/4374, 5120/5103


POTE generator: ~11/9 = 339.419
Mapping: [{{val| 1 3 6 -2 21 }}, {{val| 0 -5 -13 17 -62 }}]


Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
POTE generator: ~128/105 = 339.464


Badness: 0.022448
Optimal GPV sequence: {{Val list| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}


==== 17-limit ====
Badness: 0.031506
Subgroup: 2.3.5.7.11.13.17


Comma list: 121/120, 154/153, 169/168, 176/175, 273/272
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 3 6 -2 6 2 -1 }}, {{val| 0 -5 -13 17 -9 6 18 }}]
Comma list: 352/351, 540/539, 625/624, 847/845


POTE generator: ~11/9 = 339.366
Mapping: [{{val| 1 3 6 -2 21 17 }}, {{val| 0 -5 -13 17 -62 -47 }}]


Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
POTE generator: ~128/105 = 339.481


Badness: 0.019395
Optimal GPV sequence: {{Val list| 46ef, 53, 99ef, 152f }} <nowiki>*</nowiki>


==== 19-limit ====
<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189
Badness: 0.028008


Mapping: [{{val| 1 3 6 -2 6 2 -1 0 }}, {{val| 0 -5 -13 17 -9 6 18 15 }}]
=== Hitchcock ===
Subgroup: 2.3.5.7.11


POTE generator: ~11/9 = 339.407
Comma list: 121/120, 176/175, 2200/2187


Optimal GPV sequence: {{Val list| 7, 39h, 46, 53, 99h }}
Mapping: [{{val| 1 3 6 -2 6 }}, {{val| 0 -5 -13 17 -9 }}]


Badness: 0.017513
POTE generator: ~11/9 = 339.390


=== Catamite ===
Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
Subgroup: 2.3.5.7.11


Comma list: 441/440, 896/891, 4375/4374
Badness: 0.035187
 
Mapping: [{{val|1 3 6 -2 -7}}, {{val|0 -5 -13 17 37}}]
 
POTE generator: ~128/105 = 339.340
 
Optimal GPV sequence: {{Val list| 46, 99e, 145, 244e }}
 
Badness: 0.040976


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


Comma list: 196/195, 352/351, 364/363, 4375/4374
Comma list: 121/120, 169/168, 176/175, 325/324


Mapping: [{{val|1 3 6 -2 -7 -11}}, {{val|0 -5 -13 17 37 52}}]
Mapping: [{{val| 1 3 6 -2 6 2 }}, {{val| 0 -5 -13 17 -9 6 }}]


POTE generator: ~128/105 = 339.313
POTE generator: ~11/9 = 339.419


Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}


Badness: 0.034215
Badness: 0.022448


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


Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155
Comma list: 121/120, 154/153, 169/168, 176/175, 273/272


Mapping: [{{val|1 3 6 -2 -7 -11 -1}}, {{val|0 -5 -13 17 37 52 18}}]
Mapping: [{{val| 1 3 6 -2 6 2 -1 }}, {{val| 0 -5 -13 17 -9 6 18 }}]


POTE generator: ~17/14 = 339.313
POTE generator: ~11/9 = 339.366


Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}


Badness: 0.021193
Badness: 0.019395


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


Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475
Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189


Mapping: [{{val|1 3 6 -2 -7 -11 -1 -13}}, {{val|0 -5 -13 17 37 52 18 61}}]
Mapping: [{{val| 1 3 6 -2 6 2 -1 0 }}, {{val| 0 -5 -13 17 -9 6 18 15 }}]


POTE generator: ~17/14 = 339.325
POTE generator: ~11/9 = 339.407


Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
Optimal GPV sequence: {{Val list| 7, 39h, 46, 53, 99h }}


Badness: 0.018864
Badness: 0.017513


=== Hemiamity ===
=== Catamite ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4375/4374, 5120/5103
Comma list: 441/440, 896/891, 4375/4374


Mapping: [{{val| 2 1 -1 13 13 }}, {{val| 0 5 13 -17 -14 }}]
Mapping: [{{val|1 3 6 -2 -7}}, {{val|0 -5 -13 17 37}}]


Mapping generators: ~99/70, ~64/55
POTE generator: ~128/105 = 339.340


POTE generator: ~64/55 = 260.561
Optimal GPV sequence: {{Val list| 46, 99e, 145, 244e }}


Optimal GPV sequence: {{Val list| 14cde, 46, 106, 152, 350, 502d }}
Badness: 0.040976
 
Badness: 0.031307


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


Comma list: 352/351, 847/845, 1716/1715, 3025/3024
Comma list: 196/195, 352/351, 364/363, 4375/4374


Mapping: [{{val| 2 1 -1 13 13 20 }}, {{val| 0 5 13 -17 -14 -29 }}]
Mapping: [{{val|1 3 6 -2 -7 -11}}, {{val|0 -5 -13 17 37 52}}]


POTE generator: ~64/55 = 260.583
POTE generator: ~128/105 = 339.313


Optimal GPV sequence: {{Val list| 46, 106f, 152f, 198, 350f, 548cdff }}
Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}


Badness: 0.025784
Badness: 0.034215


== Parakleismic ==
==== 17-limit ====
{{main| Parakleismic }}
Subgroup: 2.3.5.7.11.13.17


In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo|8 14 -13}}, with the [[118edo|118EDO]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being {{multival|13 14 35 -8 19 42}} and adding 3136/3125 and 4375/4374, and the 11-limit wedgie {{multival|13 14 35 -36 -8 19 -102 42 -132 -222}} adding 385/384. For the 7-limit [[99edo|99EDO]] may be preferred, but in the 11-limit it is best to stick with 118.
Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155


Subgroup: 2.3.5
Mapping: [{{val|1 3 6 -2 -7 -11 -1}}, {{val|0 -5 -13 17 37 52 18}}]


[[Comma list]]: 1224440064/1220703125
POTE generator: ~17/14 = 339.313


[[Mapping]]: [{{val|1 5 6}}, {{val|0 -13 -14}}]
Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}


[[POTE generator]]: ~6/5 = 315.240
Badness: 0.021193


{{Val list|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


[[Badness]]: 0.043279
Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475


=== 7-limit ===
Mapping: [{{val|1 3 6 -2 -7 -11 -1 -13}}, {{val|0 -5 -13 17 37 52 18 61}}]
Subgroup: 2.3.5.7


[[Comma list]]: 3136/3125, 4375/4374
POTE generator: ~17/14 = 339.325


[[Mapping]]: [{{val|1 5 6 12}}, {{val|0 -13 -14 -35}}]
Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}


[[Wedgie]]: {{multival|13 14 35 -8 19 42}}
Badness: 0.018864


[[POTE generator]]: ~6/5 = 315.181
=== Hemiamity ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 19, 80, 99, 217, 316, 415 }}
Comma list: 3025/3024, 4375/4374, 5120/5103


[[Badness]]: 0.027431
Mapping: [{{val| 2 1 -1 13 13 }}, {{val| 0 5 13 -17 -14 }}]


=== 11-limit ===
Mapping generators: ~99/70, ~64/55
Subgroup: 2.3.5.7.11


Comma list: 385/384, 3136/3125, 4375/4374
POTE generator: ~64/55 = 260.561


Mapping: [{{val|1 5 6 12 -6}}, {{val|0 -13 -14 -35 36}}]
Optimal GPV sequence: {{Val list| 14cde, 46, 106, 152, 350, 502d }}


POTE generator: ~6/5 = 315.251
Badness: 0.031307


Optimal GPV sequence: {{Val list| 19, 99, 118 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.049711
Comma list: 352/351, 847/845, 1716/1715, 3025/3024


=== Paralytic ===
Mapping: [{{val| 2 1 -1 13 13 20 }}, {{val| 0 5 13 -17 -14 -29 }}]
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
POTE generator: ~64/55 = 260.583


Comma list: 441/440, 3136/3125, 4375/4374
Optimal GPV sequence: {{Val list| 46, 106f, 152f, 198, 350f, 548cdff }}


Mapping: [{{val|1 5 6 12 25}}, {{val|0 -13 -14 -35 -82}}]
Badness: 0.025784


POTE generator: ~6/5 = 315.220
== Parakleismic ==
{{main| Parakleismic }}


Optimal GPV sequence: {{Val list| 19e, 99e, 118, 217, 335, 552d, 887dd }}
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo|8 14 -13}}, with the [[118edo|118EDO]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being {{multival|13 14 35 -8 19 42}} and adding 3136/3125 and 4375/4374, and the 11-limit wedgie {{multival|13 14 35 -36 -8 19 -102 42 -132 -222}} adding 385/384. For the 7-limit [[99edo|99EDO]] may be preferred, but in the 11-limit it is best to stick with 118.


Badness: 0.036027
Subgroup: 2.3.5


==== 13-limit ====
[[Comma list]]: 1224440064/1220703125
Subgroup: 2.3.5.7.11.13


Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
[[Mapping]]: [{{val|1 5 6}}, {{val|0 -13 -14}}]


Mapping: [{{val|1 5 6 12 25 -16}}, {{val|0 -13 -14 -35 -82 75}}]
[[POTE generator]]: ~6/5 = 315.240


POTE generator: ~6/5 = 315.214
{{Val list|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}


Optimal GPV sequence: {{Val list| 99e, 118, 217, 552d, 769de }}
[[Badness]]: 0.043279


Badness: 0.044710
=== 7-limit ===
Subgroup: 2.3.5.7


==== Paraklein ====
[[Comma list]]: 3136/3125, 4375/4374
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
[[Mapping]]: [{{val|1 5 6 12}}, {{val|0 -13 -14 -35}}]


Comma list: 196/195, 352/351, 625/624, 729/728
[[Wedgie]]: {{multival|13 14 35 -8 19 42}}


Mapping: [{{val|1 5 6 12 25 15}}, {{val|0 -13 -14 -35 -82 -43}}]
[[POTE generator]]: ~6/5 = 315.181


POTE generator: ~6/5 = 315.225
{{Val list|legend=1| 19, 80, 99, 217, 316, 415 }}


Optimal GPV sequence: {{Val list| 19e, 99ef, 118, 217ff, 335ff }}
[[Badness]]: 0.027431


Badness: 0.037618
=== 11-limit ===
 
=== Parkleismic ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 176/175, 1375/1372, 2200/2187
Comma list: 385/384, 3136/3125, 4375/4374


Mapping: [{{val|1 5 6 12 20}}, {{val|0 -13 -14 -35 -63}}]
Mapping: [{{val|1 5 6 12 -6}}, {{val|0 -13 -14 -35 36}}]


POTE generator: ~6/5 = 315.060
POTE generator: ~6/5 = 315.251


Optimal GPV sequence: {{Val list| 19e, 80, 179, 259cd }}
Optimal GPV sequence: {{Val list| 19, 99, 118 }}


Badness: 0.055884
Badness: 0.049711


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


Comma list: 169/168, 176/175, 325/324, 1375/1372
Subgroup: 2.3.5.7.11


Mapping: [{{val|1 5 6 12 20 10}}, {{val|0 -13 -14 -35 -63 -24}}]
Comma list: 441/440, 3136/3125, 4375/4374


POTE generator: ~6/5 = 315.075
Mapping: [{{val|1 5 6 12 25}}, {{val|0 -13 -14 -35 -82}}]


Optimal GPV sequence: {{Val list| 19e, 80, 179 }}
POTE generator: ~6/5 = 315.220


Badness: 0.036559
Optimal GPV sequence: {{Val list| 19e, 99e, 118, 217, 335, 552d, 887dd }}
 
Badness: 0.036027
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== Paradigmic ===
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
Subgroup: 2.3.5.7.11


Comma list: 540/539, 896/891, 3136/3125
Mapping: [{{val|1 5 6 12 25 -16}}, {{val|0 -13 -14 -35 -82 75}}]


Mapping: [{{val|1 5 6 12 -1}}, {{val|0 -13 -14 -35 17}}]
POTE generator: ~6/5 = 315.214


POTE generator: ~6/5 = 315.096
Optimal GPV sequence: {{Val list| 99e, 118, 217, 552d, 769de }}


Optimal GPV sequence: {{Val list| 19, 61d, 80, 99e, 179e }}
Badness: 0.044710


Badness: 0.041720
==== 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]].


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


Comma list: 169/168, 325/324, 540/539, 832/825
Comma list: 196/195, 352/351, 625/624, 729/728


Mapping: [{{val|1 5 6 12 -1 10}}, {{val|0 -13 -14 -35 17 -24}}]
Mapping: [{{val|1 5 6 12 25 15}}, {{val|0 -13 -14 -35 -82 -43}}]


POTE generator: ~6/5 = 315.080
POTE generator: ~6/5 = 315.225


Optimal GPV sequence: {{Val list| 19, 61d, 80, 99e, 179e }}
Optimal GPV sequence: {{Val list| 19e, 99ef, 118, 217ff, 335ff }}


Badness: 0.035781
Badness: 0.037618


=== Semiparakleismic ===
=== Parkleismic ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 3136/3125, 4375/4374
Comma list: 176/175, 1375/1372, 2200/2187


Mapping: [{{val|2 10 12 24 19}}, {{val|0 -13 -14 -35 -23}}]
Mapping: [{{val|1 5 6 12 20}}, {{val|0 -13 -14 -35 -63}}]


POTE generator: ~6/5 = 315.181
POTE generator: ~6/5 = 315.060


Optimal GPV sequence: {{Val list| 80, 118, 198, 316, 514c, 830c }}
Optimal GPV sequence: {{Val list| 19e, 80, 179, 259cd }}


Badness: 0.034208
Badness: 0.055884
 
==== Semiparamint ====
This extension was named ''semiparakleismic'' in the earlier materials.


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


Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
Comma list: 169/168, 176/175, 325/324, 1375/1372


Mapping: [{{val|2 10 12 24 19 -1}}, {{val|0 -13 -14 -35 -23 16}}]
Mapping: [{{val|1 5 6 12 20 10}}, {{val|0 -13 -14 -35 -63 -24}}]


POTE generator: ~6/5 = 315.156
POTE generator: ~6/5 = 315.075


Optimal GPV sequence: {{Val list| 80, 118, 198 }}
Optimal GPV sequence: {{Val list| 19e, 80, 179 }}


Badness: 0.033775
Badness: 0.036559


==== Semiparawolf ====
=== Paradigmic ===
This extension was named ''gentsemiparakleismic'' in the earlier materials.  
Subgroup: 2.3.5.7.11


Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 896/891, 3136/3125


Comma list: 169/168, 325/324, 364/363, 3136/3125
Mapping: [{{val|1 5 6 12 -1}}, {{val|0 -13 -14 -35 17}}]


Mapping: [{{val|2 10 12 24 19 20}}, {{val|0 -13 -14 -35 -23 -24}}]
POTE generator: ~6/5 = 315.096


POTE generator: ~6/5 = 315.184
Optimal GPV sequence: {{Val list| 19, 61d, 80, 99e, 179e }}


Optimal GPV sequence: {{Val list| 80, 118f, 198f }}
Badness: 0.041720


Badness: 0.040467
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


== Counterkleismic ==
Comma list: 169/168, 325/324, 540/539, 832/825
{{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).
Mapping: [{{val|1 5 6 12 -1 10}}, {{val|0 -13 -14 -35 17 -24}}]


Subgroup: 2.3.5.7
POTE generator: ~6/5 = 315.080


[[Comma list]]: 4375/4374, 158203125/157351936
Optimal GPV sequence: {{Val list| 19, 61d, 80, 99e, 179e }}


[[Mapping]]: [{{val|1 -5 -4 -18}}, {{val|0 25 24 79}}]
Badness: 0.035781


[[Wedgie]]: {{multival|25 24 79 -20 55 116}}
=== Semiparakleismic ===
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~6/5 = 316.060
Comma list: 3025/3024, 3136/3125, 4375/4374


{{Val list|legend=1| 19, 205, 224, 243, 467 }}
Mapping: [{{val|2 10 12 24 19}}, {{val|0 -13 -14 -35 -23}}]


[[Badness]]: 0.090553
POTE generator: ~6/5 = 315.181


=== 11-limit ===
Optimal GPV sequence: {{Val list| 80, 118, 198, 316, 514c, 830c }}
Subgroup: 2.3.5.7.11


Comma list: 540/539, 4375/4374, 2097152/2096325
Badness: 0.034208


Mapping: [{{val|1 -5 -4 -18 19}}, {{val|0 25 24 79 -59}}]
==== Semiparamint ====
This extension was named ''semiparakleismic'' in the earlier materials.


POTE generator: ~6/5 = 316.071
Subgroup: 2.3.5.7.11.13


Optimal GPV sequence: {{Val list| 19, 205, 224 }}
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374


Badness: 0.070952
Mapping: [{{val|2 10 12 24 19 -1}}, {{val|0 -13 -14 -35 -23 16}}]


==== 13-limit ====
POTE generator: ~6/5 = 315.156
Subgroup: 2.3.5.7.11.13


Comma list: 540/539, 625/624, 729/728, 10985/10976
Optimal GPV sequence: {{Val list| 80, 118, 198 }}


Mapping: [{{val|1 -5 -4 -18 19 -15}}, {{val|0 25 24 79 -59 71}}]
Badness: 0.033775


POTE generator: ~6/5 = 316.070
==== Semiparawolf ====
This extension was named ''gentsemiparakleismic'' in the earlier materials.  


Optimal GPV sequence: {{Val list| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}
Subgroup: 2.3.5.7.11.13


Badness: 0.033874
Comma list: 169/168, 325/324, 364/363, 3136/3125


=== Counterlytic ===
Mapping: [{{val|2 10 12 24 19 20}}, {{val|0 -13 -14 -35 -23 -24}}]
Subgroup: 2.3.5.7.11


Comma list: 1375/1372, 4375/4374, 496125/495616
POTE generator: ~6/5 = 315.184


Mapping: [{{val|1 -5 -4 -18 -40}}, {{val|0 25 24 79 165}}]
Optimal GPV sequence: {{Val list| 80, 118f, 198f }}


POTE generator: ~6/5 = 316.065
Badness: 0.040467


Optimal GPV sequence: {{Val list| 19e, 205e, 224 }}
== Counterkleismic ==
{{see also| High badness temperaments #Counterhanson}}


Badness: 0.065400
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).


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


Comma list: 625/624, 729/728, 1375/1372, 10985/10976
[[Comma list]]: 4375/4374, 158203125/157351936


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


POTE generator: ~6/5 = 316.065
[[Wedgie]]: {{multival|25 24 79 -20 55 116}}


Optimal GPV sequence: {{Val list| 19e, 205e, 224 }}
[[POTE generator]]: ~6/5 = 316.060


Badness: 0.029782
{{Val list|legend=1| 19, 205, 224, 243, 467 }}


== Quincy ==
[[Badness]]: 0.090553
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 823543/819200
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val|1 2 3 3}}, {{val|0 -30 -49 -14}}]
Comma list: 540/539, 4375/4374, 2097152/2096325


[[Wedgie]]: {{multival|30 49 14 8 -62 -105}}
Mapping: [{{val|1 -5 -4 -18 19}}, {{val|0 25 24 79 -59}}]


[[POTE generator]]: ~1728/1715 = 16.613
POTE generator: ~6/5 = 316.071


{{Val list|legend=1| 72, 217, 289 }}
Optimal GPV sequence: {{Val list| 19, 205, 224 }}


[[Badness]]: 0.079657
Badness: 0.070952


=== 11-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 441/440, 4000/3993, 4375/4374
Comma list: 540/539, 625/624, 729/728, 10985/10976


Mapping: [{{val|1 2 3 3 4}}, {{val|0 -30 -49 -14 -39}}]
Mapping: [{{val|1 -5 -4 -18 19 -15}}, {{val|0 25 24 79 -59 71}}]


POTE generator: ~100/99 = 16.613
POTE generator: ~6/5 = 316.070


Optimal GPV sequence: {{Val list| 72, 217, 289 }}
Optimal GPV sequence: {{Val list| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}


Badness: 0.030875
Badness: 0.033874


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


Comma list: 364/363, 441/440, 676/675, 4375/4374
Comma list: 1375/1372, 4375/4374, 496125/495616


Mapping: [{{val|1 2 3 3 4 5}}, {{val|0 -30 -49 -14 -39 -94}}]
Mapping: [{{val|1 -5 -4 -18 -40}}, {{val|0 25 24 79 165}}]


POTE generator: ~100/99 = 16.602
POTE generator: ~6/5 = 316.065


Optimal GPV sequence: {{Val list| 72, 145, 217, 289 }}
Optimal GPV sequence: {{Val list| 19e, 205e, 224 }}


Badness: 0.023862
Badness: 0.065400


=== 17-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155
Comma list: 625/624, 729/728, 1375/1372, 10985/10976


Mapping: [{{val|1 2 3 3 4 5 5}}, {{val|0 -30 -49 -14 -39 -94 -66}}]
Mapping: [{{val|1 -5 -4 -18 -40 -15}}, {{val|0 25 24 79 165 71}}]


POTE generator: ~100/99 = 16.602
POTE generator: ~6/5 = 316.065


Optimal GPV sequence: {{Val list| 72, 145, 217, 289 }}
Optimal GPV sequence: {{Val list| 19e, 205e, 224 }}


Badness: 0.014741
Badness: 0.029782


=== 19-limit ===
== Quincy ==
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7


Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675
[[Comma list]]: 4375/4374, 823543/819200


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


POTE generator: ~100/99 = 16.594
[[Wedgie]]: {{multival|30 49 14 8 -62 -105}}


Optimal GPV sequence: {{Val list| 72, 145, 217 }}
[[POTE generator]]: ~1728/1715 = 16.613


Badness: 0.015197
{{Val list|legend=1| 72, 217, 289 }}


== Sfourth ==
[[Badness]]: 0.079657
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''


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


[[Comma list]]: 4375/4374, 64827/64000
Comma list: 441/440, 4000/3993, 4375/4374


[[Mapping]]: [{{val|1 2 3 3}}, {{val|0 -19 -31 -9}}]
Mapping: [{{val|1 2 3 3 4}}, {{val|0 -30 -49 -14 -39}}]


{{Multival|legend=1|19 31 9 5 -39 -66}}
POTE generator: ~100/99 = 16.613


[[POTE generator]]: ~49/48 = 26.287
Optimal GPV sequence: {{Val list| 72, 217, 289 }}


{{Val list|legend=1| 45, 46, 91, 137d }}
Badness: 0.030875


[[Badness]]: 0.123291
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 11-limit ===
Comma list: 364/363, 441/440, 676/675, 4375/4374
Subgroup: 2.3.5.7.11


Comma list: 121/120, 441/440, 4375/4374
Mapping: [{{val|1 2 3 3 4 5}}, {{val|0 -30 -49 -14 -39 -94}}]


Mapping: [{{val|1 2 3 3 4}}, {{val|0 -19 -31 -9 -25}}]
POTE generator: ~100/99 = 16.602


POTE generator: ~49/48 = 26.286
Optimal GPV sequence: {{Val list| 72, 145, 217, 289 }}


Optimal GPV sequence: {{Val list| 45e, 46, 91e, 137de }}
Badness: 0.023862


Badness: 0.054098
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


==== 13-limit ====
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 169/168, 325/324, 441/440
Mapping: [{{val|1 2 3 3 4 5 5}}, {{val|0 -30 -49 -14 -39 -94 -66}}]


Mapping: [{{val|1 2 3 3 4 4}}, {{val|0 -19 -31 -9 -25 -14}}]
POTE generator: ~100/99 = 16.602


POTE generator: ~49/48 = 26.310
Optimal GPV sequence: {{Val list| 72, 145, 217, 289 }}


Optimal GPV sequence: {{Val list| 45ef, 46, 91ef, 137def }}
Badness: 0.014741


Badness: 0.033067
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


=== Sfour ===
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675
Subgroup: 2.3.5.7.11


Comma list: 385/384, 2401/2376, 4375/4374
Mapping: [{{val|1 2 3 3 4 5 5 4}}, {{val|0 -30 -49 -14 -39 -94 -66 18}}]


Mapping: [{{val|1 2 3 3 3}}, {{val|0 -19 -31 -9 21}}]
POTE generator: ~100/99 = 16.594


POTE generator: ~49/48 = 26.246
Optimal GPV sequence: {{Val list| 72, 145, 217 }}


Optimal GPV sequence: {{Val list| 45, 46, 91, 137d }}
Badness: 0.015197


Badness: 0.076567
== Sfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''


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


Comma list: 196/195, 364/363, 385/384, 4375/4374
[[Comma list]]: 4375/4374, 64827/64000


Mapping: [{{val|1 2 3 3 3 3}}, {{val|0 -19 -31 -9 21 32}}]
[[Mapping]]: [{{val|1 2 3 3}}, {{val|0 -19 -31 -9}}]


POTE generator: ~49/48 = 26.239
{{Multival|legend=1|19 31 9 5 -39 -66}}


Optimal GPV sequence: {{Val list| 45, 46, 91, 137d }}
[[POTE generator]]: ~49/48 = 26.287


Badness: 0.051893
{{Val list|legend=1| 45, 46, 91, 137d }}


== Trideci ==
[[Badness]]: 0.123291
{{see also| High badness temperaments #Tridecatonic }}


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]]").
=== 11-limit ===
Subgroup: 2.3.5.7.11


Subgroup: 2.3.5.7
Comma list: 121/120, 441/440, 4375/4374


[[Comma list]]: 4375/4374, 83349/81920
Mapping: [{{val|1 2 3 3 4}}, {{val|0 -19 -31 -9 -25}}]


[[Mapping]]: [{{val|13 21 31 36}}, {{val|0 -1 -2 1}}]
POTE generator: ~49/48 = 26.286


[[POTE generator]]: ~3/2 = 699.1410
Optimal GPV sequence: {{Val list| 45e, 46, 91e, 137de }}


{{Val list|legend=1| 26, 65, 91, 156d, 247cdd }}
Badness: 0.054098


[[Badness]]: 0.184585
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 11-limit ===
Comma list: 121/120, 169/168, 325/324, 441/440
Subgroup: 2.3.5.7.11


Comma list: 245/242, 385/384, 4375/4374
Mapping: [{{val|1 2 3 3 4 4}}, {{val|0 -19 -31 -9 -25 -14}}]


Mapping: [{{val|13 21 31 36 45}}, {{val|0 -1 -2 1 0}}]
POTE generator: ~49/48 = 26.310


POTE generator: ~3/2 = 699.6179
Optimal GPV sequence: {{Val list| 45ef, 46, 91ef, 137def }}


Optimal GPV sequence: {{Val list| 26, 65, 91, 156d, 247cdde }}
Badness: 0.033067


Badness: 0.084590
=== Sfour ===
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 385/384, 2401/2376, 4375/4374
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 245/242, 325/324, 385/384
Mapping: [{{val|1 2 3 3 3}}, {{val|0 -19 -31 -9 21}}]


Mapping: [{{val|13 21 31 36 45 48}}, {{val|0 -1 -2 1 0 0}}]
POTE generator: ~49/48 = 26.246


POTE generator: ~3/2 = 699.2969
Optimal GPV sequence: {{Val list| 45, 46, 91, 137d }}


Optimal GPV sequence: {{Val list| 26, 65f, 91f, 156dff }}
Badness: 0.076567


Badness: 0.052366
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 364/363, 385/384, 4375/4374
 
Mapping: [{{val|1 2 3 3 3 3}}, {{val|0 -19 -31 -9 21 32}}]
 
POTE generator: ~49/48 = 26.239
 
Optimal GPV sequence: {{Val list| 45, 46, 91, 137d }}
 
Badness: 0.051893
 
== Trideci ==
{{see also| High badness temperaments #Tridecatonic }}


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


[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
[[Comma list]]: 4375/4374, 83349/81920


[[Mapping]]: [{{val| 1 57 38 248 }}, {{val| 0 -73 -47 -323 }}]
[[Mapping]]: [{{val|13 21 31 36}}, {{val|0 -1 -2 1}}]


[[Optimal tuning]] ([[CTE]]): ~22/13 = 910.9323
[[POTE generator]]: ~3/2 = 699.1410


{{Val list|legend=1| 494, 1125, 1619 }}
{{Val list|legend=1| 26, 65, 91, 156d, 247cdd }}


[[Badness]]: 0.234
[[Badness]]: 0.184585


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


Comma list: 4375/4374, 759375/758912, 100663296/100656875
Comma list: 245/242, 385/384, 4375/4374


Mapping: [{{val| 1 57 38 248 -14 }}, {{val| 0 -73 -47 -323 23 }}]
Mapping: [{{val|13 21 31 36 45}}, {{val|0 -1 -2 1 0}}]


Optimal tuning (CTE): ~22/13 = 910.9323
POTE generator: ~3/2 = 699.6179


Optimal GPV sequence: {{val list| 494, 1125, 1619, 2113 }}
Optimal GPV sequence: {{Val list| 26, 65, 91, 156d, 247cdde }}


Badness: 0.0678
Badness: 0.084590


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


Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
Comma list: 169/168, 245/242, 325/324, 385/384


Mapping: [{{val| 1 57 38 248 -14 -13 }}, {{val| 0 -73 -47 -323 23 22 }}]
Mapping: [{{val|13 21 31 36 45 48}}, {{val|0 -1 -2 1 0 0}}]


Optimal tuning (CTE): ~22/13 = 910.9323
POTE generator: ~3/2 = 699.2969


Optimal GPV sequence: {{val list| 494, 1125, 1619, 2113 }}
Optimal GPV sequence: {{Val list| 26, 65f, 91f, 156dff }}


Badness: 0.0271
Badness: 0.052366


[[Category:Temperament collections]]
[[Category:Temperament collections]]
Retrieved from "https://en.xen.wiki/w/Ragismic_microtemperaments"