Miscellaneous 5-limit temperaments: Difference between revisions

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[[Badness]] (Sintel): 2.88
[[Badness]] (Sintel): 2.88
== Aluminium ==
: ''For extensions, see [[Ragismic microtemperaments #Aluminium]].''
Aluminium tempers out {{monzo| 92 -39 -13 }} in the 5-limit, and sets [[135/128]] to 1/13 of an [[octave]].
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| 92 -39 -13 }}
{{Mapping|legend=1| 13 0 92 | 0 1 -3 }}
: mapping generators: ~135/128, ~3
[[Optimal tuning]]s:
* [[WE]]: ~135/128 = 92.3068{{c}}, ~3/2 = 701.9791{{c}}
: [[error map]]: {{val| -0.012 +0.012 +0.009 }}
* [[CWE]]: ~135/128 = 92.3077{{c}}, ~3/2 = 701.9897{{c}}
: error map: {{val| 0.000 +0.032 +0.033 }}
{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}
[[Badness]] (Sintel): 2.89


== Avila ==
== Avila ==
Line 553: Line 575:


Scales (Scala files): [[wizard22]]
Scales (Scala files): [[wizard22]]
== Tridecatonic ==
: ''For extensions, see [[Octagar temperaments #Tridecatonic]], [[Ragismic microtemperaments #Trideci]], and [[Varunismic temperaments #Bosonic]].''
Tridecatonic tempers out the [[devil's tridecalimma]], the comma which associates the [[10/9]] minor tone with [[13edo|2\13]].
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| -11 26 -13 }}
{{Mapping|legend=1| 13 0 -11 | 0 1 2 }}
: mapping generators: ~531441/500000, ~3
[[Optimal tuning]]s:
* [[WE]]: ~531441/500000 = 92.3166{{c}}, ~3/2 = 701.2353{{c}}
: [[error map]]: {{val| +0.116 -0.688 +0.737 }}
* [[CWE]]: ~531441/500000 = 92.3077{{c}}, ~3/2 = 701.2353{{c}}
: error map: {{val| 0.000 -0.797 +0.617 }}
{{Optimal ET sequence|legend=1| 26, 39, 65, 286, 351, 416, 481, 546 }}
[[Badness]] (Sintel): 10.3


== Merman ==
== Merman ==
Line 574: Line 618:
: ''For extensions, see [[Porwell temperaments #Hendecatonic]] and [[Marvel temperaments #Hendeca]].''  
: ''For extensions, see [[Porwell temperaments #Hendecatonic]] and [[Marvel temperaments #Hendeca]].''  


The 5-limit version of hendecatonic simply equates a stack of eleven [[16/15]]'s with the [[2/1|octave]].  
The 5-limit version of hendecatonic simply equates a stack of eleven [[16/15]]'s with the [[octave]].  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 861: Line 905:
: ''For extensions, see [[Keemic temperaments #Undeka]].''  
: ''For extensions, see [[Keemic temperaments #Undeka]].''  


Undeka has a period of 1/11 octave, like [[#Hendecatonic|hendecatonic]], but this temperament equates a stack of eleven 6/5's and three [[2/1|octave]]s, so that three periods represent [[6/5]].  
Undeka has a period of 1/11 octave, like [[#Hendecatonic|hendecatonic]], but this temperament equates a stack of eleven 6/5's and three [[octave]]s, so that three periods represent [[6/5]].  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 1,282: Line 1,326:


[[Badness]] (Sintel): 52.3
[[Badness]] (Sintel): 52.3
== Triskaidekic ==
: ''For extensions, see [[Orwellismic temperaments #Triskaidekic]].''
[[Subgroup]]: 2.3.5
[[Comma list]]: 1220703125/1073741824
{{Mapping|legend=1| 13 0 30 | 0 1 0 }}
: Mapping generators: ~128/125, ~3
[[Optimal tuning]]s:
* [[WE]]: ~128/125 = 92.589{{c}}, ~3/2 = 698.263{{c}}
: [[error map]]: {{val| +3.656 -0.036 -8.647 }}
* [[CWE]]: ~128/125 = 92.308{{c}}, ~3/2 = 698.068{{c}}
: error map: {{val| 0.000 -3.887 -17.083 }}
{{Optimal ET sequence|legend=1| 13, 26, 91cc }}
[[Badness]] (Sintel): 52.6


== Satin ==
== Satin ==

Latest revision as of 03:48, 21 July 2026

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.

Below are listed some 5-limit temperaments that do not belong to some other temperament collection, the majority of which are restrictions to the 5-limit of temperaments that emerge more fully in higher limits or subgroups; they are sorted by TE logflat badness, of which most have Sintel's > 4.7 (Smith's > 0.20). Most of these temperaments have low accuracy, high-complexity generators, or large number of generators for simple consonances. This is not an exhaustive list. Only expect to find a temperament here if you have not found it in:

Ennealimmal

For extensions, see Septiennealimmal clan #Ennealimmal.

The 5-limit version of ennealimmal tempers out the ennealimma, the amount by which a stack of nine 27/25's falls short of the octave. Since the 7-limit temperament (definable by tempering out both 2401/2400 and 4375/4374, the two smallest superparticular ratios in the 7-limit) is far more natural to think of than the 5-limit, the 5-limit temperament is only provided below for bookkeeping purposes.

Subgroup: 2.3.5

Comma list: [1 -27 18⟩

Mapping: [⟨9 1 1], ⟨0 2 3]]

mapping generators: ~27/25, ~5/3

Optimal tunings:

  • WE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3190 ¢
error map: ⟨-0.000 +0.016 -0.023]
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3191 ¢
error map: ⟨0.000 +0.017 -0.023]

Optimal ET sequence: 27, 45, 72, 99, 171, 441, 612, 1665, 2277, 2889, 3501, 6390, 16281, 22671, 29061c

Badness (Sintel): 0.403

Unidecmic

For extensions, see Gamelismic clan #Unidec.

Subgroup: 2.3.5

Comma list: 31381059609/31250000000

Mapping: [⟨2 -1 -3], ⟨0 6 11]]

mapping generators: ~177147/125000, ~25000/19683

Optimal tunings:

  • WE: ~177147/125000 = 600.0124 ¢, ~25000/19683 = 416.9619 ¢
error map: ⟨+0.025 -0.196 +0.235]
  • CWE: ~177147/125000 = 600.0000 ¢, ~25000/19683 = 416.9561 ¢
error map: ⟨0.000 -0.236 +0.172]

Optimal ET sequence: 26, 46, 72, 118, 2524, 2642, 2760

Badness (Sintel): 1.93

Quasithird

For extensions, see Ragismic microtemperaments #Quasithird.

Subgroup: 2.3.5

Comma list: [55 -64 20⟩

Mapping: [⟨4 0 -11], ⟨0 5 16]]

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

Optimal tunings:

  • WE: ~51200000/43046721 = 299.9990 ¢, ~1594323/1280000 = 380.3934 ¢
error map: ⟨-0.004 +0.012 -0.008]
  • CWE: ~51200000/43046721 = 300.0000 ¢, ~1594323/1280000 = 380.3944 ¢
error map: ⟨0.000 +0.017 -0.004]

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

Badness (Sintel): 2.33

Supersharp

For extensions, see Trienstonic clan #Octokaidecal.

Supersharp tempers out 800/729, the difference between the 27/20 wolf fourth and the 40/27 wolf fifth, splitting the octave into two 27/20~40/27 semioctaves. It generally requires a very sharp fifth, even sharper than 3\5, as a generator. This means that five steps from the Zarlino generator sequence starting with 6/5 are tempered to one and a half octaves. The only reasonable 7-limit extension adds 28/27 and 50/49 to the comma list, taking advantage of the existing semioctave.

Subgroup: 2.3.5

Comma list: 800/729

Mapping: [⟨2 0 -5], ⟨0 1 3]]

mapping generators: ~27/20, ~3

Optimal tunings:

  • WE: ~27/20 = 596.986 ¢, ~3/2 = 725.434 ¢ (~10/9 = 128.448 ¢)
error map: ⟨-6.029 +17.450 -13.027]
  • CWE: ~27/20 = 600.000 ¢, ~3/2 = 726.548 ¢ (~10/9 = 126.548 ¢)
error map: ⟨0.000 +24.593 -6.670]

Optimal ET sequence: 8, 10, 18, 28b

Badness (Sintel): 2.88

Aluminium

For extensions, see Ragismic microtemperaments #Aluminium.

Aluminium tempers out [92 -39 -13⟩ in the 5-limit, and sets 135/128 to 1/13 of an octave.

Subgroup: 2.3.5

Comma list: [92 -39 -13⟩

Mapping: [⟨13 0 92], ⟨0 1 -3]]

mapping generators: ~135/128, ~3

Optimal tunings:

  • WE: ~135/128 = 92.3068 ¢, ~3/2 = 701.9791 ¢
error map: ⟨-0.012 +0.012 +0.009]
  • CWE: ~135/128 = 92.3077 ¢, ~3/2 = 701.9897 ¢
error map: ⟨0.000 +0.032 +0.033]

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

Badness (Sintel): 2.89

Avila

For extensions, see Septisemi temperaments #Avila.

Subgroup: 2.3.5

Comma list: 729/640

Mapping: [⟨1 0 -7], ⟨0 1 6]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1210.600 ¢, ~3/2 = 667.437 ¢
error map: ⟨+10.600 -23.917 +7.711]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 663.474 ¢
error map: ⟨0.000 -38.481 -5.472]

Optimal ET sequence: 2c, 7c, 9, 20b, 29b

Badness (Sintel): 4.27

Lafa

This temperament was named by Petr Pařízek in 2011, referring to the characteristic that stacking 12 generators makes 6/1 – "l" for 12, "f" for 6[1]. Its ploidacot is therefore beta-dodecacot.

Subgroup: 2.3.5

Comma list: [77 -31 -12⟩

Mapping: [⟨1 -1 9], ⟨0 12 -31]]

mapping generators: ~2, ~[-32 13 5⟩

Optimal tunings:

  • WE: ~2 = 1199.9693 ¢, ~[-32 13 5⟩ = 258.4963 ¢
error map: ⟨-0.031 +0.031 +0.026]
  • CWE: ~2 = 1200.0000 ¢, ~[-32 13 5⟩ = 258.5031 ¢
error map: ⟨0.000 +0.082 +0.091]

Optimal ET sequence: 65, 246, 311, 376, 441, 2711, 3152, 3593, 4034, 4475, 4916, 5357, 10273bc, 15630bbcc

Badness (Sintel): 4.33

Lafayette

For extensions, see Mint temperaments #Progression.

Subgroup: 2.3.5

Comma list: 3456/3125

Mapping: [⟨1 1 2], ⟨0 5 3]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~25/24 = 138.722 ¢
  • POTE: ~2 = 1200.000 ¢, ~25/24 = 140.995 ¢

Optimal ET sequence: 8, 9, 17c

Badness (Sintel): 4.70

Neon

For extensions, see Ragismic microtemperaments #Deca, Landscape microtemperaments #Zinc, and 20th-octave temperaments #Calcium.
See also: Neon comma

Neon tempers out [21 60 -50⟩ in the 5-limit, equating 3125/2916 with one step of 10edo. It was named by Eliora in 2023 after the 10th element.

Subgroup: 2.3.5

Comma list: [21 60 -50⟩

Mapping: [⟨10 4 9], ⟨0 5 6]]

mapping generators: ~3125/2916, [10 29 -24⟩

Optimal tunings:

  • WE: ~3125/2916 = 119.9999 ¢, [10 29 -24⟩ = 284.3890 ¢
error map: ⟨-0.001 -0.011 +0.019]
  • CWE: ~3125/2916 = 120.0000 ¢, [10 29 -24⟩ = 284.3891 ¢
error map: ⟨0.000 -0.010 +0.021]

Optimal ET sequence: 80, 190, 270, 460, 730, 1730, 2460, 3190, 5650

Badness (Sintel): 4.85

Dodifo

For extensions, see Porwell temperaments #Dodifo.

Subgroup: 2.3.5

Comma list: [-67 -9 35⟩

Mapping: [⟨1 12 5], ⟨0 -35 -9]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~768/625 = 357.087 ¢
  • POTE: ~2 = 1200.000 ¢, ~768/625 = 357.088 ¢

Optimal ET sequence: 37, 84, 205, 289, 494, 783, 2060, 2843, 3626

Badness (Sintel): 5.00

Doublewide

For extensions, see Jubilismic clan #Doublewide.

Subgroup: 2.3.5

Comma list: 390625/373248

Mapping: [⟨2 1 3], ⟨0 4 3]]

mapping generators: ~625/432, ~6/5

Optimal tunings:

  • WE: ~625/432 = 600.6814 ¢, ~6/5 = 325.8892 ¢ (~144/125 = 274.7921 ¢)
error map: ⟨+1.363 +2.283 -6.602]
  • CWE: ~625/432 = 600.0000 ¢, ~6/5 = 325.8147 ¢ (~144/125 = 274.185 ¢)
error map: ⟨0.000 +1.304 -8.869]

Optimal ET sequence: 4, 14b, 18, 22, 48, 70c, 118c

Badness (Sintel): 5.32

Shibboleth

For extensions, see Gamelismic clan #Superkleismic, Starling temperaments #Gilead, and Sensamagic clan #Superthird.

Subgroup: 2.3.5

Comma list: 1953125/1889568

Mapping: [⟨1 4 5], ⟨0 -9 -10]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~6/5 = 321.773 ¢
  • POTE: ~2 = 1200.000 ¢, ~6/5 = 321.852 ¢

Optimal ET sequence: 15, 26, 41, 179c

Badness (Sintel): 5.34

Semidimfourth

For extensions, see Ragismic microtemperaments #Semidimfourth.

Subgroup: 2.3.5

Comma list: [7 41 -31⟩

Mapping: [⟨1 21 28], ⟨0 -31 -41]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~162/125 = 448.449 ¢
  • POTE: ~2 = 1200.000 ¢, ~162/125 = 448.449 ¢

Optimal ET sequence: 8, …, 91, 99, 190, 289, 388, 677, 3674, 4351, 5028, 5705, 6382, 13441c, 19823bcc

Badness (Sintel): 5.47

Mynic

For extensions, see Starling temperaments #Myna.

Subgroup: 2.3.5

Comma list: 10077696/9765625

Mapping: [⟨1 9 9], ⟨0 -10 -9]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~6/5 = 310.030 ¢
  • POTE: ~2 = 1200.000 ¢, ~6/5 = 310.140 ¢

Optimal ET sequence: 27, 31, 58, 89

Badness (Sintel): 5.86

Progress

For extensions, see Archytas clan #Progress.

Subgroup: 2.3.5

Comma list: 32768/30375

Mapping: [⟨1 0 5], ⟨0 3 -5]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~45/32 = 638.922 ¢
  • POTE: ~2 = 1200.000 ¢, ~45/32 = 638.736 ¢

Optimal ET sequence: 2, 13, 15, 32c, 47bc, 62bc

Badness (Sintel): 5.77

Abigail

For extensions, see Ragismic microtemperaments #Abigail.

Subgroup: 2.3.5

Comma list: [25 -48 22⟩

Mapping: [⟨2 7 13], ⟨0 -11 -24]]

Optimal tunings:

  • CTE: ~[-12 24 -11⟩ = 600.000 ¢, ~200000/177147 = 208.907 ¢
  • CWE: ~[-12 24 -11⟩ = 600.000 ¢, ~200000/177147 = 208.906 ¢

Optimal ET sequence: 46, 132, 178, 224, 270, 494, 1752, 2246, 2740, 4986, 7726b

Badness (Sintel): 5.97

Supermajor

For extensions, see Ragismic microtemperaments #Supermajor.

Subgroup: 2.3.5

Comma list: [-13 -46 37⟩

Mapping: [⟨1 15 19], ⟨0 -37 -46]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~625/486 = 435.081 ¢
  • CWE: ~2 = 1200.000 ¢, ~625/486 = 435.081 ¢

Optimal ET sequence: 80, 171, 593, 764, 935, 1106, 3489, 4595, 5701

Badness (Sintel): 6.19

Gamera

For extensions, see Ragismic microtemperaments #Gamera.

Subgroup: 2.3.5

Comma list: [10 -40 23⟩

Mapping: [⟨1 6 10], ⟨0 -23 -40]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~2500/2187 = 230.345 ¢
  • CWE: ~2 = 1200.000 ¢, ~2500/2187 = 230.345 ¢

Optimal ET sequence: 26, 73, 99, 224, 323, 547, 870

Badness (Sintel): 6.39

Tritriple

For extensions, see Gamelismic clan #Tritriple.

Subgroup: 2.3.5

Comma list: [31 20 -27⟩

Mapping: [⟨1 -11 -7], ⟨0 27 20]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~864/625 = 559.328 ¢
  • POTE: ~2 = 1200.000 ¢, ~864/625 = 559.332 ¢

Optimal ET sequence: 15, 88, 103, 118, 959, 1077

Badness (Sintel): 6.65

Octoid

For extensions, see Ragismic microtemperaments #Octoid.

Subgroup: 2.3.5

Comma list: [-5 -32 24⟩

Mapping: [⟨8 1 3], ⟨0 3 4]]

Mapping generators: ~2125764/1953125, ~4374/3125

Optimal tunings:

  • WE: ~2125764/1953125 = 150.001 ¢, ~4374/3125 = 584.027 ¢
  • CWE: ~2125764/1953125 = 150.000 ¢, ~4374/3125 = 584.025 ¢

Optimal ET sequence: 8, 56bcc, 64c, 72, 152, 224, 376, 976

Badness (Sintel): 6.69

Kangaroo

For extensions, see Trienstonic clan #Parakangaroo.

Subgroup: 2.3.5

Comma list: 64000/59049

Mapping: [⟨1 0 -3], ⟨0 3 10]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~40/27 = 637.879 ¢
  • POTE: ~2 = 1200.000 ¢, ~40/27 = 638.805 ¢

Optimal ET sequence: 15, 47b, 62b

Badness (Sintel): 6.94

Tertiaseptal

For extensions, see Breedsmic temperaments #Tertiaseptal.

Subgroup: 2.3.5

Comma list: [-59 5 22⟩

Mapping: [⟨1 3 2], ⟨0 -22 5]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~17578125/16777216 = 77.186 ¢
  • POTE: ~2 = 1200.000 ¢, ~17578125/16777216 = 77.191 ¢

Optimal ET sequence: 31, 109, 140, 171, 1679c, 1850c, 2021c, 2192c, 2363c, 2534c, 2705c, 2876c, 3047c

Badness (Sintel): 6.98

Untriton

For extensions, see Marvel temperaments #Untriton, Porwell temperaments #Aufo and Triwellismic temperaments #Interaufo.

Subgroup: 2.3.5

Comma list: [-51 19 9⟩

Mapping: [⟨1 6 -7], ⟨0 -9 19]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~45/32 = 588.727 ¢
  • POTE: ~2 = 1200.000 ¢, ~45/32 = 588.722 ¢

Optimal ET sequence: 51, 53, 267, 320, 373, 426, 479, 532, 2607bc

Badness (Sintel): 7.20

Nessafof

For extensions, see Porwell temperaments #Nessafof.

Subgroup: 2.3.5

Comma list: [25 15 -21⟩

Mapping: [⟨3 2 5], ⟨0 7 5]]

Optimal tunings:

  • CTE: ~78125/62208 = 400.000 ¢, ~9765625/8957952 = 157.392 ¢
  • CWE: ~78125/62208 = 400.000 ¢, ~9765625/8957952 = 157.409 ¢

Optimal ET sequence: 15, 54b, 69, 84, 99, 183, 282, 465, 1212c, 1677cc

Badness (Sintel): 7.81

Lemba

For extensions, see Jubilismic clan #Lemba.

Subgroup: 2.3.5

Comma list: 140625/131072

Mapping: [⟨2 2 5], ⟨0 3 -1]]

Mapping generators: ~375/256, ~75/64

Optimal tunings:

  • CTE: ~375/256 = 600.000 ¢, ~75/64 = 232.986 ¢
error map: ⟨0.000 -2.998 -19.300]
  • CWE: ~375/256 = 600.000 ¢, ~75/64 = 231.832 ¢
error map: ⟨0.000 -6.458 -18.146]

Optimal ET sequence: 10, 16, 26, 88cc

Badness (Sintel): 7.85

Delorean

For extensions, see Mint temperaments #Subklei.

Subgroup: 2.3.5

Comma list: 17496/15625

Mapping: [⟨1 3 4], ⟨0 -6 -7]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~6/5 = 284.817 ¢
  • POTE: ~2 = 1200.000 ¢, ~6/5 = 284.293 ¢

Optimal ET sequence: 4, 13c, 17c, 21, 38c

Badness (Sintel): 8.35

Majvam

For extensions, see No-sevens subgroup temperaments #Majvam, Cataharry temperaments #Majvamoid, and Parkleiness temperaments #Majvamic.

Subgroup: 2.3.5

Comma list: 2404631929946112/2384185791015625

Mapping: [⟨1 10 5], ⟨0 -22 -7]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~125/96 = 459.006 ¢
  • POTE: ~2 = 1200.000 ¢, ~125/96 = 458.995 ¢

Optimal ET sequence: 34, 115, 149, 183, 217, 400, 617

Badness (Sintel): 8.37

Beatles

For extensions, see Archytas clan #Beatles.

Subgroup: 2.3.5

Comma list: 524288/492075

Mapping: [⟨1 1 5], ⟨0 2 -9]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~512/405 = 356.492 ¢
  • POTE: ~2 = 1200.000 ¢, ~512/405 = 355.930 ¢

Optimal ET sequence: 10, 17c, 27, 64b, 91bc, 118bc

Badness (Sintel): 8.41

Magus

For extensions, see Sensamagic clan #Magus and Starling temperaments #Amigo.

Subgroup: 2.3.5

Comma list: 50331648/48828125

Mapping: [⟨1 -2 2], ⟨0 11 1]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~5/4 = 391.068 ¢
  • POTE: ~2 = 1200.000 ¢, ~5/4 = 391.225 ¢

Optimal ET sequence: 3, …, 40, 43, 46, 181c, 227c, 273c, 319c

Badness (Sintel): 8.45

Tritonic

For extensions, see Marvel temperaments #Tritonic.

Subgroup: 2.3.5

Comma list: 553584375/536870912

Mapping: [⟨1 4 -3], ⟨0 -5 11]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~45/32 = 580.278 ¢
  • POTE: ~2 = 1200.000 ¢, ~45/32 = 580.219 ¢

Optimal ET sequence: 29, 31, 60, 91

Badness (Sintel): 8.88

Wizard

For extensions, see Marvel temperaments #Wizard.

Subgroup: 2.3.5

Comma list: 2197265625/2147483648

Mapping: [⟨2 1 5], ⟨0 6 -1]]

mapping generators: ~42875/32768, ~9375/8192

Optimal tunings:

  • CTE: ~42875/32768 = 600.000 ¢, ~9375/8192 = 216.950 ¢
  • POTE: ~42875/32768 = 600.000 ¢, ~9375/8192 = 216.788 ¢

Optimal ET sequence: 22, 50, 72, 166, 238c, 310c, 548bcc

Badness (Sintel): 9.06

Scales (Scala files): wizard22

Tridecatonic

For extensions, see Octagar temperaments #Tridecatonic, Ragismic microtemperaments #Trideci, and Varunismic temperaments #Bosonic.

Tridecatonic tempers out the devil's tridecalimma, the comma which associates the 10/9 minor tone with 2\13.

Subgroup: 2.3.5

Comma list: [-11 26 -13⟩

Mapping: [⟨13 0 -11], ⟨0 1 2]]

mapping generators: ~531441/500000, ~3

Optimal tunings:

  • WE: ~531441/500000 = 92.3166 ¢, ~3/2 = 701.2353 ¢
error map: ⟨+0.116 -0.688 +0.737]
  • CWE: ~531441/500000 = 92.3077 ¢, ~3/2 = 701.2353 ¢
error map: ⟨0.000 -0.797 +0.617]

Optimal ET sequence: 26, 39, 65, 286, 351, 416, 481, 546

Badness (Sintel): 10.3

Merman

For extensions, see Marvel temperaments #Merman.

Subgroup: 2.3.5

Comma list: [-40 15 7⟩

Mapping: [⟨1 5 -5], ⟨0 -7 15]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~45/32 = 585.653 ¢
  • POTE: ~2 = 1200.000 ¢, ~45/32 = 585.634 ¢

Optimal ET sequence: 41, 84, 125, 209, 334c, 543bc

Badness (Sintel): 10.6

Hendecatonic

For extensions, see Porwell temperaments #Hendecatonic and Marvel temperaments #Hendeca.

The 5-limit version of hendecatonic simply equates a stack of eleven 16/15's with the octave.

Subgroup: 2.3.5

Comma list: 8796093022208/8649755859375

Mapping: [⟨11 0 43], ⟨0 1 -1]]

Mapping generators: ~16/15, ~3

Optimal tunings:

  • WE: ~16/15 = 109.0904 ¢, ~3/2 = 702.6861 ¢
error map: ⟨-0.445 +0.286 +0.613]
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 702.9082 ¢
error map: ⟨0.000 +0.953 +1.687]

Optimal ET sequence: 22, 55, 77, 99, 869bcc, 968bcc, 1067bccc, 1166bccc

Badness (Sintel): 10.7

Nusecond

For extensions, see Starling temperaments #Nusecond.

Subgroup: 2.3.5

Comma list: 51018336/48828125

Mapping: [⟨1 3 4], ⟨0 -11 -13]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3125/2916 = 154.577 ¢
  • POTE: ~2 = 1200.000 ¢, ~3125/2916 = 154.523 ¢

Optimal ET sequence: 8, 23, 31, 70, 101, 132c

Badness (Sintel): 10.9

Comic

For extensions, see Jubilismic clan #Comic.

Subgroup: 2.3.5

Comma list: 5120000/4782969

Mapping: [⟨2 1 -3], ⟨0 2 7]]

mapping generators: ~2187/1600, ~40/27

Optimal tunings:

  • WE: ~2187/1600 = 598.9674 ¢, ~40/27 = 654.2542 ¢ (~81/80 = 55.2868 ¢)
error map: ⟨-2.065 +5.521 -3.436]
  • CWE: ~2187/1600 = 600.0000 ¢, ~40/27 = 655.0201 ¢ (~81/80 = 55.0201 ¢)
error map: ⟨0.000 +8.085 -1.173]

Optimal ET sequence: 20c, 22, 86b, 108b, 130b

Badness (Sintel): 11.5

Sfourth

For extensions, see Ragismic microtemperaments #Sfourth.

Subgroup: 2.3.5

Comma list: [-5 31 -19⟩

Mapping: [⟨1 2 3], ⟨0 -19 -31]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~1594323/1562500 = 26.2323 ¢
  • POTE: ~2 = 1200.0000 ¢, ~1594323/1562500 = 26.2342 ¢

Optimal ET sequence: 45, 46, 91, 137, 183, 869, 1052, 1235, 1418c

Badness (Sintel): 11.8

Thuja

For extensions, see Buzzardsmic clan #Thuja.
See also: Hypovishnuzma

Subgroup: 2.3.5

Comma list: 254803968/244140625

Mapping: [⟨1 -4 0], ⟨0 12 5]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3125/2304 = 558.404 ¢
  • POTE: ~2 = 1200.000 ¢, ~3125/2304 = 558.592 ¢

Optimal ET sequence: 15, 43, 58

Badness (Sintel): 12.0

Sesesix

For extensions, see Horwell temperaments #Fifthplus.

Sesesix is the 22 & 171 temperament, named by Petr Pařízek in 2011 for 36 generators giving a major sixth[1].

Subgroup: 2.3.5

Comma list: [-74 13 23⟩

Mapping: [⟨1 -12 10], ⟨1 23 -13]]

mapping generators: ~2, ~6328125/4094304

Optimal tunings:

  • WE: ~2 = 1200.0899 ¢, ~6328125/4194304 = 708.8258 ¢
error map: ⟨+0.090 -0.040 -0.151]
  • CWE: ~2 = 1200.0000 ¢, ~6328125/4194304 = 708.7736 ¢
error map: ⟨0.000 -0.161 -0.371]

Optimal ET sequence: 22, 105, 127, 149, 171, 4468bcc, 4639bcc, …, 5323bccc

Badness (Sintel): 12.2

Leapday

For extensions, see Hemifamity temperaments #Leapday and Porwell temperaments #Polypyth.

Subgroup: 2.3.5

Comma list: 10737418240/10460353203

Mapping: [⟨1 0 -31], ⟨0 1 21]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 704.100 ¢
  • POTE: ~2 = 1200.000 ¢, ~3/2 = 704.179 ¢

Optimal ET sequence: 17c, 29, 46, 121, 167, 455bc, 622bbc

Badness (Sintel): 12.3

Vines

For extensions, see Starling temperaments #Vines.

Subgroup: 2.3.5

Comma list: 156728328192/152587890625

Mapping: [⟨2 7 8], ⟨0 -8 -7]]

Optimal tunings:

  • CTE: ~390625/279936 = 600.000 ¢, ~6/5 = 312.636 ¢
  • POTE: ~390625/279936 = 600.000 ¢, ~6/5 = 312.713 ¢

Optimal ET sequence: 4, …, 42, 46, 96, 142, 330c, 472cc

Badness (Sintel): 12.7

Emka

For extensions, see Hemimean clan #Emka and Horwell temperaments #Emkay.

Subgroup: 2.3.5

Comma list: [-50 -8 27⟩

Mapping: [⟨1 14 6], ⟨0 -27 -8]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~9765625/7077888 = 551.777 ¢
  • POTE: ~2 = 1200.000 ¢, ~9765625/7077888 = 551.784 ¢

Optimal ET sequence: 37, 50, 87, 137, 224

Badness (Sintel): 12.8

Misneb

For extensions, see Marvel temperaments #Misneb.

Subgroup: 2.3.5

Comma list: [-57 14 15⟩

Mapping: [⟨1 3 1], ⟨0 -15 14]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~16/15 = 113.233 ¢
  • POTE: ~2 = 1200.000 ¢, ~16/15 = 113.240 ¢

Optimal ET sequence: 21, 32, 53, 392, 445, 498, 551, 604, 657c, 710c, 1367bcc

Badness (Sintel): 14.3

Semisept

For extensions, see Hemimean clan #Semisept.

Subgroup: 2.3.5

Comma list: 782757789696/762939453125

Mapping: [⟨1 12 6], ⟨0 -17 -6]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~192/125 = 735.203 ¢
  • POTE: ~2 = 1200.000 ¢, ~192/125 = 735.146 ¢

Optimal ET sequence: 18, 31, 80, 111, 586cc, 697ccc

Badness (Sintel): 14.8

Septimin

For extensions, see Marvel temperaments #Septimin.

Subgroup: 2.3.5

Comma list: 35595703125/34359738368

Mapping: [⟨1 4 1], ⟨0 -11 6]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~75/64 = 263.571 ¢
  • POTE: ~2 = 1200.000 ¢, ~75/64 = 263.658 ¢

Optimal ET sequence: 9, 23b, 32, 41, 91, 132, 223c, 355bcc

Badness (Sintel): 15.7

Slurpee

For extensions, see Mint temperaments #Slurpee.

Subgroup: 2.3.5

Comma list: 177147/156250

Mapping: [⟨1 2 3], ⟨0 -7 -11]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~27/25 = 72.659 ¢
  • POTE: ~2 = 1200.000 ¢, ~27/25 = 72.742 ¢

Optimal ET sequence: 16, 17c, 33

Badness (Sintel): 16.2

Pentadecal

For extensions, see Keemic temperaments #Pentadecal.

Subgroup: 2.3.5

Comma list: 30517578125/29386561536

Mapping: [⟨15 0 11], ⟨0 1 1]]

Optimal tunings:

  • CTE: ~648/625 = 80.000 ¢, ~3/2 = 703.340 ¢
  • POTE: ~648/625 = 80.000 ¢, ~3/2 = 702.652 ¢

Optimal ET sequence: 15, 45, 60, 75, 135, 210c

Badness (Sintel): 16.2

Anthoine

For extensions, see Buzzardsmic clan #Anthoine.

Subgroup: 2.3.5

Comma list: 286102294921875/281474976710656

Mapping: [⟨1 8 2], ⟨0 -20 1]]

Optimal tunings:

  • WE: ~2 = 1200.303 ¢, ~5/4 = 385.024 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 384.920 ¢

Optimal ET sequence: 25, 28, 53

Badness (Sintel): 16.5

Sentinel

For extensions, see Orwellismic temperaments #Sentinel.

Subgroup: 2.3.5

Comma list: 8968066875/8589934592

Mapping: [⟨1 3 -3], ⟨0 -4 15]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~512/405 = 425.590 ¢
  • POTE: ~2 = 1200.000 ¢, ~512/405 = 425.474 ¢

Optimal ET sequence: 14, 17, 31, 79, 110, 361bbcc

Badness (Sintel): 17.0

Undeka

For extensions, see Keemic temperaments #Undeka.

Undeka has a period of 1/11 octave, like hendecatonic, but this temperament equates a stack of eleven 6/5's and three octaves, so that three periods represent 6/5.

Subgroup: 2.3.5

Comma list: 48828125/45349632

Mapping: [⟨11 0 8], ⟨0 1 1]]

Mapping generators: ~648/625, ~3

Optimal tunings:

  • WE: ~648/625 = 109.1809 ¢, ~3/2 = 704.4091 ¢
error map: ⟨+0.990 +3.444 -7.468]
  • CWE: ~648/625 = 109.0909 ¢, ~3/2 = 704.6983 ¢
error map: ⟨0.000 +2.743 -8.888]

Optimal ET sequence: 22, 77c, 99c

Badness (Sintel): 17.1

Kumonga

For extensions, see Starling temperaments #Kumonga.

Subgroup: 2.3.5

Comma list: 1289945088/1220703125

Mapping: [⟨1 4 4], ⟨0 -13 -9]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~144/125 = 223.076 ¢
  • POTE: ~2 = 1200.000 ¢, ~144/125 = 222.912 ¢

Optimal ET sequence: 16, 27, 43, 70, 183cc

Badness (Sintel): 17.1

Mirkat

For extensions, see Mirkwai clan #Mirkat.

Subgroup: 2.3.5

Comma list: [20 -39 18⟩

Mapping: [⟨3 2 1], ⟨0 6 13]]

Optimal tunings:

  • CTE: ~2000000/1594323 = 400.000 ¢, ~10/9 = 183.593 ¢
  • CWE: ~2000000/1594323 = 400.000 ¢, ~10/9 = 183.585 ¢

Optimal ET sequence: 33c, 39, 72, 111, 183, 438, 621, 1425b

Badness (Sintel): 17.6

Semidimi

For extensions, see Ragismic microtemperaments #Semidimi.

Subgroup: 2.3.5

Comma list: [-12 -73 55⟩

Mapping: [⟨1 36 48], ⟨0 -55 -73]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~162/125 = 449.127 ¢
  • POTE: ~2 = 1200.000 ¢, ~162/125 = 449.127 ¢

Optimal ET sequence: 8, …, 163, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419

Badness (Sintel): 17.7

Slender

For extensions, see Marvel temperaments #Slender.

Subgroup: 2.3.5

Comma list: [-46 10 13⟩

Mapping: [⟨1 2 2], ⟨0 -13 10]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~16875/16384 = 38.380 ¢
  • CWE: ~2 = 1200.000 ¢, ~16875/16384 = 38.400 ¢

Optimal ET sequence: 31, 94, 125, 781bcc

Badness (Sintel): 17.8

Quasitemp

For extensions, see Keemic temperaments #Quasitemp.

Subgroup: 2.3.5

Comma list: 6103515625/5804752896

Mapping: [⟨1 5 5], ⟨0 -14 -11]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3125/2592 = 292.591 ¢
  • POTE: ~2 = 1200.000 ¢, ~3125/2592 = 292.702 ¢

Optimal ET sequence: 4, …, 37, 41

Badness (Sintel): 18.0

Cypress

For extensions, see Starling temperaments #Cypress.

Subgroup: 2.3.5

Comma list: 258280326/244140625

Mapping: [⟨1 7 10], ⟨0 -12 -17]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~4374/3125 = 541.735 ¢
  • POTE: ~2 = 1200.000 ¢, ~4374/3125 = 541.726 ¢

Optimal ET sequence: 11c, 20c, 31, 113c, 144c, 175c

Badness (Sintel): 19.2

Chromo

For extensions, see Keemic temperaments #Chromo.

Subgroup: 2.3.5

Comma list: 1220703125/1146617856

Mapping: [⟨1 1 2], ⟨0 13 7]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~25/24 = 54.138 ¢
  • POTE: ~2 = 1200.000 ¢, ~25/24 = 53.922 ¢

Optimal ET sequence: 22, 45, 67c, 89c

Badness (Sintel): 19.8

Tertiosec

For extensions, see Marvel temperaments #Tertiosec.

Subgroup: 2.3.5

Comma list: [-89 21 24⟩

Mapping: [⟨3 7 5], ⟨0 -8 7]]

Optimal tunings:

  • CTE: ~[30 -7 -8⟩ = 400.000 ¢, ~16/15 = 112.275 ¢
  • POTE: ~[30 -7 -8⟩ = 400.000 ¢, ~16/15 = 112.281 ¢

Optimal ET sequence: 21, 54, 75, 96, 171

Badness (Sintel): 20.0

Fervor

For extensions, see Archytas clan #Fervor.

Subgroup: 2.3.5

Comma list: 67108864/61509375

Mapping: [⟨1 4 -2], ⟨0 -5 9]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~64/45 = 577.593 ¢
  • POTE: ~2 = 1200.000 ¢, ~64/45 = 577.705 ¢

Optimal ET sequence: 2, …, 25, 27

Badness (Sintel): 20.0

Bohpier

For extensions, see Sensamagic clan #Bohpier.

Subgroup: 2.3.5

Comma list: 1220703125/1162261467

Mapping: [⟨1 0 0], ⟨0 13 19]]

Optimal tunings:

  • CTE and POTE: ~2 = 1200.000 ¢, ~27/25 = 146.476 ¢

Optimal ET sequence: 8, …, 41, 131, 172, 213c

Badness (Sintel): 20.2

Elvis

For extensions, see Jubilismic clan #Elvis.

Subgroup: 2.3.5

Comma list: 36905625/33554432

Mapping: [⟨2 1 10], ⟨0 2 -5]]

Optimal tunings:

  • CTE: ~6075/4096 = 600.000 ¢, ~64/45 = 644.844 ¢
  • POTE: ~6075/4096 = 600.000 ¢, ~64/45 = 645.454 ¢

Optimal ET sequence: 26, 80bc, 106bbcc, 132bbcc

Badness (Sintel): 20.7

Enneadecacot

Subgroup: 2.3.5

Comma list: [79 -22 -19⟩

Mapping: [⟨1 1 3], ⟨0 19 -22]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~34171875/33554432 = 36.9607 ¢
  • POTE: ~2 = 1200.0000 ¢, ~34171875/33554432 = 36.9622 ¢

Optimal ET sequence: 32, 65, 292, 357, 422, 487, 1396c

Badness (Sintel): 20.9

Vinegar

For extensions, see Starling temperaments #Vinegar.

Subgroup: 2.3.5

Comma list: 16000000/14348907

Mapping: [⟨3 0 -5], ⟨0 2 5]]

Optimal tunings:

  • CTE: ~243/200 = 400.000 ¢, ~4000/2187 = 955.656 ¢ (~10/9 = 155.656 ¢)
  • POTE: ~243/200 = 400.000 ¢, ~4000/2187 = 956.861 ¢ (~10/9 = 156.861 ¢)

Optimal ET sequence: 15, 39c, 54, 69b

Badness (Sintel): 21.2

Oolong

For extensions, see Starling temperaments #Oolong.

Subgroup: 2.3.5

Comma list: 793437161472/762939453125

Mapping: [⟨1 6 7], ⟨0 -17 -18]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~6/5 = 311.726 ¢
  • POTE: ~2 = 1200.000 ¢, ~6/5 = 311.694 ¢

Optimal ET sequence: 23, 27, 50, 77

Badness (Sintel): 22.1

Oncle

For extensions, see Gamelismic clan #Oncle.

Subgroup: 2.3.5

Comma list: 145282683375/137438953472

Mapping: [⟨1 1 6], ⟨0 3 -19]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~4096/3645 = 232.385 ¢
  • POTE: ~2 = 1200.000 ¢, ~4096/3645 = 232.521 ¢

Optimal ET sequence: 31, 98c, 129c, 160bc, 289bbcc

Badness (Sintel): 29.2

Quasiorwell

For extensions, see Breedsmic temperaments #Quasiorwell.

Subgroup: 2.3.5

Comma list: [-93 3 38⟩

Mapping: [⟨1 31 0], ⟨0 -38 3]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~[-31 1 13⟩ = 928.896 ¢
  • CWE: ~2 = 1200.000 ¢, ~[-31 1 13⟩ = 928.893 ¢

Optimal ET sequence: 31, …, 177, 208, 239, 270, 841, 1111, 1381, 2492c

Badness (Sintel): 30.6

Diesic

For extensions, see Orwellismic temperaments #Diesic.

Subgroup: 2.3.5

Comma list: 10460353203/9765625000

Mapping: [⟨1 2 3], ⟨0 -13 -21]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~6561/6250 = 38.550 ¢
  • POTE: ~2 = 1200.000 ¢, ~6561/6250 = 38.571 ¢

Optimal ET sequence: 31, 156c

Badness (Sintel): 31.5

Quartemka

For extensions, see Gamelismic clan #Quartemka.

Subgroup: 2.3.5

Comma list: [2 -32 21⟩

Mapping: [⟨1 4 6], ⟨0 -21 -32]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~27/25 = 137.963 ¢
  • POTE: ~2 = 1200.000 ¢, ~27/25 = 137.962 ¢

Optimal ET sequence: 26, 61, 87, 287, 374, 461

Badness (Sintel): 32.4

Genocchi

Subgroup: 2.3.5

Comma list: 2359296/1953125

Mapping: [⟨1 0 2], ⟨0 9 2]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~768/625 = 210.920 ¢
  • POTE: ~2 = 1200.000 ¢, ~768/625 = 212.444 ¢

Optimal ET sequence: 6b, 11b, 17c

Badness (Sintel): 33.0

Casablanca

For extensions, see Starling temperaments #Casablanca.

Subgroup: 2.3.5

Comma list: [22 14 -19⟩

Mapping: [⟨1 -7 -14], ⟨0 19 14]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~864/625 = 542.142 ¢
  • POTE: ~2 = 1200.000 ¢, ~864/625 = 542.208 ¢

Optimal ET sequence: 11b, 20b, 31, 73, 104c, 135c

Badness (Sintel): 33.4

Archaeotherium

For extensions, see Gamelismic clan #Archaeotherium.

Subgroup: 2.3.5

Comma list: 597871125/536870912

Mapping: [⟨1 1 5], ⟨0 3 -14]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~1215/1024 = 229.947 ¢
  • POTE: ~2 = 1200.000 ¢, ~1215/1024 = 230.351 ¢

Optimal ET sequence: 21, 26, 73bc, 99bc

Badness (Sintel): 33.7

Octacoid

Subgroup: 2.3.5

Comma list: 8589934592/7688671875

Mapping: [⟨1 1 3], ⟨0 8 -9]]

Mapping generators: ~2, ~16/15

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~16/15 = 88.7331 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 88.8168 ¢

Optimal ET sequence: 13, 14, 27

Badness (Sintel): 41.1

Quasimoha

For extensions, see Breedsmic temperaments #Quasimoha.

Subgroup: 2.3.5

Comma list: [-41 23 2⟩

Mapping: [⟨1 1 9], ⟨0 2 -23]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~1048576/885735 = 348.462 ¢
  • POTE: ~2 = 1200.000 ¢, ~1048576/885735 = 348.600 ¢

Optimal ET sequence: 31, 117c, 148bc, 179bc

Badness (Sintel): 46.6

Phillips

For extensions, see Orwellismic temperaments #Phillips.

Subgroup: 2.3.5

Comma list: 9685512225/8589934592

Mapping: [⟨2 0 33], ⟨0 1 -9]]

Optimal tunings:

  • CTE: ~98415/65536 = 600.000 ¢, ~3/2 = 690.708 ¢
  • POTE: ~98415/65536 = 600.000 ¢, ~3/2 = 691.455 ¢

Optimal ET sequence: 14, 26, 66b, 92bc, 118bbc

Badness (Sintel): 52.3

Triskaidekic

For extensions, see Orwellismic temperaments #Triskaidekic.

Subgroup: 2.3.5

Comma list: 1220703125/1073741824

Mapping: [⟨13 0 30], ⟨0 1 0]]

Mapping generators: ~128/125, ~3

Optimal tunings:

  • WE: ~128/125 = 92.589 ¢, ~3/2 = 698.263 ¢
error map: ⟨+3.656 -0.036 -8.647]
  • CWE: ~128/125 = 92.308 ¢, ~3/2 = 698.068 ¢
error map: ⟨0.000 -3.887 -17.083]

Optimal ET sequence: 13, 26, 91cc

Badness (Sintel): 52.6

Satin

For extensions, see Garischismic clan #Satin.

Subgroup: 2.3.5

Comma list: [104 -70 3⟩

Mapping: [⟨1 2 12], ⟨0 -3 -70]]

mapping generators: ~2, ~[-34 23 -1⟩

Optimal tunings:

  • WE: ~2 = 1199.9008 ¢, ~[-34 23 -1⟩ = 165.8930 ¢
error map: ⟨-0.099 +0.168 -0.015]
  • CWE: ~2 = 1200.0000 ¢, ~[-34 23 -1⟩ = 165.9079 ¢
error map: ⟨0.000 +0.321 +0.130]

Optimal ET sequence: 94, 217, 528, 745, 1273, 2018b

Badness (Sintel): 66.9

Lockerbie

For extensions, see Breedsmic temperaments #Lockerbie.

Subgroup: 2.3.5

Comma list: [91 51 -74⟩

Mapping: [⟨1 -25 -16], ⟨0 74 51]]

mapping generators: ~2, ~[36 20 -29⟩

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~[36 20 -29⟩ = 431.1069 ¢
error map: ⟨0.0000 -0.0444 +0.1382]
  • CWE: ~2 = 1200.0000 ¢, ~[36 20 -29⟩ = 431.1073 ¢
error map: ⟨0.0000 -0.0174 +0.1568]

Optimal ET sequence: 103, 167, 270, 643, 913, 1183, 2636, 3819c, 5002c

Badness (Sintel): 72.4

Rarity

For extensions, see No-sevens subgroup temperaments #Rarity.

The rarity temperament is notable due to the fact that it offers a 7L 4s scale of reasonable hardness, in the 5-limit and among relatively small edos.

Subgroup: 2.3.5

Comma list: [-57 17 13⟩

Mapping: [⟨1 11 -10], ⟨0 -13 17]]

Mapping generators: ~2, ~16384/10125

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~16384/10125 = 869.392 ¢
  • CWE: ~2 = 1200.000 ¢, ~16384/10125 = 869.402 ¢

Optimal ET sequence: 11, 29, 69, 98c

Badness (Sintel): 99.2

Jorgensen

For extensions, see Hemifamity temperaments #Jorgensen.

Subgroup: 2.3.5

Comma list: [111 -70⟩

Mapping: [⟨70 111 0], ⟨0 0 1]]

mapping generators: ~[46 -29⟩, ~5

Optimal tunings:

  • WE: ~[46 -29⟩ = 17.1388 ¢, ~5/4 = 386.8829 ¢
error map: ⟨-0.285 +0.451 -0.000]
  • CWE: ~[46 -29⟩ = 17.1429 ¢, ~5/4 = 386.7543 ¢
error map: ⟨0.000 +0.902 +0.441]

Optimal ET sequence: 70, 140, 350, 490

Badness (Sintel): 203

Elven

For extensions, see Sensibeta temperaments #Elven.

Elven has a period of 1/11 octave, like hendecatonic, but this temperament equates a stack of eleven 3125/2916's with the octave.

Subgroup: 2.3.5

Comma list: [-23 -66 55⟩

Mapping: [⟨11 2 7], ⟨0 5 6]]

mapping generators: ~3125/2916, ~[10 29 -24⟩

Optimal tunings:

  • WE: ~3125/2916 = 109.0998 ¢, ~[10 29 -24⟩ = 336.8921 ¢
error map: ⟨+0.097 +0.705 -1.263]
  • CWE: ~3125/2916 = 109.0909 ¢, ~[10 29 -24⟩ = 336.8814 ¢
error map: ⟨0.000 +0.634 -1.389]

Optimal ET sequence: 121, 253, 374, 627c

Badness (Sintel): 429

References

  1. ↑ 1.0 1.1 Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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