Gamelismic clan: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus (5/)
Switch to Sintel's badness, WE & CWE tunings, per community consensus (6/)
Line 1,785: Line 1,785:


Badness (Sintel): 1.36
Badness (Sintel): 1.36
== Dee leap week ==
{{Main| Dee leap week }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 1029/1024, 2460375/2458624
{{Mapping|legend=1| 1 -5 25 5 | 0 9 -31 -3 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4835{{c}}, ~224/135 = 878.2507{{c}}
: [[error map]]: {{val| +0.484 -0.117 +0.004 -1.160 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~224/135 = 877.8926{{c}}
: error map: {{val| 0.000 -0.921 -0.985 -2.504 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~224/135 = 877.877{{c}} -->
{{Optimal ET sequence|legend=1| 41, 108, 149, 190 }}
[[Badness]] (Sintel): 2.12
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 441/440, 2460375/2458624
Mapping: {{mapping| 1 -5 25 5 -28 | 0 9 -31 -3 43 }}
Optimal tunings:
* WE: ~2 = 1200.4874{{c}}, ~224/135 = 878.2543{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~224/135 = 877.8987{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~224/135 = 877.903{{c}} -->
{{Optimal ET sequence|legend=0| 41, 108e, 149, 190 }}
Badness (Sintel): 1.35


== Unidec ==
== Unidec ==
Line 1,794: Line 1,830:
[[Comma list]]: 31381059609/31250000000
[[Comma list]]: 31381059609/31250000000


{{Mapping|legend=1| 2 5 8 | 0 -6 -11 }}
{{Mapping|legend=1| 2 -1 -3 | 0 6 11 }}
: mapping generators: ~177147/125000, ~10/9
: mapping generators: ~177147/125000, ~25000/19683


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~177147/125000 = 600.000{{c}}, ~10/9 = 183.041{{c}}
* [[WE]]: ~177147/125000 = 600.0124{{c}}, ~25000/19683 = 416.9619{{c}}
: [[error map]]: {{val| +0.025 -0.196 +0.235 }}
* [[CWE]]: ~177147/125000 = 600.0000{{c}}, ~25000/19683 = 416.9561{{c}}
: error map: {{val| 0.000 -0.236 +0.172 }}
<!-- * [[CTE]]: ~177147/125000 = 600.000{{c}}, ~25000/19683 = 416.959{{c}}
: [[error map]]: {{val| 0.000 -0.201 +0.235 }}
: [[error map]]: {{val| 0.000 -0.201 +0.235 }}
* [[POTE]]: ~177147/125000 = 600.000{{c}}, ~10/9 = 183.047{{c}}
* [[POTE]]: ~177147/125000 = 600.000{{c}}, ~25000/19683 = 416.953{{c}}
: error map: {{val| 0.000 -0.236 +0.172 }}
: error map: {{val| 0.000 -0.236 +0.172 }} -->


{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 2524, 2642, 2760 }}
{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 2524, 2642, 2760 }}


[[Badness]] (Smith): 0.082423
[[Badness]] (Sintel): 1.93


=== 7-limit ===
=== 7-limit ===
Line 1,812: Line 1,852:
[[Comma list]]: 1029/1024, 4375/4374
[[Comma list]]: 1029/1024, 4375/4374


{{Mapping|legend=1| 2 5 8 5 | 0 -6 -11 2 }}
{{Mapping|legend=1| 2 -1 -3 7 | 0 6 11 -2 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~1225/864 = 600.000{{c}}, ~10/9 = 183.060{{c}}
* [[CTE]]: ~1225/864 = 600.2429{{c}}, ~80/63 = 417.0073{{c}}
: [[error map]]: {{val| +0.486 -0.154 +0.038 -1.140 }}
* [[POTE]]: ~1225/864 = 600.0000{{c}}, ~80/63 = 416.8688{{c}}
: error map: {{val| 0.000 -0.924 -1.090 -2.503 }}
<!-- * [[CTE]]: ~1225/864 = 600.000{{c}}, ~80/63 = 416.940{{c}}
: [[error map]]: {{val| 0.000 -0.313 +0.030 -2.707 }}
: [[error map]]: {{val| 0.000 -0.313 +0.030 -2.707 }}
* [[POTE]]: ~1225/864 = 600.000{{c}}, ~10/9 = 183.161{{c}}
* [[POTE]]: ~1225/864 = 600.000{{c}}, ~80/63 = 416.839{{c}}
: error map: {{val| 0.000 -0.924 -1.090 -2.503 }}
: error map: {{val| 0.000 -0.924 -1.090 -2.503 }} -->


[[Minimax tuning]]:
[[Minimax tuning]]:
Line 1,830: Line 1,874:
{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 190 }}
{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 190 }}


[[Badness]] (Smith): 0.038393
[[Badness]] (Sintel): 0.972


=== 11-limit ===
=== 11-limit ===
Line 1,837: Line 1,881:
Comma list: 385/384, 441/440, 4375/4374
Comma list: 385/384, 441/440, 4375/4374


Mapping: {{mapping| 2 5 8 5 6 | 0 -6 -11 2 3 }}
Mapping: {{mapping| 2 -1 -3 7 9 | 0 6 11 -2 -3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000{{c}}, ~10/9 = 183.074{{c}}
* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
* CWE: ~99/70 = 600.000{{c}}, ~10/9 = 183.146{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}
<!-- * CTE: ~99/70 = 600.000{{c}}, ~14/11 = 416.926{{c}} -->


Minimax tuning:
Minimax tuning:
Line 1,850: Line 1,895:
{{Optimal ET sequence|legend=0| 26, 46, 72, 118, 190 }}
{{Optimal ET sequence|legend=0| 26, 46, 72, 118, 190 }}


Badness (Smith): 0.015479
Badness (Sintel): 0.512


==== Ekadash ====
==== Ekadash ====
Line 1,857: Line 1,902:
Comma list: 385/384, 441/440, 625/624, 729/728
Comma list: 385/384, 441/440, 625/624, 729/728


Mapping: {{mapping| 2 5 8 5 6 19 | 0 -6 -11 2 3 -38 }}
Mapping: {{mapping| 2 -1 -3 7 9 -19 | 0 6 11 -2 -3 38 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000{{c}}, ~10/9 = 183.125{{c}}
* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~10/9 = 183.187{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}
<!-- * CTE: ~99/70 = 600.000{{c}}, ~14/11 = 416.875{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~14/11 = 416.813{{c}} -->


{{Optimal ET sequence|legend=0| 46f, 72, 118, 190, 262df, 452cdef }}
{{Optimal ET sequence|legend=0| 46f, 72, 118, 190, 262df, 452cdef }}


Badness (Smith): 0.020381
Badness (Sintel): 0.842


==== Hendec ====
==== Hendec ====
Line 1,872: Line 1,919:
Comma list: 169/168, 325/324, 364/363, 385/384
Comma list: 169/168, 325/324, 364/363, 385/384


Mapping: {{mapping| 2 5 8 5 6 8 | 0 -6 -11 2 3 -2 }}
Mapping: {{mapping| 2 -1 -3 7 9 6 | 0 6 11 -2 -3 2 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~91/64 = 600.000{{c}}, ~10/9 = 183.048{{c}}
* WE: ~91/64 = 600.3825{{c}}, ~14/11 = 417.0678{{c}}
* POTE: ~91/64 = 600.000{{c}}, ~10/9 = 183.198{{c}}
* CWE: ~91/64 = 600.0000{{c}}, ~14/11 = 416.8290{{c}}
<!-- * CTE: ~91/64 = 600.000{{c}}, ~14/11 = 416.952{{c}}
* POTE: ~91/64 = 600.000{{c}}, ~14/11 = 416.802{{c}} -->


{{Optimal ET sequence|legend=0| 26, 46, 72, 190ff }}
{{Optimal ET sequence|legend=0| 26, 46, 72, 190ff }}


Badness (Smith): 0.017707
Badness (Sintel): 0.732


===== 17-limit =====
===== 17-limit =====
Line 1,887: Line 1,936:
Comma list: 169/168, 221/220, 273/272, 325/324, 364/363
Comma list: 169/168, 221/220, 273/272, 325/324, 364/363


Mapping: {{mapping| 2 5 8 5 6 8 10 | 0 -6 -11 2 3 -2 -6 }}
Mapping: {{mapping| 2 -1 -3 7 9 6 4 | 0 6 11 -2 -3 2 6 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~10/9 = 183.020{{c}}
* WE: ~17/12 = 600.3991{{c}}, ~14/11 = 417.0809{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~10/9 = 183.196{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~14/11 = 416.8330{{c}}
<!-- * CTE: ~17/12 = 600.000{{c}}, ~14/11 = 416.980{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~14/11 = 416.804{{c}} -->


{{Optimal ET sequence|legend=0| 26, 46, 72, 190ffg }}
{{Optimal ET sequence|legend=0| 26, 46, 72, 190ffg }}


Badness (Smith): 0.011676
Badness (Sintel): 0.595


== Lagaca ==
== Lagaca ==
Line 1,902: Line 1,953:
[[Comma list]]: 1029/1024, 11529602/11390625
[[Comma list]]: 1029/1024, 11529602/11390625


{{Mapping|legend=1| 2 5 2 5 | 0 -9 13 3 }}
{{Mapping|legend=1| 2 -4 15 8 | 0 9 -13 -3 }}
 
: mapping generators: ~3375/2401, ~450/343
: mapping generators: ~3375/2401, ~15/14


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~3375/2401 = 600.000{{c}}, ~15/14 = 122.031{{c}}
* [[WE]]: ~3375/2401 = 600.1355{{c}}, ~450/343 = 478.0813{{c}}
: [[error map]]: {{val| +0.271 +0.235 +0.662 -1.986 }}
* [[CWE]]: ~3375/2401 = 600.000{{c}}, ~450/343 = 477.9725{{c}}
: error map: {{val| 0.000 -0.202 +0.043 -2.743 }}
<!-- * [[CTE]]: ~3375/2401 = 600.000{{c}}, ~450/343 = 477.969{{c}}
: [[error map]]: {{val| 0.000 -0.232 +0.087 -2.734 }}
: [[error map]]: {{val| 0.000 -0.232 +0.087 -2.734 }}
* [[POTE]]: ~3375/2401 = 600.000{{c}}, ~15/14 = 122.027{{c}}
* [[POTE]]: ~3375/2401 = 600.000{{c}}, ~450/343 = 477.973{{c}}
: error map: {{val| 0.000 -0.195 +0.033 -2.746 }}
: error map: {{val| 0.000 -0.195 +0.033 -2.746 }} -->


{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }}
{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }}


[[Badness]] (Smith): 0.144345
[[Badness]] (Sintel): 3.65
 
== Dee leap week ==
{{Main|Dee leap week}}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1029/1024, 2460375/2458624
 
{{Mapping|legend=1| 1 4 -6 2 | 0 -9 31 3 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~135/112 = 322.123{{c}}
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 2460375/2458624
 
Mapping: {{mapping| 1 4 -6 2 15 | 0 -9 31 3 43 }}
 
Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~135/112 = 322.097
 
{{Optimal ET sequence|legend=0| 41, 149, 190, 231 }}


== Necromanteion ==
== Necromanteion ==

Revision as of 17:14, 15 November 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.

The 2.3.7-subgroup comma for the gamelismic clan is the gamelisma, 1029/1024, with monzo [-10 1 0 3⟩. For any member of the clan, for the rank-3 gamelismic temperament itself, and for the rank-2 2.3.7 temperament slendric (a.k.a. gamelic), this means three ~8/7 intervals give a fifth, 3/2. In fact, we find that 3/2 = (8/7)3⋅(1029/1024). From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. 36edo is a good tuning for slendric, though if the full 7-limit is desired, 72edo, 77edo, or 118edo might be preferred.

To the gamelisma itself we need to add the comma which appears next on the modified normal comma list for the full 7-limit. The second comma on the list for mothra is 81/80, for rodan 245/243, for guiron 32805/32768, for gorgo 36/35, and for gidorah 256/245. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~6/5.

Miracle adds 33075/32768 and uses the secor, half an ~8/7, as generator. Lemba adds 525/512 to the list, and has a half-octave period. Valentine adds 6144/6125 with a generator of ~21/20 and superkleismic adds 875/864 with a generator of ~6/5. Unidec adds 4375/4374, and has a generator of ~10/9 with a half-octave period. Hemithirds adds 65625/65536 with a generator half of a classical major third. Finally, tritikleismic adds 15625/15552 and has a generator of 6/5 with a 1/3-octave period.

Full 7-limit temperaments discussed elsewhere are:

The rest are considered below.

No-five subgroup extensions of slendric include radon, a 2.3.7.11 extension that may be viewed as no-five rodan, and baladic, a 2.3.7.13.17 extension, considered below. Dicussed elsewhere is gigapyth in the 2.3.7.85 subgroup.

Slendric

Main article: Slendric

Subgroup: 2.3.7

Comma list: 1029/1024

Subgroup-val mapping: [⟨1 1 3], ⟨0 3 -1]]

Gencom mapping: [⟨1 1 0 3], ⟨0 3 0 -1]]

mapping generators: ~2, ~8/7

Optimal tunings:

  • WE: ~2 = 1200.4859 ¢, ~8/7 = 233.7822 ¢
error map: ⟨+0.486 -0.123 -1.151]
  • CWE: ~2 = 1200.000 ¢, ~8/7 = 233.7474 ¢
error map: ⟨0.000 -0.713 -2.573]

Optimal ET sequence: 36, 77, 113, 190

Badness (Sintel): 0.158

Euslendric

Forms of slendric in the most optimal range for the 2.3.7 temperament (36 & 77) lack an obvious strong mapping of prime 5 or prime 11. However, slendric can extend well to the no-fives no-elevens 29-limit by tempering out 273/272, 343/342, 378/377, 392/391, 513/512, and 729/728, or a comma basis defined in terms of S-expressions as {S7/S8, S14/S16, S15/S20, S24/S26, S27, S28}. 113edo is an obvious tuning.

Subgroup: 2.3.7.13

Comma list: 729/728, 1029/1024

Sval mapping: [⟨1 1 3 0], ⟨0 3 -1 19]]

Gencom mapping: [⟨1 1 0 3 0 0], ⟨0 3 0 -1 0 19]]

Optimal tunings:

  • WE: ~2 = 1200.5057 ¢, ~8/7 = 233.7200 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.6534 ¢

Optimal ET sequence: 5, 31f, 36, 77, 113, 827bdddff

Badness (Sintel): 0.339

2.3.7.13.17 subgroup

Subgroup: 2.3.7.13.17

Comma list: 273/272, 729/728, 833/832

Sval mapping: [⟨1 1 3 0 0], ⟨0 3 -1 19 21]]

Gencom mapping: [⟨1 1 0 3 0 0 0], ⟨0 3 0 -1 0 19 21]]

Optimal tunings:

  • WE: ~2 = 1200.5282 ¢, ~8/7 = 233.6492 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.5776 ¢

Optimal ET sequence: 5g, 31fg, 36, 113, 149

Badness (Sintel): 0.332

2.3.7.13.17.19 subgroup

Subgroup: 2.3.7.13.17.19

Comma list: 273/272, 343/342, 513/512, 729/728

Sval mapping: [⟨1 1 3 0 0 6], ⟨0 3 -1 19 21 -9]]

Gencom mapping: [⟨1 1 0 3 0 0 0 6], ⟨0 3 0 -1 0 19 21 -9]]

Optimal tunings:

  • WE: ~2 = 1200.3292 ¢, ~8/7 = 233.6651 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.6106 ¢

Optimal ET sequence: 5g, 36, 77, 113, 262df

Badness (Sintel): 0.380

2.3.7.13.17.19.23 subgroup

Subgroup: 2.3.7.13.17.19.23

Comma list: 273/272, 343/342, 392/391, 513/512, 729/728

Sval mapping: [⟨1 1 3 0 0 6 9], ⟨0 3 -1 19 21 -9 -23]]

Gencom mapping: [⟨1 1 0 3 0 0 0 6 9], ⟨0 3 0 -1 0 19 21 -9 -23]]

Optimal tunings:

  • WE: ~2 = 1200.3127 ¢, ~8/7 = 233.6679 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.6091 ¢

Optimal ET sequence: 36, 77, 113, 262df

Badness (Sintel): 0.474

2.3.7.13.17.19.23.29 subgroup

Subgroup: 2.3.7.13.17.19.23.29

Comma list: 273/272, 343/342, 378/377, 392/391, 513/512, 609/608

Sval mapping: [⟨1 1 3 0 0 6 9 7], ⟨0 3 -1 19 21 -9 -23 -11]]

Gencom mapping: [⟨1 1 0 3 0 0 0 6 9 7], ⟨0 3 0 -1 0 19 21 -9 -23 -11]]

Optimal tunings:

  • WE: ~2 = 1200.2503 ¢, ~8/7 = 233.6688 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.6208 ¢

Optimal ET sequence: 36, 77, 113

Badness (Sintel): 0.473

Radon

Radon is the no-fives version of rodan, equating the diatonic major third to 14/11.

Subgroup: 2.3.7.11

Comma list: 896/891, 1029/1024

Sval mapping: [⟨1 1 3 6], ⟨0 3 -1 -13]]

Gencom mapping: [⟨1 1 0 3 6], ⟨0 3 0 -1 -13]]

Optimal tunings:

  • WE: ~2 = 1199.9708 ¢, ~8/7 = 234.3748 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.3813 ¢

Optimal ET sequence: 5, …, 36, 41, 87, 128

Badness (Sintel): 0.619

Baladic

Baladic is a 2.3.7.13.17 subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. It tempers out 169/168 (S13), which splits 7/6 in half (13/12~14/13) and one finds that the octave is therefore split in half via the interval 91/64, which is then equated to 17/12. 36edo is an excellent baladic tuning.

Subgroup: 2.3.7.13

Comma list: 169/168, 1029/1024

Sval mapping: [⟨2 2 6 7], ⟨0 3 -1 1]]

Gencom mapping: [⟨2 2 0 6 0 7], ⟨0 3 0 -1 0 1]]

mapping generators: ~91/64, ~8/7

Optimal tunings:

  • WE: ~91/64 = 600.4315 ¢, ~8/7 = 233.7724 ¢
  • CWE: ~91/64 = 600.0000 ¢, ~8/7 = 233.7039 ¢

Optimal ET sequence: 10, 26, 36, 154f, 190ff, 226ff, 262dfff

Badness (Sintel): 0.434

2.3.7.13.17 subgroup

Subgroup: 2.3.7.13.17

Comma list: 169/168, 273/272, 289/288

Sval mapping: [⟨2 2 6 7 7], ⟨0 3 -1 1 3]]

Gencom mapping: [⟨2 2 0 6 0 7 7], ⟨0 3 0 -1 0 1 3]]

Optimal tunings:

  • WE: ~17/12 = 600.4436 ¢, ~8/7 = 233.7883 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~8/7 = 233.7312 ¢

Optimal ET sequence: 10, 26, 36, 154f, 190ffg, 226ffg

Badness (Sintel): 0.253

Mothra

Main article: Mothra

Mothra tempers out 81/80 and finds the prime 5 at a stack of four fifths as does any temperament in the meantone family. It also tempers out 1728/1715, the orwellisma. It can be described as the 26 & 31. Using 31edo with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential[1], so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo in the article on strictly proper 7-tone 31edo scales.

Note that mothra is also called cynder in the 7-limit, which can be a little confusing sometimes.

Its S-expression-based comma list is {S6/S7, S7/S8, (S6/S8 = S9)}, taking advantage of the fact that 81/80 is a semiparticular.

Subgroup: 2.3.5.7

Comma list: 81/80, 1029/1024

Mapping: [⟨1 1 0 3], ⟨0 3 12 -1]]

Optimal tunings:

  • WE: ~2 = 1200.9303 ¢, ~8/7 = 232.3733 ¢
error map: ⟨+0.930 -3.905 +2.165 +1.592]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 232.2514 ¢
error map: ⟨0.000 -5.520 +0.703 -1.077]

Algebraic generator: Rabrindanath, largest real root of x8 - 3x2 + 1, or 232.0774 cents.

Minimax tuning:

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [3 0 -1/12 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Optimal ET sequence: 5, 21c, 26, 31

Badness (Sintel): 0.940

Undecimal mothra

Undecimal mothra is the extension of 7-limit cynder which tempers out 385/384 as is natural in slendric temperaments. It is the simplest extension, supported within a reasonable tuning range (between 26edo and 31edo), and is supported by the patent val of 5edo, which implies that it is better behaved as a cluster temperament. It is also notable for being supported by the just tuning of 8/7, and has a restriction to the 2.7.11 subgroup, namely amaranthine, that is a microtemperament.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 385/384

Mapping: [⟨1 1 0 3 5], ⟨0 3 12 -1 -8]]

Optimal tunings:

  • WE: ~2 = 1201.3979 ¢, ~8/7 = 232.3010 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 232.0621 ¢

Optimal ET sequence: 5, 26, 31, 88, 119be, 150be

Badness (Sintel): 0.848

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 105/104, 144/143

Mapping: [⟨1 1 0 3 5 1], ⟨0 3 12 -1 -8 14]]

Optimal tunings:

  • WE: ~2 = 1201.0985 ¢, ~8/7 = 232.0231 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 231.8425 ¢

Optimal ET sequence: 5, 26, 31, 57, 88

Badness (Sintel): 0.990

Music

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 105/104, 120/119, 144/143

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

Optimal tunings:

  • WE: ~2 = 1200.9734 ¢, ~8/7 = 231.8960 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 231.7392 ¢

Optimal ET sequence: 5g, 26, 31, 57, 88

Badness (Sintel): 1.00

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 99/98, 105/104, 120/119, 144/143, 153/152

Mapping: [⟨1 1 0 3 5 1], ⟨0 3 12 -1 -8 14 16 22]]

Optimal tunings:

  • WE: ~2 = 1200.9663 ¢, ~8/7 = 231.8393 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 231.6842 ¢

Optimal ET sequence: 26, 31, 57

Badness (Sintel): 1.05

Mosura

The S-expression-based comma list of mosura suggests it might be the most natural extension of 7-limit cynder to the 11-limit: {S6/S7, S7/S8, (S6/S8 = S9), S8/S10}.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175, 540/539

Mapping: [⟨1 1 0 3 -1], ⟨0 3 12 -1 23]]

Optimal tunings:

  • WE: ~2 = 1200.7675 ¢, ~8/7 = 232.5673 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 232.4567 ¢

Optimal ET sequence: 5e, 26e, 31, 129

Badness (Sintel): 1.04

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 144/143, 176/175, 196/195

Mapping: [⟨1 1 0 3 -1 7], ⟨0 3 12 -1 23 -17]]

Optimal tunings:

  • WE: ~2 = 1199.9347 ¢, ~8/7 = 232.6275 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 232.6392 ¢

Optimal ET sequence: 31, 67, 98

Badness (Sintel): 1.52

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 144/143, 176/175, 189/187, 196/195

Mapping: [⟨1 1 0 3 -1 7], ⟨0 3 12 -1 23 -17 -15]]

Optimal tunings:

  • WE: ~2 = 1199.7124 ¢, ~8/7 = 232.6376 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 232.6917 ¢

Optimal ET sequence: 31, 67, 98

Badness (Sintel): 1.53

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 96/95, 144/143, 153/152, 176/175, 196/195

Mapping: [⟨1 1 0 3 -1 7], ⟨0 3 12 -1 23 -17 -15 -9]]

Optimal tunings:

  • WE: ~2 = 1199.4885 ¢, ~8/7 = 232.6310 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 232.7287 ¢

Optimal ET sequence: 31, 67, 98h

Badness (Sintel): 1.50

Cyndra

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 1029/1024

Mapping: [⟨1 1 0 3 0], ⟨0 3 12 -1 18]]

Optimal tunings:

  • WE: ~2 = 1201.1585 ¢, ~8/7 = 231.5404 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 231.3850 ¢

Optimal ET sequence: 5e, 21ce, 26

Badness (Sintel): 1.84

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 78/77, 81/80, 640/637

Mapping: [⟨1 1 0 3 0 1], ⟨0 3 12 -1 18 14]]

Optimal tunings:

  • WE: ~2 = 1201.1152 ¢, ~8/7 = 231.5079 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 231.3612 ¢

Optimal ET sequence: 5e, 21cef, 26

Badness (Sintel): 1.41

Rodan

Main article: Rodan
For the 5-limit version, see Syntonic–diatonic equivalence continuum #Rodan (5-limit).

Rodan tempers out 245/243 and can be described as the 41 & 46 temperament. This temperament extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric.

Subgroup: 2.3.5.7

Comma list: 245/243, 1029/1024

Mapping: [⟨1 1 -1 3], ⟨0 3 17 -1]]

Optimal tunings:

  • WE: ~2 = 1200.2146 ¢, ~8/7 = 234.4587 ¢
error map: ⟨+0.215 +1.636 -0.731 -2.641]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.4259 ¢
error map: ⟨0.000 +1.323 -1.073 -3.252]

Minimax tuning:

[[1 0 0 0⟩, [5/3 0 1/6 -1/6⟩, [25/9 0 17/18 -17/18⟩, [25/9 0 -1/18 1/18⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5

Algebraic generator: larger root of 20x2 - 36x + 15, or (9 + √6)/10.

Optimal ET sequence: 41, 87, 128, 215d

Badness (Sintel): 0.939

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 441/440

Mapping: [⟨1 1 -1 3 6], ⟨0 3 17 -1 -13]]

Optimal tunings:

  • WE: ~2 = 1200.0553 ¢, ~8/7 = 234.4695 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.4594 ¢

Minimax tuning:

  • 11-odd-limit: ~8/7 = [4/19 2/19 0 0 -1/19⟩
[[1 0 0 0 0⟩, [31/19 6/19 0 0 -3/19⟩, [49/19 34/19 0 0 -17/19⟩, [53/19 -2/19 0 0 1/19⟩, [62/19 -26/19 0 0 13/19⟩]
unchanged-interval (eigenmonzo) basis: 2.11/9

Algebraic generator: positive root of x2 + 16x - 31, or √95 - 8.

Optimal ET sequence: 41, 87

Badness (Sintel): 0.763

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 352/351, 364/363

Mapping: [⟨1 1 -1 3 6 8], ⟨0 3 17 -1 -13 -22]]

Optimal tunings:

  • WE: ~2 = 1199.9868 ¢, ~8/7 = 234.4796 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.4822 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~8/7 = [3/14 1/14 0 0 0 -1/28⟩
unchanged-interval (eigenmonzo) basis: 2.13/9

Algebraic generator: Gatetone, positive root of 4x6 - 7x - 1. Recurrence converges slowly.

Optimal ET sequence: 41, 46, 87

Badness (Sintel): 0.762

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 154/153, 196/195, 245/243, 256/255, 273/272

Mapping: [⟨1 1 -1 3 6 8 8], ⟨0 3 17 -1 -13 -22 -20]]

Optimal tunings:

  • WE: ~2 = 1199.8331 ¢, ~8/7 = 234.4919 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.5254 ¢

Minimax tuning:

  • 17-odd-limit: ~8/7 = [3/13 1/13 0 0 0 0 -1/26⟩
unchanged-interval (eigenmonzo) basis: 2.17/9

Optimal ET sequence: 41, 46, 87

Badness (Sintel): 0.853

Aerodactyl

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 245/243, 385/384, 441/440

Mapping: [⟨1 1 -1 3 6 -1], ⟨0 3 17 -1 -13 24]]

Optimal tunings:

  • WE: ~2 = 1200.2997 ¢, ~8/7 = 234.6972 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.6439 ¢

Optimal ET sequence: 5, 41f, 46

Badness (Sintel): 1.40

Aerodino

Subgroup: 2.3.5.7.11

Comma list: 176/175, 245/243, 1029/1024

Mapping: [⟨1 1 -1 3 -3], ⟨0 3 17 -1 33]]

Optimal tunings:

  • WE: ~2 = 1199.9179 ¢, ~8/7 = 234.7123 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.7256 ¢

Optimal ET sequence: 5e, 41e, 46

Badness (Sintel): 1.79

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 176/175, 245/243, 847/845

Mapping: [⟨1 1 -1 3 -3 -1], ⟨0 3 17 -1 33 24]]

Optimal tunings:

  • WE: ~2 = 1200.0242 ¢, ~8/7 = 234.7863 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.7824 ¢

Optimal ET sequence: 5e, 41ef, 46

Badness (Sintel): 1.48

Varan

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243, 1029/1024

Mapping: [⟨1 1 -1 3 -2], ⟨0 3 17 -1 28]]

Optimal tunings:

  • WE: ~2 = 1200.3738 ¢, ~8/7 = 234.2174 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.1586 ¢

Optimal ET sequence: 5e, 36ce, 41

Badness (Sintel): 1.49

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 245/243, 352/351

Mapping: [⟨1 1 -1 3 -2 0], ⟨0 3 17 -1 28 19]]

Optimal tunings:

  • WE: ~2 = 1200.1389 ¢, ~8/7 = 234.1162 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.0946 ¢

Optimal ET sequence: 5e, 36ce, 41

Badness (Sintel): 1.33

Guiron

Guiron tempers out the schisma, and finds the prime 5 at the diminished fourth as does any temperament in the schismatic family. It can be described as the 36 & 41 temperament. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 10976/10935

Mapping: [⟨1 1 7 3], ⟨0 3 -24 -1]]

Optimal tunings:

  • WE: ~2 = 1200.3395 ¢, ~8/7 = 233.9963 ¢
error map: ⟨+0.340 +0.374 +0.151 -1.804]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.9239 ¢
error map: ⟨0.000 -0.183 -0.487 -2.750]

Minimax tuning:

[[1 0 0 0⟩, [15/8 0 -1/8 0⟩, [0 0 1 0⟩, [65/24 0 1/24 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Optimal ET sequence: 36, 41, 77, 118, 277d

Badness (Sintel): 1.20

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 10976/10935

Mapping: [⟨1 1 7 3 -2], ⟨0 3 -24 -1 28]]

Optimal tunings:

  • WE: ~2 = 1200.3453 ¢, ~8/7 = 233.9988 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.9312 ¢

Minimax tuning:

  • 11-odd-limit: ~8/7 = [7/24 0 -1/24⟩
[[1 0 0 0 0⟩, [15/8 0 -1/8 0 0⟩, [0 0 1 0 0⟩, [65/24 0 1/24 0 0⟩, [37/6 0 -7/6 0 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Optimal ET sequence: 36e, 41, 77, 118, 159, 277d

Badness (Sintel): 0.881

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384, 729/728

Mapping: [⟨1 1 7 3 -2 0], ⟨0 3 -24 -1 28 19]]

Optimal tunings:

  • WE: ~2 = 1200.1222 ¢, ~8/7 = 233.9228 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.8994 ¢

Optimal ET sequence: 36e, 41, 77, 118

Badness (Sintel): 1.18

Gorgo

For the 5-limit version, see Syntonic–diatonic equivalence continuum #Laconic.
See also: Llywelynsmic clan

Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the 16 & 21 temperament.

If we discard the inaccurate mapping of prime 3, we get shoe, so that the large commas of gorgo are explained practically entirely by the inaccurate 3, meaning that this temperament is much more accurate than its comma list suggests.

Subgroup: 2.3.5.7

Comma list: 36/35, 1029/1024

Mapping: [⟨1 1 1 3], ⟨0 3 7 -1]]

Optimal tunings:

  • WE: ~2 = 1200.9847 ¢, ~8/7 = 228.5210 ¢
error map: ⟨+0.985 -15.407 +14.318 +5.607]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 228.4371 ¢
error map: ⟨0.000 -16.644 +12.746 +2.737]

Optimal ET sequence: 5, 11c, 16, 21

Badness (Sintel): 1.54

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 1029/1024

Mapping: [⟨1 1 1 3 1], ⟨0 3 7 -1 13]]

Optimal tunings:

  • WE: ~2 = 1201.3609 ¢, ~8/7 = 227.6312 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 227.4955 ¢

Optimal ET sequence: 5e, 16, 21, 37b

Badness (Sintel): 1.64

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44, 507/500

Mapping: [⟨1 1 1 3 1 2], ⟨0 3 7 -1 13 9]]

Optimal tunings:

  • WE: ~2 = 1201.0996 ¢, ~8/7 = 227.4378 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 227.3327 ¢

Optimal ET sequence: 5e, 16, 21, 37b

Badness (Sintel): 1.35

Spartan

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 1029/1024

Mapping: [⟨1 1 1 3 5], ⟨0 3 7 -1 -8]]

Optimal tunings:

  • WE: ~2 = 1198.9344 ¢, ~8/7 = 229.3316 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 229.5124 ¢

Optimal ET sequence: 5, 16e, 21

Badness (Sintel): 2.07

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 56/55, 507/500

Mapping: [⟨1 1 1 3 5 2], ⟨0 3 7 -1 -8 9]]

Optimal tunings:

  • WE: ~2 = 1198.3002 ¢, ~8/7 = 228.7341 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 229.0044 ¢

Optimal ET sequence: 5, 16e, 21

Badness (Sintel): 1.95

Music

Gidorah

For the 5-limit version, see Syntonic–diatonic equivalence continuum #University.

Gidorah is a very low-accuracy temperament where the generator of ~8/7 is lumped together with ~6/5. 16c-, 21cc-, and 26ccc-edo are among the possible tunings.

Subgroup: 2.3.5.7

Comma list: 21/20, 144/125

Mapping: [⟨1 1 2 3], ⟨0 3 2 -1]]

Optimal tunings:

  • WE: ~2 = 1192.4932 ¢, ~8/7 = 229.3187 ¢
error map: ⟨-7.507 -21.506 +57.310 -20.665]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 229.6649 ¢
error map: ⟨0.000 -12.960 +73.016 +1.509]

Optimal ET sequence: 1b, 5

Badness (Sintel): 1.58

Oncle

For the 5-limit version, see Miscellaneous 5-limit temperaments #Oncle.

Oncle can be described as the 31 & 36c temperament.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 2430/2401

Mapping: [⟨1 1 6 3], ⟨0 3 -19 -1]]

Optimal tunings:

  • WE: ~2 = 1201.2246 ¢, ~8/7 = 232.7354 ¢
error map: ⟨+1.225 -2.524 -0.939 +2.112]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 232.4718 ¢
error map: ⟨0.000 -4.539 -3.279 -1.298]

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

Badness (Sintel): 2.24

Archaeotherium

For the 5-limit version, see Miscellaneous 5-limit temperaments #Archaeotherium.

Archaeotherium can be described as the 21 & 26 temperament.

Subgroup: 2.3.5.7

Comma list: 405/392, 1029/1024

Mapping: [⟨1 1 5 3], ⟨0 3 -14 -1]]

Optimal tunings:

  • WE: ~2 = 1202.7179 ¢, ~8/7 = 230.7800 ¢
error map: ⟨+2.718 -6.897 -3.644 +8.548]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 230.1909 ¢
error map: ⟨0.000 -11.382 -8.986 +0.983]

Optimal ET sequence: 21, 26, 47, 73bc

Badness (Sintel): 3.70

Clyndro

See also: Mavila family

Clyndro tempers out 135/128 and finds the interval class of 5 at a stack of -3 fifths as does any temperament in the mavila family. It can be described as the 11 & 16 temperament.

Subgroup: 2.3.5.7

Comma list: 135/128, 360/343

Mapping: [⟨1 1 4 3], ⟨0 3 -9 -1]]

Optimal tunings:

  • WE: ~2 = 1205.6135 ¢, ~8/7 = 227.5283 ¢
error map: ⟨+5.613 -13.757 -11.614 +20.486]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 226.3207 ¢
error map: ⟨0.000 -22.993 -23.200 +4.853]

Optimal ET sequence: 5c, 11, 16

Badness (Sintel): 4.03

11-limit

Subgroup: 2.3.5.7.11

Comma list: 33/32, 45/44, 352/343

Mapping: [⟨1 1 4 3 4], ⟨0 3 -9 -1 -3]]

Optimal tunings:

  • WE: ~2 = 1206.2134 ¢, ~8/7 = 227.6004 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 226.2421 ¢

Optimal ET sequence: 5c, 11, 16

Badness (Sintel): 2.30

Miracle

Main article: Miracle
For the 5-limit version, see Syntonic–31 equivalence continuum #Ampersand.

Miracle is one of the most important entries of this temperament clan. It tempers out 225/224, splitting the ~8/7 generator of slendric into 15/14~16/15, and can be described as the 31 & 41 temperament. It is then extremely natural to equate the neutral third, three generators up, to 11/9 and thereby extend miracle to the full 11-limit with essentially no further damage. 72edo makes for an excellent tuning.

Subgroup: 2.3.5.7

Comma list: 225/224, 1029/1024

Mapping: [⟨1 1 3 3], ⟨0 6 -7 -2]]

mapping generator: ~2, ~15/14

Optimal tunings:

  • WE: ~2 = 1200.8209 ¢, ~15/14 = 116.7550 ¢
error map: ⟨+0.821 -0.604 -1.136 +0.127]
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.6756 ¢
error map: ⟨0.000 -1.901 -3.043 -2.177]

Minimax tuning:

[[1 0 0 0⟩, [25/13 6/13 -6/13 0⟩, [25/13 -7/13 7/13 0⟩, [35/13 -2/13 2/13 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5/3
[[1 0 0 0⟩, [25/19 12/19 -6/19 0⟩, [50/19 -14/19 7/19 0⟩, [55/19 -4/19 2/19 0⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5

Tuning ranges:

  • 7-odd-limit diamond monotone: ~15/14 = [114.286, 120.000] (2\21 to 1\10)
  • 9-odd-limit diamond monotone: ~15/14 = [116.129, 120.000] (3\31 to 1\10)
  • 7- and 9-odd-limit diamond tradeoff: ~15/14 = [115.587, 116.993]

Algebraic generator: Secor59, positive root of 15x6 - 8x4 - 12

Optimal ET sequence: 10, 21, 31, 41, 72

Badness (Sintel): 0.424

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 385/384

Mapping: [⟨1 1 3 3 2], ⟨0 6 -7 -2 15]]

Optimal tunings:

  • WE: ~2 = 1200.7626 ¢, ~15/14 = 116.7069 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.6469 ¢

Minimax tuning:

  • 11-odd-limit: ~15/14 = [1/19 2/19 -1/19⟩
[[1 0 0 0 0⟩, [25/19 12/19 -6/19 0 0⟩, [50/19 -14/19 7/19 0 0⟩, [55/19 -4/19 2/19 0 0⟩, [53/19 30/19 -15/19 0 0⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5

Tuning ranges:

  • 11-odd-limit diamond monotone: ~15/14 = [116.129, 117.073] (3\31 to 4\41)
  • 11-odd-limit diamond tradeoff: ~15/14 = [115.587, 116.993]

Algebraic generator: Secor59

Optimal ET sequence: 10, 21e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde

Badness (Sintel): 0.353

Miraculous

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195, 243/242

Mapping: [⟨1 1 3 3 2 4], ⟨0 6 -7 -2 15 -3]]

Optimal tunings:

  • WE: ~2 = 1200.1267 ¢, ~15/14 = 116.7596 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.7488 ¢

Optimal ET sequence: 10, 21e, 31, 41, 72f

Badness (Sintel): 0.771

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 120/119, 144/143, 154/153, 170/169

Mapping: [⟨1 1 3 3 2 4 4], ⟨0 6 -7 -2 15 -3 1]]

Optimal tunings:

  • WE: ~2 = 1199.6759 ¢, ~15/14 = 116.7378 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.7657 ¢

Optimal ET sequence: 10, 21e, 31, 41, 72fg

Badness (Sintel): 0.870

Benediction

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 351/350, 385/384

Mapping: [⟨1 1 3 3 2 7], ⟨0 6 -7 -2 15 -34]]

Optimal tunings:

  • WE: ~2 = 1199.8601 ¢, ~15/14 = 116.6572 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.5688 ¢

Optimal ET sequence: 31, 72, 103, 175f

Badness (Sintel): 0.649

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 351/350, 375/374

Mapping: [⟨1 1 3 3 2 7 7], ⟨0 6 -7 -2 15 -34 -30]]

Optimal tunings:

  • WE: ~2 = 1200.8328 ¢, ~15/14 = 116.6661 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.5774 ¢

Optimal ET sequence: 31, 72, 103, 175f, 422bcdefffg

Badness (Sintel): 0.639

Manna

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 325/324, 385/384

Mapping: [⟨1 1 3 3 2 0], ⟨0 6 -7 -2 15 38]]

Optimal tunings:

  • WE: ~2 = 1200.7564 ¢, ~15/14 = 116.8129 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.7528 ¢

Optimal ET sequence: 31f, 41, 72, 185cf, 257cff

Badness (Sintel): 0.703

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 325/324, 385/384

Mapping: [⟨1 1 3 3 2 0 0], ⟨0 6 -7 -2 15 38 42]]

Optimal tunings:

  • WE: ~2 = 1200.7570 ¢, ~15/14 = 116.8011 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.7408 ¢

Optimal ET sequence: 31fg, 41, 72, 185cf, 257cff

Badness (Sintel): 0.748

Semimiracle

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 243/242, 385/384

Mapping: [⟨2 2 6 6 4 7], ⟨0 6 -7 -2 15 2]]

mapping generators: ~55/39, ~15/14

Optimal tunings:

  • WE: ~55/39 = 600.4844 ¢, ~15/14 = 116.7182 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~15/14 = 116.6413 ¢

Optimal ET sequence: 10, 62, 72

Badness (Sintel): 1.02

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 225/224, 243/242, 273/272

Mapping: [⟨2 2 6 6 4 7 7], ⟨0 6 -7 -2 15 2 6]]

Optimal tunings:

  • WE: ~17/12 = 600.5042 ¢, ~15/14 = 116.7264 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~15/14 = 116.6485 ¢

Optimal ET sequence: 10, 62, 72

Badness (Sintel): 0.822

Hemisecordite

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384, 847/845

Mapping: [⟨1 1 3 3 2 2], ⟨0 12 -14 -4 30 35]]

mapping generators: ~2, ~27/26

Optimal tunings:

  • WE: ~2 = 1200.6969 ¢, ~27/26 = 58.3217 ¢
  • CWE: ~2 = 1200.0000 ¢, ~27/26 = 58.2964 ¢

Optimal ET sequence: 41, 62, 103, 247c, 350bcde

Badness (Sintel): 1.06

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 385/384, 847/845

Mapping: [⟨1 1 3 3 2 2 2], ⟨0 12 -14 -4 30 35 43]]

Optimal tunings:

  • WE: ~2 = 1200.6557 ¢, ~27/26 = 58.2932 ¢
  • CWE: ~2 = 1200.0000 ¢, ~27/26 = 58.2702 ¢

Optimal ET sequence: 41, 62, 103

Badness (Sintel): 1.15

Semihemisecordite

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 289/288, 385/384, 847/845

Mapping: [⟨2 2 6 6 4 4 7], ⟨0 12 -14 -4 30 35 12]]

mapping generators: ~17/12, ~27/26

Optimal tunings:

  • WE: ~17/12 = 600.3951 ¢, ~27/26 = 58.3260 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~27/26 = 58.2974 ¢

Optimal ET sequence: 62, 144g, 206begg

Badness (Sintel): 2.39

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 209/208, 225/224, 243/242, 289/288, 361/360, 385/384

Mapping: [⟨2 2 6 6 4 4 7 8], ⟨0 12 -14 -4 30 35 12 5]]

Optimal tunings:

  • WE: ~17/12 = 600.4418 ¢, ~27/26 = 58.3255 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~27/26 = 58.2928 ¢

Optimal ET sequence: 62, 144gh, 206begghh

Badness (Sintel): 2.13

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 209/208, 225/224, 243/242, 289/288, 323/322, 361/360, 385/384

Mapping: [⟨2 2 6 6 4 4 7 8 7], ⟨0 12 -14 -4 30 35 12 5 21]]

Optimal tunings:

  • WE: ~17/12 = 600.4451 ¢, ~27/26 = 58.3264 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~27/26 = 58.2942 ¢

Optimal ET sequence: 62, 144gh, 206begghhi

Badness (Sintel): 1.89

Phicordial

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384, 2200/2197

Mapping: [⟨1 -11 17 7 -28 3], ⟨0 18 -21 -6 45 1]]

mapping generators: ~2, ~13/8

Optimal tunings:

  • WE: ~2 = 1200.7056 ¢, ~13/8 = 839.3726 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 838.8831 ¢

Optimal ET sequence: 103, 216c, 319bcde, 535bccdef

Badness (Sintel): 1.37

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 441/440, 2200/2197

Mapping: [⟨1 -11 17 7 -28 3 -5], ⟨0 18 -21 -6 45 1 13]]

Optimal tunings:

  • WE: ~2 = 1200.5918 ¢, ~13/8 = 839.2912 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 838.8809 ¢

Optimal ET sequence: 103, 216c, 319bcde

Badness (Sintel): 1.26

Revelation

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1029/1024

Mapping: [⟨1 1 3 3 5], ⟨0 6 -7 -2 -16]]

Optimal tunings:

  • WE: ~2 = 1201.3320 ¢, ~15/14 = 116.4057 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.2524 ¢

Optimal ET sequence: 10e, 21, 31

Badness (Sintel): 1.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 105/104, 512/507

Mapping: [⟨1 1 3 3 5 4], ⟨0 6 -7 -2 -16 -3]]

Optimal tunings:

  • WE: ~2 = 1200.6059 ¢, ~15/14 = 116.3263 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.2564 ¢

Optimal ET sequence: 10e, 21, 31

Badness (Sintel): 1.22

Hemimiracle

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/242, 1029/1024

Mapping: [⟨1 1 3 3 4], ⟨0 12 -14 -4 -11]]

mapping generators: ~2, ~33/32

Optimal tunings:

  • WE: ~2 = 1200.2902 ¢, ~33/32 = 58.4217 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 58.4062 ¢

Optimal ET sequence: 20, 21, 41

Badness (Sintel): 1.96

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 245/242, 512/507

Mapping: [⟨1 1 3 3 4 4], ⟨0 12 -14 -4 -11 -6]]

Optimal tunings:

  • WE: ~2 = 1199.8454 ¢, ~33/32 = 58.4220 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 58.4305 ¢

Optimal ET sequence: 20, 21, 41

Badness (Sintel): 1.78

Oracle

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 1029/1024

Mapping: [⟨1 -5 10 5 4], ⟨0 12 -14 -4 -1]]

mapping generators: ~2, ~16/11

Optimal tunings:

  • WE: ~2 = 1201.2122 ¢, ~16/11 = 658.9974 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 658.3320 ¢

Optimal ET sequence: 11, 20, 31, 82e, 113e, 144ee

Badness (Sintel): 1.41

Hemiseven

Unlike miracle which splits ~8/7, hemiseven splits ~7/4. It can be described as the 72 & 77 temperament. 149edo is an obvious tuning.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 19683/19600

Mapping: [⟨1 -2 -15 4], ⟨0 6 29 -2]]

mapping generators: ~2, ~243/160

Optimal tunings:

  • WE: ~2 = 1200.5612 ¢, ~243/160 = 717.0687 ¢
error map: ⟨+0.561 -0.665 +0.260 -0.718]
  • CWE: ~2 = 1200.0000 ¢, ~243/160 = 716.7478 ¢
error map: ⟨0.000 -1.468 -0.629 -2.321]

Optimal ET sequence: 72, 149, 221, 514bd, 735bcdd

Badness (Sintel): 1.43

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 19683/19600

Mapping: [⟨1 -2 -15 4 16], ⟨0 6 29 -2 -21]]

Optimal tunings:

  • WE: ~2 = 1200.6243 ¢, ~243/160 = 717.0969 ¢
  • CWE: ~2 = 1200.0000 ¢, ~243/160 = 716.7292 ¢

Optimal ET sequence: 72, 149, 221e, 293de

Badness (Sintel): 0.941

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 385/384, 441/440, 676/675

Mapping: [⟨1 -2 -15 4 16 -19], ⟨0 6 29 -2 -21 38]]

Optimal tunings:

  • WE: ~2 = 1200.6781 ¢, ~91/60 = 717.1496 ¢
  • CWE: ~2 = 1200.0000 ¢, ~91/60 = 716.7520 ¢

Optimal ET sequence: 72, 149, 221ef

Badness (Sintel): 0.905

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 351/350, 385/384, 441/440, 676/675

Mapping: [⟨1 -2 -15 4 16 -19 -21], ⟨0 6 29 -2 -21 38 42]]

Optimal tunings:

  • WE: ~2 = 1200.6635 ¢, ~68/45 = 717.1354 ¢
  • CWE: ~2 = 1200.0000 ¢, ~68/45 = 716.7472 ¢

Optimal ET sequence: 72, 149, 221ef

Badness (Sintel): 0.800

Valentine

Main article: Valentine
For the 5-limit version, see Syntonic–31 equivalence continuum #Valentine (5-limit).

Valentine tempers out 126/125 and 6144/6125 as well as 1029/1024. It has a generator of ~21/20, three of which make the slendric generator ~8/7. 21/20 can be stripped of its 2 and taken as 3 × 7/5. In this respect it resembles miracle, with a generator of 3 × 5/7, and casablanca, with a generator of 5 × 7/3. These three generators are the simplest in terms of the relationship of tetrads in the lattice of 7-limit tetrads. Valentine can also be described as the 31 & 46 temperament, and 77edo, 108edo, or 185edo make for excellent tunings, which also happen to be excellent tunings for starling, the rank-3 temperament tempering out 126/125. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)1/9 as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit, tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)1/10.

Valentine has a very straighforward S-expression-based comma list in the 11-limit add-23 (i.e. the 2.3.5.7.11.23 subgroup) of {(S8/S10 = S22 × S23 × S24, S11), S21, S22, S23, S24}, so it is the temperament that equalizes the 20::25 segment of the harmonic series.

Subgroup: 2.3.5.7

Comma list: 126/125, 1029/1024

Mapping: [⟨1 1 2 3], ⟨0 9 5 -3]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • WE: ~2 = 1200.0749 ¢, ~21/20 = 77.8687 ¢
error map: ⟨+0.075 -1.062 +3.179 -2.207]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 77.8673 ¢
error map: ⟨0.000 -1.149 +3.023 -2.428]

Minimax tuning:

[[1 0 0 0⟩, [5/2 3/4 0 -3/4⟩, [17/6 5/12 0 -5/12⟩, [5/2 -1/4 0 1/4⟩]
unchanged-interval (eigenmonzo) basis: 2.7/3
[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [47/21 10/21 0 -5/21⟩, [20/7 -2/7 0 1/7⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7

Algebraic generator: smaller root of x2 - 89x + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.

Optimal ET sequence: 15, 31, 46, 77, 185

Badness (Sintel): 0.786

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 126/125, 176/175

Mapping: [⟨1 1 2 3 3], ⟨0 9 5 -3 7]]

Optimal tunings:

  • WE: ~2 = 1200.3890 ¢, ~22/21 = 77.9065 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/21 = 77.9007 ¢

Minimax tuning:

  • 11-odd-limit: ~21/20 = [0 0 0 -1/10 1/10⟩
[[1 0 0 0 0⟩, [1 0 0 -9/10 9/10⟩, [2 0 0 -1/2 1/2⟩, [3 0 0 3/10 -3/10⟩, [3 0 0 -7/10 7/10⟩]
unchanged-interval (eigenmonzo) basis: 2.11/7

Algebraic generator: positive root of 4x3 + 15x2 - 21, or else Gontrand2, the smallest positive root of 4x7 - 8x6 + 5.

Optimal ET sequence: 15, 31, 46, 77

Badness (Sintel): 0.552

Valentino

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 176/175, 196/195

Mapping: [⟨1 1 2 3 3 5], ⟨0 9 5 -3 7 -20]]

Optimal tunings:

  • WE: ~2 = 1200.1967 ¢, ~22/21 = 77.9708 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/21 = 77.9594 ¢

Optimal ET sequence: 15f, 31, 46, 77

Badness (Sintel): 0.854

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 126/125, 154/153, 176/175, 196/195

Mapping: [⟨1 1 2 3 3 5 5], ⟨0 9 5 -3 7 -20 -14]]

Optimal tunings:

  • WE: ~2 = 1200.0404 ¢, ~22/21 = 78.0055 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/21 = 78.0029 ¢

Optimal ET sequence: 15f, 31, 46, 77, 123e

Badness (Sintel): 0.854

Lupercalia

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 121/120, 126/125

Mapping: [⟨1 1 2 3 3 3], ⟨0 9 5 -3 7 11]]

Optimal tunings:

  • WE: ~2 = 1199.9143 ¢, ~22/21 = 77.7039 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/21 = 77.7049 ¢

Optimal ET sequence: 15, 31

Badness (Sintel): 0.881

Dwynwen

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 126/125, 176/175

Mapping: [⟨1 1 2 3 3 2], ⟨0 9 5 -3 7 26]]

Optimal tunings:

  • WE: ~2 = 1200.1306 ¢, ~22/21 = 78.2273 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/21 = 78.2241 ¢

Optimal ET sequence: 15, 31f, 46

Badness (Sintel): 0.969

Semivalentine

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 169/168, 176/175

Mapping: [⟨2 2 4 6 6 7], ⟨0 9 5 -3 7 3]]

mapping generators: ~55/39, ~22/21

Optimal tunings:

  • WE: ~55/39 = 600.3497 ¢, ~22/21 = 77.8845 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~22/21 = 77.8715 ¢

Optimal ET sequence: 16, 30, 46, 62, 108ef

Badness (Sintel): 1.35

Hemivalentine

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 176/175, 343/338

Mapping: [⟨1 1 2 3 3 4], ⟨0 18 10 -6 14 -9]]

mapping generators: ~2, ~40/39

Optimal tunings:

  • WE: ~2 = 1199.6529 ¢, ~40/39 = 39.0323 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/39 = 39.0383 ¢

Optimal ET sequence: 30, 31, 61, 92f

Badness (Sintel): 1.94

Demivalentine

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 176/175, 676/675

Mapping: [⟨1 -8 -3 6 -4 -16], ⟨0 18 10 -6 14 37]]

mapping generators: ~2, ~13/9

Optimal tunings:

  • WE: ~2 = 1200.3929 ¢, ~13/9 = 639.1320 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/9 = 638.9325 ¢

Optimal ET sequence: 15, 47ef, 62, 77

Badness (Sintel): 1.44

Hemivalentino

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 1029/1024

Mapping: [⟨1 1 2 3 2], ⟨0 18 10 -6 45]]

Optimal tunings:

  • WE: ~2 = 1200.0816 ¢, ~45/44 = 38.9236 ¢
  • CWE: ~2 = 1200.0000 ¢, ~45/44 = 38.9228 ¢

Optimal ET sequence: 31, 92e, 123, 154, 185

Badness (Sintel): 2.03

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 243/242, 1029/1024

Mapping: [⟨1 1 2 3 2 5], ⟨0 18 10 -6 45 -40]]

Optimal tunings:

  • WE: ~2 = 1199.8782 ¢, ~45/44 = 38.9440 ¢
  • CWE: ~2 = 1200.0000 ¢, ~45/44 = 38.9472 ¢

Optimal ET sequence: 31, 123, 154

Badness (Sintel): 2.39

Hemivalentoid

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 243/242, 343/338

Mapping: [⟨1 1 2 3 2 4], ⟨0 18 10 -6 45 -9]]

Optimal tunings:

  • WE: ~2 = 1199.3614 ¢, ~45/44 = 38.9721 ¢
  • CWE: ~2 = 1200.0000 ¢, ~45/44 = 38.9839 ¢

Optimal ET sequence: 31, 92ef

Badness (Sintel): 2.39

Superkleismic

Main article: Superkleismic
For the 5-limit version, see Miscellaneous 5-limit temperaments #Shibboleth.

Superkleismic tempers out the keema, 875/864, and can be described as the 15 & 26 temperament. It splits the ~7/4 into three ~6/5 generators of around 322 cents. This is noticeably sharper than the kleismic generator, hence the name. Its ploidacot is wau-enneacot. In the 11-limit, two generator steps can be identified with ~16/11, and in the 13-limit, the same step can be treated as ~13/9. The S-expression-based comma list of 13-limit superkleismic is {S5/S6, S7/S8, S10, S12, (S21)}. Through careful observation of the equivalences therein one can derive the mapping of the full 13-limit.

Superkleismic also sets two intervals of 21/20 equal to 10/9; as 10/9 = (20/19)⋅(19/18), we can identify 21/20, 20/19, and 19/18 together to add prime 19, tempering out 361/360 (S19) and 400/399 (S20).

41edo gives an obvious tuning in all the subgroups.

Subgroup: 2.3.5.7

Comma list: 875/864, 1029/1024

Mapping: [⟨1 -5 -5 5], ⟨0 9 10 -3]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1200.7640 ¢, ~5/3 = 878.6289 ¢
error map: ⟨+0.764 +1.885 +3.844 -0.893]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.1077 ¢
error map: ⟨0.000 +1.014 -5.237 -3.149]

Optimal ET sequence: 11c, 15, 26, 41

Badness (Sintel): 1.21

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 385/384

Mapping: [⟨1 -5 -5 5 2], ⟨0 9 10 -3 2]]

Optimal tunings:

  • WE: ~2 = 1200.1691 ¢, ~5/3 = 878.2772 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.1606 ¢

Optimal ET sequence: 11c, 15, 26, 41, 179cde, 220cde, 261ccdee

Badness (Sintel): 0.848

2.3.5.7.11.19 subgroup

The following structure is preserved within the entire superkleismic tuning range between 15edo and 26edo, while extensions to 13 and 17 bifurcate and are of higher badness. These primes are of lower complexity than 13 and 17 are in the below extensions.

Subgroup: 2.3.5.7.11.19

Comma list: 100/99, 133/132, 190/189, 385/384

Mapping: [⟨1 -5 -5 5 2 -6], ⟨0 9 10 -3 2 14]]

Optimal tunings:

  • WE: ~2 = 1200.2289 ¢, ~5/3 = 878.3409 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.1840 ¢

Optimal ET sequence: 11c, 15, 26, 41, 138e

Badness (Sintel): 0.692

13-limit

Superkleismic in the 13-limit does considerably more damage than in the 11-limit, as indicated by being supported by much fewer patent vals and having higher Dirichlet badness than its 11-limit counterpart. However, this remains an obvious canonical mapping for prime 13.

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 245/242

Mapping: [⟨1 -5 -5 5 2 -8], ⟨0 9 10 -3 2 16]]

Optimal tunings:

  • WE: ~2 = 1200.0261 ¢, ~5/3 = 878.0252 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.0073 ¢

Optimal ET sequence: 11cf, 15, 26, 41

Badness (Sintel): 0.887

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 120/119, 144/143, 245/242

Mapping: [⟨1 -5 -5 5 2 -8 -12], ⟨0 9 10 -3 2 16 22]]

Optimal tunings:

  • WE: ~2 = 1200.0488 ¢, ~5/3 = 877.8872 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 877.8537 ¢

Optimal ET sequence: 11cfg, 15g, 26, 41

Badness (Sintel): 1.01

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 144/143, 133/132, 190/189

Mapping: [⟨1 -5 -5 5 2 -8 -12 -6], ⟨0 9 10 -3 2 16 22 14]]

Optimal tunings:

  • WE: ~2 = 1200.2120 ¢, ~5/3 = 878.0243 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 877.8789 ¢

Optimal ET sequence: 11cfgh, 15g, 26, 41

Badness (Sintel): 0.964

Superana

This extension (41 & 56) is the counterpart of canonical superkleismic on the other side of 41edo.

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 385/384, 441/440

Mapping: [⟨1 -5 -5 5 2 22], ⟨0 9 10 -3 2 -25]]

Optimal tunings:

  • WE: ~2 = 1199.8272 ¢, ~5/3 = 878.1538 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.2795 ¢

Optimal ET sequence: 15f, 41, 97, 138e

Badness (Sintel): 1.40

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 154/153, 196/195, 256/255, 273/272

Mapping: [⟨1 -5 -5 5 2 22 18], ⟨0 9 10 -3 2 -25 -19]]

Optimal tunings:

  • WE: ~2 = 1199.5964 ¢, ~5/3 = 878.0482 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.3444 ¢

Optimal ET sequence: 15f, 41, 56, 97g

Badness (Sintel): 1.45

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 133/132, 154/153, 190/189, 196/195, 256/255

Mapping: [⟨1 -5 -5 5 2 22 18 -6], ⟨0 9 10 -3 2 -25 -19 14]]

Optimal tunings:

  • WE: ~2 = 1199.6638 ¢, ~5/3 = 878.1109 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.3566 ¢

Optimal ET sequence: 15f, 41, 56, 97g

Badness (Sintel): 1.36

Dee leap week

Main article: Dee leap week

Subgroup: 2.3.5.7

Comma list: 1029/1024, 2460375/2458624

Mapping: [⟨1 -5 25 5], ⟨0 9 -31 -3]]

Optimal tunings:

  • WE: ~2 = 1200.4835 ¢, ~224/135 = 878.2507 ¢
error map: ⟨+0.484 -0.117 +0.004 -1.160]
  • CWE: ~2 = 1200.0000 ¢, ~224/135 = 877.8926 ¢
error map: ⟨0.000 -0.921 -0.985 -2.504]

Optimal ET sequence: 41, 108, 149, 190

Badness (Sintel): 2.12

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 2460375/2458624

Mapping: [⟨1 -5 25 5 -28], ⟨0 9 -31 -3 43]]

Optimal tunings:

  • WE: ~2 = 1200.4874 ¢, ~224/135 = 878.2543 ¢
  • CWE: ~2 = 1200.0000 ¢, ~224/135 = 877.8987 ¢

Optimal ET sequence: 41, 108e, 149, 190

Badness (Sintel): 1.35

Unidec

Main article: Unidec

5-limit (unidecmic)

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

7-limit

Subgroup: 2.3.5.7

Comma list: 1029/1024, 4375/4374

Mapping: [⟨2 -1 -3 7], ⟨0 6 11 -2]]

Optimal tunings:

  • CTE: ~1225/864 = 600.2429 ¢, ~80/63 = 417.0073 ¢
error map: ⟨+0.486 -0.154 +0.038 -1.140]
  • POTE: ~1225/864 = 600.0000 ¢, ~80/63 = 416.8688 ¢
error map: ⟨0.000 -0.924 -1.090 -2.503]

Minimax tuning:

[[1 0 0 0⟩, [47/26 0 6/13 -6/13⟩, [71/26 0 11/13 -11/13⟩, [71/26 0 -2/13 2/13⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5
[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [57/28 11/7 0 -11/14⟩, [20/7 -2/7 0 1/7⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 26, 46, 72, 118, 190

Badness (Sintel): 0.972

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 4375/4374

Mapping: [⟨2 -1 -3 7 9], ⟨0 6 11 -2 -3]]

Optimal tunings:

  • WE: ~99/70 = 600.2497 ¢, ~14/11 = 417.0085 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~14/11 = 416.8543 ¢

Minimax tuning:

[[1 0 0 0 0⟩, [10/7 6/7 0 -3/7 0⟩, [57/28 11/7 0 -11/14 0⟩, [20/7 -2/7 0 1/7 0⟩, [99/28 -3/7 0 3/14 0⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 26, 46, 72, 118, 190

Badness (Sintel): 0.512

Ekadash

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 441/440, 625/624, 729/728

Mapping: [⟨2 -1 -3 7 9 -19], ⟨0 6 11 -2 -3 38]]

Optimal tunings:

  • WE: ~99/70 = 600.2497 ¢, ~14/11 = 417.0085 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~14/11 = 416.8543 ¢

Optimal ET sequence: 46f, 72, 118, 190, 262df, 452cdef

Badness (Sintel): 0.842

Hendec

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 385/384

Mapping: [⟨2 -1 -3 7 9 6], ⟨0 6 11 -2 -3 2]]

Optimal tunings:

  • WE: ~91/64 = 600.3825 ¢, ~14/11 = 417.0678 ¢
  • CWE: ~91/64 = 600.0000 ¢, ~14/11 = 416.8290 ¢

Optimal ET sequence: 26, 46, 72, 190ff

Badness (Sintel): 0.732

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 273/272, 325/324, 364/363

Mapping: [⟨2 -1 -3 7 9 6 4], ⟨0 6 11 -2 -3 2 6]]

Optimal tunings:

  • WE: ~17/12 = 600.3991 ¢, ~14/11 = 417.0809 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~14/11 = 416.8330 ¢

Optimal ET sequence: 26, 46, 72, 190ffg

Badness (Sintel): 0.595

Lagaca

Subgroup: 2.3.5.7

Comma list: 1029/1024, 11529602/11390625

Mapping: [⟨2 -4 15 8], ⟨0 9 -13 -3]]

mapping generators: ~3375/2401, ~450/343

Optimal tunings:

  • WE: ~3375/2401 = 600.1355 ¢, ~450/343 = 478.0813 ¢
error map: ⟨+0.271 +0.235 +0.662 -1.986]
  • CWE: ~3375/2401 = 600.000 ¢, ~450/343 = 477.9725 ¢
error map: ⟨0.000 -0.202 +0.043 -2.743]

Optimal ET sequence: 10, 98, 108, 118

Badness (Sintel): 3.65

Necromanteion

Subgroup: 2.3.5.7

Comma list: 1029/1024, 5103/5000

Mapping: [⟨1 7 10 1], ⟨0 -12 -17 4]]

mapping generators: ~2, ~48/35

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~48/35 = 541.743 ¢
error map: ⟨0.000 -2.872 +4.053 -1.853]
  • POTE: ~2 = 1200.000 ¢, ~48/35 = 541.779 ¢
error map: ⟨0.000 -3.304 +3.442 -1.710]

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

Badness (Smith): 0.117680

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 243/242, 1029/1024

Mapping: [⟨1 7 10 1 17], ⟨0 -12 -17 4 -30]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~15/11 = 541.695 ¢
  • POTE: ~2 = 1200.000 ¢, ~15/11 = 541.729 ¢

Optimal ET sequence: 20ce, 31, 113c, 144c

Badness (Smith): 0.053459

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 243/242, 343/338

Mapping: [⟨1 7 10 1 17 1], ⟨0 -12 -17 4 -30 6]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~15/11 = 541.673 ¢
  • POTE: ~2 = 1200.000 ¢, ~15/11 = 541.606 ¢

Optimal ET sequence: 20ce, 31, 82cf, 113cf

Badness (Smith): 0.047015

Restles

See also: Lesser tendoneutralic

Subgroup: 2.3.5.7

Comma list: 1029/1024, 153664/151875

Mapping: [⟨1 -2 8 4], ⟨0 12 -19 -4]]

mapping generators: ~2. ~315/256

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~315/256 = 358.548 ¢
error map: ⟨0.000 +0.620 +1.275 -3.018]
  • POTE: ~2 = 1200.000 ¢, ~315/256 = 358.548 ¢
error map: ⟨0.000 +0.627 +1.265 -3.020]

Optimal ET sequence: 77, 87, 164

Badness (Smith): 0.108011

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 153664/151875

Mapping: [⟨1 -2 8 4 -7], ⟨0 12 -19 -4 35]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~27/22 = 358.575 ¢
  • POTE: ~2 = 1200.000 ¢, ~27/22 = 358.571 ¢

Optimal ET sequence: 77, 87, 164, 251d

Badness (Smith): 0.054655

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384, 676/675

Mapping: [⟨1 -2 8 4 -7 4], ⟨0 12 -19 -4 35 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~16/13 = 358.576 ¢
  • POTE: ~2 = 1200.000 ¢, ~16/13 = 358.574 ¢

Optimal ET sequence: 77, 87, 164, 251d

Badness (Smith): 0.028187

Quartemka

For the 5-limit version, see Miscellaneous 5-limit temperaments #Quartemka.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1250000/1240029

Mapping: [⟨1 4 6 2], ⟨0 -21 -32 7]]

mapping generators: ~2, ~27/25

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~27/25 = 137.971 ¢
error map: ⟨0.000 +0.658 -1.380 -3.030]
  • POTE: ~2 = 1200.000 ¢, ~27/25 = 138.006 ¢
error map: ⟨0.000 -0.075 -2.496 -2.786]

Optimal ET sequence: 26, 61, 87, 113, 200

Badness (Smith): 0.152287

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 800000/793881

Mapping: [⟨1 4 6 2 3], ⟨0 -21 -32 7 4]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~27/25 = 137.970 ¢
  • POTE: ~2 = 1200.000 ¢, ~27/25 = 137.990 ¢

Optimal ET sequence: 26, 61, 87, 200, 287d

Badness (Smith): 0.057307

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384, 2200/2197

Mapping: [⟨1 4 6 2 3 6], ⟨0 -21 -32 7 4 -20]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~13/12 = 137.971 ¢
  • POTE: ~2 = 1200.000 ¢, ~13/12 = 137.990 ¢

Optimal ET sequence: 26, 61, 87, 200

Badness (Smith): 0.028393

Tritriple

For the 5-limit version, see Miscellaneous 5-limit temperaments #Tritriple.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1959552/1953125

Mapping: [⟨1 -11 -7 7], ⟨0 27 20 -9]]

mapping generators: ~2, ~864/625

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~864/625 = 559.320 ¢
error map: ⟨0.000 -0.317 +0.085 -2.705]
  • POTE: ~2 = 1200.000 ¢, ~864/625 = 559.295 ¢
error map: ⟨0.000 -1.003 -0.423 -2.477]

Optimal ET sequence: 15, …, 88, 103, 118, 221, 339d

Badness (Smith): 0.118640

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 43923/43750

Mapping: [⟨1 -11 -7 7 -4], ⟨0 27 20 -9 16]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~242/175 = 559.327 ¢
  • POTE: ~2 = 1200.000 ¢, ~242/175 = 559.293 ¢

Optimal ET sequence: 15, …, 88, 103, 118, 221e, 339de

Badness (Smith): 0.035350

Widefourth

Subgroup: 2.3.5.7

Comma list: 1029/1024, 48828125/48771072

Mapping: [⟨1 16 8 -2], ⟨0 -33 -13 11]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3125/2304 = 524.188 ¢
error map: ⟨0.000 -0.154 -0.756 -2.759]
  • POTE: ~2 = 1200.000 ¢, ~3125/2304 = 524.210 ¢
error map: ⟨0.000 -0.892 -1.047 -2.513]

Optimal ET sequence: 16, 71, 87, 103, 190

Badness (Smith): 0.154117

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 234375/234256

Mapping: [⟨1 16 8 -2 17], ⟨0 -33 -13 11 -31]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~847/625 = 524.183 ¢
  • POTE: ~2 = 1200.000 ¢, ~847/625 = 524.210 ¢

Optimal ET sequence: 16, 71, 87, 103, 190

Badness (Smith): 0.040785

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 441/440, 625/624, 847/845

Mapping: [⟨1 16 8 -2 17 12], ⟨0 -33 -13 11 -31 -19]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~65/48 = 524.183 ¢
  • POTE: ~2 = 1200.000 ¢, ~65/48 = 524.209 ¢

Optimal ET sequence: 16, 71, 87, 103, 190

Badness (Smith): 0.021636

References

  1. ↑ 31-EDO Music Theory: Supermajor Hexatonic Scale by Zhea Erose
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