Hemifamity temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus (1/)
Switch to Sintel's badness, WE & CWE tunings, per community consensus (2/)
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: mapping generators: ~50/49, ~5
: mapping generators: ~50/49, ~5


[[Optimal tuning]] ([[POTE]]): ~50/49 = 41.379{{c}}, ~5/4 = 388.646{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~50/49 = 41.3652{{c}}, ~5/4 = 388.5128{{c}}
: [[error map]]: {{val| -0.410 +0.842 +1.378 -2.022 }}
* [[CWE]]: ~50/49 = 41.3793{{c}}, ~5/4 = 388.3030{{c}}
: error map: {{val| 0.000 +1.493 +1.989 -1.213 }}
<!-- * [[POTE]]: ~50/49 = 41.379{{c}}, ~5/4 = 388.646{{c}} -->


{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}
{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}


[[Badness]] (Smith): 0.103734
[[Badness]] (Sintel): 2.63


=== 11-limit ===
=== 11-limit ===
Line 320: Line 325:
Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}
Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}


Optimal tuning (POTE): ~45/44 = 41.379{{c}}, ~5/4 = 388.460{{c}}
Optimal tunings:
* WE: ~50/49 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}
* CWE: ~50/49 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}
<!-- * POTE: ~45/44 = 41.379{{c}}, ~5/4 = 388.460{{c}} -->


{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}
{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}


Badness (Smith): 0.034291
Badness (Sintel): 1.13


=== 13-limit ===
=== 13-limit ===
Line 333: Line 341:
Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}
Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}


Optimal tuning (POTE): ~45/44 = 41.379{{c}}, ~5/4 = 388.354{{c}}
Optimal tunings:
* WE: ~50/49 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
* CWE: ~50/49 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}
<!-- * POTE: ~45/44 = 41.379{{c}}, ~5/4 = 388.354{{c}} -->


{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232, 377cef }}
{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}


Badness (Smith): 0.018591
Badness (Sintel): 0.768


== Quanic ==
== Quanic ==
Line 347: Line 358:
: mapping generators: ~2, ~160/147
: mapping generators: ~2, ~160/147


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~160/147 = 140.493{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6159{{c}}, ~160/147 = 140.4483{{c}}
: [[error map]]: {{val| -0.384 -0.098 -0.570 +1.933 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 140.4862{{c}}
: error map: {{val| 0.000 +0.476 -0.061 +2.842 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~160/147 = 140.493{{c}} -->


{{Optimal ET sequence|legend=1| 94, 111, 205 }}
{{Optimal ET sequence|legend=1| 94, 111, 205 }}


[[Badness]] (Smith): 0.179475
[[Badness]] (Sintel): 4.54


=== 11-limit ===
=== 11-limit ===
Line 360: Line 376:
Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}
Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~88/81 = 140.489{{c}}
Optimal tunings:
* WE: ~2 = 1199.7834{{c}}, ~88/81 = 140.4635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.4850{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~88/81 = 140.489{{c}} -->


{{Optimal ET sequence|legend=0| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness (Smith): 0.058678
Badness (Sintel): 1.94


=== 13-limit ===
=== 13-limit ===
Line 373: Line 392:
Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}
Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 140.496{{c}}
Optimal tunings:
* WE: ~2 = 1199.6639{{c}}, ~13/12 = 140.4562{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4904{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~13/12 = 140.496{{c}} -->


{{Optimal ET sequence|legend=0| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness (Smith): 0.032481
Badness (Sintel): 1.34


=== 17-limit ===
=== 17-limit ===
Line 386: Line 408:
Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}
Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 140.497{{c}}
Optimal tunings:
* WE: ~2 = 1199.6699{{c}}, ~13/12 = 140.4586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4920{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~13/12 = 140.497{{c}} -->


{{Optimal ET sequence|legend=0| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness (Smith): 0.021112
Badness (Sintel): 1.08


=== 19-limit ===
=== 19-limit ===
Line 399: Line 424:
Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}
Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/12 = 140.496{{c}}
Optimal tunings:
* WE: ~2 = 1199.6745{{c}}, ~13/12 = 140.4574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4908{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~13/12 = 140.496{{c}} -->


{{Optimal ET sequence|legend=0| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness (Smith): 0.017273
Badness (Sintel): 1.05


== Septiquarter ==
== Septiquarter ==
Line 410: Line 438:
[[Comma list]]: 5120/5103, 420175/419904
[[Comma list]]: 5120/5103, 420175/419904


{{Mapping|legend=1| 1 3 10 2 | 0 -7 -38 4 }}
{{Mapping|legend=1| 1 -4 -28 6 | 0 7 38 -4 }}
: mapping generators: ~2, ~147/128
: mapping generators: ~2, ~243/140


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~147/128 = 242.453{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7212{{c}}, ~243/140 = 957.3250{{c}}
: [[error map]]: {{val| -0.279 +0.435 -0.158 +0.201 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/140 = 957.5424{{c}}
: error map: {{val| 0.000 +0.842 +0.298 +1.004 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~243/140 = 957.547{{c}} -->


{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}


[[Badness]] (Smith): 0.053760
[[Badness]] (Sintel): 1.36


=== Semiseptiquarter ===
=== Semiseptiquarter ===
Line 424: Line 457:
Comma list: 5120/5103, 9801/9800, 14641/14580
Comma list: 5120/5103, 9801/9800, 14641/14580


Mapping: {{mapping| 2 6 20 4 15 | 0 -7 -38 4 -20 }}
Mapping: {{mapping| 2 -8 -56 12 -25 | 0 7 38 -4 20 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~121/105 = 242.4511{{c}}
Optimal tunings:
* WE: ~99/70 = 599.8953{{c}}, ~210/121 = 957.3819{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5449{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~210/121 = 957.5489{{c}} -->


{{Optimal ET sequence|legend=0| 94, 198, 292, 490 }}
{{Optimal ET sequence|legend=0| 94, 198, 292, 490 }}


Badness (Smith): 0.064160
Badness (Sintel): 2.12


==== 13-limit ====
==== 13-limit ====
Line 437: Line 473:
Comma list: 352/351, 847/845, 1716/1715, 14641/14580
Comma list: 352/351, 847/845, 1716/1715, 14641/14580


Mapping: {{mapping| 2 6 20 4 15 7 | 0 -7 -38 4 -20 1 }}
Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~121/105 = 242.4448{{c}}
Optimal tunings:
* WE: ~99/70 = 599.8565{{c}}, ~210/121 = 957.3261{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5508{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~210/121 = 957.5552{{c}} -->


{{Optimal ET sequence|legend=0| 94, 198, 490f }}
{{Optimal ET sequence|legend=0| 94, 198, 490f }}


Badness (Smith): 0.034834
Badness (Sintel): 1.44


== Countriton ==
== Countriton ==

Revision as of 17:45, 25 October 2025

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

This is a collection of rank-2 temperaments tempering out the hemifamity comma (monzo: [10 -6 1 -1⟩, ratio: 5120/5103). These temperaments divide an exact or approximate septimal quartertone, 36/35 into two equal steps, each representing 81/80~64/63, the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same chain of fifths inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify 10/7 by the augmented fourth and 50/49 by the Pythagorean comma.

Temperaments belonging to this category and generated by the fifth are dominant, garibaldi, kwai, undecental, and leapday. Dominant has 5/4 mapped to M3. Garibaldi has 5/4 mapped to d4. Kwai has 5/4 mapped to 4A7. Undecental has 5/4 mapped to 5d7. Leapday has 5/4 mapped to 3A1.

Diaschismic is generated by the fifth with a semi-octave period. Hemififths has the fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma. Hemidromeda has the fourth sliced into two and 5/4 mapped to the hemifourth + 3d4. Rodan has the fifth sliced into three as does slendric. Alphatrimot has the twelfth sliced into three as does alphatricot. Monkey has the fifth sliced into four as does tetracot. Buzzard has the twelfth sliced into four as does vulture. Misty is generated by the fifth with a 1/3-octave period. Supers has the fifth sliced into three with a semi-octave period. Undim is generated by the fifth with a 1/4-octave period. Quinticosiennic and quintakwai have the fourth sliced into five. Amity has the eleventh sliced into five. Countercata has the twelfth sliced into six as does hanson. Warrior has the 6th harmonic sliced into seven as does sensi. Finally, alphaquarter has the fourth sliced into nine as does escapade.

Temperaments considered below are undecental, leapday, hemidromeda, mystery, quanic, septiquarter, countriton, artoneutral and ketchup. Discussed elsewhere are:

Undecental

Undecental adds the triwellisma to the comma list and may be described as the 29 & 70 temperament. 5/4 is mapped to the quintuple-diminished seventh or equivalently the perfect fourth minus three dieses. 58\99 is an almost perfect generator, just as the name suggests. Another interesting tuning choice is the argent fifth, 2(2 - sqrt (2)).

Subgroup: 2.3.5.7

Comma list: 5120/5103, 235298/234375

Mapping: [⟨1 0 61 71], ⟨0 1 -37 -43]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.6543 ¢, ~3/2 = 702.8370 ¢
error map: ⟨-0.346 +0.536 +0.423 -0.494]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.0465 ¢
error map: ⟨0.000 +1.092 +0.966 +0.175]

Optimal ET sequence: 29, 70, 99, 722bc, 821bc, 920bc, 1019bc

Badness (Sintel): 2.39

Leapday

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

Leapday tempers out the leapday comma, [31 -21 1⟩, in the 5-limit, mapping 5/4 to the triple-augmented unison or equivalently the minor third and two dieses. In the 7-limit it can be described as the 29 & 46 temperament, which tempers out the hemifamity and 686/675 (senga), and extends leapfrog.

It has an alternative extension called polypyth, which tempers out the same 5-limit comma as leapday, but with the porwell (6144/6125) rather than the hemifamity comma tempered out.

Subgroup: 2.3.5.7

Comma list: 686/675, 5120/5103

Mapping: [⟨1 0 -31 -21], ⟨0 1 21 15]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.7167 ¢, ~3/2 = 704.0971 ¢
error map: ⟨-0.283 +1.859 +2.559 -5.669]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2504 ¢
error map: ⟨0.000 +2.295 +2.945 -5.070]

Optimal ET sequence: 17c, 29, 46

Badness (Sintel): 2.43

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 686/675

Mapping: [⟨1 0 -31 -21 -14], ⟨0 1 21 15 11]]

Optimal tunings:

  • WE: ~2 = 1200.0731 ¢, ~3/2 = 704.2933 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2538 ¢

Optimal ET sequence: 17c, 29, 46

Badness (Sintel): 1.28

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 169/168, 352/351

Mapping: [⟨1 0 -31 -21 -14 -9], ⟨0 1 21 15 11 8]]

Optimal tunings:

  • WE: ~2 = 1200.4758 ¢, ~3/2 = 704.4930 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2346 ¢

Optimal ET sequence: 17c, 29, 46, 121def

Badness (Sintel): 1.02

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 121/120, 136/135, 154/153, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34], ⟨0 1 21 15 11 8 24]]

Optimal tunings:

  • WE: ~2 = 1200.4818 ¢, ~3/2 = 704.5121 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.2507 ¢

Optimal ET sequence: 17cg, 29g, 46, 121defg

Badness (Sintel): 0.910

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34 9], ⟨0 1 21 15 11 8 24 -3]]

Optimal tunings:

  • WE: ~2 = 1201.0192 ¢, ~3/2 = 704.7333 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1680 ¢

Optimal ET sequence: 17cg, 29g, 46, 75dfgh, 121defgh

Badness (Sintel): 1.06

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 161/160, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34 9 -5], ⟨0 1 21 15 11 8 24 -3 6]]

Optimal tunings:

  • WE: ~2 = 1200.9738 ¢, ~3/2 = 704.7120 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1695 ¢

Optimal ET sequence: 17cg, 29g, 46, 75dfgh, 121defgh

Badness (Sintel): 1.01

Leapling

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34 -37], ⟨0 1 21 15 11 8 24 26]]

Optimal tunings:

  • WE: ~2 = 1200.4745 ¢, ~3/2 = 704.4016 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1442 ¢

Optimal ET sequence: 17cgh, 29g, 46h, 75dfg

Badness (Sintel): 1.16

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 77/76, 91/90, 115/114, 121/120, 136/135, 153/152, 161/160

Mapping: [⟨1 0 -31 -21 -14 -9 -34 -37 -5], ⟨0 1 21 15 11 8 24 26 6]]

Optimal tunings:

  • WE: ~2 = 1200.5425 ¢, ~3/2 = 704.4319 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1349 ¢

Optimal ET sequence: 17cgh, 29g, 46h, 75dfg

Badness (Sintel): 1.15

Hemidromeda

Hemidromeda may be described as the 29 & 111 temperament. The name hemidromeda comes from "hemi-" (Ancient Greek for "one half") and andromeda, because the generator is 1/2 of andromeda's perfect twelfth (~3/1, about 1902.4 cents).

Subgroup: 2.3.5.7

Comma list: 5120/5103, 52734375/52706752

Mapping: [⟨1 0 38 48], ⟨0 2 -45 -57]]

mapping generator: ~2, ~12500/7203

Optimal tunings:

  • WE: ~2 = 1199.7236 ¢, ~12500/7203 = 951.1864 ¢
error map: ⟨-0.276 +0.418 -0.205 +0.282]
  • CWE: ~2 = 1200.0000 ¢, ~12500/7203 = 951.4098 ¢
error map: ⟨0.000 +0.865 +0.243 +0.813]

Optimal ET sequence: 29, 82cd, 111, 140, 251, 391, 1424bbcdd

Badness (Sintel): 2.93

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1375/1372, 5120/5103

Mapping: [⟨1 0 38 48 32], ⟨0 2 -45 -57 -36]]

Optimal tunings:

  • WE: ~2 = 1199.8767 ¢, ~400/231 = 951.3065 ¢
  • CWE: ~2 = 1200.0000 ¢, ~400/231 = 951.4063 ¢

Optimal ET sequence: 29, 82cd, 111, 140, 251, 391e

Badness (Sintel): 2.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845, 1331/1323

Mapping: [⟨1 0 38 48 32 37], ⟨0 2 -45 -57 -36 -42]]

Optimal tunings:

  • WE: ~2 = 1199.8753 ¢, ~26/15 = 951.3054 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4064 ¢

Optimal ET sequence: 29, 82cdf, 111, 140, 251, 391e

Badness (Sintel): 1.18

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 442/441, 561/560, 676/675, 715/714

Mapping: [⟨1 0 38 48 32 37 58], ⟨0 2 -45 -57 -36 -42 -68]]

Optimal tunings:

  • WE: ~2 = 1199.8770 ¢, ~26/15 = 951.3039 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4035 ¢

Optimal ET sequence: 29g, 82cdfg, 111, 140, 251, 391e

Badness (Sintel): 0.971

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560

Mapping: [⟨1 0 38 48 32 37 58 32], ⟨0 2 -45 -57 -36 -42 -68 -35]]

Optimal tunings:

  • WE: ~2 = 1199.7534 ¢, ~26/15 = 951.2024 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4020 ¢

Optimal ET sequence: 29g, 82cdfgh, 111, 140

Badness (Sintel): 1.01

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459

Mapping: [⟨1 0 38 48 32 37 58 32 18], ⟨0 2 -45 -57 -36 -42 -68 -35 -17]]

Optimal tunings:

  • WE: ~2 = 1199.9128 ¢, ~26/15 = 951.3371 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4076 ¢

Optimal ET sequence: 29g, 82cdfgh, 111, 140

Badness (Sintel): 1.10

Mystery

Main article: Mystery
For the 5-limit version, see 29th-octave temperaments #Mystery.

Mystery tempers out 50421/50000 and may be described as the 29 & 58 temperament. It has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step. 145edo or 232edo are good candidates for tunings.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 50421/50000

Mapping: [⟨29 46 0 14], ⟨0 0 1 1]]

mapping generators: ~50/49, ~5

Optimal tunings:

  • WE: ~50/49 = 41.3652 ¢, ~5/4 = 388.5128 ¢
error map: ⟨-0.410 +0.842 +1.378 -2.022]
  • CWE: ~50/49 = 41.3793 ¢, ~5/4 = 388.3030 ¢
error map: ⟨0.000 +1.493 +1.989 -1.213]

Optimal ET sequence: 29, 58, 87, 145

Badness (Sintel): 2.63

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891, 3388/3375

Mapping: [⟨29 46 0 14 33], ⟨0 0 1 1 1]]

Optimal tunings:

  • WE: ~50/49 = 41.3637 ¢, ~5/4 = 388.3136 ¢
  • CWE: ~50/49 = 41.3793 ¢, ~5/4 = 388.0598 ¢

Optimal ET sequence: 29, 58, 87, 145

Badness (Sintel): 1.13

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 676/675

Mapping: [⟨29 46 0 14 33 40], ⟨0 0 1 1 1 1]]

Optimal tunings:

  • WE: ~50/49 = 41.3623 ¢, ~5/4 = 388.1942 ¢
  • CWE: ~50/49 = 41.3793 ¢, ~5/4 = 387.9017 ¢

Optimal ET sequence: 29, 58, 87, 145, 232

Badness (Sintel): 0.768

Quanic

Subgroup: 2.3.5.7

Comma list: 5120/5103, 5832000/5764801

Mapping: [⟨1 1 -4 0], ⟨0 5 54 24]]

mapping generators: ~2, ~160/147

Optimal tunings:

  • WE: ~2 = 1199.6159 ¢, ~160/147 = 140.4483 ¢
error map: ⟨-0.384 -0.098 -0.570 +1.933]
  • CWE: ~2 = 1200.0000 ¢, ~160/147 = 140.4862 ¢
error map: ⟨0.000 +0.476 -0.061 +2.842]

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 4.54

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1331/1323, 5120/5103

Mapping: [⟨1 1 -4 0 1], ⟨0 5 54 24 21]]

Optimal tunings:

  • WE: ~2 = 1199.7834 ¢, ~88/81 = 140.4635 ¢
  • CWE: ~2 = 1200.0000 ¢, ~88/81 = 140.4850 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.94

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1331/1323

Mapping: [⟨1 1 -4 0 1 3], ⟨0 5 54 24 21 6]]

Optimal tunings:

  • WE: ~2 = 1199.6639 ¢, ~13/12 = 140.4562 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.4904 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.34

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 442/441, 540/539, 715/714, 847/845

Mapping: [⟨1 1 -4 0 1 3 -2], ⟨0 5 54 24 21 6 52]]

Optimal tunings:

  • WE: ~2 = 1199.6699 ¢, ~13/12 = 140.4586 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.4920 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.08

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714

Mapping: [⟨1 1 -4 0 1 3 -2 -5], ⟨0 5 54 24 21 6 52 79]]

Optimal tunings:

  • WE: ~2 = 1199.6745 ¢, ~13/12 = 140.4574 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.4908 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.05

Septiquarter

Subgroup: 2.3.5.7

Comma list: 5120/5103, 420175/419904

Mapping: [⟨1 -4 -28 6], ⟨0 7 38 -4]]

mapping generators: ~2, ~243/140

Optimal tunings:

  • WE: ~2 = 1199.7212 ¢, ~243/140 = 957.3250 ¢
error map: ⟨-0.279 +0.435 -0.158 +0.201]
  • CWE: ~2 = 1200.0000 ¢, ~243/140 = 957.5424 ¢
error map: ⟨0.000 +0.842 +0.298 +1.004]

Optimal ET sequence: 94, 99, 292, 391, 881bd, 1272bcd

Badness (Sintel): 1.36

Semiseptiquarter

Subgroup: 2.3.5.7.11

Comma list: 5120/5103, 9801/9800, 14641/14580

Mapping: [⟨2 -8 -56 12 -25], ⟨0 7 38 -4 20]]

Optimal tunings:

  • WE: ~99/70 = 599.8953 ¢, ~210/121 = 957.3819 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~210/121 = 957.5449 ¢

Optimal ET sequence: 94, 198, 292, 490

Badness (Sintel): 2.12

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 1716/1715, 14641/14580

Mapping: [⟨2 -8 -56 12 -25 9], ⟨0 7 38 -4 20 -1]]

Optimal tunings:

  • WE: ~99/70 = 599.8565 ¢, ~210/121 = 957.3261 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~210/121 = 957.5508 ¢

Optimal ET sequence: 94, 198, 490f

Badness (Sintel): 1.44

Countriton

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

Subgroup: 2.3.5.7

Comma list: 5120/5103, 7558272/7503125

Mapping: [⟨1 6 19 -7], ⟨0 -9 -34 20]]

mapping generators: ~2, ~1728/1225

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~1728/1225 = 588.582 ¢

Optimal ET sequence: 53, 157, 210

Badness (Smith): 0.131191

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 5120/5103, 41503/41472

Mapping: [⟨1 6 19 -7 27], ⟨0 -9 -34 20 -48]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~108/77 = 588.545 ¢

Optimal ET sequence: 53, 104c, 157

Badness (Smith): 0.084782

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845, 2197/2187

Mapping: [⟨1 6 19 -7 27 14], ⟨0 -9 -34 20 -48 -21]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~108/77 = 588.544 ¢

Optimal ET sequence: 53, 104c, 157

Badness (Smith): 0.042321

Artoneutral

Artoneutral is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11) and can be described as the 87 & 94 temperament. 181edo may be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 3828125/3779136

Mapping: [⟨1 8 18 -20], ⟨0 -9 -22 32]]

mapping generators: ~2, ~105/64

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~105/64 = 855.2452 ¢

Optimal ET sequence: 87, 94, 181

Badness (Smith): 0.157120

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 4000/3993

Mapping: [⟨1 8 18 -20 17], ⟨0 -9 -22 32 -19]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~18/11 = 855.2397 ¢

Optimal ET sequence: 87, 181

Badness (Smith): 0.045920

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 1575/1573

Mapping: [⟨1 8 18 -20 17 -2], ⟨0 -9 -22 32 -19 8]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~18/11 = 855.2369 ¢

Optimal ET sequence: 87, 181

Badness (Smith): 0.026257

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 352/351, 375/374, 385/384, 595/594

Mapping: [⟨1 8 18 -20 17 -2 44], ⟨0 -9 -22 32 -19 8 -56]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~18/11 = 855.2495 ¢

Optimal ET sequence: 87, 94, 181

Badness (Smith): 0.022749

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594

Mapping: [⟨1 8 18 -20 17 -2 44 52], ⟨0 -9 -22 32 -19 8 -56 -67]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~18/11 = 855.2534 ¢

Optimal ET sequence: 87, 94, 181

Badness (Smith): 0.019585

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483

Mapping: [⟨1 8 18 -20 17 -2 44 52 48], ⟨0 -9 -22 32 -19 8 -56 -67 -61]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~18/11 = 855.2576 ¢

Optimal ET sequence: 87, 94, 181

Badness (Smith): 0.016332

Ketchup

Subgroup: 2.3.5.7

Comma list: 5120/5103, 1071875/1062882

Mapping: [⟨2 3 4 6], ⟨0 4 15 -9]]

Optimal tuning (POTE): ~1225/864 = 600.000 ¢, ~64/63 = 25.719 ¢

mapping generators: ~1225/864, ~64/63

Optimal ET sequence: 46, 94, 140

Badness (Smith): 0.084538

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1331/1323, 2200/2187

Mapping: [⟨2 3 4 6 7], ⟨0 4 15 -9 -2]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~64/63 = 25.693 ¢

Optimal ET sequence: 46, 94, 140

Badness (Smith): 0.039555

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 1331/1323

Mapping: [⟨2 3 4 6 7 8], ⟨0 4 15 -9 -2 -14]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~66/65 = 25.697 ¢

Optimal ET sequence: 46, 94, 140

Badness (Smith): 0.024824

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 352/351, 385/384, 561/560

Mapping: [⟨2 3 4 6 7 8 8], ⟨0 4 15 -9 -2 -14 4]]

Optimal tuning (POTE): ~17/12 = 600.000 ¢, ~66/65 = 25.701 ¢

Optimal ET sequence: 46, 94, 140

Badness (Smith): 0.016591

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560

Mapping: [⟨2 3 4 6 7 8 8 9], ⟨0 4 15 -9 -2 -14 4 -12]]

Optimal tuning (POTE): ~17/12 = 600.000 ¢, ~66/65 = 25.660 ¢

Optimal ET sequence: 46, 94, 140h, 234eh

Badness (Smith): 0.018170

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 190/189, 209/208, 253/252, 289/288, 323/322, 352/351, 385/384

Mapping: [⟨2 3 4 6 7 8 8 9 9], ⟨0 4 15 -9 -2 -14 4 -12 1]]

Optimal tuning (POTE): ~17/12 = 600.000 ¢, ~66/65 = 25.661 ¢

Optimal ET sequence: 46, 94, 140h, 234ehi

Badness (Smith): 0.014033

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