Diaschismic family

From Xenharmonic Wiki
Jump to navigation Jump to search
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 diaschismic family of temperaments tempers out the diaschisma, 2048/2025.

Diaschismic

Main article: Diaschismic

The period of diaschismic is half an octave, and the generator is a fifth; the ploidacot is diploid monocot. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. 34edo is a good tuning choice, with 46edo, 56edo, 58edo, or 80edo being other possibilities. Both 12edo and 22edo support it, and retuning them to a mos of diaschismic gives two scale possibilities.

This temperament is also known as srutal in the 5-limit, but that name more strictly speaking refers to the 34d & 46 extension to the 7-limit that adds 4375/4374 to the comma list.

Subgroup: 2.3.5

Comma list: 2048/2025

Mapping: [⟨2 0 11], ⟨0 1 -2]]

mapping generators: ~45/32, ~3

Optimal tunings:

  • WE: ~45/32 = 599.4107 ¢, ~3/2 = 704.2059 ¢
error map: ⟨-1.179 +1.072 +1.150]
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.9585 ¢
error map: ⟨0.000 +3.003 +3.769]

Tuning ranges:

Optimal ET sequence: 10, 12, 22, 34, 46, 80, 206c, 286bc

Badness (Sintel): 0.467

Overview to extensions

7-limit extensions

To get the 7-limit extensions, we add another comma:

  • Septimal diaschismic adds 126/125, the starling comma, to obtain 7-limit harmony by more complex methods than pajara, but with greater accuracy.
  • Pajara derives from 64/63 and is a popular and well-known choice.
  • Srutal adds 4375/4374, the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.
  • Keen adds 875/864.

Those all keep the same half-octave period and fifth generator.

  • Bidia adds 3136/3125, the hemimean comma.
  • Echidna adds 1728/1715, the orwellisma.
  • Shrutar adds 245/243, the sensamagic comma.

Shrutar has a generator of a quartertone (which can be taken as 36/35, the septimal quartertone) and echidna has a generator of 9/7. Bidia has a quarter-octave period and a fifth generator.

Subgroup extensions

Since the diaschisma factors into (256/255)2(289/288) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as srutal archagall, documented right below. The S-expression-based comma list of this temperament is {S16, S17}.

Srutal archagall

See also: Fiventeen

Subgroup: 2.3.5.17

Comma list: 136/135, 256/255

Subgroup-val mapping: [⟨2 0 11 5], ⟨0 1 -2 1]]

mapping generators: ~17/12, ~3

Optimal tunings:

  • WE: ~45/32 = 599.5585 ¢, ~3/2 = 704.6188 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 705.1356 ¢

Optimal ET sequence: 10, 12, 22, 34, 80, 114, 194bc

Badness (Sintel): 0.212

Septimal diaschismic

Main article: Diaschismic
See also: Srutal vs diaschismic

A simpler characterization than the one given by the normal comma list is that diaschismic adds 126/125 or 5120/5103 to the set of commas, and it can also be called 46 & 58. However described, diaschismic has a 1/2-octave period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. 58edo provides an excellent tuning, but an alternative is to make 7/4 just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58edo.

Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher-limit rank-2 temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; mos of 34 notes and even more the 46-note mos will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.

Subgroup: 2.3.5.7

Comma list: 126/125, 2048/2025

Mapping: [⟨2 0 11 31], ⟨0 1 -2 -8]]

Optimal tunings:

  • WE: ~45/32 = 599.4449 ¢, ~3/2 = 703.0299 ¢
error map: ⟨-1.110 -0.035 +3.740 -1.391]
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 703.7739 ¢
error map: ⟨0.000 +1.819 +6.138 +0.983]

Tuning ranges:

Optimal ET sequence: 12, 34, 46, 58, 104c, 162c

Badness (Sintel): 0.959

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 896/891

Mapping: [⟨2 0 11 31 45], ⟨0 1 -2 -8 -12]]

Optimal tunings:

  • WE: ~45/32 = 599.4471 ¢, ~3/2 = 703.0657 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 703.7996 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [700.000, 704.348] (7\12 to 27\46)
  • 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]

Optimal ET sequence: 12, 34e, 46, 58, 104c, 162ce

Badness (Sintel): 0.828

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 364/363

Mapping: [⟨2 0 11 31 45 55], ⟨0 1 -2 -8 -12 -15]]

Optimal tunings:

  • WE: ~45/32 = 599.4451 ¢, ~3/2 = 703.0528 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 703.7813 ¢

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
  • 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
  • 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]

Optimal ET sequence: 12f, 34ef, 46, 58, 104c, 162cef

Badness (Sintel): 0.782

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 136/135, 176/175, 196/195, 256/255

Mapping: [⟨2 0 11 31 45 55 5], ⟨0 1 -2 -8 -12 -15 1]]

Optimal tunings:

  • WE: ~17/12 = 599.6253 ¢, ~3/2 = 703.3726 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 703.8520 ¢

Tuning ranges:

  • 17-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
  • 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]

Optimal ET sequence: 12f, 34ef, 46, 58, 104c

Badness (Sintel): 0.837

2.3.5.7.11.13.17.23 subgroup (Na"Naa')

Na"Naa' is a remarkable subgroup temperament of 46 & 58 with a prime harmonic of 23. It is yet to be found why it got this strange name.

Subgroup: 2.3.5.7.11.13.17.23

Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255

Subgroup-val mapping: [⟨2 0 11 31 45 55 5 63], ⟨0 1 -2 -8 -12 -15 1 -17]]

Optimal tunings:

  • WE: ~17/12 = 599.6272 ¢, ~3/2 = 703.4326 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 703.9093 ¢

Optimal ET sequence: 12i, 34efi, 46, 58i, 104ci

Badness (Sintel): 0.882

Pajara

Main article: Pajara

Pajara is closely associated with 22edo (not to mention Paul Erlich) but other tunings are possible. The 1/2-octave period serves as both a 10/7 and a 7/5. Aside from 22edo, 34 with the val ⟨34 54 79 96] and 56 with the val ⟨56 89 130 158] are interesting alternatives, with more acceptable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.

Pajara extends nicely to an 11-limit version, for which the 56edo tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.

Subgroup: 2.3.5.7

Comma list: 50/49, 64/63

Mapping: [⟨2 0 11 12], ⟨0 1 -2 -2]]

Optimal tunings:

  • WE: ~7/5 = 598.8483 ¢, ~3/2 = 705.6906 ¢
error map: ⟨-2.303 +1.432 -5.756 +10.580]
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 707.3438 ¢
error map: ⟨0.000 +5.389 -1.001 +16.487]

Tuning ranges:

Optimal ET sequence: 10, 12, 22, 34d, 56d

Badness (Sintel): 0.507

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 64/63, 99/98

Mapping: [⟨2 0 11 12 26], ⟨0 1 -2 -2 -6]]

Optimal tunings:

  • WE: ~7/5 = 598.8485 ¢, ~3/2 = 705.5285 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 707.1826 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [700.000, 709.091] (7\12 to 13\22)
  • 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.587]

Optimal ET sequence: 10e, 12, 22, 34d, 56d

Badness (Sintel): 0.673

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 64/63, 65/63, 99/98

Mapping: [⟨2 0 11 12 26 1], ⟨0 1 -2 -2 -6 2]]

Optimal tunings:

  • WE: ~7/5 = 599.9732 ¢, ~3/2 = 708.8873 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 708.9227 ¢

Optimal ET sequence: 10e, 12, 22

Badness (Sintel): 1.14

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 52/51, 64/63, 65/63, 99/98

Mapping: [⟨2 0 11 12 26 1 5], ⟨0 1 -2 -2 -6 2 1]]

Optimal tunings:

  • WE: ~7/5 = 599.8871 ¢, ~3/2 = 708.6725 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 708.8176 ¢

Optimal ET sequence: 10e, 12, 22

Badness (Sintel): 1.06

Pajarina

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 64/63, 78/77, 99/98

Mapping: [⟨2 0 11 12 26 36], ⟨0 1 -2 -2 -6 -9]]

Optimal tunings:

  • WE: ~7/5 = 598.7732 ¢, ~3/2 = 704.6889 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 706.3950 ¢

Optimal ET sequence: 12f, 22, 34d

Badness (Sintel): 0.923

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 64/63, 78/77, 85/84, 99/98

Mapping: [⟨2 0 11 12 26 36 5], ⟨0 1 -2 -2 -6 -9 1]]

Optimal tunings:

  • WE: ~7/5 = 599.0204 ¢, ~3/2 = 705.2572 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 706.5660 ¢

Optimal ET sequence: 12f, 22, 34d

Badness (Sintel): 0.936

Pajarita

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 50/49, 64/63, 66/65

Mapping: [⟨2 0 11 12 26 17], ⟨0 1 -2 -2 -6 -3]]

Optimal tunings:

  • WE: ~7/5 = 598.3048 ¢, ~3/2 = 705.4512 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 707.9238 ¢

Optimal ET sequence: 10e, 12f, 22f, 34dff

Badness (Sintel): 0.937

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 40/39, 50/49, 64/63, 66/65, 85/84

Mapping: [⟨2 0 11 12 26 17 5], ⟨0 1 -2 -2 -6 -3 1]]

Optimal tunings:

  • WE: ~7/5 = 598.6103 ¢, ~3/2 = 706.3076 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 708.2256 ¢

Optimal ET sequence: 10e, 12f, 22f

Badness (Sintel): 0.968

Pajarous

Subgroup: 2.3.5.7.11

Comma list: 50/49, 55/54, 64/63

Mapping: [⟨2 0 11 12 -9], ⟨0 1 -2 -2 5]]

Optimal tunings:

  • WE: ~7/5 = 599.4055 ¢, ~3/2 = 708.8747 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 709.5508 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = 709.091 (13\22)
  • 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.803]

Optimal ET sequence: 10, 12e, 22, 120bce, 142bce

Badness (Sintel): 0.937

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 55/54, 64/63, 65/63

Mapping: [⟨2 0 11 12 -9 1], ⟨0 1 -2 -2 5 2]]

Optimal tunings:

  • WE: ~7/5 = 599.9064 ¢, ~3/2 = 710.1289 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 710.2325 ¢

Optimal ET sequence: 10, 22

Badness (Sintel): 1.04

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 52/51, 55/54, 64/63, 65/63

Mapping: [⟨2 0 11 12 -9 1 5], ⟨0 1 -2 -2 5 2 1]]

Optimal tunings:

  • WE: ~7/5 = 599.8239 ¢, ~3/2 = 710.0128 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 710.2067 ¢

Optimal ET sequence: 10, 22, 54f, 76bdff

Badness (Sintel): 0.930

Pajaro

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 50/49, 55/54, 64/63

Mapping: [⟨2 0 11 12 -9 17], ⟨0 1 -2 -2 5 -3]]

Optimal tunings:

  • WE: ~7/5 = 598.8257 ¢, ~3/2 = 709.4266 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 710.8414 ¢

Optimal ET sequence: 10, 22f, 32f

Badness (Sintel): 1.13

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 40/39, 50/49, 55/54, 64/63, 85/84

Mapping: [⟨2 0 11 12 -9 17 5], ⟨0 1 -2 -2 5 -3 1]]

Optimal tunings:

  • WE: ~7/5 = 598.8865 ¢, ~3/2 = 709.5472 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 710.8704 ¢

Optimal ET sequence: 10, 22f, 32f

Badness (Sintel): 1.01

Pajaric

Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 56/55

Mapping: [⟨2 0 11 12 7], ⟨0 1 -2 -2 0]]

Optimal tunings:

  • WE: ~7/5 = 597.4807 ¢, ~3/2 = 702.5616 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 706.0542 ¢

Optimal ET sequence: 10, 12, 22e

Badness (Sintel): 0.787

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 45/44, 50/49, 56/55

Mapping: [⟨2 0 11 12 7 17], ⟨0 1 -2 -2 0 -3]]

Optimal tunings:

  • WE: ~7/5 = 597.1952 ¢, ~3/2 = 704.1350 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 708.1989 ¢

Optimal ET sequence: 10, 12f, 22ef

Badness (Sintel): 0.845

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 34/33, 40/39, 45/44, 50/49, 56/55

Mapping: [⟨2 0 11 12 7 17 5], ⟨0 1 -2 -2 0 -3 1]]

Optimal tunings:

  • WE: ~7/5 = 597.6509 ¢, ~3/2 = 705.7702 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 708.9719 ¢

Optimal ET sequence: 10, 12f, 22ef

Badness (Sintel): 0.896

Hemipaj

Subgroup: 2.3.5.7.11

Comma list: 50/49, 64/63, 121/120

Mapping: [⟨2 1 9 10 8], ⟨0 2 -4 -4 -1]]

mapping generators: ~2, ~16/11

Optimal tunings:

  • WE: ~7/5 = 597.6509 ¢, ~16/11 = 652.7788 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~16/11 = 653.7119 ¢

Optimal ET sequence: 2, 20, 22

Badness (Sintel): 1.29

Hemifourths

Subgroup: 2.3.5.7.11

Comma list: 50/49, 64/63, 243/242

Mapping: [⟨2 0 11 12 -1], ⟨0 2 -4 -4 5]]

mapping generators: ~2, ~55/32

Optimal tunings:

  • WE: ~7/5 = 597.6509 ¢, ~55/32 = 950.8475 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~55/32 = 953.1172 ¢

Optimal ET sequence: 10, 24d, 34d

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 64/63, 78/77, 144/143

Mapping: [⟨2 0 11 12 -1 9], ⟨0 2 -4 -4 5 -1]]

Optimal tunings:

  • WE: ~7/5 = 598.6748 ¢, ~26/15 = 950.9691 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~26/15 = 953.1052 ¢

Optimal ET sequence: 10, 24d, 34d

Badness (Sintel): 1.19

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 64/63, 78/77, 85/84, 144/143

Mapping: [⟨2 0 11 12 -1 9 5], ⟨0 2 -4 -4 5 -1 2]]

Optimal tunings:

  • WE: ~7/5 = 598.8411 ¢, ~26/15 = 951.3687 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~26/15 = 953.2169 ¢

Optimal ET sequence: 10, 24d, 34d

Badness (Sintel): 1.11

Srutal

See also: Srutal vs diaschismic

Srutal can be described as the 34d & 46 temperament, where 7/4 is located at 15 generator steps, or the double-augmented fifth (C–Gx). 80edo and 126edo are among the possible tunings. Srutal, shrutar and bidia have similar 19-limit properties, tempering out 190/189, related to rank-3 julius.

Subgroup: 2.3.5.7

Comma list: 2048/2025, 4375/4374

Mapping: [⟨2 0 11 -42], ⟨0 1 -2 15]]

Optimal tunings:

  • WE: ~45/32 = 599.4046 ¢, ~3/2 = 704.1150 ¢
error map: ⟨-1.191 +0.969 +1.289 +0.044]
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.7646 ¢
error map: ⟨0.000 +2.810 +4.157 +2.643]

Tuning ranges:

Optimal ET sequence: 34d, 46, 80, 126, 206cd, 332bcd

Badness (Sintel): 2.32

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 896/891, 1331/1323

Mapping: [⟨2 0 11 -42 -28], ⟨0 1 -2 15 11]]

Optimal tunings:

  • WE: ~45/32 = 599.4413 ¢, ~3/2 = 704.1999 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.8017 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
  • 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]

Optimal ET sequence: 34d, 46, 80, 126, 206cd

Badness (Sintel): 1.17

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 364/363

Mapping: [⟨2 0 11 -42 -28 -18], ⟨0 1 -2 15 11 8]]

Optimal tunings:

  • WE: ~45/32 = 599.5490 ¢, ~3/2 = 704.3516 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.8347 ¢

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
  • 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
  • 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]

Optimal ET sequence: 34d, 46, 80

Badness (Sintel): 1.04

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 169/168, 176/175, 221/220, 256/255

Mapping: [⟨2 0 11 -42 -28 -18 5], ⟨0 1 -2 15 11 8 1]]

Optimal tunings:

  • WE: ~17/12 = 599.6459 ¢, ~3/2 = 704.4237 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 704.8083 ¢

Tuning ranges:

  • 17-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
  • 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]

Optimal ET sequence: 34d, 46, 80, 126

Badness (Sintel): 0.947

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255

Mapping: [⟨2 0 11 -42 -28 -18 5 -55], ⟨0 1 -2 15 11 8 1 20]]

Optimal tunings:

  • WE: ~17/12 = 599.6371 ¢, ~3/2 = 704.4790 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 704.8745 ¢

Optimal ET sequence: 34dh, 46, 80

Badness (Sintel): 1.04

Srutaloo

Srutaloo adds 576/575, 736/729 or 208/207, and rhymes with skidoo.

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 256/255

Mapping: [⟨2 0 11 -42 -28 -18 5 -55 -10], ⟨0 1 -2 15 11 8 1 20 6]]

Optimal tunings:

  • WE: ~17/12 = 599.6690 ¢, ~3/2 = 704.5098 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 704.8713 ¢

Optimal ET sequence: 34dh, 46, 80

Badness (Sintel): 0.971

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 232/231, 256/255

Mapping: [⟨2 0 11 -42 -28 -18 5 -55 -10 -76], ⟨0 1 -2 15 11 8 1 20 6 27]]

Optimal tunings:

  • WE: ~17/12 = 599.6664 ¢, ~3/2 = 704.5138 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 704.8807 ¢

Optimal ET sequence: 34dhj, 46, 80

Badness (Sintel): 1.10

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 217/216, 221/220, 232/231, 256/255

Mapping: [⟨2 0 11 -42 -28 -18 5 -55 -10 -76 48], ⟨0 1 -2 15 11 8 1 20 6 27 -12]]

Optimal tunings:

  • WE: ~17/12 = 599.8115 ¢, ~3/2 = 704.5958 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 704.8086 ¢

Optimal ET sequence: 46, 80, 126

Badness (Sintel): 1.44

Keen

Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the 22 & 56 temperament. 78edo is a good tuning choice, and remains a good one in the 11-limit, where the temperament is really more interesting, adding 100/99 and 385/384 to the list of commas.

Subgroup: 2.3.5.7

Comma list: 875/864, 2048/2025

Mapping: [⟨2 0 11 -23], ⟨0 1 -2 9]]

Optimal tunings:

  • WE: ~45/32 = 599.6603 ¢, ~3/2 = 707.1707 ¢
error map: ⟨-0.679 +4.536 -3.033 -2.591]
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 707.5294 ¢
error map: ⟨0.000 +5.574 -1.373 -1.061]

Optimal ET sequence: 22, 56, 78, 134b

Badness (Sintel): 2.13

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 1232/1215

Mapping: [⟨2 0 11 -23 26], ⟨0 1 -2 9 -6]]

Optimal tunings:

  • WE: ~45/32 = 599.6286 ¢, ~3/2 = 707.1712 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 707.5984 ¢

Optimal ET sequence: 22, 56, 78

Badness (Sintel): 1.50

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 1078/1053

Mapping: [⟨2 0 11 -23 26 -18], ⟨0 1 -2 9 -6 8]]

Optimal tunings:

  • WE: ~45/32 = 599.3498 ¢, ~3/2 = 706.4009 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 707.1309 ¢

Optimal ET sequence: 22f, 34, 56f

Badness (Sintel): 1.85

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 119/117, 144/143, 154/153

Mapping: [⟨2 0 11 -23 26 -18 5], ⟨0 1 -2 9 -6 8 1]]

Optimal tunings:

  • WE: ~17/12 = 599.4053 ¢, ~3/2 = 706.4544 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 707.1243 ¢

Optimal ET sequence: 22f, 34, 56f

Badness (Sintel): 1.54

Keenic

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 100/99, 352/351, 385/384

Mapping: [⟨2 0 11 -23 26 36], ⟨0 1 -2 9 -6 -9]]

Optimal tunings:

  • WE: ~45/32 = 599.8547 ¢, ~3/2 = 707.0858 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 707.2596 ¢

Optimal ET sequence: 22, 34, 56

Badness (Sintel): 1.67

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 100/99, 136/135, 154/153, 256/255

Mapping: [⟨2 0 11 -23 26 36 5], ⟨0 1 -2 9 -6 -9 1]]

Optimal tunings:

  • WE: ~17/12 = 599.8338 ¢, ~3/2 = 707.0558 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 707.2537 ¢

Optimal ET sequence: 22, 34, 56

Badness (Sintel): 1.37

Bidia

Bidia adds 3136/3125 to the commas, splitting the period into 1/4 octave. It may be called the 12 & 56 temperament.

Subgroup: 2.3.5.7

Comma list: 2048/2025, 3136/3125

Mapping: [⟨4 0 22 43], ⟨0 1 -2 -5]]

mapping generators: ~25/21, ~3

Optimal tunings:

  • WE: ~25/21 = 299.6887 ¢, ~3/2 = 704.6318 ¢
error map: ⟨-1.245 +1.432 +0.064 +0.854]
  • CWE: ~25/21 = 300.0000 ¢, ~3/2 = 705.5070 ¢
error map: ⟨0.000 +3.552 +2.672 +3.639]

Optimal ET sequence: 12, …, 56, 68, 80, 148d

Badness (Sintel): 1.43

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 896/891, 1375/1372

Mapping: [⟨4 0 22 43 71], ⟨0 1 -2 -5 -9]]

Optimal tunings:

  • WE: ~25/21 = 299.6809 ¢, ~3/2 = 704.3367 ¢
  • CWE: ~25/21 = 600.0000 ¢, ~3/2 = 705.2170 ¢

Optimal ET sequence: 12, 56e, 68, 80

Badness (Sintel): 1.33

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 325/324, 640/637, 896/891

Mapping: [⟨4 0 22 43 71 -36], ⟨0 1 -2 -5 -9 8]]

Optimal tunings:

  • WE: ~25/21 = 299.7538 ¢, ~3/2 = 704.7222 ¢
  • CWE: ~25/21 = 600.0000 ¢, ~3/2 = 705.3241 ¢

Optimal ET sequence: 12, 68, 80, 148d, 228bcd, 376bbcddf

Badness (Sintel): 1.70

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 256/255, 325/324, 640/637

Mapping: [⟨4 0 22 43 71 -36 10], ⟨0 1 -2 -5 -9 8 1]]

Optimal tunings:

  • WE: ~25/21 = 299.7883 ¢, ~3/2 = 704.8365 ¢
  • CWE: ~25/21 = 600.0000 ¢, ~3/2 = 705.3496 ¢

Optimal ET sequence: 12, 68, 80, 148d

Badness (Sintel): 1.46

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 136/135, 176/175, 190/189, 256/255, 325/324, 640/637

Mapping: [⟨4 0 22 43 71 -36 10 17], ⟨0 1 -2 -5 -9 8 1 0]]

Optimal tunings:

  • WE: ~19/16 = 299.7967 ¢, ~3/2 = 704.8609 ¢
  • CWE: ~19/16 = 600.0000 ¢, ~3/2 = 705.3519 ¢

Optimal ET sequence: 12, 68, 80, 148d

Badness (Sintel): 1.25

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 136/135, 176/175, 190/189, 253/252, 256/255, 325/324, 640/637

Mapping: [⟨4 0 22 43 71 -36 10 17 -20], ⟨0 1 -2 -5 -9 8 1 0 6]]

Optimal tunings:

  • WE: ~19/16 = 299.7961 ¢, ~3/2 = 704.8577 ¢
  • CWE: ~19/16 = 300.0000 ¢, ~3/2 = 705.3413 ¢

Optimal ET sequence: 12, 68, 80, 148di

Badness (Sintel): 1.24

Echidna

Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the 22 & 58 temperament. 58edo or 80edo make for good tunings, or their vals can be added to ⟨138 219 321 388] (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.

Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with hedgehog; 58edo is the smallest tuning that is distinctly consistent in the 11-odd-limit and 80edo is the third smallest distinctly consistent in the 11-odd-limit.

The generator can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes 99/70 which is extremely close to 600 ¢ and is equal to it if we temper out S99. Three 11/10's then make a 4/3 (tempering out S10/S11 thus making 10/9 and 12/11 equidistant from 11/10), implying a flat tuning of 4/3.

Like most srutal extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out 176/175, tempering out 351/350 and 352/351 which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with srutal archagall, leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, of which all except 22 are consistent in the 17-odd-limit.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 2048/2025

Mapping: [⟨2 1 9 2], ⟨0 3 -6 5]]

mapping generators: ~45/32, ~9/7

Optimal tunings:

  • WE: ~45/32 = 599.3056 ¢, ~9/7 = 434.3524 ¢
error map: ⟨-1.389 +0.408 +1.322 +1.547]
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8327 ¢
error map: ⟨0.000 +2.543 +4.690 +5.338]

Optimal ET sequence: 22, 58, 80, 138cd, 218cd

Badness (Sintel): 1.47

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 896/891

Mapping: [⟨2 1 9 2 12], ⟨0 3 -6 5 -7]]

Optimal tunings:

  • WE: ~45/32 = 599.3085 ¢, ~9/7 = 434.3511 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8647 ¢

Minimax tuning:

  • 11-odd-limit: ~9/7 = [5/12 0 0 1/12 -1/12⟩
[[1 0 0 0 0⟩, [7/4 0 0 1/4 -1/4⟩, [2 0 0 -1/2 1/2⟩, [37/12 0 0 5/12 -5/12⟩, [37/12 0 0 -7/12 7/12⟩]
unchanged-interval (eigenmonzo) basis: 2.11/7

Optimal ET sequence: 22, 58, 80, 138cde, 218cde

Badness (Sintel): 0.859

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 364/363, 540/539

Mapping: [⟨2 1 9 2 12 19], ⟨0 3 -6 5 -7 -16]]

Optimal tunings:

  • WE: ~45/32 = 599.3397 ¢, ~9/7 = 434.2772 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.7864 ¢

Optimal ET sequence: 22, 36f, 58, 80, 138cde

Badness (Sintel): 0.978

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 221/220, 256/255, 540/539

Mapping: [⟨2 1 9 2 12 19 6], ⟨0 3 -6 5 -7 -16 3]]

Optimal tunings:

  • WE: ~45/32 = 599.4645 ¢, ~9/7 = 434.4282 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8340 ¢

Optimal ET sequence: 22, 36f, 58, 80, 138cde

Badness (Sintel): 1.03

Echidnic

Subgroup: 2.3.5.7

Comma list: 686/675, 1029/1024

Mapping: [⟨2 2 7 6], ⟨0 3 -6 -1]]

mapping generators: ~45/32, ~8/7

Optimal tunings:

  • WE: ~45/32 = 599.7208 ¢, ~8/7 = 234.8330 ¢
error map: ⟨-0.558 +1.986 +2.733 -5.334]
  • CWE: ~45/32 = 600.0000 ¢, ~8/7 = 234.9539 ¢
error map: ⟨0.000 +2.907 +3.963 -3.780]

Optimal ET sequence: 10, 26c, 36, 46

Badness (Sintel): 1.83

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 686/675

Mapping: [⟨2 2 7 6 3], ⟨0 3 -6 -1 10]]

Optimal tunings:

  • WE: ~45/32 = 599.8022 ¢, ~8/7 = 235.0185 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~8/7 = 235.0893 ¢

Optimal ET sequence: 10, 36e, 46, 102, 148

Badness (Sintel): 1.49

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 169/168, 385/384, 441/440

Mapping: [⟨2 2 7 6 3 7], ⟨0 3 -6 -1 10 1]]

Optimal tunings:

  • WE: ~45/32 = 599.9570 ¢, ~8/7 = 235.0708 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~8/7 = 235.0862 ¢

Optimal ET sequence: 10, 36e, 46, 102, 148f

Badness (Sintel): 1.19

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 136/135, 154/153, 169/168, 256/255

Mapping: [⟨2 2 7 6 3 7 7], ⟨0 3 -6 -1 10 1 3]]

Optimal tunings:

  • WE: ~17/12 = 599.9571 ¢, ~8/7 = 235.0709 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~8/7 = 235.0860 ¢

Optimal ET sequence: 10, 36e, 46, 102, 148f

Badness (Sintel): 0.983

Music

Shrutar

Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as 22 & 46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. 68edo makes for a good tuning, but another excellent choice is a generator of 14(1/7), making 7's just.

By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14(1/7) generator can again be used as tunings.

Subgroup: 2.3.5.7

Comma list: 245/243, 2048/2025

Mapping: [⟨2 1 9 -2], ⟨0 2 -4 7]]

mapping generators: ~45/32, ~35/24

Optimal tunings:

  • WE: ~45/32 = 599.5401 ¢, ~35/24 = 652.3108 ¢
error map: ⟨-0.920 +2.207 +0.304 -1.730]
  • CWE: ~45/32 = 600.0000 ¢, ~35/24 = 652.7736 ¢
error map: ⟨0.000 +3.592 +2.592 +0.589]

Optimal ET sequence: 22, 46, 68, 182b, 250bc

Badness (Sintel): 1.20

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 245/243

Mapping: [⟨2 1 9 -2 8], ⟨0 2 -4 7 -1]]

Optimal tunings:

  • WE: ~45/32 = 599.7721 ¢, ~16/11 = 652.4321 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~16/11 = 652.6672 ¢

Optimal ET sequence: 22, 46, 68, 114

Badness (Sintel): 0.876

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 196/195, 245/243

Mapping: [⟨2 1 9 -2 8 -10], ⟨0 2 -4 7 -1 16]]

Optimal tunings:

  • WE: ~45/32 = 599.7699 ¢, ~16/11 = 652.4035 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~16/11 = 652.6374 ¢

Optimal ET sequence: 22f, 46, 68, 114

Badness (Sintel): 1.16

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 136/135, 154/153, 176/175, 196/195

Mapping: [⟨2 1 9 -2 8 -10 6], ⟨0 2 -4 7 -1 16 2]]

Optimal tunings:

  • WE: ~17/12 = 599.7995 ¢, ~16/11 = 652.4287 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~16/11 = 652.6334 ¢

Optimal ET sequence: 22f, 46, 68, 114

Badness (Sintel): 0.953

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342

Mapping: [⟨2 1 9 -2 8 -10 6 -10], ⟨0 2 -4 7 -1 16 2 17]]

Optimal tunings:

  • WE: ~17/12 = 599.8060 ¢, ~16/11 = 652.5190 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~16/11 = 652.7164 ¢

Optimal ET sequence: 22fh, 46, 68, 114, 182bef

Badness (Sintel): 1.07

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 253/252, 343/342

Mapping: [⟨2 1 9 -2 8 -10 6 -10 -4], ⟨0 2 -4 7 -1 16 2 17 12]]

Optimal tunings:

  • WE: ~17/12 = 599.7879 ¢, ~16/11 = 652.4776 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~16/11 = 652.6926 ¢

Optimal ET sequence: 22fh, 46, 68, 114

Badness (Sintel): 1.03

Sruti

Subgroup: 2.3.5.7

Comma list: 2048/2025, 19683/19600

Mapping: [⟨2 0 11 -15], ⟨0 2 -4 13]]

mapping generators: ~45/32, ~140/81

Optimal tunings:

  • WE: ~45/32 = 599.2764 ¢, ~140/81 = 950.7284 ¢
error map: ⟨-1.447 -0.498 +2.813 +1.497]
  • CWE: ~45/32 = 600.0000 ¢, ~140/81 = 951.8227 ¢
error map: ⟨0.000 +1.690 +6.395 +4.869]

Optimal ET sequence: 24, 34d, 58, 150cd, 208ccdd, 266ccdd

Badness (Sintel): 2.97

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 243/242, 896/891

Mapping: [⟨2 0 11 -15 -1], ⟨0 2 -4 13 5]]

Optimal tunings:

  • WE: ~45/32 = 599.1951 ¢, ~121/70 = 950.5864 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~121/70 = 951.7972 ¢

Optimal ET sequence: 24, 34d, 58, 150cdee, 208ccddee, 266ccddeee

Badness (Sintel): 1.37

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 351/350, 676/675

Mapping: [⟨2 0 11 -15 -1 9], ⟨0 2 -4 13 5 -1]]

Optimal tunings:

  • WE: ~45/32 = 599.1479 ¢, ~26/15 = 950.5337 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~26/15 = 951.8314 ¢

Optimal ET sequence: 24, 34d, 58, 150cdeef, 208ccddeeff, 266ccddeeefff

Badness (Sintel): 0.983

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 144/143, 170/169, 176/175, 221/220

Mapping: [⟨2 0 11 -15 -1 9 5], ⟨0 2 -4 13 5 -1 2]]

Optimal tunings:

  • WE: ~17/12 = 599.3003 ¢, ~26/15 = 950.7465 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~26/15 = 951.8142 ¢

Optimal ET sequence: 24, 34d, 58

Badness (Sintel): 1.05

Anguirus

Subgroup: 2.3.5.7

Comma list: 49/48, 2048/2025

Mapping: [⟨2 0 11 4], ⟨0 2 -4 1]]

mapping generators: ~45/32, ~7/4

Optimal tunings:

  • WE: ~45/32 = 600.2758 ¢, ~7/4 = 953.4593 ¢
error map: ⟨+0.552 +4.964 +2.883 -14.264]
  • CWE: ~45/32 = 600.0000 ¢, ~7/4 = 953.0188 ¢
error map: ⟨0.000 +4.083 +1.611 -15.807]

Optimal ET sequence: 10, 24, 34

Badness (Sintel): 1.97

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 243/242

Mapping: [⟨2 0 11 4 -1], ⟨0 2 -4 1 5]]

Optimal tunings:

  • WE: ~45/32 = 599.9250 ¢, ~7/4 = 952.0646 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~7/4 = 952.1784 ¢

Optimal ET sequence: 10, 24, 34

Badness (Sintel): 1.63

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 243/242

Mapping: [⟨2 0 11 4 -1 9], ⟨0 2 -4 1 5 -1]]

Optimal tunings:

  • WE: ~45/32 = 599.7575 ¢, ~7/4 = 951.9241 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~7/4 = 952.2980 ¢

Optimal ET sequence: 10, 24, 34, 58d, 92ddef

Badness (Sintel): 1.27

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 49/48, 56/55, 91/90, 119/117, 154/153

Mapping: [⟨2 0 11 4 -1 9 5], ⟨0 2 -4 1 5 -1 2]]

Optimal tunings:

  • WE: ~17/12 = 599.7925 ¢, ~7/4 = 952.0004 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~7/4 = 952.3178 ¢

Optimal ET sequence: 10, 24, 34

Badness (Sintel): 1.10

Shru

Subgroup: 2.3.5.7

Comma list: 392/375, 1323/1280

Mapping: [⟨2 1 9 11], ⟨0 2 -4 -5]]

mapping generators: ~45/32, ~10/7

Optimal tunings:

  • WE: ~45/32 = 600.2519 ¢, ~10/7 = 650.4083 ¢
error map: ⟨+0.504 -0.887 +14.321 -18.096]
  • CWE: ~45/32 = 600.0000 ¢, ~10/7 = 650.1017 ¢
error map: ⟨0.000 -1.752 +13.279 -19.334]

Optimal ET sequence: 2, 22d, 24

Badness (Sintel): 3.99

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 1323/1280

Mapping: [⟨2 1 9 11 8], ⟨0 2 -4 -5 -1]]

Optimal tunings:

  • WE: ~17/12 = 600.2356 ¢, ~10/7 = 650.3856 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~10/7 = 650.1008 ¢

Optimal ET sequence: 2, 22d, 24

Badness (Sintel): 2.10

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 77/75, 105/104, 507/500

Mapping: [⟨2 1 9 11 8 15], ⟨0 2 -4 -5 -1 -7]]

Optimal tunings:

  • WE: ~45/32 = 599.9067 ¢, ~10/7 = 649.4907 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~10/7 = 649.5950 ¢

Optimal ET sequence: 2, 24

Badness (Sintel): 2.12

Quadrasruta

Subgroup: 2.3.5.7

Comma list: 2048/2025, 2401/2400

Mapping: [⟨2 0 11 8], ⟨0 4 -8 -3]]

mapping generators: ~45/32, ~21/16

Optimal tunings:

  • WE: ~45/32 = 599.4443 ¢, ~21/16 = 475.7746 ¢
error map: ⟨-1.111 +1.143 +1.377 -0.595]
  • CWE: ~45/32 = 600.0000 ¢, ~21/16 = 476.2394 ¢
error map: ⟨0.000 +3.003 +3.771 +2.456]

Optimal ET sequence: 10, …, 58, 68, 126, 446bbccd

Badness (Sintel): 1.86

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 896/891, 2401/2400

Mapping: [⟨2 0 11 8 22], ⟨0 4 -8 -3 -19]]

Optimal tunings:

  • WE: ~45/32 = 599.4648 ¢, ~21/16 = 475.6929 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~21/16 = 476.1507 ¢

Optimal ET sequence: 10e, …, 58, 126, 184c, 310bccde

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 196/195, 512/507, 676/675

Mapping: [⟨2 0 11 8 22 9], ⟨0 4 -8 -3 -19 -2]]

Optimal tunings:

  • WE: ~45/32 = 599.3787 ¢, ~21/16 = 475.6065 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~21/16 = 476.1345 ¢

Optimal ET sequence: 10e, …, 58, 126f, 184cff

Badness (Sintel): 1.18

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 170/169, 176/175, 196/195, 256/255

Mapping: [⟨2 0 11 8 22 9 5], ⟨0 4 -8 -3 -19 -2 4]]

Optimal tunings:

  • WE: ~17/12 = 599.5077 ¢, ~21/16 = 475.7713 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~21/16 = 476.1814 ¢

Optimal ET sequence: 10e, 58, 126f

Badness (Sintel): 1.21

Quadrafourths

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 2048/2025

Mapping: [⟨2 0 11 8 -1], ⟨0 4 -8 -3 10]]

Optimal tunings:

  • WE: ~45/32 = 599.2593 ¢, ~21/16 = 475.4292 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~21/16 = 476.0088 ¢

Optimal ET sequence: 10, 48c, 58, 184cee, 242ccdeee

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 243/242, 676/675

Mapping: [⟨2 0 11 8 -1 9], ⟨0 4 -8 -3 10 -2]]

Optimal tunings:

  • WE: ~45/32 = 599.2147 ¢, ~21/16 = 475.4052 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~21/16 = 476.0253 ¢

Optimal ET sequence: 10, 48c, 58, 126eef, 184ceeff, 242ccdeeeff

Badness (Sintel): 1.11

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 144/143, 170/169, 196/195, 221/220

Mapping: [⟨2 0 11 8 -1 9 5], ⟨0 4 -8 -3 10 -2 4]]

Optimal tunings:

  • WE: ~17/12 = 599.3353 ¢, ~21/16 = 475.5495 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~21/16 = 476.0691 ¢

Optimal ET sequence: 10, 48c, 58

Badness (Sintel): 1.13

Retrieved from "https://en.xen.wiki/index.php?title=Diaschismic_family&oldid=208670"