Diaschismic family

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The 5-limit parent comma for the diaschismic family is 2048/2025, the diaschisma. Its monzo is [11 -4 -2⟩, and flipping that yields ⟨⟨2 -4 -11]] for the wedgie for 5-limit diaschismic, or srutal, temperament. This tells us the period is half an octave, the GCD of 2 and -4, and that the generator is a fifth. 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.

Srutal aka diaschismic

Subgroup: 2.3.5

Comma list: 2048/2025

Mapping: [⟨2 0 11], ⟨0 1 -2]]

mapping generators: ~45/32, ~3

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.898

Tuning ranges:

  • 5-odd-limit diamond monotone: ~3/2 = [600.000 to 720.000] (1\2 to 6\10)
  • 5-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
  • 5-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 706.843]

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

Badness: 0.019915

Overview to extensions

7-limit extensions

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

  • Pajara derives from 64/63 and is a popular and well-known choice.
  • Diaschismic adds 126/125, the starling comma, to obtain 7-limit harmony by more complex methods, but with greater accuracy.
  • Srutal adds 4375/4374, the ragisma. It does no significant tuning damage, so we keep the 5-limit label srutal.
  • Keen adds 875/864.
  • Bidia adds 3136/3125, the hemimean comma.
  • Echidna adds 1728/1715, the orwellisma.
  • Shrutar adds 245/243, the sensamagic comma.

Pajara, diaschismic, srutal and keen keep the same half-octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as 36/35, the septimal quarter-tone) 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, resulting in srutal archagall.

Srutal archagall

The S-expression-based comma list of this temperament is {S16, S17}.

Subgroup: 2.3.5.17

Comma list: 136/135, 256/255

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

mapping generators: ~17/12, ~3

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 705.1272

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

Badness: 0.00575

Srutal

See also: Srutal vs diaschismic

Subgroup: 2.3.5.7

Comma list: 2048/2025, 4375/4374

Mapping: [⟨2 0 11 -42], ⟨0 1 -2 15]]

Wedgie: ⟨⟨2 -4 30 -11 42 81]]

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.814

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [703.448, 705.882] (34\58 to 20\34)
  • 7- and 9-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
  • 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 705.882]

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

Badness: 0.091504

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 tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.856

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]
  • 11-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]

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

Badness: 0.035315

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 tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.881

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]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]

Optimal ET sequence: 34d, 46, 80, 206cd, 286bcde

Badness: 0.025286

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 tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.840

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]
  • 17-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]

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

Badness: 0.018594

19-limit

Srutal, shrutar and bidia have similar 19-limit properties, tempering 190/189, related rank-3 julius.

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 tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.905

Optimal ET sequence: 34dh, 46, 80, 206cd

Badness: 0.017063

Srutaloo

Srutaloo adds 576/575, 736/729 or 208/207, 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 tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.899

Optimal ET sequence: 34dh, 46, 80, 206cd

Badness: 0.013555

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 tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.906

Optimal ET sequence: 34dhj, 46, 80, 206cd

Badness: 0.013203

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 are interesting alternatives, with more accpetable 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 56 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]]

Wedgie: ⟨⟨2 -4 -4 -11 -12 2]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.048

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [700.000, 720.000] (7\12 to 6\10)
  • 7- and 9-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.587]
  • 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 715.587]

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

Badness: 0.020033

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.885

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]
  • 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 709.091]

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

Badness: 0.020343

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.919

Optimal ET sequence: 10e, 12, 22

Badness: 0.027642

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.806

Optimal ET sequence: 10e, 12, 22

Badness: 0.020899

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.133

Optimal ET sequence: 12f, 22, 34d

Badness: 0.022327

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.410

Optimal ET sequence: 12f, 22, 34d

Badness: 0.018375

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.450

Optimal ET sequence: 10e, 12f, 22f

Badness: 0.022677

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.947

Optimal ET sequence: 10e, 12f, 22f

Badness: 0.019007

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 709.578

Tuning ranges:

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

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

Badness: 0.028349

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.240

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

Badness: 0.025176

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.221

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

Badness: 0.018249

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.818

Optimal ET sequence: 10, 22f, 32f, 54ff

Badness: 0.027355

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.866

Optimal ET sequence: 10, 22f, 32f, 54ff

Badness: 0.019844

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 705.524

Optimal ET sequence: 10, 12, 22e, 34dee

Badness: 0.023798

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.442

Optimal ET sequence: 10, 12f, 22ef

Badness: 0.020461

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 tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.544

Optimal ET sequence: 10, 12f, 22ef

Badness: 0.017592

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]]

Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 546.383

Optimal ET sequence: 20, 22, 68d, 90d

Badness: 0.038890

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]]

Optimal tuning (POTE): ~7/5 = 1\2, ~55/32 = 953.093

Optimal ET sequence: 10, 24d, 34d

Badness: 0.048885

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 tuning (POTE): ~7/5 = 1\2, ~26/15 = 953.074

Optimal ET sequence: 10, 24d, 34d

Badness: 0.028755

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 tuning (POTE): ~7/5 = 1\2, ~26/15 = 953.210

Optimal ET sequence: 10, 24d, 34d

Badness: 0.021790

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]]

Wedgie: ⟨⟨2 -4 -16 -11 -31 -26]]

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.681

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 20\34)
  • 7- and 9-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
  • 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 705.882]

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

Badness: 0.037914

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 tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.714

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]
  • 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 704.348]

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

Badness: 0.025034

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 tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.704

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]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]

Optimal ET sequence: 46, 58, 104c, 162cef

Badness: 0.018926

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 tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.812

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]
  • 17-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]

Optimal ET sequence: 46, 58, 104c

Badness: 0.016425

Na"Naa'

Na"Naa' is a remarkable subgroup temperament of 46&58 with a prime harmonic of 23.

Subgroup: 2.3.5.7.11.13.17.23

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

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

Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.870

Optimal ET sequence: 46, 58i, 104ci

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 keen, ⟨⟨2 -4 18 -12 …]], is really more interesting, adding 100/99 and 385/384 to the commas.

Subgroup: 2.3.5.7

Comma list: 875/864, 2048/2025

Mapping: [⟨2 0 11 -23], ⟨0 1 -2 9]]

Wedgie: ⟨⟨2 -4 18 -11 23 53]]

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.571

Optimal ET sequence: 22, 56, 78, 134b, 212b, 290bb

Badness: 0.083971

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 tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.609

Optimal ET sequence: 22, 56, 78, 212be, 290bbe

Badness: 0.045270

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 tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.167

Optimal ET sequence: 22f, 34, 56f

Badness: 0.044877

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 tuning (POTE): ~17/12 = 1\2, ~3/2 = 707.155

Optimal ET sequence: 22f, 34, 56f

Badness: 0.030297

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 tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.257

Optimal ET sequence: 22, 34, 56

Badness: 0.040351

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 tuning (POTE): ~17/12 = 1\2, ~3/2 = 707.252

Optimal ET sequence: 22, 34, 56

Badness: 0.026917

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]]

Wedgie: ⟨⟨4 -8 -20 -22 -43 -24]]

Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.364

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

Badness: 0.056474

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 tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.087

Optimal ET sequence: 12, 68, 80

Badness: 0.040191

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 tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.301

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

Badness: 0.041137

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 tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.334

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

Badness: 0.028631

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 tuning (POTE): ~19/16 = 1\4, ~3/2 = 705.339

Optimal ET sequence: 12, 68, 80, 148d, 376bbcddfh

Badness: 0.020590

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].

Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-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.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 2048/2025

Mapping: [⟨2 1 9 2], ⟨0 3 -6 5]]

Wedgie: ⟨⟨6 -12 10 -33 -1 57]]

Optimal tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.856

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

Badness: 0.058033

11-limit

In the 11-limit the generator of echidna 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 S99. Three 11/10's then make a 4/3 (tempering S10/S11 thus making 10/9 and 12/11 equidistant from 11/10), implying a flat 4/3.

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 tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.852

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⟩]
Eigenmonzo (unchanged-interval) basis: 2.11/7

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

Badness: 0.025987

13-limit

A surprisingly natural extension to the 13-limit is possible by observing that since we temper 176/175, tempering 351/350 and 352/351 (which sum to 176/175) is very elegant. This is notable as this mapping of 13 is supported by patent val by the three main echidna EDOs of 80, 58 and 22 (the trivial tuning), of which all except 22 are consistent in the 17-odd-limit; see the 17-limit for more details.

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 tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.756

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

Badness: 0.023679

17-limit

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). 58edo and 80edo are both interesting tunings with different advantages and both are consistent in the 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 tuning (POTE): ~17/12 = 1\2, ~9/7 = 434.816

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

Badness: 0.020273

Echidnic

Subgroup: 2.3.5.7

Comma list: 686/675, 1029/1024

Mapping: [⟨2 2 7 6], ⟨0 3 -6 -1]]

Wedgie: ⟨⟨6 -12 -2 -33 -20 29]]

Optimal tuning (POTE): ~45/32 = 1\2, ~8/7 = 234.492

Optimal ET sequence: 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bccdd

Badness: 0.072246

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 tuning (POTE): ~45/32 = 1\2, ~8/7 = 235.096

Optimal ET sequence: 10, 36e, 46, 102, 148, 342bcdd

Badness: 0.045127

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 tuning (POTE): ~45/32 = 1\2, ~8/7 = 235.088

Optimal ET sequence: 10, 46, 102, 148f, 194bcdf

Badness: 0.028874

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 tuning (POTE): ~17/12 = 1\2, ~8/7 = 235.088

Optimal ET sequence: 10, 46, 102, 148f, 194bcdf

Badness: 0.019304

Compositions

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]]

Wedgie: ⟨⟨4 -8 14 -22 11 55]]

Optimal tuning (POTE): ~45/32 = 1\2, ~35/24 = 652.811

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

Badness: 0.047377

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 tuning (POTE): ~45/32 = 1\2, ~16/11 = 652.680

Optimal ET sequence: 22, 46, 68, 114, 296bce, 410bce

Badness: 0.026489

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 tuning (POTE): ~45/32 = 1\2, ~16/11 = 652.654

Optimal ET sequence: 22f, 24f, 46, 68, 114

Badness: 0.028057

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 tuning (POTE): ~17/12 = 1\2, ~16/11 = 652.647

Optimal ET sequence: 22f, 24f, 46, 68, 114

Badness: 0.018716

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 tuning (POTE): ~17/12 = 1\2, ~16/11 = 652.730

Optimal ET sequence: 22fh, 24fh, 46, 68, 114, 182bef

Badness: 0.017540

Sruti

Subgroup: 2.3.5.7

Comma list: 2048/2025, 19683/19600

Mapping: [⟨2 0 11 -15], ⟨0 2 -4 13]]

Wedgie: ⟨⟨4 -8 26 -22 30 83]]

Optimal tuning (POTE): ~45/32 = 1\2, ~140/81 = 951.876

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

Badness: 0.117358

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 tuning (POTE): ~45/32 = 1\2, ~121/70 = 951.863

Optimal ET sequence: 24, 34d, 58

Badness: 0.041459

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 tuning (POTE): ~45/32 = 1\2, ~26/15 = 951.886

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

Badness: 0.023791

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 tuning (POTE): ~17/12 = 1\2, ~26/15 = 951.857

Optimal ET sequence: 24, 34d, 58

Badness: 0.020536

Anguirus

Subgroup: 2.3.5.7

Comma list: 49/48, 2048/2025

Mapping: [⟨2 0 11 4], ⟨0 2 -4 1]]

Wedgie: ⟨⟨4 -8 2 -22 -8 27]]

Optimal tuning (POTE): ~45/32 = 1\2, ~7/4 = 953.021

Optimal ET sequence: 10, 24, 34

Badness: 0.077955

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 tuning (POTE): ~45/32 = 1\2, ~7/4 = 952.184

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

Badness: 0.049253

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 tuning (POTE): ~45/32 = 1\2, ~7/4 = 952.309

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

Badness: 0.030829

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 tuning (POTE): ~17/12 = 1\2, ~7/4 = 952.330

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

Badness: 0.021796

Shru

Subgroup: 2.3.5.7

Comma list: 392/375, 1323/1280

Mapping: [⟨2 1 9 11], ⟨0 2 -4 -5]]

Wedgie: ⟨⟨4 -8 -10 -22 -27 -1]]

Optimal tuning (POTE): ~45/32 = 1\2, ~10/7 = 650.135

Optimal ET sequence: 2, 22d, 24

Badness: 0.157619

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 tuning (POTE): ~45/32 = 1\2, ~10/7 = 650.130

Optimal ET sequence: 2, 22d, 24

Badness: 0.063483

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 tuning (POTE): ~45/32 = 1\2, ~10/7 = 650.535

Optimal ET sequence: 22df, 24

Badness: 0.045731

Quadrasruta

Subgroup: 2.3.5.7

Comma list: 2048/2025, 2401/2400

Mapping: [⟨2 0 11 8], ⟨0 4 -8 -3]]

Wedgie: ⟨⟨8 -16 -6 -44 -32 31]]

Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.216

Optimal ET sequence: 10, 38c, 48c, 58, 68, 126

Badness: 0.073569

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 tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.118

Optimal ET sequence: 58, 126, 184c, 310bccde

Badness: 0.049018

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 tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.099

Optimal ET sequence: 58, 126f, 184cff

Badness: 0.028463

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 tuning (POTE): ~17/12 = 1\2, ~21/16 = 476.162

Optimal ET sequence: 58, 126f

Badness: 0.023820

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 tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.017

Optimal ET sequence: 10, 38c, 48c, 58

Badness: 0.049114

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 tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.028

Optimal ET sequence: 10, 38c, 48c, 58

Badness: 0.026743

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 tuning (POTE): ~17/12 = 1\2, ~21/16 = 476.077

Optimal ET sequence: 10, 38c, 48c, 58, 126eef, 184ceeff

Badness: 0.022239

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