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Diaschismic family

Revision as of 01:36, 24 July 2021 by Xenllium (talk | contribs)

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. 34 EDO is a good tuning choice, with 46 EDO, 56 EDO, 58 EDO or 80 EDO being other possibilities. Both 12 EDO and 22 EDO support it, and retuning them to a MOS of diaschismic gives two scale possibilities.

Srutal (12&34, aka diaschismic)

Subgroup: 2.3.5

Comma list: 2048/2025

Mapping: [⟨2 0 11], ⟨0 1 -2]]

POTE generator: ~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]

Template:Val list

Badness: 0.019915

Seven limit extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at.

  • Pajara derives from 64/63 and is a popular and well-known choice.
  • Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy.
  • Srutal adds [21 -15 0 1⟩. It does no significant tuning damage, so for that we keep the 5-limit label srutal.
  • Keen adds 2240/2187.
  • Echidna 1728/1715, the orwellisma.
  • Shrutar 245/243, the sensamagic comma.

Pajara, diaschismic, srutal and keen keep the same 1/2 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.

Srutal

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 ]]

POTE generator: ~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]

Template:Val list

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]]

POTE generator: ~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]

Vals: Template:Val list

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]]

POTE generator: ~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]

Vals: Template:Val list

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]]

POTE generator: ~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]

Vals: Template:Val list

Badness: 0.018594

Pajara

Main article: Pajara

Pajara is closely associated with 22 EDO (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 22 EDO, 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 12 EDO 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 ]]

POTE generator: ~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]

Template:Val list

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]]

POTE generator: ~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]

Vals: Template:Val list

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]]

POTE generator: ~3/2 = 708.919

Vals: Template:Val list

Badness: 0.027642

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]]

POTE generator: ~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

Vals: Template:Val list

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]]

POTE generator: ~3/2 = 710.240

Vals: Template:Val list

Badness: 0.025176

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]]

POTE generator ~3/2 = 710.818

Vals: Template:Val list

Badness: 0.0274

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]]

POTE generator: ~3/2 = 705.524

Vals: Template:Val list

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]]

POTE generator: ~3/2 = 707.442

Vals: Template:Val list

Badness: 0.0205

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]]

POTE generator: ~11/8 = 546.383

Vals: Template:Val list

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]]

POTE generator: ~64/55 = 246.907

Vals: Template:Val list

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]]

POTE generator: ~15/13 = 246.907

Vals: Template:Val list

Badness: 0.028755

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 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. 58 EDO 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 58 EDO.

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 two 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 ]]

POTE generator: ~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]

Template:Val list

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]]

POTE generator: ~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]

Vals: Template:Val list

Badness: 0.025034

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 364/363, 2048/2025

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

POTE generator: ~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]

Vals: Template:Val list

Badness: 0.018926

17-limit (Na"Naa')

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]]

POTE generator: ~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]

Vals: Template:Val list

Badness: 0.016425

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. 78 EDO 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 ]]

POTE generator: ~3/2 = 707.571

Template:Val list

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]]

POTE generator: ~3/2 = 707.609

Vals: Template:Val list

Badness: 0.045270

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 ]]

POTE generator: ~3/2 = 705.364

Template:Val list

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]]

POTE generator: ~3/2 = 705.087

Vals: Template:Val list

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]]

POTE generator: ~3/2 = 705.301

Vals: Template:Val list

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]]

POTE generator: ~3/2 = 705.334

Vals: Template:Val list

Badness: 0.028631

Echidna

Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the 22&58 temperament. 58 EDO or 80 EDO make for good tunings, or their vals can be add 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 28 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 ]]

POTE generator: ~9/7 = 434.856

Template:Val list

Badness: 0.058033

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]]

POTE generator: ~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⟩]
Eigenmonzos: 2, 11/7

Vals: Template:Val list

Badness: 0.025987

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]]

POTE generator: ~9/7 = 434.756

Vals: Template:Val list

Badness: 0.023679

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]]

POTE generator: ~9/7 = 434.816

Vals: Template:Val list

Badness: 0.020273

Echidnic

Subgroup: 2.3.5.7

Comma list: 686/675, 1029/1024

Mapping: [⟨2 2 7 6], ⟨0 3 -6 -1]]

POTE generator: ~8/7 = 234.492

Template:Val list

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]]

POTE generator: ~8/7 = 235.096

Vals: Template:Val list

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]]

POTE generator: ~8/7 = 235.088

Vals: Template:Val list

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]]

POTE generator: ~8/7 = 235.088

Vals: Template:Val list

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. 68 EDO makes for a good tuning, but another and excellent choice is a generator of 14(1/7), making 7s 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 3 5 5], ⟨0 2 -4 7]]

Wedgie: ⟨⟨ 4 -8 14 -22 11 55 ]]

POTE generator: ~36/35 = 52.811

Template:Val list

Badness: 0.047377

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 3 5 5 7], ⟨0 2 -4 7 -1]]

POTE generator: ~33/32 = 52.680

Vals: Template:Val list

Badness: 0.026489

13-limit

Subgroup: 2.3.5.7.11.13

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

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

POTE generator: ~33/32 = 52.654

Vals: Template:Val list

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 3 5 5 7 6 8], ⟨0 2 -4 7 -1 16 2]]

POTE generator: ~33/32 = 52.647

Vals: Template:Val list

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 3 5 5 7 6 8 7], ⟨0 2 -4 7 -1 16 2 17]]

POTE generator: ~33/32 = 52.730

Vals: Template:Val list

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 ]]

POTE generator: ~175/144 = 351.876

Template:Val list

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]]

POTE generator: ~11/9 = 351.863

Vals: Template:Val list

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]]

POTE generator: ~11/9 = 351.886

Vals: Template:Val list

Badness: 0.023791

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 ]]

POTE generator: ~8/7 = 246.979

Template:Val list

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]]

POTE generator: ~8/7 = 247.816

Vals: Template:Val list

Badness: 0.049253

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 352/351

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

POTE generator: ~8/7 = 247.691

Vals: Template:Val list

Badness: 0.030829

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 ]]

POTE generator: ~64/63 = 50.135

Template:Val list

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]]

POTE generator: ~33/32 = 50.130

Vals: Template:Val list

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]]

POTE generator: ~33/32 = 50.535

Vals: Template:Val list

Badness: 0.045731

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