Diaschismic family

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.

valid range: [600.000 to 720.000] (2 to 5)

nice range: [701.955, 706.843]

strict range: [701.955, 706.843]

POTE generator: ~3/2 = 704.898

Map: [<2 0 11|, <0 1 -2|]

EDOs: 34, 46, 80, 206c, 286bc

Seven limit children

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. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic 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. Adding 4375/4374 does no significant tuning damage, so for that we keep the 5-limit label srutal.

Srutal

Commas: 2048/2025, 4375/4374

valid range: [703.448, 705.882] (58 to 34d)

nice range: [701.955, 706.843]

strict range: [703.448, 705.882]

POTE generator: ~3/2 = 704.814

Map: [<2 0 11 -42|, <0 1 -2 15|]

Wedgie: <<2 -4 30 -11 42 81||

EDOs: 46, 80, 126, 206cd, 332bcd

Badness: 0.0915

11-limit

Commas: 176/175, 896/891, 1331/1323

valid range: [704.348, 705.882] (46 to 34d)

nice range: [701.955, 706.843]

strict range: [704.348, 705.882]

POTE generator: ~3/2 = 704.856

Map: [<2 0 11 -42 -28|, <0 1 -2 15 11|]

EDOs: 46, 80, 126, 206cd

Badness: 0.0353

13-limit

Commas: 169/168, 176/175, 325/324, 364/363

valid range: [704.348, 705.882] (46 to 34d)

nice range: [701.955, 706.843]

strict range: [704.348, 705.882]

POTE generator: ~3/2 = 704.881

Map: [<2 0 11 -42 -28 -18|, <0 1 -2 15 11 8|]

EDOs: 34d, 46, 80, 206cd, 286bcde

Badness: 0.0253

Pajara

Main article: Pajara

Pajara, with wedgie <<2 -4 -4 -11 -12 2|| is closely associated with 22et (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 22et, 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 12et 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.

Commas: 50/49, 64/63

valid range: [700.000, 720.000] (12 to 10)

nice range: [701.955, 715.587]

strict range: [701.955, 715.587]

POTE generator: 707.048

Map: [<2 0 11 12|, <0 1 -2 -2|]

EDOs: 22, 34, 56

Badness: 0.0200

11-limit

Commas: 50/49, 64/63, 99/98

valid range: [700.000, 709.091] (12 to 22)

nice range: [701.955, 715.587]

strict range: [701.955, 709.091]

POTE generator: 706.885

Map: [<2 0 11 12 26|, <0 1 -2 -2 -6|]

EDOs: 22, 34, 56, 146

Badness: 0.0203

13-limit

Commas: 50/49, 64/63, 65/63, 99/98

valid range: []

nice range: [701.955, 738.573]

strict range: []

POTE generator: ~3/2 = 708.919

Map: [<2 0 11 12 26 1|, <0 1 -2 -2 -6 2|]

EDOs: 12, 22

Badness: 0.0276

Pajarous

Commas: 50/49, 55/54, 64/63

valid range: 709.091 (22)

nice range: [701.955, 715.803]

strict range: 709.091

POTE generator: ~3/2 = 709.578

Map: [<2 0 11 12 -9|, <0 1 -2 -2 5|]

EDOs: 10, 12e, 22, 120bce, 142bce

Badness: 0.0283

13-limit

Commas: 50/49, 55/54, 64/63, 65/63

valid range: []

nice range: [701.955, 738.573]

strict range: []

POTE generator: ~3/2 = 710.240

Map: [<2 0 11 12 -9 1|, <0 1 -2 -2 5 2|]

EDOs: 10, 22, 54f, 76bdf

Badness: 0.0252

Pajaric

Commas: 45/44, 50/49, 56/55

POTE generator: ~3/2 = 705.524

Map: [<2 0 11 12 7|, <0 1 -2 -2 0|]

EDOs: 10, 12, 22e, 34de

Badness: 0.0238

13-limit

Commas: 40/39, 45/44, 50/49, 56/55

POTE generator: ~3/2 = 707.442

Map: [<2 0 11 12 7 17|, <0 1 -2 -2 0 -3|]

EDOs: 10, 12f, 22ef, 34def

Badness: 0.0205

Pajaro

Commas: 40/39, 50/49, 55/54, 64/63

POTE generator ~3/2 = 710.818

Map: [<2 0 11 12 -9 17|, <0 1 -2 -2 5 -3|]

EDOs: 10, 22f, 32f, 54f

Badness: 0.0274

Hemipaj

Commas: 50/49, 64/63, 121/120

POTE generator: ~11/8 = 546.383

Map: [<2 1 9 10 8|, <0 2 -4 -4 -1|]

EDOs: 20, 22, 68d, 90d

Badness: 0.0389

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 wedgie <<2 -4 -16 -11 -31 -26||, with a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. 58et 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 58et.

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.

Commas: 126/125, 2048/2025

POTE generator: 703.681

Map: [<2 0 11 31|, <0 1 -2 -8|]

EDOs: 46, 58, 104c, 162c

11-limit

Commas: 126/125, 176/175, 896/891

POTE generator: 703.714

Map: [<2 0 11 31 45|, <0 1 -2 -8 -12|]

EDOs: 46, 58, 104c, 162ce

13-limit

Commas: 126/125, 196/195, 364/363, 2048/2025

POTE generator: 703.704

Map: [<2 0 11 31 45 55|, <0 1 -2 -8 -12 -15|]

EDOs: 46, 58, 104c, 162cef

17-limit

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

POTE generator: 703.812

Map: [<2 0 11 31 45 55 5|, <0 1 -2 -8 -12 -15 1|]

EDOs: 46, 58, 104c

Keen

Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie <<2 -4 18 -11 23 53||. It may also be described as the 22&56 temperament. 78et 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.

Commas: 2048/2025, 875/864

POTE generator: 707.571

Map: [<2 0 11 -23|, <0 1 -2 9|]

EDOs: 22, 56, 78, 134b, 212b, 290b

11-limit

Commas: 100/99, 385/384, 1232/1215

POTE generator: 707.609

Map: [<2 0 11 -23 26|, <0 1 -2 9 -6|]

EDOs: 22, 56, 78, 212bf, 290bf

Bidia

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

Commas: 2048/2025, 3136/3125

POTE generator: ~3/2 = 705.364

Map: [<4 0 22 43|,<0 1 -2 -5|]

Wedgie: <<4 -8 -20 -22 -43 -24||

EDOs: 12, 56, 68, 80, 148d

Badness: 0.0565

11-limit

Commas: 176/175, 896/891, 1375/1372

POTE generator: ~3/2 = 705.087

Map: [<4 0 22 43 71|,<0 1 -2 -5 -9|]

EDOs: 12, 68, 80

Badness: 0.0402

13-limit

Commas: 176/175, 325/324, 640/637, 896/891

POTE generator: ~3/2 = 705.301

Map: [<4 0 22 43 71|,<0 1 -2 -5 -9|]

EDOs: 12, 68, 80, 148d, 228bcd, 376bcdf

Badness: 0.0411

Echidna

Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie <<6 -12 10 -33 -1 57|| and may be called the 22&58 temperament. 58et or 80et 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.

Commas: 2048/2025, 1728/1715

POTE generator: 434.856

Map: [<2 1 9 2|, <0 3 -6 5|]

EDOs: 22, 58, 80, 138cd, 218cd

Badness: 0.0580

11-limit

Commas: 176/175, 896/891, 540/539

11-limit minimax

[|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

Minimax generator: (224/11)^(1/12) = 434.792

POTE generator: 434.852

Map: [<2 1 9 2 12|, <0 3 -6 5 -7|]

EDOs: 22, 58, 80, 138cde, 218cde

Badness: 0.0260

13-limit

Commas: 176/175, 351/350, 364/363, 540/539

POTE generator: 434.756

Map: [<2 1 9 2 12 19|, <0 3 -6 5 -7 -16|]

EDOs: 22, 58, 80, 138cde

Badness: 0.0237

17-limit

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

POTE generator: 434.816

Map: [<2 1 9 2 12 19 6|, <0 3 -6 5 -7 -16 3|]

EDOs: 22, 58, 80, 138cde

Badness: 0.0203

Echidnic

Commas: 686/675, 1029/1024

POTE generator: 234.492

Map: [<2 2 7 6|, <0 3 -6 -1|]

EDOs: 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bcd

Badness: 0.0722

11-limit

Commas: 385/384, 441/440, 686/675

POTE generator: 235.096

Map: [<2 2 7 6 3|, <0 3 -6 -1 10|]

EDOs: 10, 46, 102, 148, 342bcd

Badness: 0.0451

13-limit

Commas: 91/90, 169/168, 385/384, 441/440

POTE generator: 235.088

Map: [<2 2 7 6 3 7|, <0 3 -6 -1 10 1|]

EDOs: 10, 46, 102, 148f, 194bcdf

Badness: 0.0289

Compositions:

http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3

(the description says "lemba" which has a similar scale structure but different mapping for 5)

Shrutar

Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. With wedgie <<4 -8 14 -22 11 55||, 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 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.

Commas: 2048/2025, 245/243

POTE generator: 52.811

Map: [<2 1 9 -2|, <0 2 -4 7|]

EDOs: 22, 46, 68, 182b, 250bc

11-limit

Commas: 2048/2025, 245/243, 121/120

POTE generator: 52.680

Map: [<2 1 9 -2 8|, <0 2 -4 7 -1|]

EDOs: 22, 46, 68, 114, 296bce, 410bce

13-limit

Commas: 121/120, 176/175, 196/195, 245/243

POTE generator: ~28/27 = 52.654

Map: [<2 1 9 -2 8 -10|, <0 2 -4 7 -1 16|]

EDOs: 22, 24, 46, 68, 114

Badness: 0.0281

17-limit

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

POTE generator: ~28/27 = 52.647

Map: [<2 1 9 -2 8 -10 6|, <0 2 -4 7 -1 16 2|]

EDOs: 22, 24, 46, 68, 114

Badness: 0.0187

19-limit

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

POTE generator: ~28/27 = 52.730

Map: [<2 1 9 -2 8 -10 6 -10|, <0 2 -4 7 -1 16 2 17|]

EDOs: 22, 24, 46, 68, 114, 182bef

Badness: 0.0175

Sruti

Commas: 2048/2025, 19683/19600

POTE generator: ~175/144 = 351.876

Map: [<2 0 11 -15|, <0 2 -4 13|]

Wedgie: <<4 -8 26 -22 30 83||

EDOs: 24, 34d, 58, 150cd, 208cd, 266cd

Badness: 0.1174

11-limit

Commas: 176/175, 243/242, 896/891

POTE generator: ~11/9 = 351.863

Map: [<2 0 11 -15 -1|, <0 2 -4 13 5|]

EDOs: 24, 34d, 58, 150cde, 208cde

Badness: 0.0415

13-limit

Commas: 144/143, 176/175, 351/350, 676/675

POTE generator: ~11/9 = 351.886

Map: [<2 0 11 -15 -1 9|, <0 2 -4 13 5 -1|]

EDOs: 24, 34d, 58, 150cdef, 208cdef

Badness: 0.0238

Anguirus

Commas: 49/48, 2048/2025

POTE generator: ~8/7 = 246.979

Map: [<2 0 11 4|, <0 2 -4 1|]

Wedgie: <<4 -8 2 -22 -8 27||

EDOs: 10, 24, 34

Badness: 0.0780

11-limit

Commas: 49/48, 56/55, 243/242

POTE generator: ~8/7 = 247.816

Map: [<2 0 11 4 -1|, <0 2 -4 1 5|]

EDOs: 10, 24, 34, 58d, 92de

Badness: 0.0493

13-limit

Commas: 49/48 56/55 91/90 352/351

POTE generator: ~8/7 = 247.691

Map: [<2 0 11 4 -1 9|, <0 2 -4 1 5 -1|]

EDOs: 10, 24, 34, 58d, 92def

Badness: 0.0308

Shru

Commas: 392/375, 1323/1280

POTE generator: ~64/63 = 50.135

Map: [<2 1 9 11|, <0 2 -4 -5|]

Wedgie: <<4 -8 -10 -22 -27 -1||

EDOs: 22d, 24

Badness: 0.1576

11-limit

Commas: 56/55, 77/75, 1323/1280

POTE generator: ~64/63 = 50.130

Map: [<2 1 9 11 8|, <0 2 -4 -5 -1|]

EDOs: 22d, 24

Badness: 0.0635

13-limit

Commas: 56/55, 77/75, 105/104, 507/500

POTE generator: ~64/63 = 50.535

Map: [<2 1 9 11 8 15|, <0 2 -4 -5 -1 -7|]

EDOs: 24

Badness: 0.0457