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.
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
- 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
Subgroup: 2.3.5.17
Comma list: 136/135, 256/255
Sval mapping: [⟨2 0 11 5], ⟨0 1 -2 1]]
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
- 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
- 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
- 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]]
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 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]]
Optimal tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.856
Optimal ET sequence: 22, 58, 80, 138cd, 218cd
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]]
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
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
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
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
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