Diaschismic family: Difference between revisions

Revision as of 14:51, 1 December 2022

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]

Template:Val list

Badness: 0.019915

Overview to extensions

7-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.
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: 256/255, 289/288

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

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

Optimal GPV sequence: Template:Val list

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]

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]]

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 GPV sequence: 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]]

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 GPV sequence: 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]]

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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]

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]]

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 GPV sequence: 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]]

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

Optimal GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: 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]]

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

Optimal GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: 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]]

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

Optimal GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: 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]]

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

Optimal GPV sequence: 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]]

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

Optimal GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 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 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 ]]

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]

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]]

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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

Template:Val list

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

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]]

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

Optimal GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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

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]]

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

Optimal GPV sequence: 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]]

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

Optimal GPV sequence: 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]]

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

Optimal GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 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 ]]

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

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 subgroup: 2.11/7

Optimal GPV sequence: 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]]

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

Optimal GPV sequence: 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]]

Optimal tuning (POTE): ~17/12 = 1\2, ~9/7 = 434.816

Optimal GPV sequence: 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]]

Wedgie: ⟨⟨ 6 -12 -2 -33 -20 29 ]]

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

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

Optimal GPV sequence: 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]]

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

Optimal GPV sequence: 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]]

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

Optimal GPV sequence: Template:Val list

Badness: 0.019304

Compositions

A Stiff Shot of Turpentine play by Peter Kosmorsky

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 and 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

Template:Val list

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 GPV sequence: 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 1 9 -2 8 -10], ⟨0 2 -4 7 -1 16]]

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

Optimal GPV sequence: 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 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 GPV sequence: 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 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 GPV sequence: 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 ]]

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

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]]

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

Optimal GPV sequence: 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]]

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

Optimal GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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

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]]

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

Optimal GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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

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]]

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

Optimal GPV sequence: 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]]

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

Optimal GPV sequence: Template:Val list

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

Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

Badness: 0.022239

@@ Line 7: / Line 7: @@
 [[Mapping]]: [{{val| 2 0 11 }}, {{val| 0 1 -2 }}]
+Mapping generators: ~45/32, ~3
 [[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 704.898
@@ Line 42: / Line 44: @@
 Sval mapping: [{{val| 2 0 11 5 }}], {{val| 0 1 -2 1 }}]
-Sval mapping generators: ~17/12, ~3
 Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 705.1272
@@ Line 52: / Line 52: @@
 == Srutal ==
-{{see also| Srutal vs diaschismic }}
+{{See also| Srutal vs diaschismic }}
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 2048/2025, 4375/4374
-[[Mapping]]: [{{val|2 0 11 -42}}, {{val|0 1 -2 15}}]
+[[Mapping]]: [{{val| 2 0 11 -42 }}, {{val| 0 1 -2 15 }}]
 {{Multival|legend=1| 2 -4 30 -11 42 81 }}
-[[POTE generator]]: ~3/2 = 704.814
+[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 704.814
 [[Tuning ranges]]:
@@ Line 80: / Line 80: @@
 Mapping: [{{val| 2 0 11 -42 -28 }}, {{val| 0 1 -2 15 11 }}]
-POTE generator: ~3/2 = 704.856
+Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.856
 Tuning ranges:
@@ Line 98: / Line 98: @@
 Mapping: [{{val| 2 0 11 -42 -28 -18 }}, {{val| 0 1 -2 15 11 8 }}]
-POTE generator: ~3/2 = 704.881
+Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.881
 Tuning ranges:
@@ Line 117: / Line 117: @@
 Mapping: [{{val| 2 0 11 -42 -28 -18 5 }}, {{val| 0 1 -2 15 11 8 1 }}]
-POTE generator: ~3/2 = 704.840
+Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.840
 Tuning ranges:
@@ Line 137: / Line 137: @@
 Mapping: [{{val| 2 0 11 -42 -28 -18 5 -55 }}, {{val| 0 1 -2 15 11 8 1 20 }}]
-POTE generator: ~3/2 = 704.905
+Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.905
 Optimal GPV sequence: {{Val list| 34dh, 46, 80, 206cd }}
@@ Line 152: / Line 152: @@
 Mapping: [{{val| 2 0 11 -42 -28 -18 5 -55 -10 }}, {{val| 0 1 -2 15 11 8 1 20 6 }}]
-POTE generator: ~3/2 = 704.899
+Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.899
 Optimal GPV sequence: {{Val list| 34dh, 46, 80, 206cd }}
@@ Line 165: / Line 165: @@
 Mapping: [{{val| 2 0 11 -42 -28 -18 5 -55 -10 -76 }}, {{val| 0 1 -2 15 11 8 1 20 6 27 }}]
-POTE generator: ~3/2 = 704.906
+Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.906
 Optimal GPV sequence: {{Val list| 34dhj, 46, 80, 206cd }}
@@ Line 172: / Line 172: @@
 == Pajara ==
-{{main| Pajara }}
+{{Main| 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 {{val| 34 54 79 96 }} and 56 with the val {{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.
@@ Line 178: / Line 178: @@
 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
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 50/49, 64/63
@@ Line 186: / Line 186: @@
 {{Multival|legend=1| 2 -4 -4 -11 -12 2 }}
-[[POTE generator]]: ~3/2 = 707.048
+[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~3/2 = 707.048
 [[Tuning ranges]]:
@@ Line 204: / Line 204: @@
 Mapping: [{{val| 2 0 11 12 26 }}, {{val| 0 1 -2 -2 -6 }}]
-POTE generator: ~3/2 = 706.885
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.885
 Tuning ranges:
@@ Line 222: / Line 222: @@
 Mapping: [{{val| 2 0 11 12 26 1 }}, {{val| 0 1 -2 -2 -6 2 }}]
-POTE generator: ~3/2 = 708.919
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.919
 Optimal GPV sequence: {{Val list| 10e, 12, 22 }}
@@ Line 235: / Line 235: @@
 Mapping: [{{val| 2 0 11 12 26 1 5 }}, {{val| 0 1 -2 -2 -6 2 1 }}]
-POTE generator: ~3/2 = 708.806
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.806
 Optimal GPV sequence: {{Val list| 10e, 12, 22 }}
@@ Line 248: / Line 248: @@
 Mapping: [{{val| 2 0 11 12 26 36 }}, {{val| 0 1 -2 -2 -6 -9 }}]
-POTE generator: ~3/2 = 706.133
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.133
 Optimal GPV sequence: {{Val list| 12f, 22, 34d }}
@@ Line 261: / Line 261: @@
 Mapping: [{{val| 2 0 11 12 26 36 5 }}, {{val| 0 1 -2 -2 -6 -9 1 }}]
-POTE generator: ~3/2 = 706.410
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.410
 Optimal GPV sequence: {{Val list| 12f, 22, 34d }}
@@ Line 274: / Line 274: @@
 Mapping: [{{val|2 0 11 12 26 17}}, {{val|0 1 -2 -2 -6 -3}}]
-POTE generator: ~3/2 = 707.450
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.450
 Optimal GPV sequence: {{Val list| 10e, 12f, 22f }}
@@ Line 287: / Line 287: @@
 Mapping: [{{val|2 0 11 12 26 17 5}}, {{val|0 1 -2 -2 -6 -3 1}}]
-POTE generator: ~3/2 = 707.947
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.947
 Optimal GPV sequence: {{Val list| 10e, 12f, 22f }}
@@ Line 300: / Line 300: @@
 Mapping: [{{val| 2 0 11 12 -9 }}, {{val| 0 1 -2 -2 5 }}]
-POTE generator: ~3/2 = 709.578
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 709.578
 Tuning ranges:
@@ Line 318: / Line 318: @@
 Mapping: [{{val| 2 0 11 12 -9 1 }}, {{val| 0 1 -2 -2 5 2 }}]
-POTE generator: ~3/2 = 710.240
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.240
 Optimal GPV sequence: {{Val list| 10, 22, 54f, 76bdff }}
@@ Line 331: / Line 331: @@
 Mapping: [{{val| 2 0 11 12 -9 1 5 }}, {{val| 0 1 -2 -2 5 2 1 }}]
-POTE generator: ~3/2 = 710.221
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.221
 Optimal GPV sequence: {{Val list| 10, 22, 54f, 76bdff }}
@@ Line 344: / Line 344: @@
 Mapping: [{{val| 2 0 11 12 -9 17 }}, {{val| 0 1 -2 -2 5 -3 }}]
-POTE generator ~3/2 = 710.818
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.818
 Optimal GPV sequence: {{Val list| 10, 22f, 32f, 54ff }}
@@ Line 357: / Line 357: @@
 Mapping: [{{val| 2 0 11 12 -9 17 5 }}, {{val| 0 1 -2 -2 5 -3 1 }}]
-POTE generator ~3/2 = 710.866
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.866
 Optimal GPV sequence: {{Val list| 10, 22f, 32f, 54ff }}
@@ Line 370: / Line 370: @@
 Mapping: [{{val| 2 0 11 12 7 }}, {{val| 0 1 -2 -2 0 }}]
-POTE generator: ~3/2 = 705.524
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 705.524
 Optimal GPV sequence: {{Val list| 10, 12, 22e, 34dee }}
@@ Line 383: / Line 383: @@
 Mapping: [{{val| 2 0 11 12 7 17 }}, {{val| 0 1 -2 -2 0 -3 }}]
-POTE generator: ~3/2 = 707.442
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.442
 Optimal GPV sequence: {{Val list| 10, 12f, 22ef }}
@@ Line 396: / Line 396: @@
 Mapping: [{{val| 2 0 11 12 7 17 5 }}, {{val| 0 1 -2 -2 0 -3 1 }}]
-POTE generator: ~3/2 = 708.544
+Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.544
 Optimal GPV sequence: {{Val list| 10, 12f, 22ef }}
@@ Line 409: / Line 409: @@
 Mapping: [{{val| 2 1 9 10 8 }}, {{val| 0 2 -4 -4 -1 }}]
-POTE generator: ~11/8 = 546.383
+Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 546.383
 Optimal GPV sequence: {{Val list| 20, 22, 68d, 90d }}
@@ Line 422: / Line 422: @@
 Mapping: [{{val| 2 0 11 12 -1 }}, {{val| 0 2 -4 -4 5 }}]
-POTE generator: ~64/55 = 246.907
+Optimal tuning (POTE): ~7/5 = 1\2, ~55/32 = 953.093
 Optimal GPV sequence: {{Val list| 10, 24d, 34d }}
@@ Line 435: / Line 435: @@
 Mapping: [{{val| 2 0 11 12 -1 9 }}, {{val| 0 2 -4 -4 5 -1 }}]
-POTE generator: ~15/13 = 246.926
+Optimal tuning (POTE): ~7/5 = 1\2, ~26/15 = 953.074
 Optimal GPV sequence: {{Val list| 10, 24d, 34d }}
@@ Line 448: / Line 448: @@
 Mapping: [{{val| 2 0 11 12 -1 9 5 }}, {{val| 0 2 -4 -4 5 -1 2 }}]
-POTE generator: ~15/13 = 246.790
+Optimal tuning (POTE): ~7/5 = 1\2, ~26/15 = 953.210
 Optimal GPV sequence: {{Val list| 10, 24d, 34d }}
@@ Line 461: / Line 461: @@
 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
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 126/125, 2048/2025
@@ Line 469: / Line 469: @@
 {{Multival|legend=1| 2 -4 -16 -11 -31 -26 }}
-[[POTE generator]]: ~3/2 = 703.681
+[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 703.681
 [[Tuning ranges]]:
@@ Line 487: / Line 487: @@
 Mapping: [{{val| 2 0 11 31 45 }}, {{val| 0 1 -2 -8 -12 }}]
-POTE generator: ~3/2 = 703.714
+Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.714
 Tuning ranges:
@@ Line 505: / Line 505: @@
 Mapping: [{{val| 2 0 11 31 45 55 }}, {{val| 0 1 -2 -8 -12 -15 }}]
-POTE generator: ~3/2 = 703.704
+Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.704
 Tuning ranges:
@@ Line 524: / Line 524: @@
 Mapping: [{{val| 2 0 11 31 45 55 5 }}, {{val| 0 1 -2 -8 -12 -15 1 }}]
-POTE generator: ~3/2 = 703.812
+Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.812
 Tuning ranges:
@@ Line 540: / Line 540: @@
 Subgroup: 2.3.5.7.11.13.17.23
-[[Comma list]]: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255
+Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255
-[[Mapping|Sval mapping]]: [{{val| 2 0 11 31 45 55 5 63 }}, {{val| 0 1 -2 -8 -12 -15 1 -17 }}]
+Sval mapping: [{{val| 2 0 11 31 45 55 5 63 }}, {{val| 0 1 -2 -8 -12 -15 1 -17 }}]
-[[POTE generator|POL2 generator]]: ~3/2 = 703.870
+Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.870
 {{Val list|legend=1| 46, 58i, 104ci }}
@@ Line 551: / Line 551: @@
 Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the 22&amp;56 temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where keen, {{multival| 2 -4 18 -12 … }}, is really more interesting, adding 100/99 and 385/384 to the commas.
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 875/864, 2048/2025
-[[Mapping]]: [{{val|2 0 11 -23}}, {{val|0 1 -2 9}}]
+[[Mapping]]: [{{val| 2 0 11 -23 }}, {{val| 0 1 -2 9 }}]
 {{Multival|legend=1| 2 -4 18 -11 23 53 }}
-[[POTE generator]]: ~3/2 = 707.571
+[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 707.571
 {{Val list|legend=1| 22, 56, 78, 134b, 212b, 290bb }}
@@ Line 572: / Line 572: @@
 Mapping: [{{val|2 0 11 -23 26}}, {{val|0 1 -2 9 -6}}]
-POTE generator: ~3/2 = 707.609
+Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.609
 Optimal GPV sequence: {{Val list| 22, 56, 78, 212be, 290bbe }}
@@ Line 585: / Line 585: @@
 Mapping: [{{val|2 0 11 -23 26 -18}}, {{val|0 1 -2 9 -6 8}}]
-POTE generator: ~3/2 = 707.167
+Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.167
 Optimal GPV sequence: {{Val list| 22f, 34, 56f }}
@@ Line 598: / Line 598: @@
 Mapping: [{{val|2 0 11 -23 26 -18 5}}, {{val|0 1 -2 9 -6 8 1}}]
-POTE generator: ~3/2 = 707.155
+Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 707.155
 Optimal GPV sequence: {{Val list| 22f, 34, 56f }}
@@ Line 611: / Line 611: @@
 Mapping: [{{val|2 0 11 -23 26 36}}, {{val|0 1 -2 9 -6 -9}}]
-POTE generator: ~3/2 = 707.257
+Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.257
 Optimal GPV sequence: {{Val list| 22, 34, 56 }}
@@ Line 624: / Line 624: @@
 Mapping: [{{val|2 0 11 -23 26 36 5}}, {{val|0 1 -2 9 -6 -9 1}}]
-POTE generator: ~3/2 = 707.252
+Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 707.252
 Optimal GPV sequence: {{Val list| 22, 34, 56 }}
@@ Line 633: / Line 633: @@
 Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the 12&amp;56 temperament.
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 2048/2025, 3136/3125
-[[Mapping]]: [{{val|4 0 22 43}}, {{val|0 1 -2 -5}}]
+[[Mapping]]: [{{val| 4 0 22 43 }}, {{val| 0 1 -2 -5 }}]
-{{Multival|legend=1|4 -8 -20 -22 -43 -24}}
+{{Multival|legend=1| 4 -8 -20 -22 -43 -24 }}
-[[POTE generator]]: ~3/2 = 705.364
+[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~3/2 = 705.364
 {{Val list|legend=1| 12, 56, 68, 80, 148d }}
@@ Line 652: / Line 652: @@
 Comma list: 176/175, 896/891, 1375/1372
-Mapping: [{{val|4 0 22 43 71}}, {{val|0 1 -2 -5 -9}}]
+Mapping: [{{val| 4 0 22 43 71 }}, {{val| 0 1 -2 -5 -9 }}]
-POTE generator: ~3/2 = 705.087
+Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.087
 Optimal GPV sequence: {{Val list| 12, 68, 80 }}
@@ Line 665: / Line 665: @@
 Comma list: 176/175, 325/324, 640/637, 896/891
-Mapping: [{{val|4 0 22 43 71 -36}}, {{val|0 1 -2 -5 -9 8 }}]
+Mapping: [{{val| 4 0 22 43 71 -36 }}, {{val| 0 1 -2 -5 -9 8 }}]
-POTE generator: ~3/2 = 705.301
+Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.301
 Optimal GPV sequence: {{Val list| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
@@ Line 678: / Line 678: @@
 Comma list: 136/135, 176/175, 256/255, 325/324, 640/637
-Mapping: [{{val|4 0 22 43 71 -36 10 }}, {{val|0 1 -2 -5 -9 8 1 }}]
+Mapping: [{{val| 4 0 22 43 71 -36 10 }}, {{val| 0 1 -2 -5 -9 8 1 }}]
-POTE generator: ~3/2 = 705.334
+Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.334
 Optimal GPV sequence: {{Val list| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
@@ Line 691: / Line 691: @@
 Comma list: 136/135, 176/175, 190/189, 256/255, 325/324, 640/637
-Mapping: [{{val|4 0 22 43 71 -36 10 17 }}, {{val|0 1 -2 -5 -9 8 1 0 }}]
+Mapping: [{{val| 4 0 22 43 71 -36 10 17 }}, {{val| 0 1 -2 -5 -9 8 1 0 }}]
-POTE generator: ~3/2 = 705.339
+Optimal tuning (POTE): ~19/16 = 1\4, ~3/2 = 705.339
 Optimal GPV sequence: {{Val list| 12, 68, 80, 148d, 376bbcddfh }}
@@ Line 704: / Line 704: @@
 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
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 1728/1715, 2048/2025
-[[Mapping]]: [{{val|2 1 9 2}}, {{val|0 3 -6 5}}]
+[[Mapping]]: [{{val| 2 1 9 2 }}, {{val| 0 3 -6 5 }}]
 {{Multival|legend=1| 6 -12 10 -33 -1 57 }}
-[[POTE generator]]: ~9/7 = 434.856
+[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~9/7 = 434.856
 {{Val list|legend=1| 22, 58, 80, 138cd, 218cd }}
@@ Line 723: / Line 723: @@
 Comma list: 176/175, 540/539, 896/891
-Mapping: [{{val|2 1 9 2 12}}, {{val|0 3 -6 5 -7}}]
+Mapping: [{{val| 2 1 9 2 12 }}, {{val| 0 3 -6 5 -7 }}]
-POTE generator: ~9/7 = 434.852
+Optimal tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.852
 Minimax tuning:
-* 11-odd-limit: ~9/7 = {{monzo|5/12 0 0 1/12 -1/12}}
+* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
-: [{{monzo|1 0 0 0 0}}, {{monzo|7/4 0 0 1/4 -1/4}}, {{monzo|2 0 0 -1/2 1/2}}, {{monzo|37/12 0 0 5/12 -5/12}}, {{monzo|37/12 0 0 -7/12 7/12}}]
+: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]
-: Eigenmonzos: 2, 11/7
+: Eigenmonzo subgroup: 2.11/7
 Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde, 218cde }}
@@ Line 741: / Line 741: @@
 Comma list: 176/175, 351/350, 364/363, 540/539
-Mapping: [{{val|2 1 9 2 12 19}}, {{val|0 3 -6 5 -7 -16}}]
+Mapping: [{{val| 2 1 9 2 12 19 }}, {{val| 0 3 -6 5 -7 -16 }}]
-POTE generator: ~9/7 = 434.756
+Optimal tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.756
 Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde }}
@@ Line 754: / Line 754: @@
 Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
-Mapping: [{{val|2 1 9 2 12 19 6}}, {{val|0 3 -6 5 -7 -16 3}}]
+Mapping: [{{val| 2 1 9 2 12 19 6 }}, {{val| 0 3 -6 5 -7 -16 3 }}]
-POTE generator: ~9/7 = 434.816
+Optimal tuning (POTE): ~17/12 = 1\2, ~9/7 = 434.816
 Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde }}
@@ Line 763: / Line 763: @@
 == Echidnic ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 686/675, 1029/1024
-[[Mapping]]: [{{val|2 2 7 6}}, {{val|0 3 -6 -1}}]
+[[Mapping]]: [{{val| 2 2 7 6 }}, {{val| 0 3 -6 -1 }}]
 {{Multival|legend=1| 6 -12 -2 -33 -20 29 }}
-[[POTE generator]]: ~8/7 = 234.492
+[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~8/7 = 234.492
 {{Val list|legend=1| 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bccdd }}
@@ Line 782: / Line 782: @@
 Comma list: 385/384, 441/440, 686/675
-Mapping: [{{val|2 2 7 6 3}}, {{val|0 3 -6 -1 10}}]
+Mapping: [{{val| 2 2 7 6 3 }}, {{val| 0 3 -6 -1 10 }}]
-POTE generator: ~8/7 = 235.096
+Optimal tuning (POTE): ~45/32 = 1\2, ~8/7 = 235.096
 Optimal GPV sequence: {{Val list| 10, 36e, 46, 102, 148, 342bcdd }}
@@ Line 795: / Line 795: @@
 Comma list: 91/90, 169/168, 385/384, 441/440
-Mapping: [{{val|2 2 7 6 3 7}}, {{val|0 3 -6 -1 10 1}}]
+Mapping: [{{val| 2 2 7 6 3 7 }}, {{val| 0 3 -6 -1 10 1 }}]
-POTE generator: ~8/7 = 235.088
+Optimal tuning (POTE): ~45/32 = 1\2, ~8/7 = 235.088
 Optimal GPV sequence: {{Val list| 10, 46, 102, 148f, 194bcdf }}
@@ Line 808: / Line 808: @@
 Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
-Mapping: [{{val|2 2 7 6 3 7 7}}, {{val|0 3 -6 -1 10 1 3}}]
+Mapping: [{{val| 2 2 7 6 3 7 7 }}, {{val| 0 3 -6 -1 10 1 3 }}]
-POTE generator: ~8/7 = 235.088
+Optimal tuning (POTE): ~17/12 = 1\2, ~8/7 = 235.088
 Optimal GPV sequence: {{Val list| 10, 46, 102, 148f, 194bcdf }}
@@ Line 820: / Line 820: @@
 == Shrutar ==
-Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as 22&amp;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<sup>(1/7)</sup>, making 7s just.
+Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as 22&amp;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<sup>(1/7)</sup>, 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<sup>(1/7)</sup> generator can again be used as tunings.
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 245/243, 2048/2025
-[[Mapping]]: [{{val| 2 3 5 5 }}, {{val| 0 2 -4 7 }}]
+[[Mapping]]: [{{val| 2 1 9 -2 }}, {{val| 0 2 -4 7 }}]
 {{Multival|legend=1| 4 -8 14 -22 11 55 }}
-[[POTE generator]]: ~36/35 = 52.811
+[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~35/24 = 652.811
 {{Val list|legend=1| 22, 46, 68, 182b, 250bc }}
@@ Line 843: / Line 843: @@
 Comma list: 121/120, 176/175, 245/243
-Mapping: [{{val| 2 3 5 5 7 }}, {{val| 0 2 -4 7 -1 }}]
+Mapping: [{{val| 2 1 9 -2 8 }}, {{val| 0 2 -4 7 -1 }}]
-POTE generator: ~33/32 = 52.680
+Optimal tuning (POTE): ~45/32 = 1\2, ~16/11 = 652.680
 Optimal GPV sequence: {{Val list| 22, 46, 68, 114, 296bce, 410bce }}
@@ Line 856: / Line 856: @@
 Comma list: 121/120, 176/175, 196/195, 245/243
-Mapping: [{{val| 2 3 5 5 7 6 }}, {{val| 0 2 -4 7 -1 16 }}]
+Mapping: [{{val| 2 1 9 -2 8 -10 }}, {{val| 0 2 -4 7 -1 16 }}]
-POTE generator: ~33/32 = 52.654
+Optimal tuning (POTE): ~45/32 = 1\2, ~16/11 = 652.654
 Optimal GPV sequence: {{Val list| 22f, 24f, 46, 68, 114 }}
@@ Line 869: / Line 869: @@
 Comma list: 121/120, 136/135, 154/153, 176/175, 196/195
-Mapping: [{{val| 2 3 5 5 7 6 8 }}, {{val| 0 2 -4 7 -1 16 2 }}]
+Mapping: [{{val| 2 1 9 -2 8 -10 6 }}, {{val| 0 2 -4 7 -1 16 2 }}]
-POTE generator: ~33/32 = 52.647
+Optimal tuning (POTE): ~17/12 = 1\2, ~16/11 = 652.647
 Optimal GPV sequence: {{Val list| 22f, 24f, 46, 68, 114 }}
@@ Line 882: / Line 882: @@
 Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342
-Mapping: [{{val| 2 3 5 5 7 6 8 7 }}, {{val| 0 2 -4 7 -1 16 2 17 }}]
+Mapping: [{{val| 2 1 9 -2 8 -10 6 -10 }}, {{val| 0 2 -4 7 -1 16 2 17 }}]
-POTE generator: ~33/32 = 52.730
+Optimal tuning (POTE): ~17/12 = 1\2, ~16/11 = 652.730
 Optimal GPV sequence: {{Val list| 22fh, 24fh, 46, 68, 114, 182bef }}
@@ Line 891: / Line 891: @@
 == Sruti ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 2048/2025, 19683/19600
@@ Line 899: / Line 899: @@
 {{Multival|legend=1| 4 -8 26 -22 30 83 }}
-POTE generator: ~175/144 = 351.876
+[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~140/81 = 951.876
 {{Val list|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}
@@ Line 912: / Line 912: @@
 Mapping: [{{val| 2 0 11 -15 -1 }}, {{val| 0 2 -4 13 5 }}]
-POTE generator: ~11/9 = 351.863
+Optimal tuning (POTE): ~45/32 = 1\2, ~121/70 = 951.863
 Optimal GPV sequence: {{Val list| 24, 34d, 58 }}
@@ Line 925: / Line 925: @@
 Mapping: [{{val| 2 0 11 -15 -1 9 }}, {{val| 0 2 -4 13 5 -1 }}]
-POTE generator: ~11/9 = 351.886
+Optimal tuning (POTE): ~45/32 = 1\2, ~26/15 = 951.886
 Optimal GPV sequence: {{Val list| 24, 34d, 58, 150cdeef, 208ccddeeff }}
@@ Line 938: / Line 938: @@
 Mapping: [{{val| 2 0 11 -15 -1 9 5 }}, {{val| 0 2 -4 13 5 -1 2 }}]
-POTE generator: ~11/9 = 351.857
+Optimal tuning (POTE): ~17/12 = 1\2, ~26/15 = 951.857
 Optimal GPV sequence: {{Val list| 24, 34d, 58 }}
@@ Line 945: / Line 945: @@
 == Anguirus ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 49/48, 2048/2025
@@ Line 953: / Line 953: @@
 {{Multival|legend=1| 4 -8 2 -22 -8 27 }}
-[[POTE generator]]: ~8/7 = 246.979
+[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~7/4 = 953.021
 {{Val list|legend=1| 10, 24, 34 }}
@@ Line 966: / Line 966: @@
 Mapping: [{{val| 2 0 11 4 -1 }}, {{val| 0 2 -4 1 5 }}]
-POTE generator: ~8/7 = 247.816
+Optimal tuning (POTE): ~45/32 = 1\2, ~7/4 = 952.184
 Optimal GPV sequence: {{Val list| 10, 24, 34, 58d, 92de }}
@@ Line 979: / Line 979: @@
 Mapping: [{{val| 2 0 11 4 -1 9 }}, {{val| 0 2 -4 1 5 -1 }}]
-POTE generator: ~8/7 = 247.691
+Optimal tuning (POTE): ~45/32 = 1\2, ~7/4 = 952.309
 Optimal GPV sequence: {{Val list| 10, 24, 34, 58d, 92ddef }}
@@ Line 992: / Line 992: @@
 Mapping: [{{val| 2 0 11 4 -1 9 5 }}, {{val| 0 2 -4 1 5 -1 2 }}]
-POTE generator: ~8/7 = 247.670
+Optimal tuning (POTE): ~17/12 = 1\2, ~7/4 = 952.330
 Optimal GPV sequence: {{Val list| 10, 24, 34, 58d, 92ddef }}
@@ Line 999: / Line 999: @@
 == Shru ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 392/375, 1323/1280
@@ Line 1,007: / Line 1,007: @@
 {{Multival|legend=1| 4 -8 -10 -22 -27 -1 }}
-[[POTE generator]]: ~64/63 = 50.135
+[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~10/7 = 650.135
 {{Val list|legend=1| 2, 22d, 24 }}
@@ Line 1,020: / Line 1,020: @@
 Mapping: [{{val| 2 1 9 11 8 }}, {{val| 0 2 -4 -5 -1 }}]
-POTE generator: ~33/32 = 50.130
+Optimal tuning (POTE): ~45/32 = 1\2, ~10/7 = 650.130
 Optimal GPV sequence: {{Val list| 2, 22d, 24 }}
@@ Line 1,033: / Line 1,033: @@
 Mapping: [{{val| 2 1 9 11 8 15 }}, {{val| 0 2 -4 -5 -1 -7 }}]
-POTE generator: ~33/32 = 50.535
+Optimal tuning (POTE): ~45/32 = 1\2, ~10/7 = 650.535
 Optimal GPV sequence: {{Val list| 22df, 24 }}
@@ Line 1,040: / Line 1,040: @@
 == Quadrasruta ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 2048/2025, 2401/2400
-[[Mapping]]: [{{val|2 0 11 8}}, {{val|0 4 -8 -3}}]
+[[Mapping]]: [{{val| 2 0 11 8 }}, {{val| 0 4 -8 -3 }}]
 {{Multival|legend=1| 8 -16 -6 -44 -32 31 }}
-[[POTE generator]]: ~15/14 = 123.784
+[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~21/16 = 476.216
 {{Val list|legend=1| 10, 38c, 48c, 58, 68, 126 }}
@@ Line 1,059: / Line 1,059: @@
 Comma list: 176/175, 896/891, 2401/2400
-Mapping: [{{val|2 0 11 8 22}}, {{val|0 4 -8 -3 -19}}]
+Mapping: [{{val| 2 0 11 8 22 }}, {{val| 0 4 -8 -3 -19 }}]
-POTE generator: ~15/14 = 123.882
+Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.118
 Optimal GPV sequence: {{Val list| 58, 126, 184c, 310bccde }}
@@ Line 1,072: / Line 1,072: @@
 Comma list: 176/175, 196/195, 512/507, 676/675
-Mapping: [{{val|2 0 11 8 22 9}}, {{val|0 4 -8 -3 -19 -2}}]
+Mapping: [{{val| 2 0 11 8 22 9 }}, {{val| 0 4 -8 -3 -19 -2 }}]
-POTE generator: ~14/13 = 123.901
+Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.099
 Optimal GPV sequence: {{Val list| 58, 126f, 184cff }}
@@ Line 1,085: / Line 1,085: @@
 Comma list: 136/135, 170/169, 176/175, 196/195, 256/255
-Mapping: [{{val|2 0 11 8 22 9 5}}, {{val|0 4 -8 -3 -19 -2 4}}]
+Mapping: [{{val| 2 0 11 8 22 9 5 }}, {{val| 0 4 -8 -3 -19 -2 4 }}]
-POTE generator: ~14/13 = 123.838
+Optimal tuning (POTE): ~17/12 = 1\2, ~21/16 = 476.162
 Optimal GPV sequence: {{Val list| 58, 126f }}
@@ Line 1,098: / Line 1,098: @@
 Comma list: 243/242, 441/440, 2048/2025
-Mapping: [{{val|2 0 11 8 -1}}, {{val|0 4 -8 -3 10}}]
+Mapping: [{{val| 2 0 11 8 -1 }}, {{val| 0 4 -8 -3 10 }}]
-POTE generator: ~15/14 = 123.983
+Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.017
 Optimal GPV sequence: {{Val list| 10, 38c, 48c, 58 }}
@@ Line 1,111: / Line 1,111: @@
 Comma list: 144/143, 196/195, 243/242, 676/675
-Mapping: [{{val|2 0 11 8 -1 9}}, {{val|0 4 -8 -3 10 -2}}]
+Mapping: [{{val| 2 0 11 8 -1 9 }}, {{val| 0 4 -8 -3 10 -2 }}]
-POTE generator: ~14/13 = 123.972
+Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.028
 Optimal GPV sequence: {{Val list| 10, 38c, 48c, 58 }}
@@ Line 1,124: / Line 1,124: @@
 Comma list: 136/135, 144/143, 170/169, 196/195, 221/220
-Mapping: [{{val|2 0 11 8 -1 9 5}}, {{val|0 4 -8 -3 10 -2 4}}]
+Mapping: [{{val| 2 0 11 8 -1 9 5 }}, {{val| 0 4 -8 -3 10 -2 4 }}]
-POTE generator: ~14/13 = 123.923
+Optimal tuning (POTE): ~17/12 = 1\2, ~21/16 = 476.077
 Optimal GPV sequence: {{Val list| 10, 38c, 48c, 58, 126eef, 184ceeff }}