5L 2s/Temperaments

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Below are some important rank-2 temperaments with optimal generator size in the diatonic (5L 2s) range (the period is always 1\1 for temperaments with this MOS structure). The temperaments are listed following the 5L 2s scale tree, in order of increasing generator size. The top-level temperaments are the most important and obvious divisions in diatonic tunings. Child temperaments are higher-complexity extensions of low-complexity parent temperaments, with new JI readings for intervals further out in the generator chain. These are finer adjustments of the major, parent temperaments, thus are less useful when the composer chooses not to use a long generator chain in the music.

Meantone

Main article: Meantone

Subgroup: 2.3.5

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.239

EDO generators: 7\12, 11\19, 18\31, 25\43, 29\50

Scales (Scala files): Meantone5, Meantone7, Meantone12

Interval table (7-note MOS, 2.3.5.7 POTE tuning)
# Cents* Approximate ratios**
0 0.00 1/1
1 696.2 3/2
2 192.5 9/8, 10/9
3 888.7 5/3
4 385.0 5/4
5 1081.2 15/8
6 577.4 25/18

* Octave-reduced

* 2.3.5, odd limit ≤ 27

Comma list: 81/80

Mapping: [⟨1 0 -4], ⟨0 1 4]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨1 4 4]]

Tuning ranges:

  • 5-odd-limit diamond monotone: ~3/2 = [685.714, 720.000] (4\7 to 3\5)
  • 5-odd-limit diamond tradeoff: ~3/2 = [694.786, 701.955]
  • 5-odd-limit diamond monotone and tradeoff: ~3/2 = [694.786, 701.955]

Optimal ET sequence: 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb

Badness: 0.007381

Flattone

Subgroup: 2.3.5.7 or 2.3.5.7.13

Period: 1\1

Optimal (POTE) generator: ~3/2 = 693.7498

EDO generators: 11\19, 15\26, 26\45, 37\64

Scales (Scala files): Flattone12

Interval table (12-note MOS, 2.3.5.7.13 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 693.7 3/2
2 187.5 9/8, 10/9
3 881.2 5/3
4 375.0 5/4, 16/13
5 1068.7 15/8, 24/13
6 562.5 18/13
7 56.2
8 750.0 20/13
9 243.7 8/7
10 937.5 12/7
11 431.2 9/7

* Octave-reduced

** 2.3.5.7.13, odd limit ≤ 27

Comma list: 65/64, 81/80, 105/104

Gencom: [2 3/2; 65/64 81/80 105/104]

Gencom mapping: [⟨1 1 0 8 6], ⟨0 1 4 -9 -4]]

Minimax tuning:

Eigenmonzos (unchanged-intervals): 2, 7/5
Eigenmonzos (unchanged-intervals): 2, 9/7

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
  • 7-odd-limit diamond tradeoff: ~3/2 = [692.353, 701.955]
  • 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
  • 7-odd-limit diamond monotone and tradeoff: ~3/2 = [692.353, 694.737]
  • 9-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]

Algebraic generator: Squarto, the positive root of 8x2 - 4x - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.

Optimal ET sequence: 7, 19, 26, 45f

RMS error: 1.742 cents

Septimal meantone

Subgroup: 2.3.5.7

Period: 1\1

Optimal (POTE) generator: 696.495

EDO generators: 7\12, 11\19, 18\31, 25\43, 29\50

Scales (Scala files): Meantone5, Meantone7, Meantone12

Interval table (12-note MOS, 2.3.5.7 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 696.5 3/2
2 193.0 9/8, 10/9
3 889.5 5/3
4 386.0 5/4
5 1082.5 15/8, 28/15
6 579.0 7/5
7 75.5 21/20, 25/24, 28/27
8 772.0 14/9, 25/16
9 268.5 7/6
10 965.0 7/4
11 461.4 21/16

* Octave-reduced

** 2.3.5.7, odd limit ≤ 27

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]

Mapping generators: ~2, ~3

Minimax tuning:

Eigenmonzos (unchanged-intervals): 2, 5

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
  • 7-odd-limit diamond tradeoff: ~3/2 = [694.786, 701.955]
  • 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
  • 7-odd-limit diamond monotone and tradeoff: ~3/2 = [694.786, 700.000]
  • 9-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]

Algebraic generator: Cybozem, the real root of 15x3 - 10x2 - 18, which comes to 503.4257 cents. The recurrence converges quickly.

Optimal ET sequence: 12, 19, 31, 81, 112b, 143b

Badness: 0.013707

Meanpop

Subgroup: 2.3.5.7.11

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.434

EDO generators: 29\50, 40\69, 47\81

Mapping: Same as septimal meantone, plus -13 gens = 11/8

Comma list: 81/80, 126/125, 385/384

Mapping: [⟨1 0 -4 -13 24], ⟨0 1 4 10 -13]]

Mapping generator: ~2, ~3

Minimax tuning:

Eigenmonzos (unchanged-intervals): 2, 5

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
  • 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
  • 11-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 696.774]

Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x3 + 6x - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

Optimal ET sequence: 12e, 19, 31, 81

Badness: 0.021543

Huygens

Subgroup: 2.3.5.7.11

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.967

EDO generators: 25\43, 43\74

Mapping: Same as septimal meantone, plus 18 gens = 11/8

Comma list: 81/80, 99/98, 126/125

Mapping: [⟨1 0 -4 -13 -25], ⟨0 1 4 10 18]]

Mapping generators: ~2, ~3

Minimax tuning:

Eigenmonzos (unchanged-intervals): 2, 11/9

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
  • 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
  • 11-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 700.000]

Algebraic generator: Traverse, the positive real root of x4 + 2x - 13, or 696.9529 cents.

Optimal ET sequence: 12, 19e, 31, 105, 136b, 167be, 198be

Badness: 0.017027

Schismic

Subgroup: 2.3.5.7.11.13.19

Period: 1\1

Optimal (POTE) generator: ~3/2 = 702.1044

EDO generators: 24\41, 31\53, 55\94

Scales: Garibaldi12, Garibaldi17

Interval table (29-note MOS, 2.3.5.7.11.13.19 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 702.10 3/2
2 204.21 9/8
3 906.31 27/16, 32/19
4 408.42
5 1110.52
6 612.63 10/7
7 114.73 15/14, 16/15
8 816.84 8/5
9 318.94 6/5
10 1021.04 9/5
11 523.15 27/20
12 25.25 81/80
13 727.36 32/21
14 229.462 8/7
15 931.57 12/7
16 433.67 9/7
17 1135.77 54/28
18 637.88 13/9
19 139.98 13/12
20 842.09 13/8
21 344.19 11/9, 39/32
22 1046.30 11/6
23 548.40 11/8, 26/19
24 50.51 33/32
25 752.61
26 254.714 22/19
27 956.82 26/15
28 458.92 13/10

* Octave-reduced

** In the 2.3.5.7.11.13.19 subgroup

Comma list: 190/189, 209/208, 225/224, 275/273, 325/324

Gencom: [2 3/2; 190/189 209/208 225/224 275/273 325/324]

Gencom mapping: [⟨1 1 7 11 -10 -8 6], ⟨0 1 -8 -14 23 20 -3]]

Optimal ET sequence: 41, 53, 94

RMS error: 0.6486 cents

Parapyth

Subgroup: 2.3.7.11.13

Period: 1\1

Optimal (POTE) generator: ~3/2 = 704.745

EDO generators: 10\17, 17\29, 27\46

Interval table (17-note MOS, 2.3.7.11.13 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 704.7 3/2
2 209.5 9/8
3 914.2 22/13
4 419.0 14/11
5 1123.7
6 628.5 13/9, (23/16)
7 133.2 13/12, 14/13
8 838.0 13/8
9 342.7 11/9
10 1047.5 11/6
11 552.2 11/8
12 56.9 28/27
13 761.7 14/9
14 266.4 7/6
15 971.2 7/4
16 475.9 21/16

* Octave-reduced

** 2.3.7.11.13, odd limit ≤ 27

Comma list: 169/168, 352/351, 364/363

Gencom: [2 3/2; 169/168 352/351 364/363]

Gencom mapping: [⟨1 1 -6 -3 -1], ⟨0 1 15 11 8]]

Optimal ET sequence: 17, 46, 63

RMS error: 0.7541 cents

Archy

Subgroup: 2.3.7

Period: 1\1

Optimal (POTE) generator: ~3/2 = 709.321

EDO generators: 10\17, 13\22, 16\27

Scales: Archy5, Archy7, Archy12

Interval table (7-note MOS, 2.3.7 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 709.3 3/2
2 218.6 9/8, 8/7
3 927.8 12/7
4 437.3 9/7
5 1146.6 27/14
6 655.9

* Octave-reduced

** 2.3.7, odd limit ≤ 27

Comma: 64/63

Gencom: [2 3/2; 64/63]

Gencom mapping: [⟨1 1 4], ⟨0 1 -2]]

Optimal ET sequence: 5, 12, 17, 22, 27, 137bd

RMS error: 1.856 cents

Supra

Subgroup: 2.3.7.11

Period: 1\1

Optimal (POTE) generator: ~3/2 = 707.192

EDO generators: 10\17, 13\22, 23\39

Scales: Supra7, Supra12

Interval table (12-note MOS, 2.3.7.11 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 707.2 3/2
2 214.4 9/8, 8/7
3 921.6 12/7
4 428.8 9/7, 14/11
5 1136.0 27/14
6 643.2 16/11
7 150.3 12/11
8 857.5 18/11
9 364.7
10 1071.9
11 579.1

* Octave-reduced

** 2.3.7.11, odd limit ≤ 27

Comma list: 64/63, 99/98

Gencom: [2 3/2; 64/63 99/98]

Gencom mapping: [⟨1 1 4 7], ⟨0 1 -2 -6]]

Optimal ET sequence: 5, 12, 17, 39c, 56d

RMS error: 1.977 cents

Superpyth

Subgroup: 2.3.5.7

Period: 1\1

Optimal (POTE) generator: ~3/2 = 710.291

EDO generators: 13\22, 16\27, 29\49

Interval table (12-note MOS, 2.3.5.7 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 710.3 3/2
2 220.6 9/8, 8/7
3 930.9 12/7
4 441.2 9/7
5 1151.5
6 661.7 40/27
7 172.0 10/9
8 882.3 5/3
9 392.6 5/4
10 1102.9 15/8
11 613.2 10/7

* Octave-reduced

** 2.3.5.7, odd limit ≤ 27

Comma list: 64/63, 245/243

Mapping: [⟨1 1 -3 4], ⟨0 1 9 -2]]

Wedgie: ⟨⟨1 9 -2 12 -6 -30]]

Vals: 5, 17, 22, 27, 49

Badness: 0.032318

Ultrapyth

Subgroup: 2.3.5.7.13

Period: 1\1

Optimal (POTE) generator: ~3/2 = 713.745

EDO generators: 19\32, 22\37

Interval table (22-note MOS, 2.3.5.7.13 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 713.7 3/2
2 227.5 9/8, 8/7
3 941.2 12/7, 26/15
4 455.0 9/7, 13/10
5 1168.7
6 682.5
7 196.2
8 910.0
9 423.7
10 1137.5
11 651.2
12 164.9 10/9
13 878.7 5/3
14 392.4 5/4
15 1106.2 15/8
16 619.9 10/7, 13/9
17 133.7 15/14, 13/12
18 847.4 13/8
19 361.2
20 1074.9 13/7
21 588.7

* Octave-reduced

** 2.3.5.7.13, odd limit ≤ 15

Comma list: 64/63, 91/90, 4394/4375

Gencom: [2 3/2; 64/63 91/90 4394/4375]

Gencom mapping: [⟨1 1 -6 4 -7], ⟨0 1 14 -2 18]]

Optimal ET sequence: 5, 32, 37

RMS error: 2.318 cents

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