5L 2s/Temperaments
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
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
Comma list: 81/80
Mapping: [⟨1 0 -4], ⟨0 1 4]]
Mapping generators: ~2, ~3
Wedgie: ⟨⟨1 4 4]]
- 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
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]]
- 7-odd-limit: ~3/2 = [8/13 0 1/13 -1/13⟩
- Eigenmonzos (unchanged-intervals): 2, 7/5
- 9-odd-limit: ~3/2 = [6/11 2/11 0 -1/11⟩
- Eigenmonzos (unchanged-intervals): 2, 9/7
- 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
Comma list: 81/80, 126/125
Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]
Mapping generators: ~2, ~3
- 7- and 9-odd-limit: ~3/2 = [0 0 1/4⟩
- Eigenmonzos (unchanged-intervals): 2, 5
- 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:
- 11-odd-limit: ~3/2 = [0 0 1/4⟩
- 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
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:
- 11-odd-limit: ~3/2 = [9/16 -1/8 0 0 1/16⟩
- 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
#Gens up | Cents [1] | Approximate ratios[2] |
---|---|---|
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 |
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
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
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
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
Comma list: 64/63, 245/243
Mapping: [⟨1 1 -3 4], ⟨0 1 9 -2]]
Wedgie: ⟨⟨1 9 -2 12 -6 -30]]
Badness: 0.032318
Ultrapyth
Subgroup: 2.3.5.7.13
Period: 1\1
Optimal (POTE) generator: ~3/2 = 713.745
#Gens up | Cents [1] | Approximate ratios[2] |
---|---|---|
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 |
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