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5L 2s

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One way of distinguishing the diatonic scale is by considering it a moment of symmetry scale produced by a chain of "fifths" (or "fourths") with the pattern of 5L 2s. Among the most well-known variants of this MOS proper are 12edo's diatonic scale along with both the Pythagorean diatonic scale and the various meantone systems. Other similar scales referred to by the term "diatonic" can be arrived at different ways- for example, through just intonation procedures, or with tetrachords; however, it should be noted that at least the majority of the other scales that fall under this category, such as the just intonation scales that use more than one size of "tone", are actually muddles or MODmuddles derived from this MOS.

↖ 4L 1s ↑ 5L 1s 6L 1s ↗
← 4L 2s 5L 2s 6L 2s →
↙ 4L 3s ↓ 5L 3s 6L 3s ↘
Scale structure
Step pattern LLLsLLs
sLLsLLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 4\7 to 3\5 (685.7 ¢ to 720.0 ¢)
Dark 2\5 to 3\7 (480.0 ¢ to 514.3 ¢)
TAMNAMS information
Name diatonic
Prefix dia-
Abbrev. dia
Related MOS scales
Parent 2L 3s
Sister 2L 5s
Daughters 7L 5s, 5L 7s
Neutralized 3L 4s
2-Flought 12L 2s, 5L 9s
Equal tunings
Equalized (L:s = 1:1) 4\7 (685.7 ¢)
Supersoft (L:s = 4:3) 15\26 (692.3 ¢)
Soft (L:s = 3:2) 11\19 (694.7 ¢)
Semisoft (L:s = 5:3) 18\31 (696.8 ¢)
Basic (L:s = 2:1) 7\12 (700.0 ¢)
Semihard (L:s = 5:2) 17\29 (703.4 ¢)
Hard (L:s = 3:1) 10\17 (705.9 ¢)
Superhard (L:s = 4:1) 13\22 (709.1 ¢)
Collapsed (L:s = 1:0) 3\5 (720.0 ¢)
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Substituting step sizes

The 5L 2s MOS scale has this generalized form.

L L s L L L s

Insert 2 for L and 1 for s and you'll get the 12edo diatonic of standard practice.

2 2 1 2 2 2 1

When L=3, s=1, you have 17edo: 3 3 1 3 3 3 1

When L=3, s=2, you have 19edo: 3 3 2 3 3 3 2

When L=4, s=1, you have 22edo: 4 4 1 4 4 4 1

When L=4, s=3, you have 26edo: 4 4 3 4 4 4 3

When L=5, s=1, you have 27edo: 5 5 1 5 5 5 1

When L=5, s=2, you have 29edo: 5 5 2 5 5 5 2

When L=5, s=3, you have 31edo: 5 5 3 5 5 5 3

When L=5, s=4, you have 33edo: 5 5 4 5 5 5 4

So you have scales where L and s are nearly equal, which approach 7edo:

1 1 1 1 1 1 1

And you have scales where s becomes so small it approaches zero, which would give us 5edo:

1 1 0 1 1 1 0 = 1 1 1 1 1

Scale tree

If 4\7 (four degrees of 7edo) is at one extreme and 3\5 (three degrees of 5edo) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12edo:

4\7
7\12
3\5

If we carry this freshman-summing out a little further, new, larger EDOs pop up in our continuum.

generator cents L s L/s comments
4\7 685.714 1 1 1.000
63\110 687.273 16 15 1.067
59\103 687.379 15 14 1.071
55\96 687.500 14 13 1.077
51\89 687.640 13 12 1.083
47\82 687.805 12 11 1.091
43\75 688.000 11 10 1.100
39\68 688.235 10 9 1.111
35\61 688.525 9 8 1.125
31\54 688.889 8 7 1.143
27\47 689.362 7 6 1.167
23\40 690.000 6 5 1.200
19\33 690.909 5 4 1.250
34\59 691.525 9 7 1.286
15\26 692.308 4 3 1.333
41\71 692.958 11 8 1.375
26\45 693.333 7 5 1.400
37\64 693.750 10 7 1.429
11\19 694.737 3 2 1.500 Optimum rank range (L/s=3/2) diatonic
51\88 695.455 14 9 1.556
695.644 π 2 1.571 LucyTuning
40\69 695.652 11 7 1.571
29\50 696.000 8 5 1.600
76\131 696.183 21 13 1.615
199\343 696.210 55 34 1.618
696.215 φ 1 1.618 Golden meantone
322\555 696.216 89 55 1.618
123\212 696.226 34 21 1.619
47\81 696.296 13 8 1.625
18\31 696.774 5 3 1.667 Meantone is in this region
43\74 697.297 12 7 1.714
697.487 √3 1 1.732
25\43 697.674 7 4 1.750
32\55 698.182 9 5 1.800
39\67 698.507 11 6 1.833
46\79 698.734 13 7 1.857
53\91 698.901 15 8 1.875
60\103 699.029 17 9 1.889
7\12 700.000 2 1 2.000 Boundary of propriety (generators smaller than this are proper)
59\101 700.990 17 8 2.125
52\89 701.124 15 7 2.143
45\77 701.299 13 6 2.167
38\65 701.539 11 5 2.200
31\53 701.887 9 4 2.250
701.955 2.260 Pythagorean (g = 3/2 ; L=9/8 ; s=256/243)
24\41 702.409 7 3 2.333
41\70 702.857 12 5 2.400
17\29 703.448 5 2 2.500
44\75 704.000 13 5 2.600
115\196 704.082 34 13 2.615
186\317 704.101 55 21 2.619
71\121 704.132 21 8 2.625
27\46 704.348 8 3 2.667
704.607 e 1 2.718
37\63 704.762 11 4 2.750
47\80 705.000 14 5 2.800
10\17 705.882 3 1 3.000
706.447 π 1 3.142
33\56 707.143 10 3 3.333
23\39 707.692 7 2 3.500
36\61 708.197 11 3 3.667
13\22 709.091 4 1 4.000 (No-5's) superpyth is in this region
29\49 710.204 9 2 4.500
16\27 711.111 5 1 5.000
19\32 712.500 6 1 6.000
22\37 713.514 7 1 7.000
25\42 714.286 8 1 8.000
28\47 714.894 9 1 9.000
31\52 715.385 10 1 10.000
34\57 715.790 11 1 11.000
37\62 716.129 12 1 12.000
40\67 716.418 13 1 13.000
43\72 716.667 14 1 14.000
46\77 716.883 15 1 15.000
49\82 717.073 16 1 16.000
3\5 720.000 1 0 -> inf

Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.

 

5L 2s contains the pentatonic MOS 2L 3s and (with the sole exception of the 5L 2s of 12edo) is itself contained in a dodecaphonic MOS: either 7L 5s or 5L 7s, depending on whether the fifth is flatter than or sharper than 7\12 (700c).

Related Muddles

See also: 5L 2s Muddles

Because the diatonic scale is so widely used, it should be no surprise that there are a number of noteworthy muddles related to this MOS.

Rank-2 temperaments

Below are some important rank-2 temperaments with optimal generator size in the 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 (12&19, 2.3.5)

Main article: Meantone

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[1] Approximate ratios[2]
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
  1. ↑ octave-reduced
  2. ↑ 2.3.5, odd limit ≤ 27
Technical data

Comma list: 81/80

Mapping: [⟨1 0 -4], ⟨0 1 4]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨1 4 4]]

Tuning ranges:

  • valid range: [685.714, 720.000] (7 to 5)
  • nice range: [694.786, 701.955] (1/3 comma to Pythagorean)
  • strict range: [694.786, 701.955]

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

Badness: 0.00736

Flattone (19&26, 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 [1] Approximate ratios[2]
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
  1. ↑ octave-reduced
  2. ↑ 2.3.5.7.13, odd limit ≤ 27
Technical data

Comma list: 81/80, 525/512

Mapping: [⟨1 0 -4 17], ⟨0 1 4 -9]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨1 4 -9 4 -17 -32]]

Minimax tuning:

[[1 0 0 0⟩, [21/13 0 1/13 -1/13⟩, [32/13 0 4/13 -4/13⟩, [32/13 0 -9/13 9/13⟩]
Eigenmonzos: 2, 7/5
[[1 0 0 0⟩, [17/11 2/11 0 -1/11⟩, [24/11 8/11 0 -4/11⟩, [34/11 -18/11 0 9/11⟩]
Eigenmonzos: 2, 9/7

Tuning ranges:

  • valid range: [692.308, 694.737] (26 to 19)
  • nice range: [692.353, 701.955]
  • strict range: [692.353, 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, 45

Badness: 0.0386

Septimal meantone (19&12, 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 [1] Approximate ratios[2]
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
  1. ↑ octave-reduced
  2. ↑ 2.3.5.7, odd limit ≤ 27
Technical data

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨1 4 10 4 13 12]]

Minimax tuning:

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [-3 0 5/2 0⟩]
Eigenmonzos: 2, 5

Tuning ranges:

  • valid range: [694.737, 700.000] (19 to 12)
  • nice range: [694.786, 701.955]
  • strict range: [694.786, 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: 0.0137

Meanpop (31&50, 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

Technical data

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:

[[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [-3 0 5/2 0 0⟩, [11 0 -13/4 0 0⟩]
Eigenmonzos: 2, 5

Tuning ranges:

  • valid range: [694.737, 696.774] (19 to 31)
  • nice range: [691.202, 701.955]
  • strict range: [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: 0.0215

Huygens (31&43, 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

Technical data

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

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

Mapping generators: ~2, ~3

Minimax tuning:

[[1 0 0 0 0⟩, [25/16 -1/8 0 0 1/16⟩, [9/4 -1/2 0 0 1/4⟩, [21/8 -5/4 0 0 5/8⟩, [25/8 -9/4 0 0 9/8⟩]
Eigenmonzos: 2, 11/9

Tuning ranges:

  • valid range: [696.774, 700.000] (31 to 12)
  • nice range: [691.202, 701.955]
  • strict range: [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: 0.0170

Schismic/Garibaldi (41&53, 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

Mapping:

  • 1 gen = 3/2
  • -8 gens = 5/4
  • -14 gens = 7/4
  • 23 gens = 11/8
  • 20 gens = 13/8
  • -3 gens = 19/16

Parapyth (29&17, 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 [1] Approximate ratios[2]
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
  1. ↑ octave-reduced
  2. ↑ 2.3.7.11.13, odd limit ≤ 27
Technical data

Period-generator mapping: [<1 0 -21 -14 -9|, <0 1 15 11 8|]

Commas: 169/168, 352/351, 364/363

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

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

EDOs: 17, 46, 63

RMS error: 0.7541 cents

Archy (17&5, 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 [1] Approximate ratios[2]
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
  1. ↑ octave-reduced
  2. ↑ 2.3.7, odd limit ≤ 27
Technical data

Period-generator mapping: [<1 2 2|, <0 -1 2|]

Comma: 64/63

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

Gencom mapping: [<1 1 0 4|, <0 1 0 -2|]

EDOs: 5, 12, 17, 22, 27, 137bc

RMS error: 1.856 cents

Supra (17&22, 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 [1] Approximate ratios[2]
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
  1. ↑ octave-reduced
  2. ↑ 2.3.7.11, odd limit ≤ 27
Technical data

Period-generator mapping: [<1 0 6 13|, <0 1 -2 -6|]

Commas: 64/63, 99/98

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

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

EDOs: 5, 12, 17, 39c, 56c

RMS error: 1.977 cents

Superpyth (22&27, 2.3.5.7)

Period: 1\1

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

EDO generators: 13\22, 18\27, 31\49

Interval table (12-note MOS, 2.3.5.7 POTE tuning)
#Gens up Cents [1] Approximate ratios[2]
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
  1. ↑ octave-reduced
  2. ↑ 2.3.5.7, odd limit ≤ 27
Technical data

Period-generator mapping: [<1 0 -12 6|, <0 1 9 -2|]

Commas: 64/63, 245/243

Wedgie: ⟨⟨1 9 -2 12 -6 -30]]

EDOs: 5, 17, 22, 27, 49

Badness: 0.0323

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