5L 2s: Difference between revisions

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=== Parapyth (29&17, 2.3.7.11.13) ===
=== Parapyth (29&17, 2.3.7.11.13) ===
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 704.745
EDO generators: [[17edo|10\17]], [[29edo|17\29]], [[46edo|27\46]]
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Mapping|Period-generator mapping]]: [&lt;1 0 -21 -14 -9|, &lt;0 1 15 11 8|]
Commas: 169/168, 352/351, 364/363
Commas: 169/168, 352/351, 364/363
[[Lp_tuning|POL2 generator]]: ~3/2 = 704.745 cents


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


Gencom mapping: [&lt;1 1 0 -6 -3 -1|, &lt;0 1 0 15 11 8|]
Gencom mapping: [&lt;1 1 0 -6 -3 -1|, &lt;0 1 0 15 11 8|]
Sval map: [&lt;1 0 -21 -14 -9|, &lt;0 1 15 11 8|]


EDOs: 17, 46, 63
EDOs: 17, 46, 63


[[Tp_tuning#T2 tuning|RMS error]]: 0.7541 cents
[[Tp_tuning#T2 tuning|RMS error]]: 0.7541 cents
</div></di>


=== Archy (17&5, 2.3.7) ===
=== Archy (17&5, 2.3.7) ===

Revision as of 11:50, 27 January 2021

One way of distinguishing the "diatonic" scale is by considering it a moment of symmetry scale produced by a chain of "fifths" (or "fourths"). This will include 12edo's diatonic scale along with the Pythagorean diatonic scale and meantone systems, while excluding just intonation scales that use more than one size of "tone".

It may be misleading to call 5L 2s "diatonic," since other scales called diatonic can be arrived at different ways (through just intonation procedures for instance, or with tetrachords). Also, a composer working with a 5L 2s scale may choose to do something very different than typical diatonic music.

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

A continuum of tunings

So 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.

5L2s.jpg

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).

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). The temperaments are listed roughly in order of increasing generator size and child temperaments are extensions of low-complexity parent temperaments.

Meantone (12&19, 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)
#Gens up 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
6 577.434 25/18
  1. ↑ octave-reduced
  2. ↑ 2.3.5, odd limit ≤ 27
Technical data

Period-generator mapping: [⟨1 0 -4], ⟨0 1 4]]

Comma: 81/80

Mapping generator: ~3

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]

EDOs: 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb

Wedgie: ⟨⟨1 4 4]]

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

Period-generator mapping: [⟨1 0 -4 17 10], ⟨0 1 4 -9 -4]]

7-limit minimax

[[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

9-limit minimax

[[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

valid range: [692.308, 694.737] (26 to 19)

nice range: [692.353, 701.955]

strict range: [692.353, 694.737]

Mapping generator: ~3

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

Wedgie: ⟨⟨1 4 -9 4 -17 -32]]

Generators: 2, 3

EDOs: 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

Period-generator mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]

Commas: 81/80, 126/125

7 and 9-limit minimax

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

Eigenmonzos: 2, 5

valid range: [694.737, 700.000] (19 to 12)

nice range: [694.786, 701.955]

strict range: [694.786, 700.000]

Mapping generator: ~3

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

Wedgie: ⟨⟨1 4 10 4 13 12]]

Vals: 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb

Badness: 0.0137

Meanpop (31&50, 2.3.5.7.11)
  • -13 gens = 11/8
  • Good EDO gens: 29\50
Huygens (31&43, 2.3.5.7.11)

Period: 1\1

POTE generator: ~3/2 = 696.967

EDO generators: 25\43

Technical data

Period-generator mapping: [<1 0 -4 -13 -25|, <0 1 4 10 18|]

Commas: 81/80, 126/125, 99/98

11-limit minimax

[[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

valid range: [696.774, 700.000] (31 to 12)

nice range: [691.202, 701.955]

strict range: [696.774, 700.000]

Mapping generator: ~3

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

Generators: 2, 3

EDOs: 12, 31, 43, 74, 105, 198be

Badness: 0.0170

Schismic/Garibaldi (12&29, 2.3.5.7)

Period: 1\1

Optimal (POTE) generator: ~3/2 = 702.085

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

Technical data

Commas: 225/224, 3125/3087

7-limit minimax tuning:

7-limit: [|1 0 0 0>, |5/3 1/15 0 -1/15>, |5/3 -8/15 0 8/15>, |5/3 -14/15 0 14/15>]

Eigenmonzos: 2, 7/6

9-limit: [|1 0 0 0>, |25/16 1/8 0 -1/16>, |5/2 -1 0 1/2>, |25/8 -7/4 0 7/8>]

Eigenmonzos: 2, 9/7

Mapping generator: ~3

Map: [<1 0 15 25|, <0 1 -8 -14|]

Wedgie: <<1 -8 -14 -15 -25 -10||

EDOs: 12, 29, 41, 53, 94, 241c, 335cd, 576cd

Badness: 0.0216

41&53, 2.3.5.7.11.13.19

  • -14 gens = 7/4
  • 23 gens = 11/8
  • 20 gens = 13/8

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

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

</di>

Archy (17&5, 2.3.7)

Comma: 64/63

POL2 generator: ~3/2 = 709.321

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

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

Map: [<1 2 2|, <0 -1 2|]

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

RMS error: 1.856 cents

Supra (17&22, 2.3.7.11)

Commas: 64/63, 99/98

POTE generator: ~3/2 = 707.192

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

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

Sval map: [<1 0 6 13|, <0 1 -2 -6|]

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

RMS error: 1.977 cents

Superpyth (22&27, 2.3.5.7)

Commas: 64/63, 245/243

POTE generator: ~3/2 = 710.291

Map: [<1 0 -12 6|, <0 1 9 -2|]

Wedgie: <<1 9 -2 12 -6 -30||

EDOs: 5, 17, 22, 27, 49

Badness: 0.0323

Ultrapyth (27&37, 2.3.7.13/10)

  • 4 gens = 13/10
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