5L 2s

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5L 2s
Pattern LLLsLLs
Period 2/1
Generator range 4\7 (685.7¢) to 3\5 (720.0¢)
Parent MOS 2L 3s
Daughter MOSes 7L 5s, 5L 7s
Sister MOS 2L 5s
Neutralized MOS 3L 4s
TAMNAMS name diatonic
Equal tunings
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¢)

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 step combination 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 whole tone – are actually JI detemperings or tempered approximations of them that both closely resemble and are derived from this MOS.

On the term diatonic

In TAMNAMS (which is the convention on all pages on scale patterns on the wiki), diatonic exclusively refers to 5L 2s. Other diatonic-based scales (specifically with 3 step sizes or more), such as Zarlino, blackdye and diasem, are called detempered (if the philosophy is RTT-based) or deregularized (RTT-agnostic) diatonic scales. The adjectives diatonic-like or diatonic-based may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate.

Substituting step sizes

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

  • L L L s L L s

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

  • 2 2 2 1 2 2 1

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

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

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

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

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

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

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

When L=5, s=4, you have 33EDO: 5 5 5 4 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 1 0 1 1 0 = 1 1 1 1 1

Tuning ranges

Parasoft to ultrasoft

"Flattone" systems, such as 26EDO.

Hyposoft

"Meantone" (more properly "septimal meantone") systems, such as 31EDO.

Hypohard

The near-just part of the region is of interest mainly for those interested in Pythagorean tuning and large, accurate EDO systems based on close-to-Pythagorean fifths, such as 41EDO and 53EDO. This class of tunings is called schismic temperament; these tunings can approximate 5-limit harmonies very accurately by tempering out a small comma called the schisma. (Technically, 12EDO tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.)

The sharp-of-just part of this range includes so-called "neogothic" or "parapyth" systems, which tune the diatonic major third slightly sharply of 81/64 (around 14/11) and the diatonic minor third slightly flatly of 32/27 (around 13/11). Good neogothic EDOs include 29EDO and 46EDO. 17EDO is often considered the sharper end of the neogothic spectrum; its major third at 423 cents is considerably more discordant than in flatter neogothic tunings.

Parahard to ultrahard

"Archy" systems such as 17EDO, 22EDO, and 27EDO.

Modes

Diatonic modes have standard names from classical music theory:

Mode UDP Name
LLLsLLs 6|0 Lydian
LLsLLLs 5|1 Ionian
LLsLLsL 4|2 Mixolydian
LsLLLsL 3|3 Dorian
LsLLsLL 2|4 Aeolian
sLLLsLL 1|5 Phrygian
sLLsLLL 0|6 Locrian

Scales

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.

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


Generator ranges:

  • Chroma-positive generator: 685.7143 cents (4\7) to 720 cents (3\5)
  • Chroma-negative generator: 480 cents (2\5) to 514.2857 cents (3\7)
Generator Cents L s L/s Comments
4\7 685.714 1 1 1.000
27\47 689.362 7 6 1.167
23\40 690.000 6 5 1.200
42\73 690.411 11 9 1.222
19\33 690.909 5 4 1.250
53\92 691.304 14 11 1.273
34\59 691.525 9 7 1.286
49\85 691.765 13 10 1.300
15\26 692.308 4 3 1.333
56\97 692.784 15 11 1.364
41\71 692.958 11 8 1.375
67\116 693.103 18 13 1.385
26\45 693.333 7 5 1.400 Flattone is in this region
63\109 693.578 17 12 1.417
37\64 693.750 10 7 1.429
48\83 693.976 13 9 1.444
11\19 694.737 3 2 1.500
51\88 695.455 14 9 1.556
40\69 695.652 11 7 1.571
69\119 695.798 19 12 1.583
29\50 696.000 8 5 1.600
66\131 696.183 21 13 1.615 Golden meantone (696.2145¢)
47\81 696.296 13 8 1.625
65\112 696.429 18 11 1.636
18\31 696.774 5 3 1.667 Meantone is in this region
61\105 697.143 17 10 1.700
43\74 697.297 12 7 1.714
68\117 697.436 19 11 1.727
25\43 697.674 7 4 1.750
57\98 697.959 16 9 1.778
32\55 698.182 9 5 1.800
39\67 698.507 11 6 1.833
7\12 700.000 2 1 2.000 Basic diatonic
(Generators smaller than this are proper)
38\65 701.539 11 5 2.200
31\53 701.887 9 4 2.250 The generator closest to a just 3/2 for EDOs less than 200
55\94 702.128 16 7 2.286 Garibaldi / Cassandra
24\41 702.409 7 3 2.333
65\111 702.703 19 8 2.375
41\70 702.857 12 5 2.400
58\99 703.030 17 7 2.428
17\29 703.448 5 2 2.500
61\104 703.846 18 7 2.571
44\75 704.000 13 5 2.600
71\121 704.132 21 8 2.625 Golden neogothic (704.0956¢)
27\46 704.348 8 3 2.667 Neogothic is in this region
64\109 704.587 19 7 2.714
37\63 704.762 11 4 2.750
47\80 705.000 14 5 2.800
10\17 705.882 3 1 3.000
43\73 706.849 13 4 3.250
33\56 707.143 10 3 3.333
56\95 707.368 17 5 3.400
23\39 707.692 7 2 3.500
59\100 708.000 18 5 3.600
36\61 708.197 11 3 3.667
49\83 708.434 15 4 3.750
13\22 709.091 4 1 4.000 Archy is in this region
42\71 709.859 13 3 4.333
29\49 710.204 9 2 4.500
45\76 710.526 14 3 4.667
16\27 711.111 5 1 5.000
35\59 711.864 11 2 5.500
19\32 712.500 6 1 6.000
22\37 713.514 7 1 7.000
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 (700¢).

Related Scales

Main article: 5L 2s MODMOSes and 5L 2s Muddles

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

Rank-2 temperaments

Main article: 5L 2s/Temperaments

Approaches to Functional Harmony

See also: Diatonic functional harmony