User:Moremajorthanmajor/5L 2s (8/3-equivalent)

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↖4L 1s⟨8/3⟩ ↑5L 1s⟨8/3⟩ 6L 1s⟨8/3⟩↗
←4L 2s⟨8/3⟩5L 2s<8/3> 6L 2s⟨8/3⟩→
↙4L 3s⟨8/3⟩ ↓5L 3s⟨8/3⟩ 6L 3s⟨8/3⟩↘
┌╥╥╥┬╥╥┬┐
│║║║│║║││
│││││││││
└┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLsLLs
sLLsLLL
Equave 8/3 (1698.0¢)
Period 8/3 (1698.0¢)
Generator size(ed8/3)
Bright 4\7 to 3\5 (970.3¢ to 1018.8¢)
Dark 2\5 to 3\7 (679.2¢ to 727.7¢)
Related MOS scales
Parent 2L 3s⟨8/3⟩
Sister 2L 5s⟨8/3⟩
Daughters 7L 5s⟨8/3⟩, 5L 7s⟨8/3⟩
Neutralized 3L 4s⟨8/3⟩
Equal tunings(ed8/3)
Equalized (L:s = 1:1) 4\7 (970.3¢)
Supersoft (L:s = 4:3) 15\26 (979.6¢)
Soft (L:s = 3:2) 11\19 (983.1¢)
Semisoft (L:s = 5:3) 18\31 (986.0¢)
Basic (L:s = 2:1) 7\12 (990.5¢)
Semihard (L:s = 5:2) 17\29 (995.4¢)
Hard (L:s = 3:1) 10\17 (998.8¢)
Superhard (L:s = 4:1) 13\22 (1003.4¢)
Collapsed (L:s = 1:0) 3\5 (1018.8¢)

5L 2s⟨8/3⟩ is a 8/3-equivalent (non-octave) moment of symmetry scale containing 5 large steps and 2 small steps, repeating every interval of 8/3 (1698.0¢). Generators that produce this scale range from 970.3¢ to 1018.8¢, or from 679.2¢ to 727.7¢. Among the most well-known variants of this 17/12 diatonic MOS proper are 17ed8/3s 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 12ed8/3 diatonic.

  • 2 2 2 1 2 2 1

When L=3, s=1, you have 17ED8/3 of standard practice: 3 3 3 1 3 3 1

When L=3, s=2, you have 19ed8/3: 3 3 3 2 3 3 2

When L=4, s=1, you have 22ed8/3: 4 4 4 1 4 4 1

When L=4, s=3, you have 26ed8/3: 4 4 4 3 4 4 3

When L=5, s=1, you have 27ed8/3: 5 5 5 1 5 5 1

When L=5, s=2, you have 29ed8/3: 5 5 5 2 5 5 2

When L=5, s=3, you have 31ed8/3: 5 5 5 3 5 5 3

When L=5, s=4, you have 33ed8/3: 5 5 5 4 5 5 4

So you have scales where L and s are nearly equal, which approach 7ed8/3:

  • 1 1 1 1 1 1 1

And you have scales where s becomes so small it approaches zero, which would give us 5ed8/3:

  • 1 1 1 0 1 1 0 = 1 1 1 1 1

Tuning ranges

Parasoft to ultrasoft

"17/12 Flattone" systems, such as 26ed8/3.

Hyposoft

"17/12 Meantone" (more properly "septimal meantone") systems, such as 31ed8/3.

Hypohard

The near-just part of the region is of interest mainly for those interested in 17/12 Pythagorean tuning and large, accurate EDO systems based on close-to-Pythagorean fifths, such as 41ED8/3 and 53ED8/3. This class of tunings is called 17/12 schismic temperament; these tunings can approximate 517/12-limit harmonies very accurately by tempering out 17/12 of a small comma called the schisma. (Technically, 12ED8/3 tempers out the 17/12 schisma and thus is a 17/12 schismic tuning, but it is nowhere near as accurate as 17/12 schismic tunings can be.)

The sharp-of-just part of this range includes so-called "17/12 neogothic" or "17/12 parapyth" systems, which tune the 17/12 diatonic major third slightly sharply of 11/7 (around 128/81) and the diatonic minor third slightly flatly of 14/11 (around 81/64). Good 17/12 neogothic ED8/3s include 29ed8/3 and 46ed8/3. 17ed8/3 is often considered the sharper end of the neogothic spectrum; its major third at 800 cents is considerably more discordant than in flatter 17/12 neogothic tunings.

Parahard to ultrahard

"17/12 Archy" systems such as 17ed8/3, 22ed8/3, and 27ed8/3.

Modes

17/12 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 7ED8/3) is at one extreme and 3\5 (three degrees of 5ED8/3) 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 12ED8/3.

If we carry this freshman-summing out a little further, new, larger ED8/3s pop up in our continuum.


Generator ranges:

  • Chroma-positive generator: 970.311 cents (4\7) to 1018.827 cents (3\5)
  • Chroma-negative generator: 679.218 cents (2\5) to 727.734 cents (3\7)


Scale Tree and Tuning Spectrum of 5L 2s⟨8/3⟩
Generator(ed8/3) Cents Step Ratio Comments
Bright Dark L:s Hardness
4\7 970.311 727.734 1:1 1.000 Equalized 5L 2s⟨8/3⟩
23\40 976.376 721.669 6:5 1.200
19\33 977.662 720.383 5:4 1.250
34\59 978.534 719.511 9:7 1.286
15\26 979.641 718.404 4:3 1.333 Supersoft 5L 2s⟨8/3⟩
41\71 980.561 717.484 11:8 1.375
26\45 981.093 716.952 7:5 1.400
37\64 981.682 716.363 10:7 1.429
11\19 983.079 714.966 3:2 1.500 Soft 5L 2s⟨8/3⟩
40\69 984.374 713.671 11:7 1.571
29\50 984.866 713.179 8:5 1.600
47\81 985.285 712.760 13:8 1.625
18\31 985.962 712.083 5:3 1.667 Semisoft 5L 2s⟨8/3⟩
43\74 986.702 711.343 12:7 1.714
25\43 987.235 710.810 7:4 1.750
32\55 987.953 710.092 9:5 1.800
7\12 990.526 707.519 2:1 2.000 Basic 5L 2s⟨8/3⟩
Scales with tunings softer than this are proper
31\53 993.196 704.849 9:4 2.250
24\41 993.978 704.067 7:3 2.333
41\70 994.569 703.476 12:5 2.400
17\29 995.406 702.639 5:2 2.500 Semihard 5L 2s⟨8/3⟩
44\75 996.186 701.859 13:5 2.600
27\46 996.679 701.366 8:3 2.667
37\63 997.265 700.780 11:4 2.750
10\17 998.850 699.195 3:1 3.000 Hard 5L 2s⟨8/3⟩
33\56 1000.634 697.411 10:3 3.333
23\39 1001.411 696.634 7:2 3.500
36\61 1002.125 695.920 11:3 3.667
13\22 1003.390 694.655 4:1 4.000 Superhard 5L 2s⟨8/3⟩
29\49 1004.965 693.080 9:2 4.500
16\27 1006.249 691.796 5:1 5.000
19\32 1008.214 689.831 6:1 6.000
3\5 1018.827 679.218 1:0 → ∞ Collapsed 5L 2s⟨8/3⟩

Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include 17/12 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 17/12 Pythagorean tuning itself (well approximated by 31\53) as well as 17/12 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 5-limit than 7-limit intervals: 6:5 and 5:4 as opposed to 7:6 and 9:7.

5L2s.jpg

5L 2s contains the pentatonic MOS 2L 3s and (with the sole exception of the 5L 2s of 12EDXI) 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 (988.2353¢ normalized).

Related Scales

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

Approaches to Functional Harmony

See also: Diatonic functional harmony