User:Moremajorthanmajor/5L 2s (perfect eleventh equivalent)

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One way of distinguishing the 17/12 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 17EDXIs 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 12EDXI diatonic.

  • 2 2 2 1 2 2 1

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

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

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

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

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

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

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

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

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

  • 1 1 1 1 1 1 1

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

  • 1 1 1 0 1 1 0 = 1 1 1 1 1

Tuning ranges

Parasoft to ultrasoft

"17/12 Flattone" systems, such as 26EDXI.

Hyposoft

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

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 41EDXI and 53EDXI. 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, 12EDXI 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 13/11 (around 32/27). Good 17/12 neogothic EDXIs include 29EDXI and 46EDXI. 17EDXI 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 17EDXI, 22EDXI, and 27EDXI.

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

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

Generator ranges:

  • Chroma-positive generator: 960 cents (4\7, normalized) to 1028.5714 cents (3\5, normalized)
  • Chroma-negative generator: [[1]] cents (2\5, normalized) to 720 cents (3\7, normalized)
Generator Normalized L s L/s Comments
4\7 960.000¢ 1 1 1.000
27\47 967.164¢ 7 6 1.167
23\40 968.421¢ 6 5 1.200
42\73 969.231¢ 11 9 1.222
19\33 970.213¢ 5 4 1.250
34\59 971.429¢ 9 7 1.286
15\26 972.972¢ 4 3 1.333
56\97 973.913¢ 15 11 1.364
41\71 974.2574¢ 11 8 1.375
67\116 974.54¢ 18 13 1.385
26\45 975.000¢ 7 5 1.400 17/12 Flattone is in this region
63\109 975.484¢ 17 12 1.417
37\64 975.824¢ 10 7 1.429
48\83 976.271¢ 13 9 1.444
11\19 977.7¢ 3 2 1.500
51\88 979.200¢ 14 9 1.556
40\69 979.592¢ 11 7 1.571
29\50 980.282¢ 8 5 1.600
76\131 980.645¢ 21 13 1.615 17/12 Golden meantone (980.7157¢)
47\81 980.870¢ 13 8 1.625
18\31 981.81¢ 5 3 1.667 17/12 Meantone is in this region
43\74 982.857¢ 12 7 1.714
25\43 983.607¢ 7 4 1.750
32\55 984.615¢ 9 5 1.800
39\67 985.263¢ 11 6 1.833
46\79 985.714¢ 13 7 1.857
53\91 986.047¢ 15 8 1.875
60\103 986.301¢ 17 9 1.889
67\115 986.503¢ 19 10 1.900
74\127 986.6¢ 21 11 1.909
81\139 986.802¢ 23 12 1.917
88\151 986.916¢ 25 13 1.923
95\163 987.013¢ 27 14 1.929
7\12 988.235¢ 2 1 2.000 Basic 17/12 diatonic
(Generators smaller than this are proper)
45\77 990.826¢ 13 6 2.167
38\65 991.304¢ 11 5 2.200
31\53 992.000¢ 9 4 2.250 The generator closest to 17/12 of a just 3/2 for EDXIs less than 200
55\94 992.481¢ 16 7 2.286 17/12 Garibaldi / Cassandra
24\41 993.103¢ 7 3 2.333
41\70 993.93¢ 12 5 2.400
58\99 994.2857¢ 17 7 2.428
17\29 995.122¢ 5 2 2.500
44\75 996.226¢ 13 5 2.600
71\121 996.491¢ 21 8 2.625 17/12 Golden neogothic (996.3946¢)
27\46 996.923¢ 8 3 2.667 17/12 Neogothic is in this region
37\63 997.753¢ 11 4 2.750
47\80 998.230¢ 14 5 2.800
57\97 998.540¢ 17 6 2.833
67\114 998.758¢ 20 7 2.857
77\131 998.918¢ 23 8 2.875
87\148 999.043¢ 26 9 2.889
10\17 1000.000¢ 3 1 3.000
43\73 1001.942¢ 13 4 3.250
33\56 1002.532¢ 10 3 3.333
23\39 1003.63¢ 7 2 3.500
36\61 1004.651¢ 11 3 3.667
49\83 1005.128¢ 15 4 3.750
13\22 1006.452¢ 4 1 4.000 17/12 Archy is in this region
55\93 1007.634¢ 17 4 4.250
42\71 1008.000¢ 13 3 4.333
29\49 1008.696¢ 9 2 4.500
45\76 1009.346¢ 14 3 4.667
16\27 1010.526¢ 5 1 5.000
51\86 1011.570¢ 16 3 5.333
35\59 1012.048¢ 11 2 5.500
19\32 1013.3¢ 6 1 6.000
41\69 1014.432¢ 13 2 6.500
22\37 1015.385¢ 7 1 7.000
3\5 1028.571¢ 1 0 → inf

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 (988.235¢ 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

See also

5L 2s (8/3-equivalent)