4L 3s

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4L 3s
Pattern LLsLsLs
Period 2/1
Generator range 1\4 (300.0¢) to 2\7 (342.9¢)
Parent MOS 3L 1s
Daughter MOSes 7L 4s, 4L 7s
Sister MOS 3L 4s
TAMNAMS name smitonic
Equal tunings
Supersoft (L:s = 4:3) 7\25 (336.0¢)
Soft (L:s = 3:2) 5\18 (333.3¢)
Semisoft (L:s = 5:3) 8\29 (331.0¢)
Basic (L:s = 2:1) 3\11 (327.3¢)
Semihard (L:s = 5:2) 7\26 (323.1¢)
Hard (L:s = 3:1) 4\15 (320.0¢)
Superhard (L:s = 4:1) 5\19 (315.8¢)

4L 3s refers to an MOS scale with four large steps and three small steps, one mode of which is LLsLsLs. It is generated by any interval between 1\4edo (one degree of 4edo, or 300¢) and 2\7edo (two degrees of 7edo, or approx. 342.857¢).

4L 3s can be thought of as a warped diatonic scale, because it has one large step of diatonic (5L 2s, LLsLLLs) replaced with a small step (yielding LLsLsLs).

Standing assumptions

The TAMNAMS system is used in this article to name 4L 3s intervals and step size ratios and step ratio ranges.

The notation used in this article is LsLsLsL = JKLMNOPJ unless specified otherwise. We denote raising and lowering by a chroma (L − s) by & "amp" and @ "at". (Mnemonics: & "and" means additional pitch. @ "at" rhymes with "flat".)

Thus the 11edo gamut is as follows:

J J&/K@ K L L&/M@ M N N&/O@ O P P&/J@ J

Names

The TAMNAMS MOS naming system (used in this article) uses the name smitonic smy-TON-ik /smaɪˈtɒnɪk/ for this pattern. The name is derived from 'sharp minor third', since the central range of the spectrum, 4\15 = 320¢ to 7\18 = 333.33¢, has minor third generators that are significantly sharp of 6/5.

Intervals

Note: In TAMNAMS, a k-step interval class in smitonic may be called a "k-step", "k-mosstep", or "k-smistep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.

Generators Notation (1/1 = J) TAMNAMS name In L's and s's Generators Notation of 2/1 inverse TAMNAMS name In L's and s's
The 7-note MOS has the following intervals (from some root):
0 J perfect unison 0L + 0s 0 J octave 4L + 3s
1 L perfect 2-mosstep 1L + 1s -1 O perfect 5-mosstep 3L + 2s
2 N minor 4-mosstep 2L + 2s -2 M major 3-mosstep 2L + 1s
3 P minor 6-mosstep 3L + 3s -3 K major 1-mosstep 1L + 0s
4 K@ minor 1-mosstep 0L + 1s -4 Q& major 6-mosstep 4L + 2s
5 M@ minor 3-mosstep 1L + 2s -5 N& major 4-mosstep 3L + 1s
6 O@ diminished 5-mosstep 2L + 3s -6 L& augmented 2-mosstep 2L + 0s
The chromatic 11-note MOS (either 7L 4s, 4L 7s, or 11edo) also has the following intervals (from some root):
7 J@ diminished 7-mosstep 5L + 2s -7 J& augmented mosunison; chroma 1L - 1s
8 L@ diminished 2-mosstep 0L + 2s -8 O& augmented 5-mosstep 4L + 1s
9 N@ diminished 4-mosstep 1L + 3s -9 M& augmented 3-mosstep 3L + 0s
10 P@ diminished 6-mosstep 2L + 4s -10 K& augmented 1-mosstep 2L - 1s

Low harmonic entropy scales

There are two notable harmonic entropy minima:

  • Kleismic temperament, in which the generator is 6/5 and 6 of them make a 3/1 (making the diminished 5-mosstep 3/2)
  • Myna, in which the generator is also 6/5 but now 10 of them make a 6/1 (so no 4/3's or 3/2's appear in this scale).

Tuning ranges

Simple tunings

Degree 11edo (basic) 15edo (hard) 18edo (soft) Note name on J Approximate ratios #Gens up
unison 0\11, 0.0 0\15, 0.0 0\18, 0.0 J 0
minor 1-mosstep 1\11, 109.1 1\15, 80.0 2\18, 133.3 K@ +4
major 1-mosstep 2\11, 218.2 3\15, 240.0 3\18, 200.0 K 8/7 -3
perf. 2-mosstep 3\11, 327.3 4\15, 320.0 5\18, 333.3 L 77/64, 6/5 +1
aug. 2-mosstep 4\11, 436.4 6\15, 480.0 6\18, 400.0 L& -6
minor 3-mosstep 4\11, 436.4 5\15, 400.0 7\18, 466.7 M@ 14/11 +5
major 3-mosstep 5\11, 545.5 7\15, 560.0 8\18, 533.3 M 11/8 -2
minor 4-mosstep 6\11, 656.6 8\15, 640.0 10\18, 666.7 N 16/11 +2
major 4-mosstep 7\11, 763.6 10\15, 800.0 11\18, 733.3 N& 11/7 -5
dim. 5-mosstep 7\11, 763.6 9\15, 720.0 12\18, 800.0 O@ +6
perf. 5-mosstep 8\11, 872.7 11\15, 880.0 13\18, 866.7 O 5/3 -1
minor 6-mosstep 9\11, 981.8 12\15, 960.0 15\18, 1000.0 P 7/4 +3
major 6-mosstep 10\11, 1090.9 14\15, 1120.0 16\18, 1066.7 P& -4

Parasoft

Parasoft smitonic tunings have step ratios between 4/3 and 3/2, which implies a generator sharper than 5\18 = 333.3¢ and flatter than 7\25 = 336.0¢.

Parasoft smitonic can be considered "meantone smitonic". This is because these tunings share the following features with meantone diatonic tunings:

  • The large step is a "meantone", somewhere between near-10/9 (as in 32edo) and near-9/8 (as in 18edo).
  • The augmented 2-mosstep (made of two large steps) is a roughly meantone-sized major third, thus is a stand-in for the classical diatonic major third.

Parasoft smitonic tunings have both minor fifths and major fifths about equally off a just fifth, and they have otherwise fairly convincing versions of both diatonic structure and tertian harmony, provided you frequently modify using the comma-like chromas. For this reason, parasoft might be the most accessible smitonic tuning range.

Parasoft smitonic EDOs include 18edo, 25edo, and 43edo.

  • 18edo can be used to make large and small steps more distinct (the step ratio is 3/2, thus 18edo smitonic is distorted 19edo diatonic), or for its nearly pure 9/8. It also makes rising fifths (733.3c, a perfect 5-mosstep) and falling fifths (666.7c, a major 4-mosstep) almost equally off from a just fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
  • 25edo can be used to make the augmented 2-mosstep a good 5/4 (384¢).

The sizes of the generator, large step and small step of smitonic are as follows in various parasoft smitonic tunings.

18edo (soft) 25edo (supersoft) 43edo Optimized (2.9.5 POTE dual-3 sixix) tuning
generator (g) 5\18, 333.3 7\25, 336.0 12\43, 334.9 335.84
L (octave - 3g) 3\18, 200.0 4\25, 192.0 7\43, 195.3 193.16
s (4g - octave) 2\18, 133.3 3\25, 144.0 5\43, 139.5 143.36

Intervals

Sortable table of the extended generator chain (-13 to 13 generators) in parasoft smitonic tunings. The several different interval flavors separated by the chroma shows that parasoft smitonic is a useful cluster MOS, though many of the intervals lack simple JI interpretations.

Degree 18edo (soft) 25edo (supersoft) 43edo Note name on J Approximate ratios #Gens up
unison 0\18, 0.0 0\25, 0.0 0\43, 0.0 J 1/1 0
chroma 1\18, 66.7 1\25, 48.0 2\43, 55.8 J& -7
dim. 1-mosstep 1\18, 66.7 2\25, 96.0 3\43, 83.7 K@@ +11
minor 1-mosstep 2\18, 133.3 3\25, 144.0 5\43, 139.5 K@ 13/12 +4
major 1-mosstep 3\18, 200.0 4\25, 192.0 7\43, 195.3 K 9/8, 10/9 -3
aug. 1-mosstep 4\18, 266.7 5\25, 240.0 9\43, 251.2 K& -10
dim. 2-mosstep 4\18, 266.7 6\25, 288.0 10\43, 279.1 L@ +8
perf. 2-mosstep 5\18, 333.3 7\25, 336.0 12\43, 334.9 L 17/14, 40/33 +1
aug. 2-mosstep 6\18, 400.0 8\25, 384.4 14\43, 390.7 L& 5/4 -6
doubly aug. 2-mosstep 7\18, 466.7 9\25, 432.0 16\43, 446.5 L&& -13
dim. 3-mosstep 6\18, 400.0 9\25, 432.0 15\43, 418.6 M@@ +12
minor 3-mosstep 7\18, 466.7 10\25, 480.0 17\43, 474.4 M@ 21/16 +5
major 3-mosstep 8\18, 533.3 11\25, 528.0 19\43, 530.2 M 19/14, 34/25 -2
aug. 3-mosstep 9\18, 600.0 12\25, 576.0 21\43, 586.0 M& 7/5 -9
dim. 4-mosstep 9\18, 600.0 13\25, 624.0 22\43, 614.0 N@ 10/7 +9
minor 4-mosstep 10\18, 666.7 14\25, 672.0 24\43, 669.8 N 28/19, 25/17 +2
major 4-mosstep 11\18, 733.3 15\25, 720.0 26\43, 725.6 N& 32/21 -5
aug. 4-mosstep 12\18, 800.0 16\25, 768.0 28\43, 781.4 N&& -12
doubly dim. 5-mosstep 11\18, 733.3 16\25, 768.0 27\43, 753.5 O@@ +13
dim. 5-mosstep 12\18, 800.0 17\25, 816.0 29\43, 809.3 O@ 8/5 +6
perf. 5-mosstep 13\18, 866.7 18\25, 864.0 31\43, 865.1 O 28/17, 33/20 -1
aug. 5-mosstep 14\18, 933.3 19\25, 912.0 33\43, 920.9 O& -8
dim. 6-mosstep 14\18, 933.3 20\25, 960.0 34\34, 948.8 P@ +10
minor 6-mosstep 15\18, 1000.0 21\25, 1008.0 36\43, 1004.7 P 16/9, 9/5 +3
major 6-mosstep 16\18, 1066.7 22\25, 1056.0 38\43, 1060.5 P& 24/13 -4
aug. 6-mosstep 17\18, 1133.3 23\25, 1104.0 40\43, 1116.3 P& -11
dim. mosoctave 17\18, 1133.3 24\25, 1152.0 41\43, 1144.2 J@ +7

Hyposoft

Hyposoft tunings of smitonic have step ratios between 3/2 and 2/1 which implies that the generator is a supraminor third sharper than 3\11 = 327.27¢ and flatter than 5\18 = 333.33¢.

The large step is a sharper major second in these tunings than in parasoft tunings. These tunings could be considered "neogothic smitonic" or "archy smitonic", in analogy to parasoft smitonic being meantone smitonic.

11edo (basic) 18edo (soft) 29edo (semisoft)
generator (g) 3\11, 327.27 5\18, 333.33 8\29, 331.03
L (octave - 3g) 2\11, 218.18 3\18, 200.00 5\29, 206.90
s (4g - octave) 1\11, 109.09 2\18, 133.33 3\29, 124.14

Intervals

Sortable table of major and minor intervals in hyposoft smitonic tunings (11edo and 18edo not shown):

Degree 29edo (semisoft) Note name on J Approximate ratios (for 29edo) #Gens up
unison 0\29, 0.0 J 1/1 0
minor 1-mosstep 3\29, 124.1 K@ 14/13 +4
major 1-mosstep 5\29, 206.9 K 9/8 -3
perf. 2-mosstep 8\29, 331.0 L 23/19, 40/33 +1
aug. 2-mosstep 10\29, 413.8 L& 14/11 -6
minor 3-mosstep 11\29, 455.2 M@ 13/10 +5
major 3-mosstep 13\29, 537.9 M 15/11 -2
minor 4-mosstep 16\29, 662.1 N 19/13, 22/15 +2
major 4-mosstep 18\26, 744.8 N& 20/13 -5
dim. 5-mosstep 19\29, 786.2 O@ 11/7 +6
perf. 5-mosstep 21\29, 869.0 O 33/20, 38/23 -1
minor 6-mosstep 24\29, 993.1 P 16/9 +3
major 6-mosstep 26\28, 1075.9 P& 13/7 -4

Hypohard

Hypohard tunings have step ratios between 2 and 3, implying a generator sharper than 4\15 = 320¢ and flatter than 3\11 = 327.27¢. The large step tends to approximate 8/7, and the major 3-mosstep (2 large steps + 1 small step) tends to approximate 11/8; 26edo is stellar in both of these approximations. This set of JI approximations is associated with orgone temperament.

Hypohard smitonic edos include 11edo, 15edo, 26edo, and 37edo. The sizes of the generator, large step and small step of smitonic are as follows in various hypohard smitonic tunings.

11edo (basic) 15edo (hard) 26edo (semihard) Some JI approximations
generator (g) 3\11, 327.27 4\15, 320.00 7\26, 323.08 77/64, 6/5
L (octave - 3g) 2\11, 218.18 3\15, 240.00 5\26, 230.77 8/7
s (4g - octave) 1\11, 109.09 1\15, 80.00 2\26, 92.31 128/121, (16/15)

Intervals

Sortable table of major and minor intervals in hypohard smitonic tunings:

Degree 11edo (basic) 15edo (hard) 26edo (semihard) Note name on J Approximate ratios #Gens up
unison 0\11, 0.0 0\15, 0.0 0\26, 0.0 J 1/1 0
minor 1-mosstep 1\11, 109.1 1\15, 80.0 2\26, 92.3 K@ +4
major 1-mosstep 2\11, 218.2 3\15, 240.0 5\26, 230.8 K 8/7 -3
perf. 2-mosstep 3\11, 327.3 4\15, 320.0 7\26, 323.1 L 77/64, 6/5 +1
aug. 2-mosstep 4\11, 436.4 6\15, 480.0 10\26, 461.5 L& -6
minor 3-mosstep 4\11, 436.4 5\15, 400.0 9\26, 415.4 M@ 14/11 +5
major 3-mosstep 5\11, 545.5 7\15, 560.0 12\26, 553.9 M 11/8 -2
minor 4-mosstep 6\11, 656.6 8\15, 640.0 14\26, 646.2 N 16/11 +2
major 4-mosstep 7\11, 763.6 10\15, 800.0 17\26, 784.62 N& 11/7 -5
dim. 5-mosstep 7\11, 763.6 9\15, 720.0 16\26, 738.5 O@ +6
perf. 5-mosstep 8\11, 872.7 11\15, 880.0 19\26, 876.9 O 5/3 -1
minor 6-mosstep 9\11, 981.8 12\15, 960.0 21\26, 969.2 P 7/4 +3
major 6-mosstep 10\11, 1090.9 14\15, 1120.0 24\26, 1107.7 P& -4

Parahard

In parahard smitonic (step ratio between 3 and 4, thus with generator between 5\19, 315.79¢ and 4\15, 320¢), the generator is close to a pure 6/5 minor third, and 6 minor thirds are used to reach a perfect fifth. The parahard range contains very accurate edos such as 53edo and 72edo, and has very accurate approximations to many low-overtone JI intervals, namely basic 5-limit ratios and some ratios involving 13. However, the 7-note MOS only has one perfect fifth, so extending the chain to bigger MOSes, such as the 4L 7s 11-note MOS, is suggested for getting 5-limit harmony.

This set of JI approximations is associated with kleismic temperament (we're specifically describing the 2.3.5.13 extension of it called cata).

EDOs that have parahard smitonic include 15edo, 19edo, 34edo, and 53edo.

The sizes of the generator, large step and small step of smitonic are as follows in various parahard smitonic tunings (not including 15edo).

19edo (superhard) 34edo 53edo JI intervals represented
generator (g) 5\19, 315.79 9\34, 317.65 14\53, 316.98 6/5
L (octave - 3g) 4\19, 252.63 7\34, 247.06 11\53, 249.06 15/13
s (4g - octave) 1\19, 63.16 2\34, 70.59 3\53, 67.92 25/24, 26/25

Intervals

Sortable table of major and minor intervals in parahard smitonic tunings:

Degree 19edo (superhard) 34edo 53edo Note name on J Approximate ratios #Gens up
unison 0\19, 0.0 0\34, 0.0 0\53, 0.0 J 1/1 0
minor 1-mosstep 1\19, 63.2 2\34, 70.6 3\53, 67.9 K@ 25/24, 26/25 +4
major 1-mosstep 4\19, 252.6 7\34, 247.1 11\53, 249.1 K 15/13 -3
perf. 2-mosstep 5\19, 315.8 9\34, 317.6 14\53, 317.0 L 6/5 +1
aug. 2-mosstep 8\19, 505.3 14\34, 494.1 22\53, 498.1 L& 4/3 -6
minor 3-mosstep 6\19, 378.9 11\34, 388.2 17\53, 384.9 M@ 5/4 +5
major 3-mosstep 9\19, 568.4 16\34, 564.7 25\53, 566.0 M 18/13 -2
minor 4-mosstep 10\19, 631.6 18\34, 635.3 28\53, 634.0 N 13/9 +2
major 4-mosstep 16\19, 821.1 23\34, 811.8 39\53, 815.0 N& 8/5 -5
dim. 5-mosstep 11\19, 694.7 20\34, 705.9 31\53, 701.9 O@ 3/2 +6
perf. 5-mosstep 14\19, 884.2 25\34, 882.4 39\53, 883.0 O 5/3 -1
minor 6-mosstep 15\19, 947.4 27\34, 952.9 42\53, 950.9 P 26/15 +3
major 6-mosstep 18\19, 1136.8 32\34, 1129.4 50\53, 1132.1 P& 25/13 -4

Modes

A naming scheme proposed by Alexandru Ianu (User:Ayceman)[1], relating to the Almsivi in Morrowind (TES):

Mode UDP Name
LLsLsLs 6|0 Nerevarine
LsLLsLs 5|1 Vivecan
LsLsLLs 4|2 Lorkhanic
LsLsLsL 3|3 Sothic
sLLsLsL 2|4 Kagrenacan
sLsLLsL 1|5 Almalexian
sLsLsLL 0|6 Dagothic

Approaches

Temperaments

Main article: 4L 3s/Temperaments

4L 3s has several temperament interpretations (see main article for mappings and optimal generator tunings):

  1. With generator size between 5\18 (333.3c) and 11\39 (338.5c): Sixix, corresponding to a L/s ratio between 3/2 and 6/5.
  2. With generator size between 4\15 (320.0c) and 3\11 (327.3c): Orgone, corresponding to a L/s ratio between 3 and 2.
  3. With generator size between 5\19 (315.8c) and 4\15 (320.0c): Kleismic, corresponding to a L/s ratio between 4 and 3.

There are also other temperaments in the 4L 3s range, particularly amity and myna, but 7 notes in the generator chain are not enough to contain the concordant chords optimized by these temperaments; you would need to use a MODMOS or use a larger MOS gamut, if you restrict to a rank-2 approach.

Scales

Music

Scale tree

The spectrum looks like this:

Generator Cents L s L/s Comments
Chroma-positive Chroma-negative
5\7 857.143 342.857 1 1 1.000
28\39 861.538 338.462 6 5 1.200 Amity/hitchcock↑
23\32 862.500 337.500 5 4 1.250 Sixix
41\57 863.158 336.842 9 7 1.286
18\25 864.000 336.000 4 3 1.333
49\68 864.706 335.294 11 8 1.375
31\43 865.116 334.884 7 5 1.400
17\58 865.574 334.426 10 7 1.428
13\18 866.667 333.333 3 2 1.500 L/s = 3/2
47\65 867.692 332.308 11 7 1.571
34\47 868.085 331.915 8 5 1.600
55\76 868.421 331.579 13 8 1.625 Golden smitonic (?)
21\29 868.966 331.034 5 3 1.667
50\69 869.565 330.435 12 7 1.714
29\40 870.000 330.000 7 4 1.750
37\51 870.588 329.422 9 5 1.800
8\11 872.727 327.273 2 1 2.000 Basic smitonic
(Generators smaller than this are proper)
35\48 875.000 325.000 9 4 2.250
27\37 875.676 324.324 7 3 2.333
46\63 876.190 323.810 12 5 2.400
19\26 876.923 323.077 5 2 2.500 Orgone is in this region
49\67 877.612 322.388 13 5 2.600 Golden superkleismic
30\41 878.049 321.951 8 3 2.667 Superkleismic
41\56 878.571 321.429 11 4 2.750
11\15 880.000 320.000 3 1 3.000 L/s = 3/1
36\49 881.633 318.367 10 3 3.333
25\34 882.353 317.647 7 2 3.500
39\53 883.019 316.981 11 3 3.667 Hanson/keemun is in this region
14\19 884.211 315.789 4 1 4.000
31\42 885.714 314.286 9 2 4.500
17\23 886.957 313.043 5 1 5.000
20\27 888.889 311.111 6 1 6.000 Oolong, myna↓
3\4 900.000 300.000 1 0 → inf

References

  1. Description of Sylvian Moon Dance mentioning the naming proposal https://musescore.com/user/36772625/scores/6700443 – The theme relates to the mystical nature of the Tribunal and TES lore, which fits smitonic.