4L 3s

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↖ 3L 2s ↑4L 2s 5L 2s ↗
← 3L 3s4L 3s 5L 3s →
↙ 3L 4s ↓4L 4s 5L 4s ↘
┌╥╥┬╥┬╥┬┐
│║║│║│║││
│││││││││
└┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLsLsLs
sLsLsLL
Equave 2/1 (1200.0¢)
Period 2/1 (1200.0¢)
Generator size
Bright 5\7 to 3\4 (857.1¢ to 900.0¢)
Dark 1\4 to 2\7 (300.0¢ to 342.9¢)
TAMNAMS information
Name smitonic
Prefix smi-
Abbrev. smi
Related MOS scales
Parent 3L 1s
Sister 3L 4s
Daughters 7L 4s, 4L 7s
Neutralized 1L 6s
2-Flought 11L 3s, 4L 10s
Equal tunings
Equalized (L:s = 1:1) 5\7 (857.1¢)
Supersoft (L:s = 4:3) 18\25 (864.0¢)
Soft (L:s = 3:2) 13\18 (866.7¢)
Semisoft (L:s = 5:3) 21\29 (869.0¢)
Basic (L:s = 2:1) 8\11 (872.7¢)
Semihard (L:s = 5:2) 19\26 (876.9¢)
Hard (L:s = 3:1) 11\15 (880.0¢)
Superhard (L:s = 4:1) 14\19 (884.2¢)
Collapsed (L:s = 1:0) 3\4 (900.0¢)

4L 3s, named smitonic in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 3 small steps, repeating every octave. Generators that produce this scale range from 857.1¢ to 900¢, or from 300¢ to 342.9¢. 4L 3s can be seen as a warped diatonic scale, where one large step of diatonic (5L 2s) is replaced with a small step.

Name

TAMNAMS suggests the temperament-agnostic name smitonic smy-TON-ik /smaɪˈtɒnɪk/ for this scale. The name is derived from 'sharp minor third', since the central range for the dark generator (320¢ to 333.3¢) is significantly sharp of 6/5 (just minor 3rd, 315.6¢).

Notation

This article assumes TAMNAMS for naming step ratios, intervals, and scale degrees, and diamond-MOS notation for note names.

Intervals and degrees

Names for this scale's intervals (mossteps) and scale degrees (mosdegrees) are based on the number of large and small steps from the root, starting at 0 (0-mosstep and 0-mosdegree) for the unison, per TAMNAMS. Ordinal names, such as mos-1st for the unison, are discouraged for non-diatonic MOS scales.

Being a moment-of-symmetry scale, every interval class of 4L 3s, except for the unison and octave, has two varieties – large and small – whose relative qualities are denoted as major or minor, or augmented, perfect, and diminished for the generators.

Intervals of 4L 3s
Intervals Steps subtended Range in cents Average of HE
(from HE Calc)
Min of HE
Generic[1] Specific[2] Abbrev.[3]
0-smistep Perfect 0-smistep P0ms 0 0.0¢ ~2.4654 nats ~2.4654 nats
1-smistep Minor 1-smistep m1ms s 0.0¢ to 171.4¢ ~4.6810 nats ~4.6286 nats
Major 1-smistep M1ms L 171.4¢ to 300.0¢ ~4.5848 nats ~4.5794 nats
2-smistep Perfect 2-smistep P2ms L + s 300.0¢ to 342.9¢ ~4.5630 nats ~4.5379 nats
Augmented 2-smistep A2ms 2L 342.9¢ to 600.0¢ ~4.5325 nats ~4.3884 nats
3-smistep Minor 3-smistep m3ms L + 2s 300.0¢ to 514.3¢ ~4.5669 nats ~4.5066 nats
Major 3-smistep M3ms 2L + s 514.3¢ to 600.0¢ ~4.6137 nats ~4.5864 nats
4-smistep Minor 4-smistep m4ms 2L + 2s 600.0¢ to 685.7¢ ~4.6297 nats ~4.6044 nats
Major 4-smistep M4ms 3L + s 685.7¢ to 900.0¢ ~4.5720 nats ~4.3710 nats
5-smistep Diminished 5-smistep d5ms 2L + 3s 600.0¢ to 857.1¢ ~4.5116 nats ~4.1942 nats
Perfect 5-smistep P5ms 3L + 2s 857.1¢ to 900.0¢ ~4.4841 nats ~4.4208 nats
6-smistep Minor 6-smistep m6ms 3L + 3s 900.0¢ to 1028.6¢ ~4.5678 nats ~4.5308 nats
Major 6-smistep M6ms 4L + 2s 1028.6¢ to 1200.0¢ ~4.6276 nats ~4.6077 nats
7-smistep Perfect 7-smistep P7ms 4L + 3s 1200.0¢ ~3.3273 nats ~3.3273 nats

  1. Generic intervals are denoted solely by the number of steps they subtend.
  2. Specific intervals denote whether an interval is major, minor, augmented, perfect, or diminished.
  3. Abbreviations can be further shortened to 'ms' if context allows.

Note names

For this article, note names are based on diamond-MOS notation, where the naturals JKLMNOP are applied to the step pattern LsLsLsL and the accidentals & (pronounced "am" or "amp") and @ (pronounced "at") are used to represent sharps and flats respectively. Thus, the basic gamut for 4L 3s is the following:

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

Theory

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.
  • Myna temperament, in which the generator is also 6/5 but 10 of them make a 6/1, resulting in the intervals 4/3 and 3/2 being absent.

Temperament interpretations

Main article: 4L 3s/Temperaments

4L 3s has the following temperament interpretations:

  • Sixix, with generators around 338.6¢.
  • Orgone, with generators around 323.4¢.
  • Kleismic, with generators around 317¢.

Other temperaments, such as amity and myna, require more than 7 pitches to contain the concordant chords optimized by these temperaments. If restricted to a rank-2 approach, a MODMOS or a larger MOS gamut is necessary to access these pitches.

Tuning ranges

Simple tunings

The basic tuning for 4L 3s has a large and small step size of 2 and 1 respectively, which is supported by 11edo. Other small edos include 15edo and 18edo.

Scale degree of 4L 3s
Scale degree 11edo (Basic, L:s = 2:1) 15edo (Hard, L:s = 3:1) 18edo (Soft, L:s = 3:2) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents
Perfect 0-smidegree (unison) 0 0 0 0 0 0 1/1 (exact)
Minor 1-smidegree 1 109.1 1 80 2 133.3
Major 1-smidegree 2 218.2 3 240 3 200 8/7
Perfect 2-smidegree 3 327.3 4 320 5 333.3 6/5, 77/64
Augmented 2-smidegree 4 436.4 6 480 6 400
Minor 3-smidegree 4 436.4 5 400 7 466.7 14/11
Major 3-smidegree 5 545.5 7 560 8 533.3 11/8
Minor 4-smidegree 6 654.5 8 640 10 666.7 16/11
Major 4-smidegree 7 763.6 10 800 11 733.3 11/7
Diminished 5-smidegree 7 763.6 9 720 12 800
Perfect 5-smidegree 8 872.7 11 880 13 866.7 5/3
Minor 6-smidegree 9 981.8 12 960 15 1000 7/4
Major 6-smidegree 10 1090.9 14 1120 16 1066.7
Perfect 7-smidegree (octave) 11 1200 15 1200 18 1200 2/1 (exact)

Parasoft tunings

Parasoft smitonic tunings (4:3 to 3:2) can be considered "meantone smitonic" since it has the following features of meantone diatonic tunings:

  • The major 1-mosstep, or large step, is around 10/9 to 9/8, thus making it a "meantone".
  • The augmented 2-mosstep is around the size of a meantone-sized major 3rd and can be used as a stand-in for such.

These tunings have a major 4-mosstep and minor 4-mosstep that are about equally off a just 3/2 (702¢), 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.

Edos include 18edo, 25edo, and 43edo. Some key considerations include:

  • 18edo can be used to make the large and small steps more distinct, or can be considered a distorted 19edo diatonic.
    • 18edo has a major 1-mosstep that is close to 9/8 (203¢).
    • 18edo's major and minor 4-mossteps are both equally off from 12edo's diatonic perfect 5th (700¢) by 33.3¢.
    • 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
  • The augmented 2-mosstep of 25edo is very close to 5/4 (386¢).
  • The various interval flavors separated by a chroma shows that parasoft smitonic is a useful cluster MOS. However, many of these intervals lack simple JI interpretations.
Scale degree of 4L 3s
Scale degree 18edo (Soft, L:s = 3:2) 25edo (Supersoft, L:s = 4:3) 43edo (L:s = 7:5) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents
Perfect 0-smidegree (unison) 0 0 0 0 0 0 1/1 (exact)
Augmented 0-smidegree 1 66.7 1 48 2 55.8
Diminished 1-smidegree 1 66.7 2 96 3 83.7
Minor 1-smidegree 2 133.3 3 144 5 139.5 13/12
Major 1-smidegree 3 200 4 192 7 195.3 9/8, 10/9
Augmented 1-smidegree 4 266.7 5 240 9 251.2
Diminished 2-smidegree 4 266.7 6 288 10 279.1
Perfect 2-smidegree 5 333.3 7 336 12 334.9 17/14, 40/33
Augmented 2-smidegree 6 400 8 384 14 390.7 5/4
2× Augmented 2-smidegree 7 466.7 9 432 16 446.5
Diminished 3-smidegree 6 400 9 432 15 418.6
Minor 3-smidegree 7 466.7 10 480 17 474.4 21/16
Major 3-smidegree 8 533.3 11 528 19 530.2 19/14, 34/25
Augmented 3-smidegree 9 600 12 576 21 586 7/5
Diminished 4-smidegree 9 600 13 624 22 614 10/7
Minor 4-smidegree 10 666.7 14 672 24 669.8 28/19, 25/17
Major 4-smidegree 11 733.3 15 720 26 725.6 32/21
Augmented 4-smidegree 12 800 16 768 28 781.4
2× Diminished 5-smidegree 11 733.3 16 768 27 753.5
Diminished 5-smidegree 12 800 17 816 29 809.3 8/5
Perfect 5-smidegree 13 866.7 18 864 31 865.1 28/17, 33/20
Augmented 5-smidegree 14 933.3 19 912 33 920.9
Diminished 6-smidegree 14 933.3 20 960 34 948.8
Minor 6-smidegree 15 1000 21 1008 36 1004.7 16/9, 9/5
Major 6-smidegree 16 1066.7 22 1056 38 1060.5 24/13
Augmented 6-smidegree 17 1133.3 23 1104 40 1116.3
Diminished 7-smidegree 17 1133.3 24 1152 41 1144.2
Perfect 7-smidegree (octave) 18 1200 25 1200 43 1200 2/1 (exact)

Hyposoft tunings

Hyposoft smitonic tunings (3:2 to 2:1) are characterized by generators that are a supraminor 3rd, between 327¢ and 333¢. By analogy of parasoft tunings being called "meantone smitonic", these tunings can be considered "neogothic smitonic" or "archy smitonic".

Edos include 11edo (not shown), 18edo, and 29edo.

Scale degree of 4L 3s
Scale degree 18edo (Soft, L:s = 3:2) 29edo (Semisoft, L:s = 5:3) Approx. JI Ratios
Steps Cents Steps Cents
Perfect 0-smidegree (unison) 0 0 0 0 1/1 (exact)
Minor 1-smidegree 2 133.3 3 124.1 14/13
Major 1-smidegree 3 200 5 206.9 9/8
Perfect 2-smidegree 5 333.3 8 331 23/19, 40/33
Augmented 2-smidegree 6 400 10 413.8 14/11
Minor 3-smidegree 7 466.7 11 455.2 13/10
Major 3-smidegree 8 533.3 13 537.9 15/11
Minor 4-smidegree 10 666.7 16 662.1 19/13, 22/15
Major 4-smidegree 11 733.3 18 744.8 20/13
Diminished 5-smidegree 12 800 19 786.2 11/7
Perfect 5-smidegree 13 866.7 21 869 33/20, 38/23
Minor 6-smidegree 15 1000 24 993.1 16/9
Major 6-smidegree 16 1066.7 26 1075.9 13/7
Perfect 7-smidegree (octave) 18 1200 29 1200 2/1 (exact)

Hypohard tunings

Hypohard smitonic tunings (2:1 to 3:1) have generators between 320¢ and 327¢. The major 1-mosstep, or large step, tends to approximate 8/7 (231¢) and the major 3-mosstep tends to approximate 11/8 (551¢). 26edo approximates these two intervals very well. These JI approximations are associated with orgone temperament.

Other hypohard edos include 11edo (not shown), 15edo and 37edo.

Scale degree of 4L 3s
Scale degree 15edo (Hard, L:s = 3:1) 26edo (Semihard, L:s = 5:2) 37edo (L:s = 7:3) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents
Perfect 0-smidegree (unison) 0 0 0 0 0 0 1/1 (exact)
Minor 1-smidegree 1 80 2 92.3 3 97.3
Major 1-smidegree 3 240 5 230.8 7 227 8/7
Perfect 2-smidegree 4 320 7 323.1 10 324.3 6/5, 77/64
Augmented 2-smidegree 6 480 10 461.5 14 454.1
Minor 3-smidegree 5 400 9 415.4 13 421.6 14/11
Major 3-smidegree 7 560 12 553.8 17 551.4 11/8
Minor 4-smidegree 8 640 14 646.2 20 648.6 16/11
Major 4-smidegree 10 800 17 784.6 24 778.4 11/7
Diminished 5-smidegree 9 720 16 738.5 23 745.9
Perfect 5-smidegree 11 880 19 876.9 27 875.7 5/3
Minor 6-smidegree 12 960 21 969.2 30 973 7/4
Major 6-smidegree 14 1120 24 1107.7 34 1102.7
Perfect 7-smidegree (octave) 15 1200 26 1200 37 1200 2/1 (exact)

Parahard tunings

Parahard smitonic tunings (3:1 to 4:1) have generators between 315.9¢ and 320¢, putting it close to a pure 6/5 (316¢). Stacking six generators and octave-reducing approximates 3/2 (702¢), a diatonic perfect 5th, represented by the diminished 5-mosstep.

This 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, 4L 3s only has one interval of 3/2, so it's suggested to use a larger MOS, such as 4L 7s, to achieve 5-limit harmony.

These JI approximations are associated with kleismic temperament, though the 2.3.5.13 extension described here is called cata.

Parahard edos smaller than 53edo include 15edo (not shown), 19edo, and 34edo.

Scale degree of 4L 3s
Scale degree 19edo (Superhard, L:s = 4:1) 34edo (L:s = 7:2) 53edo (L:s = 11:3) 72edo (L:s = 15:4) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents Steps Cents
Perfect 0-smidegree (unison) 0 0 0 0 0 0 0 0 1/1 (exact)
Minor 1-smidegree 1 63.2 2 70.6 3 67.9 4 66.7 25/24, 26/25
Major 1-smidegree 4 252.6 7 247.1 11 249.1 15 250 15/13
Perfect 2-smidegree 5 315.8 9 317.6 14 317 19 316.7 6/5
Augmented 2-smidegree 8 505.3 14 494.1 22 498.1 30 500 4/3
Minor 3-smidegree 6 378.9 11 388.2 17 384.9 23 383.3 5/4
Major 3-smidegree 9 568.4 16 564.7 25 566 34 566.7 18/13
Minor 4-smidegree 10 631.6 18 635.3 28 634 38 633.3 13/9
Major 4-smidegree 13 821.1 23 811.8 36 815.1 49 816.7 8/5
Diminished 5-smidegree 11 694.7 20 705.9 31 701.9 42 700 3/2
Perfect 5-smidegree 14 884.2 25 882.4 39 883 53 883.3 5/3
Minor 6-smidegree 15 947.4 27 952.9 42 950.9 57 950 26/15
Major 6-smidegree 18 1136.8 32 1129.4 50 1132.1 68 1133.3 25/13
Perfect 7-smidegree (octave) 19 1200 34 1200 53 1200 72 1200 2/1 (exact)

Modes

Alexandru Ianu (Ayceman)[1] has proposed the following mode names relating to the Almsivi in Morrowind (TES):


Modes of 4L 3s
UDP Rotational order Step pattern Mode names
6|0 1 LLsLsLs Nerevarine
5|1 6 LsLLsLs Vivecan
4|2 4 LsLsLLs Lorkhanic
3|3 2 LsLsLsL Sothic
2|4 7 sLLsLsL Kagrenacan
1|5 5 sLsLLsL Almalexian
0|6 3 sLsLsLL Dagothic

Scale degrees

Scale degree qualities of 4L 3s modes
UDP Rotational Order Step pattern Mode names Scale degree (smidegree)
0 1 2 3 4 5 6 7
6|0 1 LLsLsLs Nerevarine Perf. Maj. Aug. Maj. Maj. Perf. Maj. Perf.
5|1 6 LsLLsLs Vivecan Perf. Maj. Perf. Maj. Maj. Perf. Maj. Perf.
4|2 4 LsLsLLs Lorkhanic Perf. Maj. Perf. Maj. Min. Perf. Maj. Perf.
3|3 2 LsLsLsL Sothic Perf. Maj. Perf. Maj. Min. Perf. Min. Perf.
2|4 7 sLLsLsL Kagrenacan Perf. Min. Perf. Maj. Min. Perf. Min. Perf.
1|5 5 sLsLLsL Almalexian Perf. Min. Perf. Min. Min. Perf. Min. Perf.
0|6 3 sLsLsLL Dagothic Perf. Min. Perf. Min. Min. Dim. Min. Perf.

Scales

Subset and superset scales

4L 3s has a parent scale of 3L 1s, a tetratonic scale, meaning 1L 3s is a subset. 4L 3s also has two child scales, which are supersets of 4L 3s:

  • 7L 4s, a smitonic chromatic scale produced using soft-of-basic step ratios.
  • 4L 7s, a smitonic chromatic scale produced using hard-of-basic step ratios.

11edo, the equalized form of both 7L 4s and 4L 7s, is also a superset of 4L 3s.

Scala files

Scale tree

Scale Tree and Tuning Spectrum of 4L 3s
Generator(edo) Cents Step Ratio Comments
Bright Dark L:s Hardness
5\7 857.143 342.857 1:1 1.000 Equalized 4L 3s
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 Supersoft 4L 3s
49\68 864.706 335.294 11:8 1.375
31\43 865.116 334.884 7:5 1.400
44\61 865.574 334.426 10:7 1.429
13\18 866.667 333.333 3:2 1.500 Soft 4L 3s
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 4L 3s (868.3282¢)
21\29 868.966 331.034 5:3 1.667 Semisoft 4L 3s
50\69 869.565 330.435 12:7 1.714
29\40 870.000 330.000 7:4 1.750
37\51 870.588 329.412 9:5 1.800
8\11 872.727 327.273 2:1 2.000 Basic 4L 3s
Scales with tunings softer 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 Hyperkleismic
19\26 876.923 323.077 5:2 2.500 Semihard 4L 3s
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 Hard 4L 3s
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
14\19 884.211 315.789 4:1 4.000 Superhard 4L 3s
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 → ∞ Collapsed 4L 3s

Music

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.