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== Scale properties ==
== Scale properties ==
{{TAMNAMS use}}
{{TAMNAMS use}}
{{MOS scale properties}}
 
=== Intervals ===
{{MOS intervals}}
 
=== Generator chain ===
{{MOS genchain}}
 
=== Modes ===
{{MOS mode degrees}}


==== Proposed names ====
==== Proposed names ====

Revision as of 19:00, 3 March 2025

↖ 3L 2s ↑ 4L 2s 5L 2s ↗
← 3L 3s 4L 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 ¢)
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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 as the name of 4L 3s. The name derives from "sharp minor third", referring to the generator's quality.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 4L 3s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-smistep Perfect 0-smistep P0smis 0 0.0 ¢
1-smistep Minor 1-smistep m1smis s 0.0 ¢ to 171.4 ¢
Major 1-smistep M1smis L 171.4 ¢ to 300.0 ¢
2-smistep Perfect 2-smistep P2smis L + s 300.0 ¢ to 342.9 ¢
Augmented 2-smistep A2smis 2L 342.9 ¢ to 600.0 ¢
3-smistep Minor 3-smistep m3smis L + 2s 300.0 ¢ to 514.3 ¢
Major 3-smistep M3smis 2L + s 514.3 ¢ to 600.0 ¢
4-smistep Minor 4-smistep m4smis 2L + 2s 600.0 ¢ to 685.7 ¢
Major 4-smistep M4smis 3L + s 685.7 ¢ to 900.0 ¢
5-smistep Diminished 5-smistep d5smis 2L + 3s 600.0 ¢ to 857.1 ¢
Perfect 5-smistep P5smis 3L + 2s 857.1 ¢ to 900.0 ¢
6-smistep Minor 6-smistep m6smis 3L + 3s 900.0 ¢ to 1028.6 ¢
Major 6-smistep M6smis 4L + 2s 1028.6 ¢ to 1200.0 ¢
7-smistep Perfect 7-smistep P7smis 4L + 3s 1200.0 ¢

Generator chain

Generator chain of 4L 3s
Bright gens Scale degree Abbrev.
10 Augmented 1-smidegree A1smid
9 Augmented 3-smidegree A3smid
8 Augmented 5-smidegree A5smid
7 Augmented 0-smidegree A0smid
6 Augmented 2-smidegree A2smid
5 Major 4-smidegree M4smid
4 Major 6-smidegree M6smid
3 Major 1-smidegree M1smid
2 Major 3-smidegree M3smid
1 Perfect 5-smidegree P5smid
0 Perfect 0-smidegree
Perfect 7-smidegree		 P0smid
P7smid
−1 Perfect 2-smidegree P2smid
−2 Minor 4-smidegree m4smid
−3 Minor 6-smidegree m6smid
−4 Minor 1-smidegree m1smid
−5 Minor 3-smidegree m3smid
−6 Diminished 5-smidegree d5smid
−7 Diminished 7-smidegree d7smid
−8 Diminished 2-smidegree d2smid
−9 Diminished 4-smidegree d4smid
−10 Diminished 6-smidegree d6smid

Modes

Scale degrees of the modes of 4L 3s
UDP Cyclic
order 		Step
pattern 		Scale degree (smidegree)
0 1 2 3 4 5 6 7
6|0 1 LLsLsLs Perf. Maj. Aug. Maj. Maj. Perf. Maj. Perf.
5|1 6 LsLLsLs Perf. Maj. Perf. Maj. Maj. Perf. Maj. Perf.
4|2 4 LsLsLLs Perf. Maj. Perf. Maj. Min. Perf. Maj. Perf.
3|3 2 LsLsLsL Perf. Maj. Perf. Maj. Min. Perf. Min. Perf.
2|4 7 sLLsLsL Perf. Min. Perf. Maj. Min. Perf. Min. Perf.
1|5 5 sLsLLsL Perf. Min. Perf. Min. Min. Perf. Min. Perf.
0|6 3 sLsLsLL Perf. Min. Perf. Min. Min. Dim. Min. Perf.

Proposed names

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

Modes of 4L 3s
UDP Cyclic
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

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 it takes 10 of them to make a 6/1, meaning that a larger MOS than 4L 3s is required to reach 3/2 or 4/3.

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

Todo: Populate

Populate with JI ratios from prior edits of this page.

Simple tunings

The simplest tunings are those with step ratios 2:1, 3:1, and 3:2, producing 11edo, 15edo, and 18edo, respectively.

Simple Tunings of 4L 3s
Scale degree Abbrev. Basic (2:1)
11edo 		Hard (3:1)
15edo 		Soft (3:2)
18edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-smidegree P0smid 0\11 0.0 0\15 0.0 0\18 0.0
Minor 1-smidegree m1smid 1\11 109.1 1\15 80.0 2\18 133.3
Major 1-smidegree M1smid 2\11 218.2 3\15 240.0 3\18 200.0
Perfect 2-smidegree P2smid 3\11 327.3 4\15 320.0 5\18 333.3
Augmented 2-smidegree A2smid 4\11 436.4 6\15 480.0 6\18 400.0
Minor 3-smidegree m3smid 4\11 436.4 5\15 400.0 7\18 466.7
Major 3-smidegree M3smid 5\11 545.5 7\15 560.0 8\18 533.3
Minor 4-smidegree m4smid 6\11 654.5 8\15 640.0 10\18 666.7
Major 4-smidegree M4smid 7\11 763.6 10\15 800.0 11\18 733.3
Diminished 5-smidegree d5smid 7\11 763.6 9\15 720.0 12\18 800.0
Perfect 5-smidegree P5smid 8\11 872.7 11\15 880.0 13\18 866.7
Minor 6-smidegree m6smid 9\11 981.8 12\15 960.0 15\18 1000.0
Major 6-smidegree M6smid 10\11 1090.9 14\15 1120.0 16\18 1066.7
Perfect 7-smidegree P7smid 11\11 1200.0 15\15 1200.0 18\18 1200.0

Parasoft tunings

Parasoft smitonic tunings 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 ¢).
Parasoft Tunings of 4L 3s
Scale degree Abbrev. Supersoft (4:3)
25edo 		7:5
43edo 		Soft (3:2)
18edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-smidegree P0smid 0\25 0.0 0\43 0.0 0\18 0.0
Minor 1-smidegree m1smid 3\25 144.0 5\43 139.5 2\18 133.3
Major 1-smidegree M1smid 4\25 192.0 7\43 195.3 3\18 200.0
Perfect 2-smidegree P2smid 7\25 336.0 12\43 334.9 5\18 333.3
Augmented 2-smidegree A2smid 8\25 384.0 14\43 390.7 6\18 400.0
Minor 3-smidegree m3smid 10\25 480.0 17\43 474.4 7\18 466.7
Major 3-smidegree M3smid 11\25 528.0 19\43 530.2 8\18 533.3
Minor 4-smidegree m4smid 14\25 672.0 24\43 669.8 10\18 666.7
Major 4-smidegree M4smid 15\25 720.0 26\43 725.6 11\18 733.3
Diminished 5-smidegree d5smid 17\25 816.0 29\43 809.3 12\18 800.0
Perfect 5-smidegree P5smid 18\25 864.0 31\43 865.1 13\18 866.7
Minor 6-smidegree m6smid 21\25 1008.0 36\43 1004.7 15\18 1000.0
Major 6-smidegree M6smid 22\25 1056.0 38\43 1060.5 16\18 1066.7
Perfect 7-smidegree P7smid 25\25 1200.0 43\43 1200.0 18\18 1200.0

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.


Hyposoft Tunings of 4L 3s
Scale degree Abbrev. Soft (3:2)
18edo 		Semisoft (5:3)
29edo 		7:4
40edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-smidegree P0smid 0\18 0.0 0\29 0.0 0\40 0.0
Minor 1-smidegree m1smid 2\18 133.3 3\29 124.1 4\40 120.0
Major 1-smidegree M1smid 3\18 200.0 5\29 206.9 7\40 210.0
Perfect 2-smidegree P2smid 5\18 333.3 8\29 331.0 11\40 330.0
Augmented 2-smidegree A2smid 6\18 400.0 10\29 413.8 14\40 420.0
Minor 3-smidegree m3smid 7\18 466.7 11\29 455.2 15\40 450.0
Major 3-smidegree M3smid 8\18 533.3 13\29 537.9 18\40 540.0
Minor 4-smidegree m4smid 10\18 666.7 16\29 662.1 22\40 660.0
Major 4-smidegree M4smid 11\18 733.3 18\29 744.8 25\40 750.0
Diminished 5-smidegree d5smid 12\18 800.0 19\29 786.2 26\40 780.0
Perfect 5-smidegree P5smid 13\18 866.7 21\29 869.0 29\40 870.0
Minor 6-smidegree m6smid 15\18 1000.0 24\29 993.1 33\40 990.0
Major 6-smidegree M6smid 16\18 1066.7 26\29 1075.9 36\40 1080.0
Perfect 7-smidegree P7smid 18\18 1200.0 29\29 1200.0 40\40 1200.0

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.


Hypohard Tunings of 4L 3s
Scale degree Abbrev. 7:3
37edo 		Semihard (5:2)
26edo 		Hard (3:1)
15edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-smidegree P0smid 0\37 0.0 0\26 0.0 0\15 0.0
Minor 1-smidegree m1smid 3\37 97.3 2\26 92.3 1\15 80.0
Major 1-smidegree M1smid 7\37 227.0 5\26 230.8 3\15 240.0
Perfect 2-smidegree P2smid 10\37 324.3 7\26 323.1 4\15 320.0
Augmented 2-smidegree A2smid 14\37 454.1 10\26 461.5 6\15 480.0
Minor 3-smidegree m3smid 13\37 421.6 9\26 415.4 5\15 400.0
Major 3-smidegree M3smid 17\37 551.4 12\26 553.8 7\15 560.0
Minor 4-smidegree m4smid 20\37 648.6 14\26 646.2 8\15 640.0
Major 4-smidegree M4smid 24\37 778.4 17\26 784.6 10\15 800.0
Diminished 5-smidegree d5smid 23\37 745.9 16\26 738.5 9\15 720.0
Perfect 5-smidegree P5smid 27\37 875.7 19\26 876.9 11\15 880.0
Minor 6-smidegree m6smid 30\37 973.0 21\26 969.2 12\15 960.0
Major 6-smidegree M6smid 34\37 1102.7 24\26 1107.7 14\15 1120.0
Perfect 7-smidegree P7smid 37\37 1200.0 26\26 1200.0 15\15 1200.0

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, through the 2.3.5.13 extension known as cata.

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


Parahard Tunings of 4L 3s
Scale degree Abbrev. 7:2
34edo 		11:3
53edo 		Superhard (4:1)
19edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-smidegree P0smid 0\34 0.0 0\53 0.0 0\19 0.0
Minor 1-smidegree m1smid 2\34 70.6 3\53 67.9 1\19 63.2
Major 1-smidegree M1smid 7\34 247.1 11\53 249.1 4\19 252.6
Perfect 2-smidegree P2smid 9\34 317.6 14\53 317.0 5\19 315.8
Augmented 2-smidegree A2smid 14\34 494.1 22\53 498.1 8\19 505.3
Minor 3-smidegree m3smid 11\34 388.2 17\53 384.9 6\19 378.9
Major 3-smidegree M3smid 16\34 564.7 25\53 566.0 9\19 568.4
Minor 4-smidegree m4smid 18\34 635.3 28\53 634.0 10\19 631.6
Major 4-smidegree M4smid 23\34 811.8 36\53 815.1 13\19 821.1
Diminished 5-smidegree d5smid 20\34 705.9 31\53 701.9 11\19 694.7
Perfect 5-smidegree P5smid 25\34 882.4 39\53 883.0 14\19 884.2
Minor 6-smidegree m6smid 27\34 952.9 42\53 950.9 15\19 947.4
Major 6-smidegree M6smid 32\34 1129.4 50\53 1132.1 18\19 1136.8
Perfect 7-smidegree P7smid 34\34 1200.0 53\53 1200.0 19\19 1200.0

Scales

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
Orgone
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.
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