4L 3s

↖ 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

Equal tunings

Equalized (L:s = 1:1)

5\7 (857.1¢)

Supersoft (L:s = 4:3)

18\25 (864.0¢)

13\18 (866.7¢)

21\29 (869.0¢)

8\11 (872.7¢)

19\26 (876.9¢)

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 as the name of 4L 3s. The name derives from "sharp minor third", referring to the generator's quality.

Scale properties

Intervals

The intervals of 4L 3s are named after the number of mossteps (L and s) they subtend. Each interval, apart from the root and octave (perfect 0-smistep and perfect 7-smistep), has two varieties, or sizes, each. Interval varieties are named major and minor for the large and small sizes, respectively, and augmented, perfect, and diminished for the scale's generators.

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

Generator chain

A chain of bright generators, each a perfect 5-smistep, produces the following scale degrees. A chain of 7 bright generators contains the scale degrees of one of the modes of 4L 3s. Expanding the chain to 11 scale degrees produces the modes of either 7L 4s (for soft-of-basic tunings) or 4L 7s (for hard-of-basic tunings).

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)
UDP	Cyclic order	Step pattern	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		Approx. ratios*
Scale degree	Abbrev.	Steps	¢	Steps	¢	Steps	¢	Approx. ratios*
Perfect 0-smidegree	P0smid	0\11	0.0	0\15	0.0	0\18	0.0	1/1
Minor 1-smidegree	m1smid	1\11	109.1	1\15	80.0	2\18	133.3	16/15, 14/13
Major 1-smidegree	M1smid	2\11	218.2	3\15	240.0	3\18	200.0	9/8, 8/7
Perfect 2-smidegree	P2smid	3\11	327.3	4\15	320.0	5\18	333.3	6/5, 11/9
Augmented 2-smidegree	A2smid	4\11	436.4	6\15	480.0	6\18	400.0	14/11, 9/7
Minor 3-smidegree	m3smid	4\11	436.4	5\15	400.0	7\18	466.7	14/11, 9/7
Major 3-smidegree	M3smid	5\11	545.5	7\15	560.0	8\18	533.3	11/8, 18/13
Minor 4-smidegree	m4smid	6\11	654.5	8\15	640.0	10\18	666.7	13/9, 16/11
Major 4-smidegree	M4smid	7\11	763.6	10\15	800.0	11\18	733.3	14/9, 11/7
Diminished 5-smidegree	d5smid	7\11	763.6	9\15	720.0	12\18	800.0	14/9, 11/7
Perfect 5-smidegree	P5smid	8\11	872.7	11\15	880.0	13\18	866.7	18/11, 5/3
Minor 6-smidegree	m6smid	9\11	981.8	12\15	960.0	15\18	1000.0	7/4, 16/9
Major 6-smidegree	M6smid	10\11	1090.9	14\15	1120.0	16\18	1066.7	13/7, 15/8
Perfect 7-smidegree	P7smid	11\11	1200.0	15\15	1200.0	18\18	1200.0	2/1

* Ratios shown are within the 50-integer limit. Automatic search may be inexact. Other interpretations are possible.

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		Approx. ratios*
Scale degree	Abbrev.	Steps	¢	Steps	¢	Steps	¢	Approx. ratios*
Perfect 0-smidegree	P0smid	0\25	0.0	0\43	0.0	0\18	0.0	1/1
Minor 1-smidegree	m1smid	3\25	144.0	5\43	139.5	2\18	133.3	16/15, 14/13, 12/11, 11/10
Major 1-smidegree	M1smid	4\25	192.0	7\43	195.3	3\18	200.0	10/9, 9/8
Perfect 2-smidegree	P2smid	7\25	336.0	12\43	334.9	5\18	333.3	6/5, 11/9, 16/13
Augmented 2-smidegree	A2smid	8\25	384.0	14\43	390.7	6\18	400.0	5/4, 14/11
Minor 3-smidegree	m3smid	10\25	480.0	17\43	474.4	7\18	466.7	4/3
Major 3-smidegree	M3smid	11\25	528.0	19\43	530.2	8\18	533.3	11/8
Minor 4-smidegree	m4smid	14\25	672.0	24\43	669.8	10\18	666.7	16/11
Major 4-smidegree	M4smid	15\25	720.0	26\43	725.6	11\18	733.3	3/2
Diminished 5-smidegree	d5smid	17\25	816.0	29\43	809.3	12\18	800.0	11/7, 8/5
Perfect 5-smidegree	P5smid	18\25	864.0	31\43	865.1	13\18	866.7	13/8, 18/11, 5/3
Minor 6-smidegree	m6smid	21\25	1008.0	36\43	1004.7	15\18	1000.0	16/9, 9/5
Major 6-smidegree	M6smid	22\25	1056.0	38\43	1060.5	16\18	1066.7	20/11, 11/6, 13/7, 15/8
Perfect 7-smidegree	P7smid	25\25	1200.0	43\43	1200.0	18\18	1200.0	2/1

* Ratios shown are within the 50-integer limit. Automatic search may be inexact. Other interpretations are possible.

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		Approx. ratios*
Scale degree	Abbrev.	Steps	¢	Steps	¢	Steps	¢	Approx. ratios*
Perfect 0-smidegree	P0smid	0\18	0.0	0\29	0.0	0\40	0.0	1/1
Minor 1-smidegree	m1smid	2\18	133.3	3\29	124.1	4\40	120.0	16/15, 14/13, 12/11
Major 1-smidegree	M1smid	3\18	200.0	5\29	206.9	7\40	210.0	10/9, 9/8, 8/7
Perfect 2-smidegree	P2smid	5\18	333.3	8\29	331.0	11\40	330.0	6/5, 11/9, 16/13
Augmented 2-smidegree	A2smid	6\18	400.0	10\29	413.8	14\40	420.0	5/4, 14/11, 9/7
Minor 3-smidegree	m3smid	7\18	466.7	11\29	455.2	15\40	450.0	9/7
Major 3-smidegree	M3smid	8\18	533.3	13\29	537.9	18\40	540.0	11/8, 18/13
Minor 4-smidegree	m4smid	10\18	666.7	16\29	662.1	22\40	660.0	13/9, 16/11
Major 4-smidegree	M4smid	11\18	733.3	18\29	744.8	25\40	750.0	14/9
Diminished 5-smidegree	d5smid	12\18	800.0	19\29	786.2	26\40	780.0	14/9, 11/7, 8/5
Perfect 5-smidegree	P5smid	13\18	866.7	21\29	869.0	29\40	870.0	13/8, 18/11, 5/3
Minor 6-smidegree	m6smid	15\18	1000.0	24\29	993.1	33\40	990.0	7/4, 16/9, 9/5
Major 6-smidegree	M6smid	16\18	1066.7	26\29	1075.9	36\40	1080.0	11/6, 13/7, 15/8
Perfect 7-smidegree	P7smid	18\18	1200.0	29\29	1200.0	40\40	1200.0	2/1

* Ratios shown are within the 50-integer limit. Automatic search may be inexact. Other interpretations are possible.

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		Approx. ratios*
Scale degree	Abbrev.	Steps	¢	Steps	¢	Steps	¢	Approx. ratios*
Perfect 0-smidegree	P0smid	0\37	0.0	0\26	0.0	0\15	0.0	1/1
Minor 1-smidegree	m1smid	3\37	97.3	2\26	92.3	1\15	80.0	16/15
Major 1-smidegree	M1smid	7\37	227.0	5\26	230.8	3\15	240.0	9/8, 8/7
Perfect 2-smidegree	P2smid	10\37	324.3	7\26	323.1	4\15	320.0	6/5, 11/9
Augmented 2-smidegree	A2smid	14\37	454.1	10\26	461.5	6\15	480.0	9/7
Minor 3-smidegree	m3smid	13\37	421.6	9\26	415.4	5\15	400.0	14/11, 9/7
Major 3-smidegree	M3smid	17\37	551.4	12\26	553.8	7\15	560.0	11/8, 18/13
Minor 4-smidegree	m4smid	20\37	648.6	14\26	646.2	8\15	640.0	13/9, 16/11
Major 4-smidegree	M4smid	24\37	778.4	17\26	784.6	10\15	800.0	14/9, 11/7
Diminished 5-smidegree	d5smid	23\37	745.9	16\26	738.5	9\15	720.0	14/9
Perfect 5-smidegree	P5smid	27\37	875.7	19\26	876.9	11\15	880.0	18/11, 5/3
Minor 6-smidegree	m6smid	30\37	973.0	21\26	969.2	12\15	960.0	7/4, 16/9
Major 6-smidegree	M6smid	34\37	1102.7	24\26	1107.7	14\15	1120.0	15/8
Perfect 7-smidegree	P7smid	37\37	1200.0	26\26	1200.0	15\15	1200.0	2/1

* Ratios shown are within the 50-integer limit. Automatic search may be inexact. Other interpretations are possible.

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.

Parahard Tunings of 4L 3s
Scale degree	Abbrev.	7:2 34edo		11:3 53edo		Superhard (4:1) 19edo		Approx. ratios*
Scale degree	Abbrev.	Steps	¢	Steps	¢	Steps	¢	Approx. ratios*
Perfect 0-smidegree	P0smid	0\34	0.0	0\53	0.0	0\19	0.0	1/1
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	8/7, 7/6
Perfect 2-smidegree	P2smid	9\34	317.6	14\53	317.0	5\19	315.8	6/5
Augmented 2-smidegree	A2smid	14\34	494.1	22\53	498.1	8\19	505.3	4/3
Minor 3-smidegree	m3smid	11\34	388.2	17\53	384.9	6\19	378.9	5/4
Major 3-smidegree	M3smid	16\34	564.7	25\53	566.0	9\19	568.4	11/8, 18/13, 7/5
Minor 4-smidegree	m4smid	18\34	635.3	28\53	634.0	10\19	631.6	10/7, 13/9, 16/11
Major 4-smidegree	M4smid	23\34	811.8	36\53	815.1	13\19	821.1	8/5
Diminished 5-smidegree	d5smid	20\34	705.9	31\53	701.9	11\19	694.7	3/2
Perfect 5-smidegree	P5smid	25\34	882.4	39\53	883.0	14\19	884.2	5/3
Minor 6-smidegree	m6smid	27\34	952.9	42\53	950.9	15\19	947.4	12/7, 7/4
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	2/1

* Ratios shown are within the 50-integer limit. Automatic search may be inexact. Other interpretations are possible.

Scales

Scale tree

Scale Tree and Tuning Spectrum of 4L 3s
Generator^(edo)						Cents		Step ratio		Comments
Generator^(edo)						Bright	Dark	L:s	Hardness	Comments
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

City of the Asleep, "An Amputated Elliptic Knob of the Cryptocurve Regenerates" (Various orgone edos)
ks26, Ghost Bridge (11edo)
Alexandru Ianu, Sylvian Moon Dance (11edo) (sheet music)

References

↑ 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.

[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.

[1]

4L 3s

Contents

Name

Scale properties

Intervals

Generator chain

Modes

Proposed names

Theory

Low harmonic entropy scales

Temperament interpretations

Tuning ranges

Simple tunings

Parasoft tunings

Hyposoft tunings

Hypohard tunings

Parahard tunings

Scales

Scale tree

Music

References

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