User:Ganaram inukshuk/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
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 ¢)

This is a test page. For the main page, see 4L 3s.

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 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 and diamond-MOS notation for naming intervals, scale degrees, and note names.

Interval names

Names for intervals and scale degrees are based on TAMNAMS, where names for mossteps (intervals) and mosdegrees (scale degrees) are numbered starting at 0 for the unison. 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 (or sizes), whose relative qualities are denoted as major or minor, or augmented, perfect, and diminished for the generators.

TODO: make this into a template

Interval class	Large variety		Small variety
	Size	Quality	Size	Quality
0-mosstep (unison)	0	Perfect	0	Perfect
1-mosstep	L	Major	s	Minor
2-mosstep	L+s	Augmented	2L	Perfect
3-mosstep	L + 2s	Major	2L + s	Minor
4-mosstep	2L + 2s	Perfect	3L + 1s	Minor
5-mosstep	2L + 3s	Perfect	3L + 2s	Diminished
6-mosstep	2L + 3s	Major	3L + 3s	Minor
7-mosstep (octave)	4L + 3s	Perfect	4L + 3s	Perfect

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 ((6/5)^6, tempered, equals 3/1), and myna, in which the generator is also 6/5 but 10 of them make a 6/1 ((6/5)^10, tempered, equals 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:

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

There are also other temperaments in the 4L 3s range, particularly amity and myna, but 7 pitches are not enough to contain the concordant chords optimized by these temperaments; a MODMOS or a larger MOS gamut is necessary to access these pitches, if restricted to a rank-2 approach.

Step ratios

The basic tuning for 4L 3s has a large and small step size of 2 and 1 respectively, which is supported by 11edo. Raising or lowering certain nominals by a chroma produces the following scale degrees.

User:MOS degrees is deprecated. Please use Template:MOS tunings instead.

Scale degree of 4L 3s

Scale degree	11edo (Basic, L:s = 2:1)		Approx. JI Ratios
	Steps	Cents
Perfect 0-smidegree (unison)	0	0	1/1 (exact)
Minor 1-smidegree	1	109.1
Major 1-smidegree	2	218.2
Perfect 2-smidegree	3	327.3
Augmented 2-smidegree	4	436.4
Minor 3-smidegree	4	436.4
Major 3-smidegree	5	545.5
Minor 4-smidegree	6	654.5
Major 4-smidegree	7	763.6
Diminished 5-smidegree	7	763.6
Perfect 5-smidegree	8	872.7
Minor 6-smidegree	9	981.8
Major 6-smidegree	10	1090.9
Perfect 7-smidegree (octave)	11	1200	2/1 (exact)

Simple tunings

User:MOS degrees is deprecated. Please use Template:MOS tunings instead.

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
Perfect 2-smidegree	3	327.3	4	320	5	333.3
Augmented 2-smidegree	4	436.4	6	480	6	400
Minor 3-smidegree	4	436.4	5	400	7	466.7
Major 3-smidegree	5	545.5	7	560	8	533.3
Minor 4-smidegree	6	654.5	8	640	10	666.7
Major 4-smidegree	7	763.6	10	800	11	733.3
Diminished 5-smidegree	7	763.6	9	720	12	800
Perfect 5-smidegree	8	872.7	11	880	13	866.7
Minor 6-smidegree	9	981.8	12	960	15	1000
Major 6-smidegree	10	1090.9	14	1120	16	1066.7
Perfect 7-smidegree (octave)	11	1200	15	1200	18	1200	2/1 (exact)

Soft tunings

User:MOS degrees is deprecated. Please use Template:MOS tunings instead.

Scale degree of 4L 3s

Scale degree	18edo (Soft, L:s = 3:2)		25edo (Supersoft, L:s = 4:3)		29edo (Semisoft, L:s = 5:3)		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	2	133.3	3	144	3	124.1
Major 1-smidegree	3	200	4	192	5	206.9
Perfect 2-smidegree	5	333.3	7	336	8	331
Augmented 2-smidegree	6	400	8	384	10	413.8
Minor 3-smidegree	7	466.7	10	480	11	455.2
Major 3-smidegree	8	533.3	11	528	13	537.9
Minor 4-smidegree	10	666.7	14	672	16	662.1
Major 4-smidegree	11	733.3	15	720	18	744.8
Diminished 5-smidegree	12	800	17	816	19	786.2
Perfect 5-smidegree	13	866.7	18	864	21	869
Minor 6-smidegree	15	1000	21	1008	24	993.1
Major 6-smidegree	16	1066.7	22	1056	26	1075.9
Perfect 7-smidegree (octave)	18	1200	25	1200	29	1200	2/1 (exact)

Hard tunings

User:MOS degrees is deprecated. Please use Template:MOS tunings instead.

Scale degree of 4L 3s

Scale degree	15edo (Hard, L:s = 3:1)		19edo (Superhard, L:s = 4:1)		26edo (Semihard, L:s = 5:2)		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	80	1	63.2	2	92.3
Major 1-smidegree	3	240	4	252.6	5	230.8
Perfect 2-smidegree	4	320	5	315.8	7	323.1
Augmented 2-smidegree	6	480	8	505.3	10	461.5
Minor 3-smidegree	5	400	6	378.9	9	415.4
Major 3-smidegree	7	560	9	568.4	12	553.8
Minor 4-smidegree	8	640	10	631.6	14	646.2
Major 4-smidegree	10	800	13	821.1	17	784.6
Diminished 5-smidegree	9	720	11	694.7	16	738.5
Perfect 5-smidegree	11	880	14	884.2	19	876.9
Minor 6-smidegree	12	960	15	947.4	21	969.2
Major 6-smidegree	14	1120	18	1136.8	24	1107.7
Perfect 7-smidegree (octave)	15	1200	19	1200	26	1200	2/1 (exact)

Modes

Modes of 4L 3s

UDP	Cyclic order	Step pattern
6|0	1	LLsLsLs
5|1	6	LsLLsLs
4|2	4	LsLsLLs
3|3	2	LsLsLsL
2|4	7	sLLsLsL
1|5	5	sLsLLsL
0|6	3	sLsLsLL