User:Ganaram inukshuk/4L 3s

The following is a draft for a proposed rewrite of the following page: 4L 3s The primary changes are as follows: see what the page looks like with proposed changes (scale characteristics section, new intro wording); see what it looks like with a quickstart guide The original page can be compared with this page here.

This page has a quickstart guide. Information on how to start composing or playing can be found at the page linked above.

↖ 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 ¢)

4L 3s, named smitonic, is a moment-of-symmetry scale characterized by having 4 large steps and 3 small steps distributed as evenly as possible within the octave. Generators that produce this scale range from (bright-gen-range), or from (dark-gen-range).

4L 3s can be seen as a warped diatonic scale, where one large step of diatonic (5L 2s) is replaced with a small step.

Scale characteristics

This article assumes TAMNAMS conventions for interval, degree, and scale naming.

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¢).

Intervals sizes

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 (using large/small terminology instead of major/minor and aug/perfect/dim)

Intervals		Size		Abbrev.
Generic	Specific	Steps	Range in cents
0-smistep (root)	Perfect 0-smistep	0	0.0¢	P0ms
1-smistep	Small 1-smistep	s	0.0¢ to 171.4¢	s1ms
	Large 1-smistep	L	171.4¢ to 300.0¢	L1ms
2-smistep	Small 2-smistep	L + s	300.0¢ to 342.9¢	s2ms
	Large 2-smistep	2L	342.9¢ to 600.0¢	L2ms
3-smistep	Small 3-smistep	L + 2s	300.0¢ to 514.3¢	s3ms
	Large 3-smistep	2L + s	514.3¢ to 600.0¢	L3ms
4-smistep	Small 4-smistep	2L + 2s	600.0¢ to 685.7¢	s4ms
	Large 4-smistep	3L + s	685.7¢ to 900.0¢	L4ms
5-smistep	Small 5-smistep	2L + 3s	600.0¢ to 857.1¢	s5ms
	Large 5-smistep	3L + 2s	857.1¢ to 900.0¢	L5ms
6-smistep	Small 6-smistep	3L + 3s	900.0¢ to 1028.6¢	s6ms
	Large 6-smistep	4L + 2s	1028.6¢ to 1200.0¢	L6ms
7-smistep (octave)	Perfect 7-smistep	4L + 3s	1200.0¢	P7ms

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 ¢

Modes and scale degrees

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

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.

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.

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	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.

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)		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.

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

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.

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

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.

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

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)

Scales

Move to quickstart guide.

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)
• A fugue in 18edo smitonic functional harmony (WIP)