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User:Ganaram inukshuk/4L 3s

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

Interval and degree names

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 (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.
  • 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.

Step ratios

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
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)

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.
    • 18edo smitonic can also 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 3/2 (702¢).
    • 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¢).
  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)
Minor 1-smidegree 2 133.3 3 144 5 139.5
Major 1-smidegree 3 200 4 192 7 195.3
Perfect 2-smidegree 5 333.3 7 336 12 334.9
Augmented 2-smidegree 6 400 8 384 14 390.7
Minor 3-smidegree 7 466.7 10 480 17 474.4
Major 3-smidegree 8 533.3 11 528 19 530.2
Minor 4-smidegree 10 666.7 14 672 24 669.8
Major 4-smidegree 11 733.3 15 720 26 725.6
Diminished 5-smidegree 12 800 17 816 29 809.3
Perfect 5-smidegree 13 866.7 18 864 31 865.1
Minor 6-smidegree 15 1000 21 1008 36 1004.7
Major 6-smidegree 16 1066.7 22 1056 38 1060.5
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, 18edo, and 29edo. (11edo and 18edo not shown in table below.)

  User:MOS degrees is deprecated. Please use Template:MOS tunings instead.
Scale degree of 4L 3s
Scale degree 29edo (Semisoft, L:s = 5:3) Approx. JI Ratios
Steps Cents
Perfect 0-smidegree (unison) 0 0 1/1 (exact)
Minor 1-smidegree 3 124.1
Major 1-smidegree 5 206.9
Perfect 2-smidegree 8 331
Augmented 2-smidegree 10 413.8
Minor 3-smidegree 11 455.2
Major 3-smidegree 13 537.9
Minor 4-smidegree 16 662.1
Major 4-smidegree 18 744.8
Diminished 5-smidegree 19 786.2
Perfect 5-smidegree 21 869
Minor 6-smidegree 24 993.1
Major 6-smidegree 26 1075.9
Perfect 7-smidegree (octave) 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, 15edo, 26edo, and 37edo.

  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) 26edo (Semihard, L:s = 5:2) 37edo (L:s = 7:3) 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 1 109.1 1 80 2 92.3 3 97.3
Major 1-smidegree 2 218.2 3 240 5 230.8 7 227
Perfect 2-smidegree 3 327.3 4 320 7 323.1 10 324.3
Augmented 2-smidegree 4 436.4 6 480 10 461.5 14 454.1
Minor 3-smidegree 4 436.4 5 400 9 415.4 13 421.6
Major 3-smidegree 5 545.5 7 560 12 553.8 17 551.4
Minor 4-smidegree 6 654.5 8 640 14 646.2 20 648.6
Major 4-smidegree 7 763.6 10 800 17 784.6 24 778.4
Diminished 5-smidegree 7 763.6 9 720 16 738.5 23 745.9
Perfect 5-smidegree 8 872.7 11 880 19 876.9 27 875.7
Minor 6-smidegree 9 981.8 12 960 21 969.2 30 973
Major 6-smidegree 10 1090.9 14 1120 24 1107.7 34 1102.7
Perfect 7-smidegree (octave) 11 1200 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¢), six of which and octave-reducing approximates 3/2 (702¢), a diatonic perfect 5th.

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.

Other edos smaller than 53edo include 15edo, 19edo, and 34edo.

  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) 34edo (L:s = 7:2) 53edo (L:s = 11:3) 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 80 1 63.2 2 70.6 3 67.9
Major 1-smidegree 3 240 4 252.6 7 247.1 11 249.1
Perfect 2-smidegree 4 320 5 315.8 9 317.6 14 317
Augmented 2-smidegree 6 480 8 505.3 14 494.1 22 498.1
Minor 3-smidegree 5 400 6 378.9 11 388.2 17 384.9
Major 3-smidegree 7 560 9 568.4 16 564.7 25 566
Minor 4-smidegree 8 640 10 631.6 18 635.3 28 634
Major 4-smidegree 10 800 13 821.1 23 811.8 36 815.1
Diminished 5-smidegree 9 720 11 694.7 20 705.9 31 701.9
Perfect 5-smidegree 11 880 14 884.2 25 882.4 39 883
Minor 6-smidegree 12 960 15 947.4 27 952.9 42 950.9
Major 6-smidegree 14 1120 18 1136.8 32 1129.4 50 1132.1
Perfect 7-smidegree (octave) 15 1200 19 1200 34 1200 53 1200 2/1 (exact)

Tuning examples

(add tuning examples)

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

Approaches

Scales

Scala files

Music

(add music)

Tuning spectrum

Template:Scale tree

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