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User:Ganaram inukshuk/4L 3s

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Revision as of 20:40, 27 June 2023 by Ganaram inukshuk (talk | contribs) (→Temperament interpretations: Being consistent with the list types)
↖ 3L 2s ↑ 4L 2s 5L 2s ↗
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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 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).
  • Myna temperament, 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:

  • 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.
  • 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.
  • 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

Tuning spectrum

Template:Scale tree

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