User:Ganaram inukshuk/4L 3s
| ↖3L 2s | ↑4L 2s | 5L 2s↗ |
| ←3L 3s | 4L 3s | 5L 3s→ |
| ↙3L 4s | ↓4L 4s | 5L 4s↘ |
┌╥╥┬╥┬╥┬┐ │║║│║│║││ │││││││││ └┴┴┴┴┴┴┴┘
sLsLsLL
4L 7s
- 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 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
- This article assumes TAMNAMS for naming step ratios, intervals, and scale degrees.
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.
Except for the unison and octave, all interval classes have two varieties or sizes, denoted using the terms major and minor for the large and small sizes, respectively. The exception to this rule are the generators, which use the terms augmented, perfect, and diminished instead.
| Interval class | Specific intervals | Size (in ascending order) |
|---|---|---|
| 0-smistep | Perfect 0-smistep (unison) | 0 |
| 1-smistep | Minor 1-smistep | s |
| Major 1-smistep | L | |
| 2-smistep | Perfect 2-smistep | L + s |
| Augmented 2-smistep | 2L | |
| 3-smistep | Minor 3-smistep | L + 2s |
| Major 3-smistep | 2L + s | |
| 4-smistep | Minor 4-smistep | 2L + 2s |
| Major 4-smistep | 3L + s | |
| 5-smistep | Diminished 5-smistep | 2L + 3s |
| Perfect 5-smistep | 3L + 2s | |
| 6-smistep | Minor 6-smistep | 3L + 3s |
| Major 6-smistep | 4L + 2s | |
| 7-smistep (octave) | Perfect 7-smistep (octave) | 4L + 3s |
Different forms of this MOS are generally built using one specific interval for each interval class, while the step pattern LLsLsLs, or some rotation thereof, is formed between consecutive scale degrees. Alternatively, MODMOS scales can be formed by deviating from the step pattern, such as with LLLssLs, or by including alterations outside the specific sizes described.
Notation
Names for notes in this scale can be denoted using the following notation schemes.
Ups and downs notation
- Main article: Ups and downs notation
(Insert ups-and-downs example here)
Diamond-mos notation
- Main article: Diamond-mos notation
Diamond-mos can be used by assigning the note names JKLMNOP to one of the modes and using the symbols "&" and "@" to denote generalized sharps and flats, respectively. Using the symmetric mode (LsLsLsL), the basic 10edo gamut is as follows:
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.
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.
| 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, 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¢).
| 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 (not shown), 18edo, and 29edo.
| 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 | |
| Major 1-smidegree | 3 | 200 | 5 | 206.9 | |
| Perfect 2-smidegree | 5 | 333.3 | 8 | 331 | |
| Augmented 2-smidegree | 6 | 400 | 10 | 413.8 | |
| Minor 3-smidegree | 7 | 466.7 | 11 | 455.2 | |
| Major 3-smidegree | 8 | 533.3 | 13 | 537.9 | |
| Minor 4-smidegree | 10 | 666.7 | 16 | 662.1 | |
| Major 4-smidegree | 11 | 733.3 | 18 | 744.8 | |
| Diminished 5-smidegree | 12 | 800 | 19 | 786.2 | |
| Perfect 5-smidegree | 13 | 866.7 | 21 | 869 | |
| Minor 6-smidegree | 15 | 1000 | 24 | 993.1 | |
| Major 6-smidegree | 16 | 1066.7 | 26 | 1075.9 | |
| 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.
| 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 | |
| Perfect 2-smidegree | 4 | 320 | 7 | 323.1 | 10 | 324.3 | |
| Augmented 2-smidegree | 6 | 480 | 10 | 461.5 | 14 | 454.1 | |
| Minor 3-smidegree | 5 | 400 | 9 | 415.4 | 13 | 421.6 | |
| Major 3-smidegree | 7 | 560 | 12 | 553.8 | 17 | 551.4 | |
| Minor 4-smidegree | 8 | 640 | 14 | 646.2 | 20 | 648.6 | |
| Major 4-smidegree | 10 | 800 | 17 | 784.6 | 24 | 778.4 | |
| Diminished 5-smidegree | 9 | 720 | 16 | 738.5 | 23 | 745.9 | |
| Perfect 5-smidegree | 11 | 880 | 19 | 876.9 | 27 | 875.7 | |
| Minor 6-smidegree | 12 | 960 | 21 | 969.2 | 30 | 973 | |
| 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.
| 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 | |
| Major 1-smidegree | 4 | 252.6 | 7 | 247.1 | 11 | 249.1 | 15 | 250 | |
| Perfect 2-smidegree | 5 | 315.8 | 9 | 317.6 | 14 | 317 | 19 | 316.7 | |
| Augmented 2-smidegree | 8 | 505.3 | 14 | 494.1 | 22 | 498.1 | 30 | 500 | |
| Minor 3-smidegree | 6 | 378.9 | 11 | 388.2 | 17 | 384.9 | 23 | 383.3 | |
| Major 3-smidegree | 9 | 568.4 | 16 | 564.7 | 25 | 566 | 34 | 566.7 | |
| Minor 4-smidegree | 10 | 631.6 | 18 | 635.3 | 28 | 634 | 38 | 633.3 | |
| Major 4-smidegree | 13 | 821.1 | 23 | 811.8 | 36 | 815.1 | 49 | 816.7 | |
| Diminished 5-smidegree | 11 | 694.7 | 20 | 705.9 | 31 | 701.9 | 42 | 700 | |
| Perfect 5-smidegree | 14 | 884.2 | 25 | 882.4 | 39 | 883 | 53 | 883.3 | |
| Minor 6-smidegree | 15 | 947.4 | 27 | 952.9 | 42 | 950.9 | 57 | 950 | |
| Major 6-smidegree | 18 | 1136.8 | 32 | 1129.4 | 50 | 1132.1 | 68 | 1133.3 | |
| Perfect 7-smidegree (octave) | 19 | 1200 | 34 | 1200 | 53 | 1200 | 72 | 1200 | 2/1 (exact) |
Modes
| UDP | Rotational 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 |
Scales
Subset and superset scales
4L 3s has a parent scale of 1L 3s, a tetratonic scale, meaning 1L 3s is a subset. 4L 3s also has two child scales, which are supersets of 4L 3s:
- 7L 4s, a smitonic chromatic scale produced using soft-of-basic step ratios.
- 4L 7s, a smitonic chromatic scale produced using hard-of-basic step ratios.
11edo, the equalized form of both 7L 4s and 4L 7s, is also a superset of 4L 3s.
Scala files
Scale tree
| Generator(edo) | Cents | Step Ratio | Comments | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Bright | Dark | L:s | Hardness | |||||||
| 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 (Generators smaller 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
(add music)
See also
Approaches: