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 ↘ |
┌╥╥┬╥┬╥┬┐ │║║│║│║││ │││││││││ └┴┴┴┴┴┴┴┘
sLsLsLL
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 naming conventions for the following: step ratios, intervals, scale degrees, modes, step patterns. For more information, see the respective links.
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
| 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 |
Modes and scale degrees
Alexandru Ianu (Ayceman)[1] has proposed the following mode names relating to the Almsivi in Morrowind (TES):
| 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.
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.
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.
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.
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.
Scales
Move to quickstart guide.
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 | |||||
| 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
- 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)
References
- ↑ Description of Sylvian Moon Dance mentioning the naming proposal https://musescore.com/user/36772625/scores/6700443 – The theme relates to the mystical nature of the Tribunal and TES lore, which fits smitonic.