5L 4s: Difference between revisions
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| 8\37, 259.46 | | 8\37, 259.46 |
Revision as of 09:14, 31 March 2021
User:IlL/Template:RTT restriction
↖ 4L 3s | ↑ 5L 3s | 6L 3s ↗ |
← 4L 4s | 5L 4s | 6L 4s → |
↙ 4L 5s | ↓ 5L 5s | 6L 5s ↘ |
┌╥╥┬╥┬╥┬╥┬┐ │║║│║│║│║││ │││││││││││ └┴┴┴┴┴┴┴┴┴┘
sLsLsLsLL
5L 4s, or semiquartal, refers to the structure of MOS scales with generators ranging from 1\5 (one degree of 5edo = 240¢) to 2\9 (two degrees of 9edo = 266.7¢). In the case of 9edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's).
Semiquartal tunings can be divided into two major ranges:
- Semaphore generated by semifourths flatter than 3\14 (257.14¢). This implies a diatonic fifth.
- The generator could be viewed as a 15/13, and the resulting "ultramajor" chords and "inframinor" triads could be viewed as approximating 10:13:15 and 26:30:39. See Arto and Tendo Theory.
- Bug, generated by semifourths sharper than 3\14 (257.14¢). This implies a "mavila" or superdiatonic fifth.
Notation
This article uses diamond MOS notation, with the convention JKLMNOPQR = LLSLSLSLS and pitch standard J = C4 = 261.6255653 Hz. The accidentals & and @ are used for raising and lowering by the chroma = L − S, respectively.
Tuning ranges
Semaphore
We view semaphore as any 5L 4s tuning where two semifourth generators make a diatonic (5L 2s) fourth, i.e. any tuning where the semifourth is between 1\5 (240¢) or 3\14 (257.14¢). One important sub-range of semaphore is given by stipulating that two semifourth generators must make a meantone fourth; i.e. that four fifths should approximate a 5/4 major third. This is supported by 19edo (4\19), 24edo (5\24), 43edo (9\43), and 62edo (13\62).
The sizes of the generator, large step and small step of 5L 4s are as follows in various semaphore tunings.
14edo | 19edo | 24edo | 29edo | |
---|---|---|---|---|
generator (g) | 3\14, 257.14 | 4\19, 252.63 | 5\24, 250. | 6\29, 248.28 |
L (octave - 4g) | 171.43 | 189.47 | 200.00 | 206.90 |
s (5g - octave) | 85.71 | 63.16 | 50.00 | 41.38 |
Bug
For convenience's sake, we view bug as any 5L 4s tuning where two semifourth generators make a superdiatonic (7L 2s) fourth, i.e. any tuning where the semifourth is between 3\14 (257.14¢) and 2\9 (266.67¢). 23edo's 5\23 (260.87¢) is an example of a bug generator.
The sizes of the generator, large step and small step of 5L 4s are as follows in various bug tunings.
23edo | 32edo | 37edo | |
---|---|---|---|
generator (g) | 5\23, 260.87 | 7\32, 262.50 | 8\37, 259.46 |
L (octave - 4g) | 156.52 | 150.00 | 162.16 |
s (5g - octave) | 104.35 | 112.50 | 97.30 |
Intervals
Modes
TODO: names
- LLsLsLsLs
- LsLLsLsLs
- LsLsLLsLs
- LsLsLsLLs
- LsLsLsLsL
- sLLsLsLsL
- sLsLLsLsL
- sLsLsLLsL
- sLsLsLsLL
One can think of 5L 4s modes as being built from two pentachords (division of the perfect fourth into four intervals) plus a whole tone. The possible pentachords are LsLs, sLLs, and sLsL.
Chords
Primodal theory
Nejis
14nejis
- 95:100:105:110:116:122:128:135:141:148:156:164:172:180:190 (uses /19 prime family intervals while being pretty close to equal)
Samples
- Rin's UFO Ride by Starshine
- File:Dream EP 14edo Sketch.mp3 is a short swing ditty in 14edo semiquartal, in the 212121221 mode.
- File:19edo Semaphore Fugue.mp3 is an unfinished fugue in 19edo semiquartal, in the 212121221 mode.
Scale tree
Generator | Cents | Comments | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1\5 | 240 | |||||||||||
12\59 | 244.068 | |||||||||||
11\54 | 244.444 | |||||||||||
10\49 | 244.898 | |||||||||||
9\44 | 245.455 | |||||||||||
8\39 | 246.154 | |||||||||||
7\34 | 247.059 | |||||||||||
6\29 | 248.276 | |||||||||||
11\53 | 249.057 | |||||||||||
5\24 | 250 | L/s = 4 | ||||||||||
9\43 | 251.163 | |||||||||||
4\19 | 252.632 |
L/s = 3 | ||||||||||
11\52 | 253.813 | |||||||||||
29\137 | 254.015 | |||||||||||
76\359 | 254.039 | |||||||||||
199\940 | 254.043 | |||||||||||
123\581 | 254.045 | |||||||||||
47\222 | 254.054 | |||||||||||
18\85 | 254.118 | |||||||||||
7\33 | 254.5455 | |||||||||||
10\47 | 255.319 | |||||||||||
13\61 | 255.734 | |||||||||||
16\75 | 256.000 | |||||||||||
3\14 | 257.143 | Boundary of propriety (generators
larger than this are proper) | ||||||||||
11\51 | 258.8235 | |||||||||||
258.957 | ||||||||||||
8\37 | 259.459 | |||||||||||
21\97 | 259.794 | |||||||||||
55\254 | 259.843 | |||||||||||
144\665 | 259.850 | |||||||||||
233\1076 | 259.851 | Golden bug | ||||||||||
89\411 | 259.854 | |||||||||||
34\157 | 259.873 | |||||||||||
13\60 | 260 | |||||||||||
260.246 | ||||||||||||
5\23 | 260.870 | Optimum rank range (L/s=3/2) bug | ||||||||||
7\32 | 262.5 | |||||||||||
9\41 | 263.415 | |||||||||||
11\50 | 264 | |||||||||||
13\59 | 264.407 | |||||||||||
15\68 | 264.706 | |||||||||||
17\77 | 264.935 | |||||||||||
19\86 | 265.116 | |||||||||||
21\95 | 265.263 | |||||||||||
2\9 | 266.667 |