5L 4s

From Xenharmonic Wiki
Revision as of 18:28, 4 April 2021 by Inthar (talk | contribs) (→Modes)
Jump to navigation Jump to search

User:IlL/Template:RTT restriction

↖ 4L 3s ↑ 5L 3s 6L 3s ↗
← 4L 4s 5L 4s 6L 4s →
↙ 4L 5s ↓ 5L 5s 6L 5s ↘ 
┌╥╥┬╥┬╥┬╥┬┐
│║║│║│║│║││
│││││││││││
└┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLsLsLsLs
sLsLsLsLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 7\9 to 4\5 (933.3 ¢ to 960.0 ¢)
Dark 1\5 to 2\9 (240.0 ¢ to 266.7 ¢)
TAMNAMS information
Name semiquartal
Prefix cthon-
Abbrev. ct
Related MOS scales
Parent 4L 1s
Sister 4L 5s
Daughters 9L 5s, 5L 9s
Neutralized 1L 8s
2-Flought 14L 4s, 5L 13s
Equal tunings
Equalized (L:s = 1:1) 7\9 (933.3 ¢)
Supersoft (L:s = 4:3) 25\32 (937.5 ¢)
Soft (L:s = 3:2) 18\23 (939.1 ¢)
Semisoft (L:s = 5:3) 29\37 (940.5 ¢)
Basic (L:s = 2:1) 11\14 (942.9 ¢)
Semihard (L:s = 5:2) 26\33 (945.5 ¢)
Hard (L:s = 3:1) 15\19 (947.4 ¢)
Superhard (L:s = 4:1) 19\24 (950.0 ¢)
Collapsed (L:s = 1:0) 4\5 (960.0 ¢)

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:

  1. 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.
  2. 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¢).

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, 250.00 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

Parahard

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 can be considered the 19edo (4\19)-to-24edo (5\24) range, i.e. parahard semiquartal, which also contains 43edo (9\43) and 62edo (13\62).

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. 514.29¢ to 533.33¢), 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

The following names are taken from a fictional semaphore-based culture.

  • LLsLsLsLs Tsimma'ian
  • LsLLsLsLs Tavulian
  • LsLsLLsLs Lynkaesian
  • LsLsLsLLs Bonzhian
  • LsLsLsLsL Tjatupian
  • sLLsLsLsL Zierokian
  • sLsLLsLsL Da'aemian
  • sLsLsLLsL Pahnachian
  • sLsLsLsLL Apozian

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

  1. 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

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
Retrieved from "https://en.xen.wiki/index.php?title=5L_4s&oldid=65646"