5L 4s: Difference between revisions

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== Scale tree ==
== Scale tree ==
{| class="wikitable"
 
|-
{| class="wikitable center-all"
! colspan="11" | Generator
! colspan="6" rowspan="2" | Generator
! | Cents
! colspan="2" | Cents
! | Comments
! rowspan="2" | L
|-
! rowspan="2" | s
| | 1\5
! rowspan="2" | L/s
| |
! rowspan="2" | Comments
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 240
| style="text-align:center;" |  
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 12\59
| | 244.068
| style="text-align:center;" |  
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 11\54
| |
| | 244.444
| style="text-align:center;" |  
|-
| |
| |
| |
| |
| |
| |
| |
| |
| | 10\49
| |
| |
| | 244.898
| style="text-align:center;" |  
|-
| |
| |
| |
| |
| |
| |
| |
| | 9\44
| |
| |
| |
| | 245.455
| style="text-align:center;" |  
|-
|-
| |
! Chroma-positive
| |
! Chroma-negative
| |
| |
| |
| |
| | 8\39
| |
| |
| |
| |
| | 246.154
| style="text-align:center;" |
|-
|-
| |  
| 7\9 || || || || || || 933.333 || 266.667 || 1 || 1 || 1.000 ||  
| |  
| |  
| |  
| |  
| | 7\34
| |  
| |
| |
| |  
| |  
| | 247.059
| style="text-align:center;" |  
|-
|-
| |  
| || || || || || 39\50 || 936.000 || 264.000 || 6 || 5 || 1.200 ||  
| |  
| |  
| |  
| | 6\29
| |  
| |  
| |  
| |
| |
| |  
| | 248.276
| style="text-align:center;" |  
|-
|-
| |  
| || || || || 32\41 || || 936.585 || 263.415 || 5 || 4 || 1.250 || Septimin
| |  
| |  
| |  
| |
| | 11\53
| |  
| |  
| |  
| |  
| |  
| | 249.057
| style="text-align:center;" |  
|-
|-
| |  
| || || || || || 57\73 || 936.986 || 263.014 || 9 || 7 || 1.286 ||
| |  
| |  
| | 5\24
| |  
| |
| |
| |  
| |  
| |  
| |  
| | 250
| style="text-align:center;" | L/s = 4
|-
|-
| |  
| || || || 25\32 || || || 937.500 || 262.500 || 4 || 3 || 1.333 || Beep
| |  
| |  
| |  
| |  
| | 9\43
| |  
| |
| |
| |  
| |  
| | 251.163
| |  
|-
|-
| |  
| || || || || || 68\87 || 937.931 || 262.069 || 11 || 8 || 1.375 ||
| |  
| | 4\19
| |  
| |  
| |  
| |  
| |
| |
| |  
| |  
| | 252.632
| style="text-align:center;" |  
 
L/s = 3
|-
|-
| |  
| || || || || 43\55 || || 938.182 || 261.818 || 7 || 5 || 1.400 ||
| |  
| |  
| |  
| | 11\52
| |  
| |  
| |  
| |
| |  
| |  
| | 253.813
| |  
|-
|-
| |  
| || || || || || 61\78 || 938.462 || 261.538 || 10 || 7 || 1.428 ||
| |  
| |  
| |  
| |  
| |
| | 29\137
| |  
| |  
| |  
| |  
| | 254.015
| |  
|-
|-
| |  
| || || 18\23 || || || || 939.130 || 260.870 || 3 || 2 || 1.500 || L/s = 3/2, bug
| |  
| |  
| |  
| |  
| |  
| |  
| |
| | 76\359
| |  
| |  
| | 254.039
| |  
|-
|-
| |  
| || || || || || 65\83 || 939.759 || 260.241 || 11 || 7 || 1.571 ||
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| |
| | 199\940
| | 254.043
| |  
|-
|-
| |  
| || || || || 47\60 || || 940.000 || 260.000 || 8 || 5 || 1.600 ||  
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| |
| | 123\581
| |  
| | 254.045
| |  
|-
|-
| |  
| || || || || || 76\97 || 940.206 || 259.794 || 13 || 8 || 1.625 || Golden bug
| |  
| |  
| |  
| |  
| |
| |
| | 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
| style="text-align:center;" | Boundary of propriety (generators
 
larger than this are proper)
|-
|-
| |  
| || || || 29\37 || || || 940.541 || 259.459 || 5 || 3 || 1.667 ||
| |  
| |  
| |
| | 11\51
| |  
| |  
| |  
| |  
| |  
| |  
| | 258.8235
| |  
|-
|-
| |  
| || || || || || 69\88 || 940.909 || 259.091 || 12 || 7 || 1.714 ||  
| |  
| |  
| |  
| |  
| |  
| |  
| |
| |
| |  
| |  
| | 258.957
| |  
|-
|-
| |  
| || || || || 40\51 || || 941.176 || 258.824 || 7 || 4 || 1.750 ||
| |  
| |  
| | 8\37
| |  
| |  
| |  
| |
| |
| |  
| |  
| | 259.459
| style="text-align:center;" |  
|-
|-
| |  
| || || || || || 51\65 || 941.538 || 258.462 || 9 || 5 || 1.800 ||
| |  
| |  
| |  
| |  
| | 21\97
| |  
| |  
| |
| |  
| |  
| | 259.794
| style="text-align:center;" |  
|-
|-
| |  
| || 11\14 || || || || || 942.857 || 257.143 || 2 || 1 || 2.000 || Basic semiquartal<br>(Generators smaller than this are proper)
| |  
| |  
| |  
| |  
| |  
| |  
| | 55\254
| |
| |  
| |  
| | 259.843
| style="text-align:center;" |  
|-
|-
| |  
| || || || || || 48\61 || 944.262 || 255.738 || 9 || 4 || 2.250 ||
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| |
| | 144\665
| |  
| | 259.850
| style="text-align:center;" |  
|-
|-
| |  
| || || || || 37\47 || || 944.681 || 255.319 || 7 || 3 || 2.333 ||
| |  
| |  
| |  
| |  
| |  
| |  
| |
| |
| |  
| | 233\1076
| | 259.851
| style="text-align:center;" | Golden [[5L 4s#Bug|bug]]
|-
|-
| |  
| || || || || || 63\80 || 945.000 || 255.000 || 12 || 5 || 2.400 ||
| |  
| |  
| |  
| |  
| |  
| |  
| |
| | 89\411
| |  
| |  
| | 259.854
| style="text-align:center;" |  
|-
|-
| |  
| || || || 26\33 || || || 945.455 || 254.545 || 5 || 2 || 2.500 ||
| |  
| |  
| |  
| |  
| |  
| | 34\157
| |
| |
| |  
| |  
| | 259.873
| style="text-align:center;" |  
|-
|-
| |  
| || || || || || 67\85 || 945.882 || 254.118 || 13 || 5 || 2.600 || Unnamed golden tuning
| |  
| |  
| |  
| | 13\60
| |  
| |  
| |  
| |
| |
| |  
| | 260
| style="text-align:center;" |  
|-
|-
| |  
| || || || || 41\52 || || 946.154 || 253.846 || 8 || 3 || 2.667 ||
| |  
| |  
| |  
| |  
| |  
| |  
| |
| |
| |  
| |  
| | 260.246
| |  
|-
|-
| |  
| || || || || || 56\71 || 946.479 || 253.521 || 11 || 4 || 2.750 ||
| |  
| | 5\23
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| |
| | 260.870
| style="text-align:center;" | Optimum rank range (L/s=3/2) bug
|-
|-
| |  
| || || 15\19 || || || || 947.368 || 252.632 || 3 || 1 || 3.000 || L/s = 3/1, godzilla
| |  
| |
| | 7\32
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| | 262.5
| style="text-align:center;" |
|-
|-
| |  
| || || || || || 49\62 || 948.387 || 251.613 || 10 || 3 || 3.333 ||
| |  
| |  
| |  
| | 9\41
| |  
| |  
| |
| |
| |  
| |  
| | 263.415
| style="text-align:center;" |  
|-
|-
| |  
| || || || || 34\43 || || 948.837 || 251.163 || 7 || 2 || 3.500 ||
| |  
| |  
| |  
| |
| | 11\50
| |  
| |  
| |  
| |  
| |  
| | 264
| style="text-align:center;" |  
|-
|-
| |  
| || || || || || 53\67 || 949.254 || 250.746 || 11 || 3 || 3.667 || Semaphore
| |  
| |  
| |  
| |  
| |
| | 13\59
| |  
| |  
| |  
| |  
| | 264.407
| style="text-align:center;" |  
|-
|-
| |  
| || || || 19\24 || || || 950.000 || 250.000 || 4 || 1 || 4.000 ||
| |  
| |  
| |  
| |  
| |  
| |  
| | 15\68
| |
| |  
| |  
| | 264.706
| style="text-align:center;" |  
|-
|-
| |  
| || || || || || 42\53 || 950.943 || 249.057 || 9 || 2 || 4.500 ||
| |  
| |  
| |  
| |  
| |  
| |  
| |
| | 17\77
| |  
| |  
| | 264.935
| style="text-align:center;" |  
|-
|-
| |  
| || || || || 23\29 || || 951.724 || 248.276 || 5 || 1 || 5.000 ||  
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| |
| | 19\86
| |  
| | 265.116
| style="text-align:center;" |  
|-
|-
| |  
| || || || || || 27\34 || 952.941 || 247.059 || 6 || 1 || 6.000 ||
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| |
| | 21\95
| | 265.263
| style="text-align:center;" |  
|-
|-
| | 2\9
| 4\5 || || || || || || 960.000 || 240.000 || 1 || 0 || → inf ||
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| |  
| | 266.667
| style="text-align:center;" |  
|}
|}


[[Category:Scales]]
[[Category:Scales]]
[[Category:MOS scales]]
[[Category:MOS scales]]
[[Category:9-tone scales]]
[[Category:Abstract MOS patterns]]
[[Category:Abstract MOS patterns]]
[[Category:Semiquartal| ]] <!-- main article -->

Revision as of 09:31, 12 April 2021

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 is "hard-of-center" semiquartal, 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 is "soft-of-center" semiquartal, 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 Lynkaedian
  • LsLsLsLLs Bonzhian
  • LsLsLsLsL Tjatupian
  • sLLsLsLsL Zierokian
  • sLsLLsLsL Da'aemian
  • sLsLsLLsL Pahnachian
  • sLsLsLsLL Iporian

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 L s L/s Comments
Chroma-positive Chroma-negative
7\9 933.333 266.667 1 1 1.000
39\50 936.000 264.000 6 5 1.200
32\41 936.585 263.415 5 4 1.250 Septimin
57\73 936.986 263.014 9 7 1.286
25\32 937.500 262.500 4 3 1.333 Beep
68\87 937.931 262.069 11 8 1.375
43\55 938.182 261.818 7 5 1.400
61\78 938.462 261.538 10 7 1.428
18\23 939.130 260.870 3 2 1.500 L/s = 3/2, bug
65\83 939.759 260.241 11 7 1.571
47\60 940.000 260.000 8 5 1.600
76\97 940.206 259.794 13 8 1.625 Golden bug
29\37 940.541 259.459 5 3 1.667
69\88 940.909 259.091 12 7 1.714
40\51 941.176 258.824 7 4 1.750
51\65 941.538 258.462 9 5 1.800
11\14 942.857 257.143 2 1 2.000 Basic semiquartal
(Generators smaller than this are proper)
48\61 944.262 255.738 9 4 2.250
37\47 944.681 255.319 7 3 2.333
63\80 945.000 255.000 12 5 2.400
26\33 945.455 254.545 5 2 2.500
67\85 945.882 254.118 13 5 2.600 Unnamed golden tuning
41\52 946.154 253.846 8 3 2.667
56\71 946.479 253.521 11 4 2.750
15\19 947.368 252.632 3 1 3.000 L/s = 3/1, godzilla
49\62 948.387 251.613 10 3 3.333
34\43 948.837 251.163 7 2 3.500
53\67 949.254 250.746 11 3 3.667 Semaphore
19\24 950.000 250.000 4 1 4.000
42\53 950.943 249.057 9 2 4.500
23\29 951.724 248.276 5 1 5.000
27\34 952.941 247.059 6 1 6.000
4\5 960.000 240.000 1 0 → inf
Retrieved from "https://en.xen.wiki/index.php?title=5L_4s&oldid=66368"