4L 4s

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↖ 3L 3s ↑4L 3s 5L 3s ↗
← 3L 4s4L 4s 5L 4s →
↙ 3L 5s ↓4L 5s 5L 5s ↘
┌╥┬╥┬╥┬╥┬┐
│║│║│║│║││
││││││││││
└┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLsLsLs
sLsLsLsL
Equave 2/1 (1200.0¢)
Period 1\4 (300.0¢)
Generator size
Bright 1\8 to 1\4 (150.0¢ to 300.0¢)
Dark 0\4 to 1\8 (0.0¢ to 150.0¢)
TAMNAMS information
Name tetrawood
Prefix tetrawd-
Abbrev. ttw
Related MOS scales
Parent none
Sister 4L 4s (self)
Daughters 8L 4s, 4L 8s
Neutralized 8edo
2-Flought 12L 4s, 4L 12s
Equal tunings
Equalized (L:s = 1:1) 1\8 (150.0¢)
Supersoft (L:s = 4:3) 4\28 (171.4¢)
Soft (L:s = 3:2) 3\20 (180.0¢)
Semisoft (L:s = 5:3) 5\32 (187.5¢)
Basic (L:s = 2:1) 2\12 (200.0¢)
Semihard (L:s = 5:2) 5\28 (214.3¢)
Hard (L:s = 3:1) 3\16 (225.0¢)
Superhard (L:s = 4:1) 4\20 (240.0¢)
Collapsed (L:s = 1:0) 1\4 (300.0¢)

4L 4s, named tetrawood in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 4 small steps, with a period of 1 large step and 1 small step that repeats every 300.0¢, or 4 times every octave. Generators that produce this scale range from 150¢ to 300¢, or from 0¢ to 150¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period. The minimum harmonic entropy scale with this MOS pattern is diminished[8], the familiar octatonic scale of 12edo.

In addition to the true MOS form, LsLsLsLs, there are four near-MOS forms – LLsLsLss, LLsLssLs, LLssLLss, and LLssLsLs – in which the period and its multiples are the only intervals with more than two varieties. The near-MOS forms are only proper if the dark generator is larger than 1\12 (100¢).

Intervals

Intervals of 4L 4s
Intervals Steps subtended Range in cents Average of HE
(from HE Calc)
Min of HE
Generic[1] Specific[2] Abbrev.[3]
0-tetrawdstep Perfect 0-tetrawdstep P0ms 0 0.0¢ ~2.4654 nats ~2.4654 nats
1-tetrawdstep Minor 1-tetrawdstep m1ms s 0.0¢ to 150.0¢ ~4.6945 nats ~4.6426 nats
Major 1-tetrawdstep M1ms L 150.0¢ to 300.0¢ ~4.5904 nats ~4.5841 nats
2-tetrawdstep Perfect 2-tetrawdstep P2ms L + s 300.0¢ ~4.5654 nats ~4.5654 nats
3-tetrawdstep Minor 3-tetrawdstep m3ms L + 2s 300.0¢ to 450.0¢ ~4.5631 nats ~4.4856 nats
Major 3-tetrawdstep M3ms 2L + s 450.0¢ to 600.0¢ ~4.5125 nats ~4.3665 nats
4-tetrawdstep Perfect 4-tetrawdstep P4ms 2L + 2s 600.0¢ ~4.5799 nats ~4.5799 nats
5-tetrawdstep Minor 5-tetrawdstep m5ms 2L + 3s 600.0¢ to 750.0¢ ~4.4080 nats ~4.1319 nats
Major 5-tetrawdstep M5ms 3L + 2s 750.0¢ to 900.0¢ ~4.5955 nats ~4.5709 nats
6-tetrawdstep Perfect 6-tetrawdstep P6ms 3L + 3s 900.0¢ ~4.4980 nats ~4.4980 nats
7-tetrawdstep Minor 7-tetrawdstep m7ms 3L + 4s 900.0¢ to 1050.0¢ ~4.5684 nats ~4.5338 nats
Major 7-tetrawdstep M7ms 4L + 3s 1050.0¢ to 1200.0¢ ~4.6332 nats ~4.6079 nats
8-tetrawdstep Perfect 8-tetrawdstep P8ms 4L + 4s 1200.0¢ ~3.3273 nats ~3.3273 nats

  1. Generic intervals are denoted solely by the number of steps they subtend.
  2. Specific intervals denote whether an interval is major, minor, augmented, perfect, or diminished.
  3. Abbreviations can be further shortened to 'ms' if context allows.

Modes

Modes of 4L 4s
UDP Rotational order Step pattern
4|0(4) 1 LsLsLsLs
0|4(4) 2 sLsLsLsL

Scale tree

Generator Cents L s L/s Comments
Chroma-positive Chroma-negative
1\8 150.000 150.000 1 1 1.000
6\44 163.636 136.364 6 5 1.200 Fourfives↑
5\36 166.667 133.333 5 4 1.250
9\64 168.750 105.000 9 7 1.286
4\28 171.429 128.571 4 3 1.333
11\76 173.684 126.316 11 8 1.375
7\48 175.000 125.000 7 5 1.400
10\68 176.471 123.529 10 7 1.428
3\20 180.000 60.000 3 2 1.500 L/s = 3/2
11\72 183.333 116.667 11 7 1.571
8\52 184.615 115.385 8 5 1.600
13\84 185.714 114.286 13 8 1.625 Golden diminished
5\32 187.500 112.500 5 3 1.667
12\76 189.474 110.526 12 7 1.714
7\44 190.909 109.091 7 4 1.750
9\56 192.857 107.143 9 5 1.800
2\12 200.000 100.000 2 1 2.000 Basic tetrawood
Diminished is optimal around here
9\52 207.692 92.308 9 4 2.250
7\40 210.000 90.000 7 3 2.333
12\68 211.765 88.235 12 5 2.400
5\28 214.286 85.714 5 2 2.500
13\72 216.667 83.333 13 5 2.600 Unnamed golden tuning
8\44 218.182 81.818 8 3 2.667
11\60 220.000 80.000 11 4 2.750
3\16 225.000 75.000 3 1 3.000 L/s = 3/1
10\52 230.769 69.231 10 3 3.333
7\36 233.333 66.667 7 2 3.500
11\56 235.714 64.286 11 3 3.667
4\20 240.000 60.000 4 1 4.000
9\44 245.455 54.545 9 2 4.500
5\24 250.000 50.000 5 1 5.000
6\28 257.143 42.857 6 1 6.000 Quadritikleismic↓
1\4 300.000 0.000 1 0 → inf