4L 4s: Difference between revisions

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m →Scale tree: Add {{ToDo:Convert manually-generated table to MOS tuning spectrum template without losing information from the existing table}}
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| Pattern = LsLsLsLs
| Pattern = LsLsLsLs
}}
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{{MOS intro}}
{{MOS intro}}


The minimum harmonic entropy scale with this [[MOS]] pattern is [[diminished]][8], the familiar [[octatonic scale]] of 12edo.
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¢).
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{{c}}).


== Intervals ==
== Intervals ==
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== Scale tree ==
== Scale tree ==
{{todo|inline=1|complete table|Convert manually-generated scale tree table to MOS tuning spectrum without losing information from the existing table}}
{{todo|inline=1|complete table|comment=Convert manually-generated scale tree table to mos tuning spectrum without losing information from the existing table.}}
{| class="wikitable center-all"
{| class="wikitable center-all"
! colspan="6" rowspan="2" | Generator
! colspan="6" rowspan="2" | Generator

Revision as of 11:05, 21 August 2025

↖ 3L 3s ↑ 4L 3s 5L 3s ↗
← 3L 4s 4L 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
Generic Specific Abbrev.
0-tetrawdstep Perfect 0-tetrawdstep P0ttws 0 0.0 ¢
1-tetrawdstep Minor 1-tetrawdstep m1ttws s 0.0 ¢ to 150.0 ¢
Major 1-tetrawdstep M1ttws L 150.0 ¢ to 300.0 ¢
2-tetrawdstep Perfect 2-tetrawdstep P2ttws L + s 300.0 ¢
3-tetrawdstep Minor 3-tetrawdstep m3ttws L + 2s 300.0 ¢ to 450.0 ¢
Major 3-tetrawdstep M3ttws 2L + s 450.0 ¢ to 600.0 ¢
4-tetrawdstep Perfect 4-tetrawdstep P4ttws 2L + 2s 600.0 ¢
5-tetrawdstep Minor 5-tetrawdstep m5ttws 2L + 3s 600.0 ¢ to 750.0 ¢
Major 5-tetrawdstep M5ttws 3L + 2s 750.0 ¢ to 900.0 ¢
6-tetrawdstep Perfect 6-tetrawdstep P6ttws 3L + 3s 900.0 ¢
7-tetrawdstep Minor 7-tetrawdstep m7ttws 3L + 4s 900.0 ¢ to 1050.0 ¢
Major 7-tetrawdstep M7ttws 4L + 3s 1050.0 ¢ to 1200.0 ¢
8-tetrawdstep Perfect 8-tetrawdstep P8ttws 4L + 4s 1200.0 ¢

Modes

Scale degrees of the modes of 4L 4s
UDP Cyclic
order 		Step
pattern 		Scale degree (tetrawddegree)
0 1 2 3 4 5 6 7 8
4|0(4) 1 LsLsLsLs Perf. Maj. Perf. Maj. Perf. Maj. Perf. Maj. Perf.
0|4(4) 2 sLsLsLsL Perf. Min. Perf. Min. Perf. Min. Perf. Min. Perf.

Scale tree

Todo: complete table

Convert manually-generated scale tree table to mos tuning spectrum without losing information from the existing table.

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
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