3L 3s: Difference between revisions

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The true MOS is LsLsLs but interesting near-MOSs include LLsLss and LLssLs. The MOS is always proper but the other forms are only proper if the generator is larger than 1\9 of an octave (so not augmented).
{{MOS intro}}


Out of all ''[[Rothenberg propriety|proper]]'' six-note MOS scales, this augmented scale probably has the lowest harmonic entropy.
In addition to the true MOS (LsLsLs or sLsLsL), there are also near-MOS patterns of LLsLss and LLssLs, which are only proper if the generator is larger than [[9edo|1\9]].


== Scale tree ==
Out of all ''[[Rothenberg propriety|proper]]'' six-note MOS scales, this augmented scale probably has the lowest harmonic entropy{{Clarify}}.
 
== Intervals ==
{{MOS intervals}}
 
== Modes==
{{MOS mode degrees}}
 
==Scale tree==
{{todo|inline=1|complete table|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
! colspan="2" | Cents
! colspan="2" |Cents
! rowspan="2" | L
! rowspan="2" |L
! rowspan="2" | s
! rowspan="2" |s
! rowspan="2" | L/s
! rowspan="2" |L/s
! rowspan="2" | Comments
! rowspan="2" | Comments
|-
|-
Line 25: Line 34:
! Chroma-negative
! Chroma-negative
|-
|-
| 1\6 || || || || || || 200.000 || 200.000 || 1 || 1 || 1.000 ||  
|1\6|| || || || || ||200.000||200.000 || 1||1||1.000||
|-
|-
| || || || || || 6\33 || 218.182 || 181.818 || 6 || 5 || 1.200 ||  
| || || || || ||6\33||218.182|| 181.818||6||5||1.200||
|-
|-
| || || || || 5\27 || || 222.222 || 177.778 || 5 || 4 || 1.250 ||  
| || || || ||5\27|| ||222.222|| 177.778||5||4||1.250||
|-
|-
| || || || || || 9\48 || 225.000 || 175.000 || 9 || 7 || 1.286 || Oodako
| || || || || ||9\48||225.000|| 175.000||9||7||1.286||Oodako
|-
|-
| || || || 4\21 || || || 228.571 || 171.429 || 4 || 3 || 1.333 ||  
| || || ||4\21|| || ||228.571|| 171.429||4||3||1.333||
|-
|-
| || || || || || 11\57 || 231.579 || 168.421 || 11 || 8 || 1.375 ||  
| || || || || ||11\57||231.579||168.421 || 11 ||8||1.375||  
|-
|-
| || || || || 7\36 || || 233.333 || 166.667 || 7 || 5 || 1.400 ||  
| || || || ||7\36|| ||233.333|| 166.667||7||5||1.400||
|-
|-
| || || || || || 10\51 || 235.294 || 164.706 || 10 || 7 || 1.428 ||  
| || || || || ||10\51||235.294||164.706 || 10 ||7||1.428||  
|-
|-
| || || 3\15 || || || || 240.000 || 160.000 || 3 || 2 || 1.500 ||  
| || ||3\15 || || || ||240.000|| 160.000|| 3||2||1.500||
|-
|-
| || || || || || 11\54 || 244.444 || 155.556 || 11 || 7 || 1.571 ||  
| || || || || ||11\54||244.444||155.556 || 11 ||7||1.571||  
|-
|-
| || || || || 8\39 || || 246.154 || 153.846 || 8 || 5 || 1.600 || Triforce
| || || || ||8\39|| ||246.154|| 153.846||8||5||1.600||Triforce
|-
|-
| || || || || || 13\63 || 247.619 || 152.381 || 13 || 8 || 1.625 || Unnamed golden tuning
| || || || || ||13\63||247.619||152.381 || 13 ||8||1.625|| Unnamed golden tuning
|-
|-
| || || || 5\24 || || || 250.000 || 150.000 || 5 || 3 || 1.667 ||  
| || || ||5\24|| || ||250.000|| 150.000||5||3||1.667||
|-
|-
| || || || || || 12\57 || 252.632 || 147.368 || 12 || 7 || 1.714 ||  
| || || || || ||12\57||252.632||147.368 || 12 ||7||1.714||  
|-
|-
| || || || || 7\33 || || 254.545 || 145.455 || 7 || 4 || 1.750 ||  
| || || || ||7\33|| ||254.545|| 145.455||7||4||1.750||
|-
|-
| || || || || || 9\42 || 257.143 || 142.857 || 9 || 5 || 1.800 ||
| || || || || ||9\42||257.143|| 142.857||9||5||1.800||
|-
|-
| || 2\9 || || || || || 266.667 || 133.333 || 2 || 1 || 2.000 || Basic triwood
| ||2\9 || || || || ||266.667||133.333 || 2||1||2.000||Basic triwood
|-
|-
| || || || || || 9\39 || 276.923 || 123.077 || 9 || 4 || 2.250 ||  
| || || || || || 9\39||276.923|| 123.077||9||4||2.250||
|-
|-
| || || || || 7\30 || || 280.000 || 120.000 || 7 || 3 || 2.333 || Deflated (optimal around here)
| || || || ||7\30|| ||280.000|| 120.000||7||3||2.333||Deflated (optimal around here)
|-
|-
| || || || || || 12\51 || 282.353 || 117.647 || 12 || 5 || 2.400 ||
| || || || || ||12\51||282.353||117.647 || 12 ||5||2.400||
|-
|-
| || || || 5\21 || || || 285.714 || 114.286 || 5 || 2 || 2.500 ||  
| || || || 5\21|| || ||285.714|| 114.286||5||2||2.500||
|-
|-
| || || || || || 13\54 || 288.889 || 111.111 || 13 || 5 || 2.600 || Unnamed golden tuning
| || || || || ||13\54||288.889||111.111 || 13 ||5||2.600|| Unnamed golden tuning
|-
|-
| || || || || 8\33 || || 290.909 || 109.091 || 8 || 3 || 2.667 ||  
| || || || ||8\33|| ||290.909|| 109.091||8||3||2.667||
|-
|-
| || || || || || 11\45 || 293.333 || 106.667 || 11 || 4 || 2.750 || August
| || || || || ||11\45||293.333||106.667 || 11 ||4||2.750|| August
|-
|-
| || || 3\12 || || || || 300.000 || 100.000 || 3 || 1 || 3.000 || Trug (optimal around here)
| || ||3\12|| || || ||300.000|| 100.000|| 3||1||3.000||Trug (optimal around here)
|-
|-
| || || || || || 10\39 || 307.692 || 92.308 || 10 || 3 || 3.333 || Augene
| || || || || ||10\39||307.692||92.308||10||3||3.333||Augene
|-
|-
| || || || || 7\27 || || 311.111 || 88.889 || 7 || 2 || 3.500 ||  
| || || || ||7\27|| ||311.111|| 88.889||7 ||2 || 3.500||
|-
|-
| || || || || || 11\42 || 314.286 || 85.714 || 11 || 3 || 3.667 ||  
| || || || || ||11\42||314.286||85.714||11||3||3.667||
|-
|-
| || || || 4\15 || || || 320.000 || 80.000 || 4 || 1 || 4.000 || Inflated
| || || ||4\15 || || ||320.000|| 80.000||4||1 || 4.000||Inflated
|-
|-
| || || || || || 9\33 || 327.273 || 72.727 || 9 || 2 || 4.500 ||  
| || || || || ||9\33 ||327.273|| 72.727||9 || 2|| 4.500||
|-
|-
| || || || || 5\18 || || 333.333 || 66.667 || 5 || 1 || 5.000 ||  
| || || || ||5\18|| ||333.333|| 66.667||5 ||1 || 5.000||
|-
|-
| || || || || || 6\21 || 342.857 || 57.143 || 6 || 1 || 6.000 || Hemiug↓, hemiaug↓
| || || || || ||6\21||342.857|| 57.143||6 || 1|| 6.000||Hemiug↓, hemiaug↓
|-
|-
| 1\3 || || || || || || 400.000 || 0.000 || 1 || 0 || → inf ||
|1\3 || || || || || ||400.000||0.000||1 || 0||→ inf||
|}
|}


[[Category:Triwood| ]] <!-- main article -->
[[Category:Triwood| ]]
[[Category:6-tone scales]]
<!-- main article -->

Latest revision as of 22:27, 1 August 2025

↖ 2L 2s ↑ 3L 2s 4L 2s ↗
← 2L 3s 3L 3s 4L 3s →
↙ 2L 4s ↓ 3L 4s 4L 4s ↘ 
┌╥┬╥┬╥┬┐
│║│║│║││
││││││││
└┴┴┴┴┴┴┘
Scale structure
Step pattern LsLsLs
sLsLsL
Equave 2/1 (1200.0 ¢)
Period 1\3 (400.0 ¢)
Generator size
Bright 1\6 to 1\3 (200.0 ¢ to 400.0 ¢)
Dark 0\3 to 1\6 (0.0 ¢ to 200.0 ¢)
TAMNAMS information
Name triwood
Prefix triwd-
Abbrev. tw
Related MOS scales
Parent none
Sister 3L 3s (self)
Daughters 6L 3s, 3L 6s
Neutralized 6edo
2-Flought 9L 3s, 3L 9s
Equal tunings
Equalized (L:s = 1:1) 1\6 (200.0 ¢)
Supersoft (L:s = 4:3) 4\21 (228.6 ¢)
Soft (L:s = 3:2) 3\15 (240.0 ¢)
Semisoft (L:s = 5:3) 5\24 (250.0 ¢)
Basic (L:s = 2:1) 2\9 (266.7 ¢)
Semihard (L:s = 5:2) 5\21 (285.7 ¢)
Hard (L:s = 3:1) 3\12 (300.0 ¢)
Superhard (L:s = 4:1) 4\15 (320.0 ¢)
Collapsed (L:s = 1:0) 1\3 (400.0 ¢)

3L 3s, named triwood in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 3 small steps, with a period of 1 large step and 1 small step that repeats every 400.0 ¢, or 3 times every octave. Generators that produce this scale range from 200 ¢ to 400 ¢, or from 0 ¢ to 200 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

In addition to the true MOS (LsLsLs or sLsLsL), there are also near-MOS patterns of LLsLss and LLssLs, which are only proper if the generator is larger than 1\9.

Out of all proper six-note MOS scales, this augmented scale probably has the lowest harmonic entropy[clarification needed].

Intervals

Intervals of 3L 3s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-triwdstep Perfect 0-triwdstep P0tws 0 0.0 ¢
1-triwdstep Minor 1-triwdstep m1tws s 0.0 ¢ to 200.0 ¢
Major 1-triwdstep M1tws L 200.0 ¢ to 400.0 ¢
2-triwdstep Perfect 2-triwdstep P2tws L + s 400.0 ¢
3-triwdstep Minor 3-triwdstep m3tws L + 2s 400.0 ¢ to 600.0 ¢
Major 3-triwdstep M3tws 2L + s 600.0 ¢ to 800.0 ¢
4-triwdstep Perfect 4-triwdstep P4tws 2L + 2s 800.0 ¢
5-triwdstep Minor 5-triwdstep m5tws 2L + 3s 800.0 ¢ to 1000.0 ¢
Major 5-triwdstep M5tws 3L + 2s 1000.0 ¢ to 1200.0 ¢
6-triwdstep Perfect 6-triwdstep P6tws 3L + 3s 1200.0 ¢

Modes

Scale degrees of the modes of 3L 3s
UDP Cyclic
order 		Step
pattern 		Scale degree (triwddegree)
0 1 2 3 4 5 6
3|0(3) 1 LsLsLs Perf. Maj. Perf. Maj. Perf. Maj. Perf.
0|3(3) 2 sLsLsL 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\6 200.000 200.000 1 1 1.000
6\33 218.182 181.818 6 5 1.200
5\27 222.222 177.778 5 4 1.250
9\48 225.000 175.000 9 7 1.286 Oodako
4\21 228.571 171.429 4 3 1.333
11\57 231.579 168.421 11 8 1.375
7\36 233.333 166.667 7 5 1.400
10\51 235.294 164.706 10 7 1.428
3\15 240.000 160.000 3 2 1.500
11\54 244.444 155.556 11 7 1.571
8\39 246.154 153.846 8 5 1.600 Triforce
13\63 247.619 152.381 13 8 1.625 Unnamed golden tuning
5\24 250.000 150.000 5 3 1.667
12\57 252.632 147.368 12 7 1.714
7\33 254.545 145.455 7 4 1.750
9\42 257.143 142.857 9 5 1.800
2\9 266.667 133.333 2 1 2.000 Basic triwood
9\39 276.923 123.077 9 4 2.250
7\30 280.000 120.000 7 3 2.333 Deflated (optimal around here)
12\51 282.353 117.647 12 5 2.400
5\21 285.714 114.286 5 2 2.500
13\54 288.889 111.111 13 5 2.600 Unnamed golden tuning
8\33 290.909 109.091 8 3 2.667
11\45 293.333 106.667 11 4 2.750 August
3\12 300.000 100.000 3 1 3.000 Trug (optimal around here)
10\39 307.692 92.308 10 3 3.333 Augene
7\27 311.111 88.889 7 2 3.500
11\42 314.286 85.714 11 3 3.667
4\15 320.000 80.000 4 1 4.000 Inflated
9\33 327.273 72.727 9 2 4.500
5\18 333.333 66.667 5 1 5.000
6\21 342.857 57.143 6 1 6.000 Hemiug↓, hemiaug↓
1\3 400.000 0.000 1 0 → inf
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