3L 3s: Difference between revisions

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==Scale tree==
==Scale tree==
{{todo|inline=1|complete table|Convert manually-generated scale tree table to MOS tuning spectrum}}
{{MOS tuning spectrum}}
{| class="wikitable center-all"
! colspan="6" rowspan="2" |Generator
! colspan="2" |Cents
! rowspan="2" |L
! rowspan="2" |s
! rowspan="2" |L/s
! rowspan="2" | 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||
|}
 
[[Category:Triwood| ]]
[[Category:Triwood| ]]
[[Category:6-tone scales]]
[[Category:6-tone scales]]
  <!-- main article -->
  <!-- main article -->

Revision as of 15:30, 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 ¢)
View • Talk • Edit

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

Scale tree and tuning spectrum of 3L 3s
Generator(edo) Cents Step ratio Comments(always proper)
Bright Dark L:s Hardness
1\6 200.000 200.000 1:1 1.000 Equalized 3L 3s
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
4\21 228.571 171.429 4:3 1.333 Supersoft 3L 3s
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 1.429
3\15 240.000 160.000 3:2 1.500 Soft 3L 3s
11\54 244.444 155.556 11:7 1.571
8\39 246.154 153.846 8:5 1.600
13\63 247.619 152.381 13:8 1.625
5\24 250.000 150.000 5:3 1.667 Semisoft 3L 3s
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 3L 3s
9\39 276.923 123.077 9:4 2.250
7\30 280.000 120.000 7:3 2.333
12\51 282.353 117.647 12:5 2.400
5\21 285.714 114.286 5:2 2.500 Semihard 3L 3s
13\54 288.889 111.111 13:5 2.600
8\33 290.909 109.091 8:3 2.667
11\45 293.333 106.667 11:4 2.750
3\12 300.000 100.000 3:1 3.000 Hard 3L 3s
10\39 307.692 92.308 10:3 3.333
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 Superhard 3L 3s
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
1\3 400.000 0.000 1:0 → ∞ Collapsed 3L 3s
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