3L 3s: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Added TAMNAMS name
New scale tree
Line 9: Line 9:
}}
}}


{| class="wikitable"
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).
 
Out of all ''[[Rothenberg propriety|proper]]'' six-note MOS scales, this augmented scale probably has the lowest harmonic entropy.
 
== Scale tree ==
{| 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 || L/s = 3/2
|-
| || || || || || 11\54 || 244.444 || 155.556 || 11 || 7 || 1.571 ||
|-
| || || || || 8\39 || || 246.154 || 153.846 || 8 || 5 || 1.600 || Triforce
|-
|-
! colspan="5" | Generator
| || || || || || 13\63 || 247.619 || 152.381 || 13 || 8 || 1.625 || Unnamed golden tuning<br>Trug (optimal around here)
! | Cents
! | Comments
|-
|-
| | 0\3
| || || || 5\24 || || || 250.000 || 150.000 || 5 || 3 || 1.667 ||  
| |  
| |  
| |  
| |  
| | 0
| style="text-align:center;" |  
|-
|-
| |  
| || || || || || 12\57 || 252.632 || 147.368 || 12 || 7 || 1.714 ||  
| |  
| |  
| | 1\15
| |  
| | 80
| style="text-align:center;" | L/s = 4
|-
|-
| |  
| || || || || 7\33 || || 254.545 || 145.455 || 7 || 4 || 1.750 ||  
| |  
| |  
| |  
| | 2\27
| | 88.88
| style="text-align:center;" | Optimal augmented is around here
|-
|-
| |  
| || || || || || 9\42 || 257.143 || 142.857 || 9 || 5 || 1.800 ||
| |  
| |  
| |  
| |  
| | 400/(1+pi)
| |  
|-
|-
| |  
| || 2\9 || || || || || 266.667 || 133.333 || 2 || 1 || 2.000 || Basic 3L 3s
| |  
| | 1\12
| |  
| |  
| | 100
| style="text-align:center;" | L/s = 3
|-
|-
| |  
| || || || || || 9\39 || 276.923 || 123.077 || 9 || 4 || 2.250 ||  
| |  
| |  
| |  
| |  
| | 400/(1+e)
| |  
|-
|-
| |  
| || || || || 7\30 || || 280.000 || 120.000 || 7 || 3 || 2.333 || Deflated (optimal around here)
| |  
| |  
| |  
| | 3\33
| | 109.09
| |  
|-
|-
| |  
| || || || || || 12\51 || 282.353 || 117.647 || 12 || 5 || 2.400 ||
| |  
| |  
| |  
| |  
| | 400/(2+phi)
| |  
|-
|-
| |  
| || || || 5\21 || || || 285.714 || 114.286 || 5 || 2 || 2.500 ||  
| |  
| |  
| | 2\21
| |  
| | 114.29
| |  
|-
|-
| |  
| || || || || || 13\54 || 288.889 || 111.111 || 13 || 5 || 2.600 || Unnamed golden tuning
| | 1\9
|-
| |  
| || || || || 8\33 || || 290.909 || 109.091 || 8 || 3 || 2.667 ||
| |  
|-
| |  
| || || || || || 11\45 || 293.333 || 106.667 || 11 || 4 || 2.750 || August
| | 133.33
|-
| style="text-align:center;" | Boundary of propriety for near-MOS
| || || 3\12 || || || || 300.000 || 100.000 || 3 || 1 || 3.000 || L/s = 3/1
 
|-
Optimum rank range (L/s=2/1) for MOS
| || || || || || 10\39 || 307.692 || 92.308 || 10 || 3 || 3.333 || Augene
|-
|-
| |  
| || || || || 7\27 || || 311.111 || 88.889 || 7 || 2 || 3.500 ||  
| |  
| |  
| |  
| |  
| | 400/(1+sqrt(3))
| |  
|-
|-
| |  
| || || || || || 11\42 || 314.286 || 85.714 || 11 || 3 || 3.667 ||  
| |  
| |  
| | 3\24
| |  
| | 150
| style="text-align:center;" |  
|-
|-
| |  
| || || || 4\15 || || || 320.000 || 80.000 || 4 || 1 || 4.000 || Inflated
| |  
| |  
| |  
| |  
| | 400/(1+phi)
| style="text-align:center;" |  
|-
|-
| |  
| || || || || || 9\33 || 327.273 || 72.727 || 9 || 2 || 4.500 ||  
| |  
| |  
| |  
| | 5\39
| | 153.85
| style="text-align:center;" |  
|-
|-
| |  
| || || || || 5\18 || || 333.333 || 66.667 || 5 || 1 || 5.000 ||  
| |  
| |  
| |  
| |  
| | 400/(1+pi/2)
| |  
|-
|-
| |  
| || || || || || 6\21 || 342.857 || 57.143 || 6 || 1 || 6.000 || Hemiug↓, hemiaug↓
| |  
| | 2\15
| |  
| |  
| | 160
| style="text-align:center;" |  
|-
|-
| | 1\6
| 1\3 || || || || || || 400.000 || 0.000 || 1 || 0 || → inf ||
| |  
| |  
| |  
| |  
| | 200
| style="text-align:center;" |  
|}
|}
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).
Out of all '''[[Rothenberg_propriety|proper]]''' six-note MOS scales, this augmented scale probably has the lowest harmonic entropy.


[[Category:Abstract MOS patterns]]
[[Category:Abstract MOS patterns]]
[[Category:Triwood| ]] <!-- main article -->

Revision as of 10:11, 3 March 2022

↖ 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

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).

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

Scale tree

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 L/s = 3/2
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
Trug (optimal around here)
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 3L 3s
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 L/s = 3/1
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
Retrieved from "https://en.xen.wiki/index.php?title=3L_3s&oldid=88913"