4L 4s: Difference between revisions

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The minimum harmonic entropy scale with this [[MOSScales|MOS]] pattern is [[Jubilismic_clan|diminished]][8], basically the familiar [[Octatonic_scale|octatonic scale]] of 12edo.
The minimum harmonic entropy scale with this [[MOS]] pattern is [[diminished]][8], basically the familiar [[octatonic scale]] of 12edo.


There are 4 different near-MOSes such that multiples of the period are the only generic intervals with more than two specific representatives: LLsLsLss, LLsLssLs, LLssLLss, LLssLsLs. However, none are strictly proper in 12edo. (They are only proper if the generator is larger than 100 cents, which unfortunately is the wrong direction for good diminished tunings.)
There are 4 different near-MOSes such that multiples of the period are the only generic intervals with more than two specific representatives: LLsLsLss, LLsLssLs, LLssLLss, LLssLsLs. However, none are strictly proper in 12edo. (They are only proper if the generator is larger than 100 cents, which unfortunately is the wrong direction for good diminished tunings.)


{| class="wikitable"
== 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
|-
|-
! colspan="5" | Generator
! Chroma-positive
! | Cents
! Chroma-negative
! | Comments
|-
|-
| | 0\4
| 1\8 || || || || || || 150.000 || 150.000 || 1 || 1 || 1.000 ||  
| |  
| |  
| |  
| |  
| | 0
| style="text-align:center;" |  
|-
|-
| |  
| || || || || || 6\44 || 163.636 || 136.364 || 6 || 5 || 1.200 || Fourfives↑
| |  
| |  
| |  
| | 1\24
| | 50
| |  
|-
|-
| |  
| || || || || 5\36 || || 166.667 || 133.333 || 5 || 4 || 1.250 ||
| |  
| |  
| | 1\20
| |  
| | 60
| style="text-align:center;" | L/s = 4
|-
|-
| |  
| || || || || || 9\64 || 168.750 || 105.000 || 9 || 7 || 1.286 ||  
| |  
| |  
| |  
| |  
| | 300/(1+pi)
| |  
|-
|-
| |  
| || || || 4\28 || || || 171.429 || 128.571 || 4 || 3 || 1.333 ||
| |  
| | 1\16
| |  
| |  
| | 75
| style="text-align:center;" | L/s = 3
|-
|-
| |  
| || || || || || 11\76 || 173.684 || 126.316 || 11 || 8 || 1.375 ||  
| |  
| |  
| |  
| |  
| | 300/(1+e)
| |  
|-
|-
| |  
| || || || || 7\48 || || 175.000 || 125.000 || 7 || 5 || 1.400 ||  
| |  
| |  
| |  
| | 3\44
| | 81.82
| |  
|-
|-
| |  
| || || || || || 10\68 || 176.471 || 123.529 || 10 || 7 || 1.428 ||  
| |  
| |  
| | 2\28
| |  
| | 85.71
| |  
|-
|-
| |  
| || || 3\20 || || || || 180.000 || 60.000 || 3 || 2 || 1.500 || L/s = 3/2
| | 1\12
|-
| |  
| || || || || || 11\72 || 183.333 || 116.667 || 11 || 7 || 1.571 ||
| |  
|-
| |  
| || || || || 8\52 || || 184.615 || 115.385 || 8 || 5 || 1.600 ||
| | 100
|-
| style="text-align:center;" | Diminished is around here
| || || || || || 13\84 || 185.714 || 114.286 || 13 || 8 || 1.625 || Golden diminished
 
|-
Optimum rank range (L/s=2/1) for MOS
| || || || 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<br>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 ||
| |  
| |  
| |  
| |  
| | 300/(1+sqrt(3))
| |  
|-
|-
| |  
| || || || || 7\36 || || 233.333 || 66.667 || 7 || 2 || 3.500 ||  
| |  
| |  
| | 3\32
| |  
| | 112.5
| style="text-align:center;" |  
|-
|-
| |  
| || || || || || 11\56 || 235.714 || 64.286 || 11 || 3 || 3.667 ||  
| |  
| |  
| |  
| | 5\52
| | 115.38
| style="text-align:center;" | Golden diminished
|-
|-
| |  
| || || || 4\20 || || || 240.000 || 60.000 || 4 || 1 || 4.000 ||
| |  
| |  
| |  
| |  
| | 300/(1+pi/2)
| |  
|-
|-
| |  
| || || || || || 9\44 || 245.455 || 54.545 || 9 || 2 || 4.500 ||  
| |  
| | 2\20
| |  
| |  
| | 120
| style="text-align:center;" |  
|-
|-
|
| || || || || 5\24 || || 250.000 || 50.000 || 5 || 1 || 5.000 ||  
|
|
|3\28
|
|128.57
|
|-
|-
|
| || || || || || 6\28 || 257.143 || 42.857 || 6 || 1 || 6.000 || Quadritikleismic↓
|
|
|
|4\36
|133.33
|
|-
|-
| | 1\8
| 1\4 || || || || || || 300.000 || 0.000 || 1 || 0 || → inf ||
| |  
| |  
| |  
| |  
| | 150
| style="text-align:center;" |  
|}
|}


[[Category:Scale theory]]
[[Category:Abstract MOS patterns]]
[[Category:Abstract MOS patterns]]
[[Category:Tetrawood]] <!-- main article -->

Revision as of 12:58, 13 January 2022

↖ 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 ¢)

The minimum harmonic entropy scale with this MOS pattern is diminished[8], basically the familiar octatonic scale of 12edo.

There are 4 different near-MOSes such that multiples of the period are the only generic intervals with more than two specific representatives: LLsLsLss, LLsLssLs, LLssLLss, LLssLsLs. However, none are strictly proper in 12edo. (They are only proper if the generator is larger than 100 cents, which unfortunately is the wrong direction for good diminished tunings.)

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
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