2L 5s: Difference between revisions

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== Scale tree ==
== Scale tree ==
{| class="wikitable"
{| class="wikitable center-all"
! colspan="6" | Generator
! Cents
! L
! s
! L/s
! Comments
|-
|-
! colspan="7" | generator in degrees of an [[EDO|edo]]
| 3\7 || || || || || || 514.286 || 1 || 1 || 1.000 ||
! | generator in cents
! | tetrachord
! | L in cents
! | s in cents
! | L to s ratio
! | comments
|-
|-
| style="text-align:center;" | 3\7
| || || || || || 16\37 || 518.919 || 6 || 5 || 1.200 ||  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| |
| |  
| style="text-align:center;" | 514.3
| style="text-align:center;" | 1 1 1
| style="text-align:center;" | 171.4
| style="text-align:center;" | 171.4
| style="text-align:center;" | 1.00
| style="text-align:center;" |  
|-
|-
|
| || || || || 13\30 || || 520.000 || 5 || 4 || 1.250 ||  
|
|
|
|
|
|19\44
|518.2
|6 6 7
|190.9
|163.6
|1.17
|
|-
|-
|
| || || || || || 23\53 || 520.755 || 9 || 7 || 1.286 ||  
|
|
|
|
|16\37
|
|518.9
|5 5 6
|194.6
|162.2
|1.20
|
|-
|-
| style="text-align:center;" |  
| || || || 10\23 || || || 521.739 || 4 || 3 || 1.333 ||
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |
| style="text-align:center;" | 13\30
| |  
| |  
| style="text-align:center;" | 520.0
| style="text-align:center;" | 4 4 5
| style="text-align:center;" | 200.0
| style="text-align:center;" | 160.0
| style="text-align:center;" | 1.25
| style="text-align:center;" | Mavila extends from here...
|-
|-
| style="text-align:center;" |  
| || || || || || 27\62 || 522.581 || 11 || 8 || 1.375 ||
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" | 10\23
| style="text-align:center;" |  
| |  
| |
| style="text-align:center;" | 521.7
| style="text-align:center;" | 3 3 4
| style="text-align:center;" | 208.7
| style="text-align:center;" | 156.5
| style="text-align:center;" | 1.33
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" |  
| || || || || 17\39 || || 523.077 || 7 || 5 || 1.400 ||
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" | 17\39
| |
| |  
| style="text-align:center;" | 523.1
| style="text-align:center;" | 5 5 7
| style="text-align:center;" | 215.4
| style="text-align:center;" | 153.8
| style="text-align:center;" | 1.40
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" |  
| || || || || || 24\55 || 523.636 || 10 || 7 || 1.428 ||
| style="text-align:center;" |  
| style="text-align:center;" | 7\16
| style="text-align:center;" |  
| style="text-align:center;" |  
| |
| |  
| style="text-align:center;" | 525.0
| style="text-align:center;" | 2 2 3
| style="text-align:center;" | 225.0
| style="text-align:center;" | 150.0
| style="text-align:center;" | 1.50
| style="text-align:center;" | Mavila in Armodue
 
Optimum rank range (L/s=3/2)
|-
|-
| |  
| || || 7\16 || || || || 525.000 || 3 || 2 || 1.500 || L/s = 3/2, mavila is in this region
| |  
| |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 526.3
| style="text-align:center;" | <span style="display: block; text-align: center;">2 2 pi</span>
| style="text-align:center;" | 231.5
| style="text-align:center;" | 147.4
| style="text-align:center;" | <span style="display: block; text-align: center;">pi/2</span>
| |
|-
|-
| style="text-align:center;" |  
| || || || || || 25\57 || 526.316 || 11 || 7 || 1.571 ||
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" | 18\41
| |
| |
| style="text-align:center;" | 526.8
| style="text-align:center;" | 5 5 8
| style="text-align:center;" | 234.1
| style="text-align:center;" | 146.3
| style="text-align:center;" | 1.60
| style="text-align:center;" |  
|-
|-
| |  
| || || || || 18\41 || || 526.829 || 8 || 5 || 1.600 ||  
| |  
| |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 1200*5/(13-phi)
| style="text-align:center;" | 1 1 phi
| style="text-align:center;" | 235.7
| style="text-align:center;" | 145.7
| style="text-align:center;" | phi
| style="text-align:center;" | Golden mavila
|-
|-
| |  
| || || || || || 29\66 || 527.273 || 13 || 8 || 1.625 || Golden mavila
| |  
| |  
| |  
| |
| | 29\66
| |
| style="text-align:center;" | 527.3
| style="text-align:center;" | 8 8 13
| style="text-align:center;" | 236.4
| style="text-align:center;" | 145.5
| style="text-align:center;" | 1.625
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" |  
| || || || 11\25 || || || 528.000 || 5 || 3 || 1.667 ||
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" | 11\25
| style="text-align:center;" |  
| |  
| |
| style="text-align:center;" | 528.0
| style="text-align:center;" | 3 3 5
| style="text-align:center;" | 240.0
| style="text-align:center;" | 144.0
| style="text-align:center;" | 1.67
| style="text-align:center;" |  
|-
|-
| |  
| || || || || || 26\59 || 528.814 || 12 || 7 || 1.714 ||  
| |  
| |  
| |  
| |  
| |  
| |
| style="text-align:center;" | 529.1
| style="text-align:center;" | <span style="background-color: #ffffff;"><span style="line-height: 1.5;">1 1 </span>√3 </span>
| style="text-align:center;" | 245.6
| style="text-align:center;" | 141.8
| style="text-align:center;" | <span style="background-color: #ffffff; display: block; text-align: center;">√3</span>
| |  
|-
|-
| style="text-align:center;" |  
| || || || || 15\34 || || 529.412 || 7 || 4 || 1.750 ||
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" | 15\34
| |
| |  
| style="text-align:center;" | 529.4
| style="text-align:center;" | 4 4 7
| style="text-align:center;" | 247.1
| style="text-align:center;" | 141.2
| style="text-align:center;" | 1.75
| style="text-align:center;" | ...to somewhere around here
|-
|-
| style="text-align:center;" |  
| || || || || || 19\43 || 530.233 || 9 || 5 || 1.800 ||
| style="text-align:center;" | 4\9
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| |  
| |
| style="text-align:center;" | 533.3
| style="text-align:center;" | 1 1 2
| style="text-align:center;" | 266.7
| style="text-align:center;" | 133.3
| style="text-align:center;" | 2.00
| style="text-align:center;" | Boundary of propriety (generators
 
smaller than this are proper)
|-
|-
| style="text-align:center;" |  
| || 4\9 || || || || || 533.333 || 2 || 1 || 2.000 || Basic antidiatonic<br>(Generators smaller than this are proper)
| style="text-align:center;" |
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" | 13\29
| |  
| |  
| style="text-align:center;" | 537.9
| style="text-align:center;" | 3 3 7
| style="text-align:center;" | 289.7
| style="text-align:center;" | 124.1
| style="text-align:center;" | 2.33
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" |  
| || || || || || 17\38 || 536.842 || 9 || 4 || 2.250 ||
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" | 9\20
| style="text-align:center;" |  
| |  
| |
| style="text-align:center;" | 540.0
| style="text-align:center;" | 2 2 5
| style="text-align:center;" | 300.0
| style="text-align:center;" | 120.0
| style="text-align:center;" | 2.50
| style="text-align:center;" |  
|-
|-
| |  
| || || || || 13\29 || || 537.931 || 7 || 3 || 2.333 ||  
| |  
| |  
| |  
| |  
| |  
| |
| style="text-align:center;" | 541.4
| style="text-align:center;" | <span style="display: block; text-align: center;">1 1 phi+1</span>
| style="text-align:center;" | 306.9
| style="text-align:center;" | 117.2
| style="text-align:center;" | <span style="display: block; text-align: center;">1 1 phi+1</span>
| |  
|-
|-
| style="text-align:center;" |  
| || || || || || 22\49 || 538.776 || 12 || 5 || 2.400 ||
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" | 14\31
| |  
| |
| style="text-align:center;" | 541.9
| style="text-align:center;" | 3 3 8
| style="text-align:center;" | 309.7
| style="text-align:center;" | 116.1
| style="text-align:center;" | 2.66
| style="text-align:center;" |  
|-
|-
| |  
| || || || 9\20 || || || 540.000 || 5 || 2 || 2.500 ||  
| |  
| |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 542<span style="line-height: 1.5;">.5</span>
| style="text-align:center;" | 1 1 e
| style="text-align:center;" | 321.55
| style="text-align:center;" | 115.0
| style="text-align:center;" | e
| style="text-align:center;" | L/s = e
|-
|-
| style="text-align:center;" |  
| || || || || || 23\51 || 541.176 || 13 || 5 || 2.600 || Unnamed golden tuning
| style="text-align:center;" |  
| style="text-align:center;" | 5\11
| style="text-align:center;" |  
| style="text-align:center;" |  
| |  
| |
| style="text-align:center;" | 545.5
| style="text-align:center;" | 1 1 3
| style="text-align:center;" | 327.3
| style="text-align:center;" | 109.1
| style="text-align:center;" | 3.00
| style="text-align:center;" | L/s = 3
|-
|-
| |  
| || || || || 14\31 || || 541.935 || 8 || 3 || 2.667 ||
| |  
| |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 546.8
| style="text-align:center;" | 1 1 pi
| style="text-align:center;" | 334.1
| style="text-align:center;" | 106.35
| style="text-align:center;" | pi
| style="text-align:center;" | <span style="display: block; text-align: center;">L/s = pi</span>
|-
|-
| style="text-align:center;" |  
| || || || || || 19\42 || 542.857 || 11 || 4 || 2.750 ||
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" | 11\24
| |  
| |  
| style="text-align:center;" | 550.0
| style="text-align:center;" | 2 2 7
| style="text-align:center;" | 350.0
| style="text-align:center;" | 100.0
| style="text-align:center;" | 3.50
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" |
| || || 5\11 || || || || 545.455 || 3 || 1 || 3.000 || L/s = 3/1
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" | 6\13
| style="text-align:center;" |  
| |  
| |  
| style="text-align:center;" | 553.8
| style="text-align:center;" | 1 1 4
| style="text-align:center;" | 369.2
| style="text-align:center;" | 92.3
| style="text-align:center;" | 4.00
| style="text-align:center;" | Thuja is optimal around here
 
L/s = 4
|-
|-
| style="text-align:center;" |  
| || || || || || 16\35 || 548.571 || 10 || 3 || 3.333 ||
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" | 7\15
| |
| |
| style="text-align:center;" | 560.0
| style="text-align:center;" | 1 1 5
| style="text-align:center;" | 400.0
| style="text-align:center;" | 80.0
| style="text-align:center;" | 5.00
| style="text-align:center;" | ie. (11/8)^5 = 5/1
|-
|-
| style="text-align:center;" |  
| || || || || 11\24 || || 550.000 || 7 || 2 || 3.500 ||
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |
| style="text-align:center;" | 8\17
| style="text-align:center;" |  
| style="text-align:center;" | 564.7
| style="text-align:center;" | 1 1 6
| style="text-align:center;" | 423.5
| style="text-align:center;" | 70.6
| style="text-align:center;" | 6.00
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" |  
| || || || || || 17\37 || 551.351 || 11 || 3 || 3.667 ||
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |
| style="text-align:center;" | 9\19
| style="text-align:center;" | 568.4
| style="text-align:center;" | 1 1 7
| style="text-align:center;" | 442.1
| style="text-align:center;" | 63.2
| style="text-align:center;" | 7.00
| style="text-align:center;" | Liese/Triton is around here
|-
|-
| style="text-align:center;" | 1\2
| || || || 6\13 || || || 553.846 || 4 || 1 || 4.000 ||
| style="text-align:center;" |  
|-
| style="text-align:center;" |  
| || || || || || 13\28 || 557.143 || 9 || 2 || 4.500 ||
| style="text-align:center;" |  
|-
| style="text-align:center;" |  
| || || || || 7\15 || || 560.000 || 5 || 1 || 5.000 ||  
| |  
|-
| |  
| || || || || || 8\17 || 564.706 || 6 || 1 || 6.000 || Liese↓, triton↓
| style="text-align:center;" | 600.0
|-
| style="text-align:center;" | 0 0 1
| 1\2 || || || || || || 600.000 || 1 || 0 || → inf ||
| style="text-align:center;" | 600.0
| style="text-align:center;" | 0
| style="text-align:center;" | —
| style="text-align:center;" |  
|}
|}


Revision as of 14:04, 11 April 2021

User:IlL/Template:RTT restriction

↖ 1L 4s ↑ 2L 4s 3L 4s ↗
← 1L 5s 2L 5s 3L 5s →
↙ 1L 6s ↓ 2L 6s 3L 6s ↘ 
┌╥┬┬╥┬┬┬┐
│║││║││││
│││││││││
└┴┴┴┴┴┴┴┘
Scale structure
Step pattern LssLsss
sssLssL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 3\7 to 1\2 (514.3 ¢ to 600.0 ¢)
Dark 1\2 to 4\7 (600.0 ¢ to 685.7 ¢)
TAMNAMS information
Name antidiatonic
Prefix pel-
Abbrev. pel
Related MOS scales
Parent 2L 3s
Sister 5L 2s
Daughters 7L 2s, 2L 7s
Neutralized 4L 3s
2-Flought 9L 5s, 2L 12s
Equal tunings
Equalized (L:s = 1:1) 3\7 (514.3 ¢)
Supersoft (L:s = 4:3) 10\23 (521.7 ¢)
Soft (L:s = 3:2) 7\16 (525.0 ¢)
Semisoft (L:s = 5:3) 11\25 (528.0 ¢)
Basic (L:s = 2:1) 4\9 (533.3 ¢)
Semihard (L:s = 5:2) 9\20 (540.0 ¢)
Hard (L:s = 3:1) 5\11 (545.5 ¢)
Superhard (L:s = 4:1) 6\13 (553.8 ¢)
Collapsed (L:s = 1:0) 1\2 (600.0 ¢)

2L 5s or antidiatonic refers to the structure of octave-equivalent MOS scales with generators ranging from 3\7 (3 degrees of 7edo = 514.29¢) to 1\2 (one degree of 2edo = 600¢). In the case of 7edo, L and s are the same size; in the case of 2edo, s becomes so small it disappears (and all that remains are the two equal L's).

While antidiatonic is closely associated with mavila, not every 2L 5s scale is an instance of "mavila", since some of them extend to 2L 7s scales (like the 2L 5s generated by 11edo's 6\11 = 656.5657¢), not 7L 2s mavila superdiatonic scales.

Notation

Diamond MOS notation, &/@ = raise and lower by one chroma. We'll write this using CDEFGABC is C Antiionian (ssLsssL); C = 261.6256 Hz. The chain of mavila fifths becomes ... E& B& F C G D A E B F@ C@ ... Note that 7 fifths up flattens a note by a chroma, rather than sharpening it as in diatonic (5L 2s).

Scale tree

Generator Cents L s L/s Comments
3\7 514.286 1 1 1.000
16\37 518.919 6 5 1.200
13\30 520.000 5 4 1.250
23\53 520.755 9 7 1.286
10\23 521.739 4 3 1.333
27\62 522.581 11 8 1.375
17\39 523.077 7 5 1.400
24\55 523.636 10 7 1.428
7\16 525.000 3 2 1.500 L/s = 3/2, mavila is in this region
25\57 526.316 11 7 1.571
18\41 526.829 8 5 1.600
29\66 527.273 13 8 1.625 Golden mavila
11\25 528.000 5 3 1.667
26\59 528.814 12 7 1.714
15\34 529.412 7 4 1.750
19\43 530.233 9 5 1.800
4\9 533.333 2 1 2.000 Basic antidiatonic
(Generators smaller than this are proper)
17\38 536.842 9 4 2.250
13\29 537.931 7 3 2.333
22\49 538.776 12 5 2.400
9\20 540.000 5 2 2.500
23\51 541.176 13 5 2.600 Unnamed golden tuning
14\31 541.935 8 3 2.667
19\42 542.857 11 4 2.750
5\11 545.455 3 1 3.000 L/s = 3/1
16\35 548.571 10 3 3.333
11\24 550.000 7 2 3.500
17\37 551.351 11 3 3.667
6\13 553.846 4 1 4.000
13\28 557.143 9 2 4.500
7\15 560.000 5 1 5.000
8\17 564.706 6 1 6.000 Liese↓, triton↓
1\2 600.000 1 0 → inf

Musical Examples

Mike Battaglia has "translated" several common practice pieces into mavila antidiatonic by using Graham Breed's Lilypond code to tune the generators flat. Musical examples are provided in 9-EDO, 16-EDO, 23-EDO, and 25-EDO, for comparison. Note that the melodic and/or intonational properties differ slightly for each tuning.

9-EDO: Provided ID could not be validated.

16-EDO: Provided ID could not be validated.

23-EDO: Provided ID could not be validated.

25-EDO: Provided ID could not be validated.

Retrieved from "https://en.xen.wiki/index.php?title=2L_5s&oldid=66206"