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Also reverting this, just to be safe.
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{{Infobox MOS
2L 5s refers to a Moment of Symmetry scale with two large steps and five small steps. Common names for such a tuning are mavila and anti-diatonic. The generator is a sharp fourth (or flat fifth), falling between 3\7 (3 degrees of [[7edo|7edo]]) and 1\2 (1 degree of [[2edo|2edo]] — the half-octave tritone that appears in every even-numbered edo).
| Name = antidiatonic
| Periods = 1
| nLargeSteps = 2
| nSmallSteps = 5
| Equalized = 3
| Paucitonic = 1
| Pattern = sLsssLs
| Neutral = 4L 3s
}}


'''2L 5s''' or '''antidiatonic''' refers to the structure of octave-equivalent [[MOS]] scales with generators ranging from 3\7 (3 degrees of [[7edo|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).
The word "mavila" is used in different ways by different folks. Not every user of the word would consider every 2L 5s scale an instance of "mavila." In particular, between 13\29 and 14\31, and centered on 9\20, is the albitonic scale for the 2.7.11.13 subgroup temperament [[Chromatic_pairs#Score|score]], which is not intended to be treated as having any kind of fifth, flat or otherwise.


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.
In terms of harmonic entropy, the most significant minimum is at [[Meantone_family|Liese]]/Triton, in which the generator is about 7/5 and three of them make a 3/1.


== Notation ==
{| class="wikitable"
We'll use the convention DEFGABCD (D Antidorian, sLsssLs); D = 293.665 Hz, &/@ = raise and lower by one chroma.  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 ==
{| class="wikitable center-all"
! colspan="6" | Generator
! Cents
! L
! s
! L/s
! Comments
|-
|-
| 3\7 || || || || || || 514.286 || 1 || 1 || 1.000 ||
! colspan="7" | generator in degrees of an [[EDO|edo]]
! | generator in cents
! | tetrachord
! | L in cents
! | s in cents
! | L to s ratio
! | comments
|-
|-
| || || || || || 16\37 || 518.919 || 6 || 5 || 1.200 ||  
| style="text-align:center;" | 3\7
| 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
|
|-
|-
| || || || 10\23 || || || 521.739 || 4 || 3 || 1.333 ||
| style="text-align:center;" |  
| 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...
|-
|-
| || || || || || 27\62 || 522.581 || 11 || 8 || 1.375 ||
| style="text-align:center;" |  
| 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;" |  
|-
|-
| || || || || 17\39 || || 523.077 || 7 || 5 || 1.400 ||
| style="text-align:center;" |  
| 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;" |  
|-
|-
| || || || || || 24\55 || 523.636 || 10 || 7 || 1.428 ||
| style="text-align:center;" |  
| 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>
| |
|-
|-
| || || || || || 25\57 || 526.316 || 11 || 7 || 1.571 ||
| style="text-align:center;" |  
| 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;" |  
|-
|-
| || || || 11\25 || || || 528.000 || 5 || 3 || 1.667 ||
| style="text-align:center;" |  
| 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>
| |  
|-
|-
| || || || || 15\34 || || 529.412 || 7 || 4 || 1.750 ||
| style="text-align:center;" |  
| 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
|-
|-
| || || || || || 19\43 || 530.233 || 9 || 5 || 1.800 ||
| style="text-align:center;" |  
| 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)
|-
|-
| || 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;" |  
| 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;" |  
|-
|-
| || || || || || 17\38 || 536.842 || 9 || 4 || 2.250 ||
| style="text-align:center;" |  
| 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>
| |  
|-
|-
| || || || || || 22\49 || 538.776 || 12 || 5 || 2.400 ||
| style="text-align:center;" |  
| 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
|-
|-
| || || || || || 23\51 || 541.176 || 13 || 5 || 2.600 || Unnamed golden tuning
| style="text-align:center;" |  
| 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>
|-
|-
| || || || || || 19\42 || 542.857 || 11 || 4 || 2.750 ||
| style="text-align:center;" |  
| 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;" |  
|-
|-
| || || 5\11 || || || || 545.455 || 3 || 1 || 3.000 || L/s = 3/1
| style="text-align:center;" |  
|-
| style="text-align:center;" |  
| || || || || || 16\35 || 548.571 || 10 || 3 || 3.333 ||
| style="text-align:center;" |  
|-
| style="text-align:center;" | 6\13
| || || || || 11\24 || || 550.000 || 7 || 2 || 3.500 ||
| style="text-align:center;" |  
|-
| |  
| || || || || || 17\37 || 551.351 || 11 || 3 || 3.667 ||
| |  
| 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
|-
|-
| || || || 6\13 || || || 553.846 || 4 || 1 || 4.000 ||
| style="text-align:center;" |  
| 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
|-
|-
| || || || || || 13\28 || 557.143 || 9 || 2 || 4.500 ||
| style="text-align:center;" |  
| 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;" |  
|-
|-
| || || || || 7\15 || || 560.000 || 5 || 1 || 5.000 ||  
| style="text-align:center;" |  
| 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
|-
|-
| || || || || || 8\17 || 564.706 || 6 || 1 || 6.000 || Liese↓, triton↓
| style="text-align:center;" | 1\2
|-
| style="text-align:center;" |  
| 1\2 || || || || || || 600.000 || 1 || 0 || → inf ||
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| |  
| |  
| style="text-align:center;" | 600.0
| style="text-align:center;" | 0 0 1
| style="text-align:center;" | 600.0
| style="text-align:center;" | 0
| style="text-align:center;" | —
| style="text-align:center;" |  
|}
|}
== 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: <soundcloud>https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-9-edo</soundcloud>
16-EDO: <soundcloud>https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-16-edo</soundcloud>
23-EDO: <soundcloud>https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments</soundcloud>
25-EDO: <soundcloud>https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-25-edo</soundcloud>
[[Category:Scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]

Revision as of 05:01, 14 April 2021

2L 5s refers to a Moment of Symmetry scale with two large steps and five small steps. Common names for such a tuning are mavila and anti-diatonic. The generator is a sharp fourth (or flat fifth), falling between 3\7 (3 degrees of 7edo) and 1\2 (1 degree of 2edo — the half-octave tritone that appears in every even-numbered edo).

The word "mavila" is used in different ways by different folks. Not every user of the word would consider every 2L 5s scale an instance of "mavila." In particular, between 13\29 and 14\31, and centered on 9\20, is the albitonic scale for the 2.7.11.13 subgroup temperament score, which is not intended to be treated as having any kind of fifth, flat or otherwise.

In terms of harmonic entropy, the most significant minimum is at Liese/Triton, in which the generator is about 7/5 and three of them make a 3/1.

generator in degrees of an edo generator in cents tetrachord L in cents s in cents L to s ratio comments
3\7 514.3 1 1 1 171.4 171.4 1.00
19\44 518.2 6 6 7 190.9 163.6 1.17
16\37 518.9 5 5 6 194.6 162.2 1.20
13\30 520.0 4 4 5 200.0 160.0 1.25 Mavila extends from here...
10\23 521.7 3 3 4 208.7 156.5 1.33
17\39 523.1 5 5 7 215.4 153.8 1.40
7\16 525.0 2 2 3 225.0 150.0 1.50 Mavila in Armodue

Optimum rank range (L/s=3/2)

526.3 2 2 pi 231.5 147.4 pi/2
18\41 526.8 5 5 8 234.1 146.3 1.60
1200*5/(13-phi) 1 1 phi 235.7 145.7 phi Golden mavila
29\66 527.3 8 8 13 236.4 145.5 1.625
11\25 528.0 3 3 5 240.0 144.0 1.67
529.1 1 1 √3 245.6 141.8 √3
15\34 529.4 4 4 7 247.1 141.2 1.75 ...to somewhere around here
4\9 533.3 1 1 2 266.7 133.3 2.00 Boundary of propriety (generators

smaller than this are proper)

13\29 537.9 3 3 7 289.7 124.1 2.33
9\20 540.0 2 2 5 300.0 120.0 2.50
541.4 1 1 phi+1 306.9 117.2 1 1 phi+1
14\31 541.9 3 3 8 309.7 116.1 2.66
542.5 1 1 e 321.55 115.0 e L/s = e
5\11 545.5 1 1 3 327.3 109.1 3.00 L/s = 3
546.8 1 1 pi 334.1 106.35 pi L/s = pi
11\24 550.0 2 2 7 350.0 100.0 3.50
6\13 553.8 1 1 4 369.2 92.3 4.00 Thuja is optimal around here

L/s = 4

7\15 560.0 1 1 5 400.0 80.0 5.00 ie. (11/8)^5 = 5/1
8\17 564.7 1 1 6 423.5 70.6 6.00
9\19 568.4 1 1 7 442.1 63.2 7.00 Liese/Triton is around here
1\2 600.0 0 0 1 600.0 0 —
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