2L 5s: Difference between revisions

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Line 6: Line 6:
| Equalized = 3
| Equalized = 3
| Paucitonic = 1
| Paucitonic = 1
| Pattern = sLsssLs
| Pattern = LssLsss
| Neutral = 4L 3s
| Neutral = 4L 3s
}}
}}


'''2L 5s''' 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).
'''2L 5s''' 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 temperament]] and [[7L 2s]], 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 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 temperament]] and [[7L 2s]], 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 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.)


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


== Names ==
== Names ==
It is often called '''antidiatonic''', because it has the step pattern of diatonic ([[5L 2s]]), but with the large and small steps switched. [[TAMNAMS]] uses this name as well.
It is often called '''antidiatonic''', because it has the step pattern of diatonic ([[5L 2s]]), but with the large and small steps switched. [[TAMNAMS]] uses this name as well.
== Notation ==
== Notation ==
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]]).
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]]).
== Intervals ==
== Intervals ==
Note: In TAMNAMS, a k-step interval class in antidiatonic may be called a "k-step", "k-mosstep", or "k-pelstep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.
Note: In TAMNAMS, a k-step interval class in antidiatonic may be called a "k-step", "k-mosstep", or "k-pelstep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.
== Modes ==
* 6|0 LssLsss (anti-locrian)
* 5|1 LsssLss (anti-phrygian)
* 4|2 sLssLss (anti-aeolian)
* 3|3 sLsssLs (anti-dorian)
* 2|4 ssLssLs (anti-mixolydian)
* 1|5 ssLsssL (anti-ionian)
* 0|6 sssLssL (anti-lydian)


== Scale tree ==
== Scale tree ==
Generator ranges:
* Chroma-positive generator: 514.2857 cents (3\7) to 600 cents (1\2)
* Chroma-negative generator: 600 cents (1\2) to 685.7143 cents (4\7)
{| class="wikitable center-all"
{| class="wikitable center-all"
! colspan="6" | Generator
! colspan="6" | Generator
Line 34: Line 49:
| 3\7 || || || || || || 514.286 || 1 || 1 || 1.000 ||  
| 3\7 || || || || || || 514.286 || 1 || 1 || 1.000 ||  
|-
|-
| || || || || || 16\37 || 518.919 || 6 || 5 || 1.200 ||  
| || || || || || 16\37 || 518.919 || 6 || 5 || 1.200 || [[Gravity]]↑
|-
|-
| || || || || 13\30 || || 520.000 || 5 || 4 || 1.250 ||  
| || || || || 13\30 || || 520.000 || 5 || 4 || 1.250 ||  
Line 48: Line 63:
| || || || || || 24\55 || 523.636 || 10 || 7 || 1.428 ||
| || || || || || 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
| || || 7\16 || || || || 525.000 || 3 || 2 || 1.500 || [[Mavila]] is in this region
|-
|-
| || || || || || 25\57 || 526.316 || 11 || 7 || 1.571 ||
| || || || || || 25\57 || 526.316 || 11 || 7 || 1.571 ||
Line 54: Line 69:
| || || || || 18\41 || || 526.829 || 8 || 5 || 1.600 ||  
| || || || || 18\41 || || 526.829 || 8 || 5 || 1.600 ||  
|-
|-
| || || || || || 29\66 || 527.273 || 13 || 8 || 1.625 || Golden mavila
| || || || || || 29\66 || 527.273 || 13 || 8 || 1.625 || Golden mavila (527.1497¢)
|-
|-
| || || || 11\25 || || || 528.000 || 5 || 3 || 1.667 ||
| || || || 11\25 || || || 528.000 || 5 || 3 || 1.667 ||
Line 60: Line 75:
| || || || || || 26\59 || 528.814 || 12 || 7 || 1.714 ||  
| || || || || || 26\59 || 528.814 || 12 || 7 || 1.714 ||  
|-
|-
| || || || || 15\34 || || 529.412 || 7 || 4 || 1.750 ||
| || || || || 15\34 || || 529.412 || 7 || 4 || 1.750 ||  
|-
|-
| || || || || || 19\43 || 530.233 || 9 || 5 || 1.800 ||
| || || || || || 19\43 || 530.233 || 9 || 5 || 1.800 || [[Mabila]] / [[Amavil]]
|-
|-
| || 4\9 || || || || || 533.333 || 2 || 1 || 2.000 || Basic antidiatonic<br>(Generators smaller than this are proper)
| || 4\9 || || || || || 533.333 || 2 || 1 || 2.000 || Basic antidiatonic <br>(Generators smaller than this are proper)
|-
|-
| || || || || || 17\38 || 536.842 || 9 || 4 || 2.250 ||
| || || || || || 17\38 || 536.842 || 9 || 4 || 2.250 ||
Line 74: Line 89:
| || || || 9\20 || || || 540.000 || 5 || 2 || 2.500 || Score
| || || || 9\20 || || || 540.000 || 5 || 2 || 2.500 || Score
|-
|-
| || || || || || 23\51 || 541.176 || 13 || 5 || 2.600 || Unnamed golden tuning
| || || || || || 23\51 || 541.176 || 13 || 5 || 2.600 || Unnamed golden tuning (541.3837¢)
|-
|-
| || || || || 14\31 || || 541.935 || 8 || 3 || 2.667 || Casablanca is around here
| || || || || 14\31 || || 541.935 || 8 || 3 || 2.667 || [[Casablanca]] is around here
|-
|-
| || || || || || 19\42 || 542.857 || 11 || 4 || 2.750 ||
| || || || || || 19\42 || 542.857 || 11 || 4 || 2.750 ||
|-
|-
| || || 5\11 || || || || 545.455 || 3 || 1 || 3.000 || L/s = 3/1
| || || 5\11 || || || || 545.455 || 3 || 1 || 3.000 ||  
|-
|-
| || || || || || 16\35 || 548.571 || 10 || 3 || 3.333 ||
| || || || || || 16\35 || 548.571 || 10 || 3 || 3.333 ||
Line 86: Line 101:
| || || || || 11\24 || || 550.000 || 7 || 2 || 3.500 ||
| || || || || 11\24 || || 550.000 || 7 || 2 || 3.500 ||
|-
|-
| || || || || || 17\37 || 551.351 || 11 || 3 || 3.667 || Freivald, emka
| || || || || || 17\37 || 551.351 || 11 || 3 || 3.667 || [[Freivald]] / [[emka]]
|-
|-
| || || || 6\13 || || || 553.846 || 4 || 1 || 4.000 ||
| || || || 6\13 || || || 553.846 || 4 || 1 || 4.000 ||
Line 92: Line 107:
| || || || || || 13\28 || 557.143 || 9 || 2 || 4.500 ||
| || || || || || 13\28 || 557.143 || 9 || 2 || 4.500 ||
|-
|-
| || || || || 7\15 || || 560.000 || 5 || 1 || 5.000 || Thuja is around here
| || || || || 7\15 || || 560.000 || 5 || 1 || 5.000 || [[Thuja]] is around here
|-
|-
| || || || || || 8\17 || 564.706 || 6 || 1 || 6.000 || Liese↓, triton↓
| || || || || || 8\17 || 564.706 || 6 || 1 || 6.000 || [[Liese]]↓, [[triton]]↓
|-
|-
| 1\2 || || || || || || 600.000 || 1 || 0 || → inf ||
| 1\2 || || || || || || 600.000 || 1 || 0 || → inf ||

Revision as of 13:42, 13 February 2022

↖ 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 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 temperament and 7L 2s, 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 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.

Names

It is often called antidiatonic, because it has the step pattern of diatonic (5L 2s), but with the large and small steps switched. TAMNAMS uses this name as well.

Notation

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).

Intervals

Note: In TAMNAMS, a k-step interval class in antidiatonic may be called a "k-step", "k-mosstep", or "k-pelstep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.

Modes

  • 6|0 LssLsss (anti-locrian)
  • 5|1 LsssLss (anti-phrygian)
  • 4|2 sLssLss (anti-aeolian)
  • 3|3 sLsssLs (anti-dorian)
  • 2|4 ssLssLs (anti-mixolydian)
  • 1|5 ssLsssL (anti-ionian)
  • 0|6 sssLssL (anti-lydian)

Scale tree

Generator ranges:

  • Chroma-positive generator: 514.2857 cents (3\7) to 600 cents (1\2)
  • Chroma-negative generator: 600 cents (1\2) to 685.7143 cents (4\7)
Generator Cents L s L/s Comments
3\7 514.286 1 1 1.000
16\37 518.919 6 5 1.200 Gravity↑
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 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 (527.1497¢)
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 Mabila / Amavil
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 Score
23\51 541.176 13 5 2.600 Unnamed golden tuning (541.3837¢)
14\31 541.935 8 3 2.667 Casablanca is around here
19\42 542.857 11 4 2.750
5\11 545.455 3 1 3.000
16\35 548.571 10 3 3.333
11\24 550.000 7 2 3.500
17\37 551.351 11 3 3.667 Freivald / emka
6\13 553.846 4 1 4.000
13\28 557.143 9 2 4.500
7\15 560.000 5 1 5.000 Thuja is around here
8\17 564.706 6 1 6.000 Liese↓, triton↓
1\2 600.000 1 0 → inf
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