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2L 5s: Difference between revisions

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{{Infobox MOS
{{Infobox MOS}}
| Name = antidiatonic
{{MOS intro}}
| Periods = 1
==Name==
| nLargeSteps = 2
[[TAMNAMS]] suggests the temperament-agnostic name '''antidiatonic''' for this scale, adopted from the common use of the term to refer to diatonic ([[5L 2s]]) but with the large and small steps switched.
| nSmallSteps = 5
==Notation==
| Equalized = 3
:''This article assumes [[TAMNAMS]] for naming step ratios, intervals, and scale degrees, and [[Diamond-mos notation|diamond-MOS notation]] for note names.''<!-- May be worth adding other notation schemes, such as that where standard notation is read is antidiatonic. -->
| Collapsed = 1
===Intervals and degrees===
| Pattern = LssLsss
Names for this scale's intervals (mossteps) and scale degrees (mosdegrees) are based on the number of large and small steps from the root, starting at 0 (0-mosstep and 0-mosdegree) for the unison, per TAMNAMS. Ordinal names, such as mos-1st for the unison, are discouraged for non-diatonic MOS scales.
| Neutral = 4L 3s
}}{{MOS intro}}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|Liese]]/Triton, in which the generator is about 7/5 and three of them make a 3/1.
Being a moment-of-symmetry scale, every [[interval class]] of 2L 5s, except for the unison and octave, has two [[Interval variety|varieties]] – large and small – whose [[Interval quality|relative qualities]] are denoted as major or minor, or augmented, perfect, and diminished for the generators.
{| class="wikitable"
|+
2L 5s interval varieties
! rowspan="2" |Interval class
! colspan="2" |Large variety
! colspan="2" |Small variety
|-
! Size
! Quality
!Size
!Quality
|-
|'''0-mosstep (unison)'''
|0
|Perfect
|0
|Perfect
|-
| 1-mosstep
| L
|Major
|s
|Minor
|-
| 2-mosstep
|L + s
|Major
|2s
|Minor
|-
|3-mosstep
|L + 2s
|Perfect
|3s
|Diminished
|-
|4-mosstep
| 2L + 2s
| Augmented
| L + 3s
|Perfect
|-
|5-mosstep
|2L + 3s
|Major
|L + 4s
|Minor
|-
|6-mosstep
|2L + 4s
| Major
|L + 5s
|Minor
|-
|'''7-mosstep (octave)'''
|2L + 5s
|Perfect
|2L + 5s
|Perfect
|}
===Note names ===
For this article, note names are based on diamond-MOS notation, where the naturals JKLMNOP are applied to the step pattern sLsssLs and the accidentals & (pronounced "am" or "amp") and @ (pronounced "at") are used to represent sharps and flats respectively. Thus, the basic gamut for 2L 5s is the following:
 
{{MOS gamut}}
==Theory==
Antidiatonic is similar to diatonic (5L 2s) except interval classes are flipped. For example, there are natural, harmonic, and melodic major scales instead of minor scales, and its locrian scale, called "antilocrian", has an augmented "fifth" instead of a diminished fifth. The flatter the "fifth" (or 4-mosstep) gets, the less the scale resembles diatonic.
 
Certain temperaments associated with this MOS, such as [[score]], do not have intervals that resemble a diatonic 5th, flattened or otherwise.


The antidiatonic scale is similar to the normal diatonic scale except interval classes are flipped. For example, there are natural, harmonic, and melodic major scales instead of minor scales, and the antilocrian scale has an augmented fifth instead of a diminished fifth. The flatter the "fifth" gets, the less the scale resembles the diatonic scale, with mavila scales generated by flat fifths appearing like a flipped diatonic, but casablanca and triton scales generated by subfifths or even tritones much less so.
===Low harmonic entropy scales===
== Names ==
There is one notable harmonic entropy minimum: [[Liese]]/triton, in which the generator is [[7/5]] (582.5¢) and three of them make a [[3/1]] (1902¢).
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 ==
===Temperament interpretations===
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]]).
2L 5s has several rank-2 temperament interpretations, such as:


== Intervals ==
*[[Chromatic pairs#Score|Score]], with generators around 540.1¢.
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.
*[[Liese]], with generators around 632.4¢.
*[[Mavila]], with generators around 679.8¢.


== Modes ==
==Modes==
Modes of the antidiatonic scale are usually named as "anti-" combined with the opposite mode of the diatonic scale, e.g. 4|2 being called "antiaeolian". [[User:CompactStar|CompactStar]] also gave original names based on regions of France to mirror how modes of the diatonic scale are named on regions of Greece and Turkey.
Modes of the antidiatonic scale are usually named as "anti-" combined with the opposite mode of the diatonic scale, e.g. 4|2 being called "antiaeolian". [[User:CompactStar|CompactStar]] also gave original names based on regions of France to mirror how modes of the diatonic scale are named on regions of Greece and Turkey.


{{MOS modes|Mode Names=Anti-locrian; Anti-phrygian; Anti-aeolian; Anti-dorian; Anti-mixolydian; Anti-ionian; Anti-lydian|Table Headers=CompactStar's names|Table Entries=Corsican; Breton; Burgundian; Picardian; Norman; Provencal; Alsatian}}
{{MOS modes|Mode Names=Anti-locrian; Anti-phrygian; Anti-aeolian; Anti-dorian; Anti-mixolydian; Anti-ionian; Anti-lydian|Table Headers=CompactStar's names|Table Entries=Corsican; Breton; Burgundian; Picardian; Norman; Provencal; Alsatian}}


== Scale tree ==
==Scale tree==
Generator ranges:  
Generator ranges:  
* Chroma-positive generator: 514.2857 cents (3\7) to 600 cents (1\2)
* 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)
*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
! Cents
!Cents
! L
!L
! s
!s
! L/s
!L/s
! Comments
!Comments
|-
|-
| 3\7 || || || || || || 514.286 || 1 || 1 || 1.000 ||  
|3\7|| || || || || || 514.286||1||1||1.000||
|-
|-
| || || || || || 16\37 || 518.919 || 6 || 5 || 1.200 || [[Gravity]]↑
| || || || || ||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||
|-
|-
| || || || || || 23\53 || 520.755 || 9 || 7 || 1.286 ||  
| || || || || ||23\53 ||520.755||9||7||1.286||
|-
|-
| || || || 10\23 || || || 521.739 || 4 || 3 || 1.333 ||
| || || ||10\23|| || ||521.739||4||3||1.333 ||
|-
|-
| || || || || || 27\62 || 522.581 || 11 || 8 || 1.375 ||
| || || || || ||27\62 ||522.581||11||8||1.375||
|-
|-
| || || || || 17\39 || || 523.077 || 7 || 5 || 1.400 ||
| || || || ||17\39|| ||523.077 ||7 ||5||1.400 ||
|-
|-
| || || || || || 24\55 || 523.636 || 10 || 7 || 1.428 ||
| || || || || ||24\55 || 523.636||10 || 7||1.428||
|-
|-
| || || 7\16 || || || || 525.000 || 3 || 2 || 1.500 || [[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||
|-
|-
| || || || || 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 (527.1497¢)
| || || || || || 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 ||
|-
|-
| || || || || || 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 || [[Mabila]] / [[Amavil]]
| || || || || ||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||
|-
|-
| || || || || 13\29 || || 537.931 || 7 || 3 || 2.333 ||  
| || || || ||13\29|| ||537.931 ||7||3||2.333||
|-
|-
| || || || || || 22\49 || 538.776 || 12 || 5 || 2.400 ||
| || || || || ||22\49||538.776||12 || 5||2.400||
|-
|-
| || || || 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 (541.3837¢)
| || || || || ||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 ||  
| || ||5\11|| || || ||545.455||3||1||3.000||
|-
|-
| || || || || || 16\35 || 548.571 || 10 || 3 || 3.333 ||
| || || || || || 16\35||548.571 ||10||3||3.333||
|-
|-
| || || || || 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||
|-
|-
| || || || || || 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||
|}
|}


[[Category:7-tone scales]]
[[Category:7-tone scales]]
[[Category:Antidiatonic| ]] <!-- main article -->
[[Category:Antidiatonic| ]] <!-- main article -->
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