2L 5s: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Ganaram inukshuk (talk | contribs)
autopatrolled, emailconfirmed
4,810 edits
→Scale tree: Replaced scale tree with template, added soft-of-basic tunings, added to-do
Ganaram inukshuk (talk | contribs)
autopatrolled, emailconfirmed
4,810 edits
Changed note names back to CDEFGAB and b/#, as recommended by CompactStar; since antidiatonic and diatonic intervals line up so closely, this article now uses ordinal notation instead
Tags: Visual edit Mobile edit Mobile web edit Advanced mobile edit
Line 5: Line 5:
[[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.
[[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.
==Notation==
==Notation==
:''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. -->
:''This article assumes [[TAMNAMS]] for naming step ratios, with TAMNAMS-prefixed ordinal notation used for intervals and scale degrees.''
===Intervals and degrees===
===Intervals and degrees===
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.
Since diatonic and antidiatonic interval classes are so similar to one another, names for this scale's intervals are the same as that as diatonic intervals, prefixed with pel- to distinguish them from diatonic intervals.


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.
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.
{{MOS intervals}}
{{MOS intervals|Interval Notation=ordinals|MOS Prefix=pel}}
===Note names ===
===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:
The most common way of notating this scale is to use the same note names (CDEFGAB) and accidentals (# and b) as that of diatonic, but read as antidiatonic instead. There are, however, two ways of notating accidentals:


{{MOS gamut}}
* Harmonic notation, where the sharps and flats of diatonic switch roles (sharp flattens and flats sharpen). This article will use this interpretation of sharps and flats.
* Melodic notation, where the meaning of sharps and flats are preserved (sharps sharpen and flats flatten).
 
In any case, the naturals DEFGABC are applied to the step pattern sLsssLs, the antidorian mode on D. Thus, the basic gamut for 2L 5s is the following:
 
{{MOS gamut|Notation=DEFGABC; b; #}}
==Theory==
==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. Additionally, there are temperaments associated with this MOS, such as [[score]], that do not have intervals that resemble a diatonic 5th, flattened or otherwise.
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 gets, the less the scale resembles diatonic. Additionally, there are temperaments associated with this MOS, such as [[score]], that do not have intervals that resemble a diatonic 5th, flattened or otherwise.


===Low harmonic entropy scales===
===Low harmonic entropy scales===
Line 32: Line 37:
=== Simple tunings ===
=== Simple tunings ===
The simplest tunings are those with step ratios 2:1, 3:1, and 3:2, producing [[9edo]], [[11edo]], and [[16edo]].
The simplest tunings are those with step ratios 2:1, 3:1, and 3:2, producing [[9edo]], [[11edo]], and [[16edo]].
{{MOS degrees|Step Ratio=2/1; 3/1; 3/2|Genchain Extend=7}}
{{MOS degrees|Step Ratio=2/1; 3/1; 3/2|Genchain Extend=7|Notation=DEFGABC; b; #|MOS Prefix=pel|Degree Notation=ordinal}}
=== Soft-of-basic tunings ===
=== Soft-of-basic tunings ===
{{Main|Mavila}}
{{Main|Mavila}}
Much of the range for soft-of-basic antidiatonic tunings (1:1 to 2:1) corresponds to mavila temperament. Edos include 9edo (not shown), [[16edo]], and [[23edo]].
Much of the range for soft-of-basic antidiatonic tunings (1:1 to 2:1) corresponds to mavila temperament. Edos include 9edo (not shown), [[16edo]], and [[23edo]].
{{MOS degrees|Step Ratio=3/2; 4/3|Genchain Extend=3}}
{{MOS degrees|Step Ratio=3/2; 4/3|Genchain Extend=3|Notation=DEFGABC; b; #|MOS Prefix=pel|Degree Notation=ordinal}}
==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.

Revision as of 00:59, 4 July 2023

Todo: expand

Add hard-of-basic tunings, include JI ratio approximations

↖ 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, named antidiatonic in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 2 large steps and 5 small steps, repeating every octave. Generators that produce this scale range from 514.3 ¢ to 600 ¢, or from 600 ¢ to 685.7 ¢.

Name

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.

Notation

This article assumes TAMNAMS for naming step ratios, with TAMNAMS-prefixed ordinal notation used for intervals and scale degrees.

Intervals and degrees

Since diatonic and antidiatonic interval classes are so similar to one another, names for this scale's intervals are the same as that as diatonic intervals, prefixed with pel- to distinguish them from diatonic intervals.

Being a moment-of-symmetry scale, every interval class of 2L 5s, except for the unison and octave, has two varieties – large and small – whose relative qualities are denoted as major or minor, or augmented, perfect, and diminished for the generators.

Intervals of 2L 5s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-pelstep Perfect 0-pelstep P0pels 0 0.0 ¢
1-pelstep Minor 1-pelstep m1pels s 0.0 ¢ to 171.4 ¢
Major 1-pelstep M1pels L 171.4 ¢ to 600.0 ¢
2-pelstep Minor 2-pelstep m2pels 2s 0.0 ¢ to 342.9 ¢
Major 2-pelstep M2pels L + s 342.9 ¢ to 600.0 ¢
3-pelstep Diminished 3-pelstep d3pels 3s 0.0 ¢ to 514.3 ¢
Perfect 3-pelstep P3pels L + 2s 514.3 ¢ to 600.0 ¢
4-pelstep Perfect 4-pelstep P4pels L + 3s 600.0 ¢ to 685.7 ¢
Augmented 4-pelstep A4pels 2L + 2s 685.7 ¢ to 1200.0 ¢
5-pelstep Minor 5-pelstep m5pels L + 4s 600.0 ¢ to 857.1 ¢
Major 5-pelstep M5pels 2L + 3s 857.1 ¢ to 1200.0 ¢
6-pelstep Minor 6-pelstep m6pels L + 5s 600.0 ¢ to 1028.6 ¢
Major 6-pelstep M6pels 2L + 4s 1028.6 ¢ to 1200.0 ¢
7-pelstep Perfect 7-pelstep P7pels 2L + 5s 1200.0 ¢

Note names

The most common way of notating this scale is to use the same note names (CDEFGAB) and accidentals (# and b) as that of diatonic, but read as antidiatonic instead. There are, however, two ways of notating accidentals:

  • Harmonic notation, where the sharps and flats of diatonic switch roles (sharp flattens and flats sharpen). This article will use this interpretation of sharps and flats.
  • Melodic notation, where the meaning of sharps and flats are preserved (sharps sharpen and flats flatten).

In any case, the naturals DEFGABC are applied to the step pattern sLsssLs, the antidorian mode on D. Thus, the basic gamut for 2L 5s is the following:

D, E, Eb/F#, F, G, A, B, Bb/C#, C, D

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 gets, the less the scale resembles diatonic. Additionally, there are temperaments associated with this MOS, such as score, that do not have intervals that resemble a diatonic 5th, flattened or otherwise.

Low harmonic entropy scales

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¢).

Temperament interpretations

2L 5s has several rank-2 temperament interpretations, such as:

  • Mavila, with generators around 679.8¢.
  • Casablanca, with generators around 657.8¢.
  • Liese, with generators around 632.4¢.

Tuning ranges

Simple tunings

The simplest tunings are those with step ratios 2:1, 3:1, and 3:2, producing 9edo, 11edo, and 16edo.

MOS degrees is deprecated. Please use Template:MOS tunings instead.
Scale degree of 2L 5s
Scale degree On D 9edo (Basic, L:s = 2:1) 11edo (Hard, L:s = 3:1) 16edo (Soft, L:s = 3:2) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents
Perfect 0-peldegree (unison) D 0 0 0 0 0 0 1/1 (exact)
Minor 1-peldegree E 1 133.3 1 109.1 2 150
Major 1-peldegree Eb 2 266.7 3 327.3 3 225
Minor 2-peldegree F# 2 266.7 2 218.2 4 300
Major 2-peldegree F 3 400 4 436.4 5 375
Diminished 3-peldegree G# 3 400 3 327.3 6 450
Perfect 3-peldegree G 4 533.3 5 545.5 7 525
Perfect 4-peldegree A 5 666.7 6 654.5 9 675
Augmented 4-peldegree Ab 6 800 8 872.7 10 750
Minor 5-peldegree B 6 800 7 763.6 11 825
Major 5-peldegree Bb 7 933.3 9 981.8 12 900
Minor 6-peldegree C# 7 933.3 8 872.7 13 975
Major 6-peldegree C 8 1066.7 10 1090.9 14 1050
Perfect 7-peldegree (octave) D 9 1200 11 1200 16 1200 2/1 (exact)

Soft-of-basic tunings

Main article: Mavila

Much of the range for soft-of-basic antidiatonic tunings (1:1 to 2:1) corresponds to mavila temperament. Edos include 9edo (not shown), 16edo, and 23edo.

MOS degrees is deprecated. Please use Template:MOS tunings instead.
Scale degree of 2L 5s
Scale degree On D 16edo (Soft, L:s = 3:2) 23edo (Supersoft, L:s = 4:3) Approx. JI Ratios
Steps Cents Steps Cents
Perfect 0-peldegree (unison) D 0 0 0 0 1/1 (exact)
Minor 1-peldegree E 2 150 3 156.5
Major 1-peldegree Eb 3 225 4 208.7
Minor 2-peldegree F# 4 300 6 313
Major 2-peldegree F 5 375 7 365.2
Diminished 3-peldegree G# 6 450 9 469.6
Perfect 3-peldegree G 7 525 10 521.7
Perfect 4-peldegree A 9 675 13 678.3
Augmented 4-peldegree Ab 10 750 14 730.4
Minor 5-peldegree B 11 825 16 834.8
Major 5-peldegree Bb 12 900 17 887
Minor 6-peldegree C# 13 975 19 991.3
Major 6-peldegree C 14 1050 20 1043.5
Perfect 7-peldegree (octave) D 16 1200 23 1200 2/1 (exact)

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". 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 2L 5s
UDP Cyclic
order		 Step
pattern
6|0 1 LssLsss
5|1 4 LsssLss
4|2 7 sLssLss
3|3 3 sLsssLs
2|4 6 ssLssLs
1|5 2 ssLsssL
0|6 5 sssLssL

Scale tree

Template:Scale tree

Retrieved from "https://en.xen.wiki/index.php?title=2L_5s&oldid=118300"