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The step ratios shown above form a continuum of step ratios. The section Tuning spectrum shows how this is made, as well as a larger spectrum.
The step ratios shown above form a continuum of step ratios. The section Tuning spectrum shows how this is made, as well as a larger spectrum.


==Temperament interpretations==
===Rank-2 temperament interpretations===


: ''Main article: [[5L 2s/Temperaments]]''
: ''Main article: [[5L 2s/Temperaments]]''
Line 144: Line 144:
5L 2s has several rank-2 temperament interpretations, such as:
5L 2s has several rank-2 temperament interpretations, such as:


* [[Meantone]], with generators around 696.2¢. These temperaments flatten the perfect 5th (702¢) to produce 5/4 (386¢) for a major 3rd.
* [[Meantone]], with generators around 696.2¢. This includes:
** [[Flattone]], with generators around 693.7¢. These temperaments have major 3rds that are flatter than 5/4.
** [[Flattone]], with generators around 693.7¢.
*[[Schismic]], with generators around 702¢. These temperaments have perfect 5ths that are close to just, producing 81/64 (407¢) for a major 3rd.
*[[Schismic]], with generators around 702¢.
*[[Parapyth]], with generators around 704.7¢. These temperaments have major 3rds that are sharper than 81/64.
*[[Parapyth]], with generators around 704.7¢.
*[[Archy]], with generators around 709.3¢. These temperaments have perfect 5ths that are significantly sharp.
*[[Archy]], with generators around 709.3¢. This includes:
**Supra, with generators around 707.2¢
**Supra, with generators around 707.2¢
**Superpyth, with generators around 710.3¢
**Superpyth, with generators around 710.3¢
Line 154: Line 154:


== Step ratio ranges==
== Step ratio ranges==
This section describes step ratio ranges for 5L 2s and what rank-2 temperaments they correspond to. Although the two may have closely correspond with one another, the purpose of step ratios is to describe 5L 2s without necessarily using [[regular temperament theory]].
=== Simple step ratios ===
=== Simple step ratios ===
17edo and 19edo, produced using step ratios of 3:1 and 3:2 respectively, are the smallest edos that offer a greater variety of pitches than 12edo. Note that any enharmonic equivalences that 12edo has no longer hold for either 17edo or 19edo, as shown in the table below.{{MOS degrees|Scale Signature=5L 2s|Step Ratio=2/1; 3/1; 3/2|Genchain Extend=7}}
17edo and 19edo, produced using step ratios of 3:1 and 3:2 respectively, are the smallest edos that offer a greater variety of pitches than 12edo. Note that any enharmonic equivalences that 12edo has no longer hold for either 17edo or 19edo, as shown in the table below.{{MOS degrees|Scale Signature=5L 2s|Step Ratio=2/1; 3/1; 3/2|Genchain Extend=7}}
===Soft step ratios===
=== Parasoft step ratios ===
:''Main article: [[Meantone]] and [[Flattone]]''
:''Main article: [[Flattone]]''
Most of the soft step ratio range (1:1 to 2:1) correspond to meantone temperaments. More specifically, the hyposoft range (3:2 to 2:1) corresponds to meantone and the parasoft range (4:3 and 3:2) corresponds to flattone.
Parasoft step ratios (4:3 and 3:2) correspond to flattone temperaments, characterized by flattened perfect 5ths (flat of 702¢) to produce major 3rds that are flatter than 5/4 (386¢).{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/2; 4/3; 7/5; 10/7|Genchain Extend=0, 5}}
===Hyposoft step ratios===
:''Main article: [[Meantone]]''
Hyposoft step ratios (3:2 to 2:1) correspond to meantone temperaments, characterized by flattened perfect 5ths (flat of 702¢) to produce diatonic major 3rds that approximate 5/4 (386¢).{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/2; 5/3; 7/4; 8/5|Genchain Extend=0, 5}}
===Hypohard step ratios ===
:''Main article: [[Pythagorean tuning]] and [[Schismatic family#Schismatic aka Helmholtz|schismatic temperament]]''
The range of hypohard step ratios can be divided into a minihard range (between 2:1 to 5:2) and quasihard range (between 5:2 to 3:1).
==== Minihard step ratios ====
Minihard step ratios correspond to Pythagorean tuning and schismatic temperament, characterized by having a perfect 5th that is as close to just (701.96¢) as possible, resulting in a major 3rd of 81/64 (407¢).{{MOS degrees|Scale Signature=5L 2s|Step Ratio=7/3; 9/4|Genchain Extend=0, 5}}
 
==== Quasihard step ratios ====
Quasihard step ratios correspond to "neogothic" or "parapyth" systems whose perfect 5th is sharper than just, resulting in major 3rds that are sharper than 81/64.


==== Parasoft step ratios ====
17edo is considered to be on the sharper end of the neogothic spectrum, with a major 3rd that is more discordant than flatter neogothic tunings.{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/1; 5/2; 8/3|Genchain Extend=0, 5}}
{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/2; 4/3; 7/5; 10/7|Genchain Extend=0, 5}}
===Parahard and ultrahard step ratios===
====Hyposoft step ratios====
:''Main article: [[Archy]]''
{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/2; 5/3; 7/4; 8/5|Genchain Extend=0, 5}}
The parahard and ultrahard ranges (3:1 to 1:1) correspond to archy systems, with perfect 5ths that are significantly sharper than than 702¢.{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/1; 4/1; 5/1; 6/1|Genchain Extend=0, 5}}
===Hard step ratios ===
:''Main article: [[Pythagorean tuning]], [[Schismatic family#Schismatic aka Helmholtz|schismatic temperament]], and [[Archy]]''
The range of hard step ratios can be divided into a hypohard range (2:1 to 3:1), parahard range (3:1 to 4:1), and ultrahard range (4:1 to 1:0).
==== Hypohard step ratios ====
The flatter end of the hypohard region corresponds to Pythagorean tuning and schismic temperament, with perfect 5ths that are close to 702¢.{{MOS degrees|Scale Signature=5L 2s|Step Ratio=7/3; 9/4|Genchain Extend=0, 5}}
The sharper end of the hypohard region corresponds to "neogothic" or "parapyth" systems, with perfect 5ths that are sharper than 702¢.{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/1; 5/2; 8/3|Genchain Extend=0, 5}}
====Parahard and ultrahard step ratios====
The parahard and ultrahard ranges correspond to archy systems, with perfect 5ths that are significantly sharper than than 702¢.{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/1; 4/1; 5/1; 6/1|Genchain Extend=0, 5}}


== Modes==
== Modes==
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{{SB tree|Depth=3}}
{{SB tree|Depth=3}}


This process can be repeated to produce a finer continuum of step ratios as shown below, with each ratio producing a different edo.{{Scale tree|5L 2s|depth=6|Comments=7/5:[[Flattone]] is in this region;21/13:[[Golden meantone]] (696.2145¢);5/3:[[Meantone]] is in this region;2/1:(Generators smaller than this are proper);9/4:The generator closest to a just [[3/2]] for EDOs less than 200;16/7:[[Garibaldi]] / [[Cassandra]];21/8:Golden neogothic (704.0956¢);8/3:[[Neogothic]] is in this region;4/1:[[Archy]] is in this region|tuning=5L 2s}}
This process can be repeated to produce a finer, larger continuum of step ratios as shown below, with each ratio producing a different edo.{{Scale tree|5L 2s|depth=6|Comments=7/5:[[Flattone]] is in this region;21/13:[[Golden meantone]] (696.2145¢);5/3:[[Meantone]] is in this region;2/1:(Generators smaller than this are proper);9/4:The generator closest to a just [[3/2]] for EDOs less than 200;16/7:[[Garibaldi]] / [[Cassandra]];21/8:Golden neogothic (704.0956¢);8/3:[[Neogothic]] is in this region;4/1:[[Archy]] is in this region|tuning=5L 2s}}


==See also==
==See also==


*[[Diatonic functional harmony]]
*[[Diatonic functional harmony]]

Revision as of 01:04, 1 July 2023

↖ 4L 1s ↑ 5L 1s 6L 1s ↗
← 4L 2s 5L 2s 6L 2s →
↙ 4L 3s ↓ 5L 3s 6L 3s ↘ 
┌╥╥╥┬╥╥┬┐
│║║║│║║││
│││││││││
└┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLsLLs
sLLsLLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 4\7 to 3\5 (685.7 ¢ to 720.0 ¢)
Dark 2\5 to 3\7 (480.0 ¢ to 514.3 ¢)
TAMNAMS information
Name diatonic
Prefix dia-
Abbrev. dia
Related MOS scales
Parent 2L 3s
Sister 2L 5s
Daughters 7L 5s, 5L 7s
Neutralized 3L 4s
2-Flought 12L 2s, 5L 9s
Equal tunings
Equalized (L:s = 1:1) 4\7 (685.7 ¢)
Supersoft (L:s = 4:3) 15\26 (692.3 ¢)
Soft (L:s = 3:2) 11\19 (694.7 ¢)
Semisoft (L:s = 5:3) 18\31 (696.8 ¢)
Basic (L:s = 2:1) 7\12 (700.0 ¢)
Semihard (L:s = 5:2) 17\29 (703.4 ¢)
Hard (L:s = 3:1) 10\17 (705.9 ¢)
Superhard (L:s = 4:1) 13\22 (709.1 ¢)
Collapsed (L:s = 1:0) 3\5 (720.0 ¢)
This is a test page. For the main page, see 5L 2s.

5L 2s, named diatonic in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 5 large steps and 2 small steps, repeating every octave. Generators that produce this scale range from 685.7 ¢ to 720 ¢, or from 480 ¢ to 514.3 ¢.

Name

TAMNAMS suggests the name diatonic for this scale, which commonly refers to a scale with 5 whole steps and 2 small steps.

Under TAMNAMS and for all scale pattern pages on the wiki, the term diatonic exclusively refers to 5L 2s. Other diatonic-based scales, such as Zarlino, blackdye and diasem, are called detempered diatonic scales (for an RTT-based philosophy) or deregularized diatonic scales (for an RTT-agnostic philosophy). The terms diatonic-like or diatonic-based may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate.

Notation

Intervals

Intervals are identical to that of standard notation. As such, the usual interval qualities of major/minor and augmented/perfect/diminished apply here.

Interval class Large variety Small variety
Size Quality Size Quality
1st (unison) 0 Perfect 0 Perfect
2nd L Major s Minor
3rd 2L Major L + s Minor
4th 3L Augmented 2L + 1s Perfect
5th 3L + 1s Perfect 2L + 2s Diminished
6th 4L + 1s Major 3L + 2s Minor
7th 5L + 1s Major 4L + 2s Minor
8th (octave) 5L + 2s Perfect 5L + 2s Perfect

Note names

Note names are identical to that of standard notation. Thus, the basic (12edo) gamut for 5L 2s is the following:

J, J&/K@, K, L, L&/M@, M, M&/N@, N, N&/O@, O, P, P&/J@, J

Theory

Generalizing whole and half steps

The familiar pattern of 5 whole steps and 2 half steps, commonly written as WWHWWWH for the major scale, has step sizes of 2 (whole step) and 1 (small step), producing 12edo. This can be generalized to form the pattern LLsLLLs with whole-number step sizes for L and s, where L is greater than s. The terms "large step" and "small step" are preferred as most step size pairings cannot be interpreted as "whole" and "half" steps.

Different edos are produced by using different ratios of step sizes. A few examples are shown below.

Step ratio (L:s) Step pattern EDO Selected multiples
1:1 1 1 1 1 1 1 1 7edo 14edo, 21edo, etc.
4:3 4 4 3 4 4 4 3 26edo
3:2 3 3 2 3 3 3 2 19edo 38edo
5:3 5 5 3 5 5 5 3 31edo
2:1 2 2 1 2 2 2 1 12edo (standard tuning) 24edo, 36edo, etc.
5:2 5 5 2 5 5 5 2 29edo
3:1 3 3 1 3 3 3 1 17edo 34edo
4:1 4 4 1 4 4 4 1 22edo
1:0 1 1 0 1 1 1 0 5edo 10edo, 15edo, etc.

Edos that are multiples of the examples above can be reached by entering non-simplified step ratios. For example, edos that are multiples of 12 are reached by using larger values whose ratio simplifies to 2:1, such as 4:2 for 24edo.

The step ratios 1:1 and 1:0 represent the limits for valid step ratios. A step ratio that approaches 1:1, where the large and small step are equal to one another, approaches 7edo, and a step ratio that approaches 1:0, where the small step "collapses" to zero, approaches 5edo.

The step ratios shown above form a continuum of step ratios. The section Tuning spectrum shows how this is made, as well as a larger spectrum.

Rank-2 temperament interpretations

Main article: 5L 2s/Temperaments

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

  • Meantone, with generators around 696.2¢. This includes:
    • Flattone, with generators around 693.7¢.
  • Schismic, with generators around 702¢.
  • Parapyth, with generators around 704.7¢.
  • Archy, with generators around 709.3¢. This includes:
    • Supra, with generators around 707.2¢
    • Superpyth, with generators around 710.3¢
    • Ultrapyth, with generators around 713.7¢.

Step ratio ranges

Simple step ratios

17edo and 19edo, produced using step ratios of 3:1 and 3:2 respectively, are the smallest edos that offer a greater variety of pitches than 12edo. Note that any enharmonic equivalences that 12edo has no longer hold for either 17edo or 19edo, as shown in the table below.

User:MOS degrees is deprecated. Please use Template:MOS tunings instead.
Scale degree of 5L 2s
Scale degree 12edo (Basic, L:s = 2:1) 17edo (Hard, L:s = 3:1) 19edo (Soft, L:s = 3:2) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents
Perfect 0-diadegree (unison) 0 0 0 0 0 0 1/1 (exact)
Minor 1-diadegree 1 100 1 70.6 2 126.3
Major 1-diadegree 2 200 3 211.8 3 189.5
Minor 2-diadegree 3 300 4 282.4 5 315.8
Major 2-diadegree 4 400 6 423.5 6 378.9
Perfect 3-diadegree 5 500 7 494.1 8 505.3
Augmented 3-diadegree 6 600 9 635.3 9 568.4
Diminished 4-diadegree 6 600 8 564.7 10 631.6
Perfect 4-diadegree 7 700 10 705.9 11 694.7
Minor 5-diadegree 8 800 11 776.5 13 821.1
Major 5-diadegree 9 900 13 917.6 14 884.2
Minor 6-diadegree 10 1000 14 988.2 16 1010.5
Major 6-diadegree 11 1100 16 1129.4 17 1073.7
Perfect 7-diadegree (octave) 12 1200 17 1200 19 1200 2/1 (exact)

Parasoft step ratios

Main article: Flattone

Parasoft step ratios (4:3 and 3:2) correspond to flattone temperaments, characterized by flattened perfect 5ths (flat of 702¢) to produce major 3rds that are flatter than 5/4 (386¢).

User:MOS degrees is deprecated. Please use Template:MOS tunings instead.
Scale degree of 5L 2s
Scale degree 19edo (Soft, L:s = 3:2) 26edo (Supersoft, L:s = 4:3) 45edo (L:s = 7:5) 64edo (L:s = 10:7) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents Steps Cents
Perfect 0-diadegree (unison) 0 0 0 0 0 0 0 0 1/1 (exact)
Minor 1-diadegree 2 126.3 3 138.5 5 133.3 7 131.3
Major 1-diadegree 3 189.5 4 184.6 7 186.7 10 187.5
Minor 2-diadegree 5 315.8 7 323.1 12 320 17 318.8
Major 2-diadegree 6 378.9 8 369.2 14 373.3 20 375
Perfect 3-diadegree 8 505.3 11 507.7 19 506.7 27 506.2
Augmented 3-diadegree 9 568.4 12 553.8 21 560 30 562.5
Diminished 4-diadegree 10 631.6 14 646.2 24 640 34 637.5
Perfect 4-diadegree 11 694.7 15 692.3 26 693.3 37 693.8
Minor 5-diadegree 13 821.1 18 830.8 31 826.7 44 825
Major 5-diadegree 14 884.2 19 876.9 33 880 47 881.2
Minor 6-diadegree 16 1010.5 22 1015.4 38 1013.3 54 1012.5
Major 6-diadegree 17 1073.7 23 1061.5 40 1066.7 57 1068.8
Perfect 7-diadegree (octave) 19 1200 26 1200 45 1200 64 1200 2/1 (exact)

Hyposoft step ratios

Main article: Meantone

Hyposoft step ratios (3:2 to 2:1) correspond to meantone temperaments, characterized by flattened perfect 5ths (flat of 702¢) to produce diatonic major 3rds that approximate 5/4 (386¢).

User:MOS degrees is deprecated. Please use Template:MOS tunings instead.
Scale degree of 5L 2s
Scale degree 19edo (Soft, L:s = 3:2) 31edo (Semisoft, L:s = 5:3) 43edo (L:s = 7:4) 50edo (L:s = 8:5) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents Steps Cents
Perfect 0-diadegree (unison) 0 0 0 0 0 0 0 0 1/1 (exact)
Minor 1-diadegree 2 126.3 3 116.1 4 111.6 5 120
Major 1-diadegree 3 189.5 5 193.5 7 195.3 8 192
Minor 2-diadegree 5 315.8 8 309.7 11 307 13 312
Major 2-diadegree 6 378.9 10 387.1 14 390.7 16 384
Perfect 3-diadegree 8 505.3 13 503.2 18 502.3 21 504
Augmented 3-diadegree 9 568.4 15 580.6 21 586 24 576
Diminished 4-diadegree 10 631.6 16 619.4 22 614 26 624
Perfect 4-diadegree 11 694.7 18 696.8 25 697.7 29 696
Minor 5-diadegree 13 821.1 21 812.9 29 809.3 34 816
Major 5-diadegree 14 884.2 23 890.3 32 893 37 888
Minor 6-diadegree 16 1010.5 26 1006.5 36 1004.7 42 1008
Major 6-diadegree 17 1073.7 28 1083.9 39 1088.4 45 1080
Perfect 7-diadegree (octave) 19 1200 31 1200 43 1200 50 1200 2/1 (exact)

Hypohard step ratios

Main article: Pythagorean tuning and schismatic temperament

The range of hypohard step ratios can be divided into a minihard range (between 2:1 to 5:2) and quasihard range (between 5:2 to 3:1).

Minihard step ratios

Minihard step ratios correspond to Pythagorean tuning and schismatic temperament, characterized by having a perfect 5th that is as close to just (701.96¢) as possible, resulting in a major 3rd of 81/64 (407¢).

User:MOS degrees is deprecated. Please use Template:MOS tunings instead.
Scale degree of 5L 2s
Scale degree 41edo (L:s = 7:3) 53edo (L:s = 9:4) Approx. JI Ratios
Steps Cents Steps Cents
Perfect 0-diadegree (unison) 0 0 0 0 1/1 (exact)
Minor 1-diadegree 3 87.8 4 90.6
Major 1-diadegree 7 204.9 9 203.8
Minor 2-diadegree 10 292.7 13 294.3
Major 2-diadegree 14 409.8 18 407.5
Perfect 3-diadegree 17 497.6 22 498.1
Augmented 3-diadegree 21 614.6 27 611.3
Diminished 4-diadegree 20 585.4 26 588.7
Perfect 4-diadegree 24 702.4 31 701.9
Minor 5-diadegree 27 790.2 35 792.5
Major 5-diadegree 31 907.3 40 905.7
Minor 6-diadegree 34 995.1 44 996.2
Major 6-diadegree 38 1112.2 49 1109.4
Perfect 7-diadegree (octave) 41 1200 53 1200 2/1 (exact)

Quasihard step ratios

Quasihard step ratios correspond to "neogothic" or "parapyth" systems whose perfect 5th is sharper than just, resulting in major 3rds that are sharper than 81/64.

17edo is considered to be on the sharper end of the neogothic spectrum, with a major 3rd that is more discordant than flatter neogothic tunings.

User:MOS degrees is deprecated. Please use Template:MOS tunings instead.
Scale degree of 5L 2s
Scale degree 17edo (Hard, L:s = 3:1) 29edo (Semihard, L:s = 5:2) 46edo (L:s = 8:3) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents
Perfect 0-diadegree (unison) 0 0 0 0 0 0 1/1 (exact)
Minor 1-diadegree 1 70.6 2 82.8 3 78.3
Major 1-diadegree 3 211.8 5 206.9 8 208.7
Minor 2-diadegree 4 282.4 7 289.7 11 287
Major 2-diadegree 6 423.5 10 413.8 16 417.4
Perfect 3-diadegree 7 494.1 12 496.6 19 495.7
Augmented 3-diadegree 9 635.3 15 620.7 24 626.1
Diminished 4-diadegree 8 564.7 14 579.3 22 573.9
Perfect 4-diadegree 10 705.9 17 703.4 27 704.3
Minor 5-diadegree 11 776.5 19 786.2 30 782.6
Major 5-diadegree 13 917.6 22 910.3 35 913
Minor 6-diadegree 14 988.2 24 993.1 38 991.3
Major 6-diadegree 16 1129.4 27 1117.2 43 1121.7
Perfect 7-diadegree (octave) 17 1200 29 1200 46 1200 2/1 (exact)

Parahard and ultrahard step ratios

Main article: Archy

The parahard and ultrahard ranges (3:1 to 1:1) correspond to archy systems, with perfect 5ths that are significantly sharper than than 702¢.

User:MOS degrees is deprecated. Please use Template:MOS tunings instead.
Scale degree of 5L 2s
Scale degree 17edo (Hard, L:s = 3:1) 22edo (Superhard, L:s = 4:1) 27edo (L:s = 5:1) 32edo (L:s = 6:1) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents Steps Cents
Perfect 0-diadegree (unison) 0 0 0 0 0 0 0 0 1/1 (exact)
Minor 1-diadegree 1 70.6 1 54.5 1 44.4 1 37.5
Major 1-diadegree 3 211.8 4 218.2 5 222.2 6 225
Minor 2-diadegree 4 282.4 5 272.7 6 266.7 7 262.5
Major 2-diadegree 6 423.5 8 436.4 10 444.4 12 450
Perfect 3-diadegree 7 494.1 9 490.9 11 488.9 13 487.5
Augmented 3-diadegree 9 635.3 12 654.5 15 666.7 18 675
Diminished 4-diadegree 8 564.7 10 545.5 12 533.3 14 525
Perfect 4-diadegree 10 705.9 13 709.1 16 711.1 19 712.5
Minor 5-diadegree 11 776.5 14 763.6 17 755.6 20 750
Major 5-diadegree 13 917.6 17 927.3 21 933.3 25 937.5
Minor 6-diadegree 14 988.2 18 981.8 22 977.8 26 975
Major 6-diadegree 16 1129.4 21 1145.5 26 1155.6 31 1162.5
Perfect 7-diadegree (octave) 17 1200 22 1200 27 1200 32 1200 2/1 (exact)

Modes

Diatonic modes have standard names from classical music theory:


Modes of 5L 2s
UDP Cyclic
order		 Step
pattern		 Mode names
6|0 1 LLLsLLs Lydian
5|1 5 LLsLLLs Ionian (major)
4|2 2 LLsLLsL Mixolydian
3|3 6 LsLLLsL Dorian
2|4 3 LsLLsLL Aeolian (minor)
1|5 7 sLLLsLL Phrygian
0|6 4 sLLsLLL Locrian

Each mode has the following scale degrees, reached by raising or lowering certain naturals by a chroma.

TODO: Make this into a template
Mode Scale degree (on C)
UDP Step pattern 1st 2nd 3rd 4th 5th 6th 7th 8th
6|0 LLLsLLs Perfect (C) Major (D) Major (E) Augmented (F#) Perfect (G) Major (A) Major (B) Perfect (C)
5|1 LLsLLLs Perfect (C) Major (D) Major (E) Perfect (F) Perfect (G) Major (A) Major (B) Perfect (C)
4|2 LLsLLsL Perfect (C) Major (D) Major (E) Perfect (F) Perfect (G) Major (A) Minor (Bb) Perfect (C)
3|3 LsLLLsL Perfect (C) Major (D) Minor (Eb) Perfect (F) Perfect (G) Major (A) Minor (Bb) Perfect (C)
2|4 LsLLsLL Perfect (C) Major (D) Minor (Eb) Perfect (F) Perfect (G) Minor (Ab) Minor (Bb) Perfect (C)
1|5 sLLLsLL Perfect (C) Minor (Db) Minor (Eb) Perfect (F) Perfect (G) Minor (Ab) Minor (Bb) Perfect (C)
0|6 sLLsLLL Perfect (C) Minor (Db) Minor (Eb) Perfect (F) Diminished (Gb) Minor (Ab) Minor (Bb) Perfect (C)

Scales

Subset and superset scales

5L 2s has a parent scale of 2L 3s, meaning 5L 2s contains 2L 3s as a subset. 5L 2s also has two child scales that both contain 5L 2s as a subset: either 7L 5s (if the step ratio is less than 2:1) or 5L 7s (if the step ratio is greater than 2:1). A step ratio exactly 2:1 will produce 12edo, an equalized form of 5L 7s and 7L 5s.

MODMOS scales and muddles

Main article: 5L 2s MODMOSes
 and 5L 2s Muddles

Scala files

Tuning spectrum

A spectrum of step ratios can be produced by starting with the ratios 1:1 and 1:0 and repeatedly finding the mediants between adjacent ratios. The first three iterations are shown below.

Ratios
1/1
2/1
1/0
Ratios
1/1
3/2
2/1
3/1
1/0
Ratios
1/1
4/3
3/2
5/3
2/1
5/2
3/1
4/1
1/0

This process can be repeated to produce a finer, larger continuum of step ratios as shown below, with each ratio producing a different edo.Template:Scale tree

See also

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