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User:Ganaram inukshuk/5L 2s

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Revision as of 08:05, 29 June 2023 by Ganaram inukshuk (talk | contribs) (→Temperament interpretations: generators now reference the temperaments subpage's optimal generators)
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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 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
4:3 4 4 3 4 4 4 3 26edo
3:2 3 3 2 3 3 3 2 19edo
5:3 5 5 3 5 5 5 3 31edo
2:1 2 2 1 2 2 2 1 12edo (standard tuning)
5:2 5 5 2 5 5 5 2 29edo
3:1 3 3 1 3 3 3 1 17edo
4:1 4 4 1 4 4 4 1 22edo

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 and 12:6 for 72edo. The step sizes may be called whole and half in this case.

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, yielding the step ratios previously mentioned.

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

Larger edos, such as 53edo (step ratio 9:4), can be reached by repeatedly expanding the tuning spectrum. A larger tuning spectrum can be found in this page's tuning spectrum section.

The step ratios 1:1 and 1:0 represent the extremes of the tuning spectrum. 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 size of the small step approaches 0 relative to the size of the large step, approaches 5edo.

Temperament interpretations

Main article: 5L 2s/Temperaments

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

  • Flattone, with a generator around 693.7¢.
  • Meantone, with a generator around 696.2¢.
  • Schismic, with a generator around 702¢.
  • Parapyth, with a generator around 704.7¢.
  • Archy, with a generator around 709.3¢.

Tuning ranges

Parasoft

Main article: Flattone

Parasoft tunings (step ratios between 4:3 to 3:2) correspond to flattone temperaments, characterized by flattening the perfect 5th (3/2) so it makes the diatonic major 3rd flatter than 5/4 (386¢). Compatible edos include 26edo, 45edo, and 64edo.

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

Hyposoft

Main article: Meantone

Hyposoft tunings (step ratio between 3:2 to 2:1) correspond to meantone temperaments, characterized by flattening the perfect 5th (3/2) to achieve a diatonic major 3rd that approximates 5/4 (386¢). Compatible edos include 19edo, 31edo, and 50edo.

  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) 50edo (L:s = 8:5) 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 2 126.3 3 116.1 5 120
Major 1-diadegree 3 189.5 5 193.5 8 192
Minor 2-diadegree 5 315.8 8 309.7 13 312
Major 2-diadegree 6 378.9 10 387.1 16 384
Perfect 3-diadegree 8 505.3 13 503.2 21 504
Augmented 3-diadegree 9 568.4 15 580.6 24 576
Diminished 4-diadegree 10 631.6 16 619.4 26 624
Perfect 4-diadegree 11 694.7 18 696.8 29 696
Minor 5-diadegree 13 821.1 21 812.9 34 816
Major 5-diadegree 14 884.2 23 890.3 37 888
Minor 6-diadegree 16 1010.5 26 1006.5 42 1008
Major 6-diadegree 17 1073.7 28 1083.9 45 1080
Perfect 7-diadegree (octave) 19 1200 31 1200 50 1200 2/1 (exact)

Hypohard

Main articles: Pythagorean tuning and schismatic temperament

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

Minihard

Minihard tunings 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 81/64 (407.8¢) for its major 3rd. Compatible edos include 41edo, and 53edo.

  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

Quasihard tunings correspond to "neogothic" or "parapyth" systems whose perfect 5th is sharper than just, resulting in major 3rds that are sharper than 81/64. Compatible edos include 29edo and 46edo.

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

Parahard and ultrahard

Parahard and ultrahard tunings (step ratio 3:1 or sharper) correspond to "archy" systems. Compatible edos include 17edo, 22edo, and 27edo.

  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) 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 1 54.5 1 44.4
Major 1-diadegree 3 211.8 4 218.2 5 222.2
Minor 2-diadegree 4 282.4 5 272.7 6 266.7
Major 2-diadegree 6 423.5 8 436.4 10 444.4
Perfect 3-diadegree 7 494.1 9 490.9 11 488.9
Augmented 3-diadegree 9 635.3 12 654.5 15 666.7
Diminished 4-diadegree 8 564.7 10 545.5 12 533.3
Perfect 4-diadegree 10 705.9 13 709.1 16 711.1
Minor 5-diadegree 11 776.5 14 763.6 17 755.6
Major 5-diadegree 13 917.6 17 927.3 21 933.3
Minor 6-diadegree 14 988.2 18 981.8 22 977.8
Major 6-diadegree 16 1129.4 21 1145.5 26 1155.6
Perfect 7-diadegree (octave) 17 1200 22 1200 27 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

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

Template:Scale tree

See also

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