User:Ganaram inukshuk/5L 2s

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↖4L 1s ↑5L 1s 6L 1s↗
←4L 2s5L 2s 6L 2s→
↙4L 3s ↓5L 3s 6L 3s↘
 
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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
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¢.

Among the most well-known forms of this scale are the diatonic scale of 12edo, the Pythagorean diatonic scale, and scales produced by meantone systems.

Name

TAMNAMS suggests the temperament-agnostic 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.

The term diatonic may also refer to scales produced using tetrachords, just intonation, or in general have more than one size of whole tone. Such scales, such as Zarlino, blackdye and diasem, are specifically 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 such scales, depending on what's contextually the most appropriate.

Intervals

This article assumes TAMNAMS for naming step ratios, mossteps, and mosdegrees.

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 (the 0-mosstep and 0-mosdegree) for the unison.

Except for the unison and octave, all interval classes have two varieties or sizes, denoted using the terms major and minor for the large and small sizes, respectively. The exception to this rule are the generators, which use the terms augmented, perfect, and diminished instead.

5L 2s interval varieties
Interval class Specific intervals Size (in ascending order)
0-diastep Perfect 0-diastep (unison) 0
1-diastep Minor 1-diastep s
Major 1-diastep L
2-diastep Minor 2-diastep L + s
Major 2-diastep 2L
3-diastep Perfect 3-diastep 2L + s
Augmented 3-diastep 3L
4-diastep Diminished 4-diastep 2L + 2s
Perfect 4-diastep 3L + s
5-diastep Minor 5-diastep 3L + 2s
Major 5-diastep 4L + s
6-diastep Minor 6-diastep 4L + 2s
Major 6-diastep 5L + s
7-diastep (octave) Perfect 7-diastep (octave) 5L + 2s

A 7-note scale using these intervals will typically use scale degrees that represents one size from each interval class, with the true MOS upholding the step pattern of LLLsLLs, or some rotation thereof. MODMOS scales may be formed this way without upholding the step pattern, thereby creating a non-MOS pattern such as LLLLsLs, or may include alterations that exceed the two varieties typical of a MOS scale.

Notation

See 5L 2s/Notation

Theory

Introduction to step sizes

Main article: Scale tree and TAMNAMS#Step ratio spectrum

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 (half step), producing 12edo. This can be generalized into the form LLsLLLs, with whole-number sizes for the large steps and small steps, denoted as "L" and "s" respectively.

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.

All step ratios lie on a spectrum from 1:1 to 1:0, referred to on the wiki as a scale tree. 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.

TAMNAMS has names for regions of this spectrum based on whether they are "soft" (between 1:1 and 2:1) or "hard" (between 2:1 and 1:0).

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¢.

Tuning ranges

Simple tunings

17edo and 19edo 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.


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 tunings

Main article: Flattone

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

Edos include 19edo, 26edo, 45edo, and 64edo.


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 tunings

Main article: Meantone

Hyposoft tunings (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¢).

Edos include 19edo, 31edo, 43edo, and 50edo.


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 tunings

Main article: Pythagorean tuning and schismatic temperament

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

Minihard tunings

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 a major 3rd of 81/64 (407¢).

Edos include 41edo and 53edo.


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 tunings

Quasihard tunings correspond to "neogothic" or "parapyth" systems whose perfect 5th is slightly sharper than just, resulting in major 3rds that are sharper than 81/64 and minor 3rds that are slightly flat of 32/27 (294¢).

Edos include 17edo, 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.


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 tunings

Main article: Archy

Parahard (3:1 to 4:1) and ultrahard tunings (4:1 to 1:0) correspond to archy systems, with perfect 5ths that are significantly sharper than than 702¢.

Edos include 17edo, 22edo, 27edo, and 32edo, among others.


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 Rotational 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)

Generator chain

Explain how the scale can also be generated by stacking 6 generating intervals in any combination of up or down from the root. Also explain how this can be extended further to 11 generators to produce chromatic scales.

Scales

Subset and superset scales

5L 2s has a parent scale of 2L 3s, a pentatonic scale, meaning 2L 3s is a subset. 5L 2s also has the two child scales, which are supersets of 5L 2s:

  • 7L 5s, a chromatic scale produced using soft-of-basic step ratios.
  • 5L 7s, a chromatic scale produced using hard-of-basic step ratios.

12edo contains 5L 2s as the equalized form of both 5L 7s and 7L 5s.

MODMOS scales and muddles

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

Scala files

Scale tree

Scale Tree and Tuning Spectrum of 5L 2s
Generator(edo) Cents Step Ratio Comments
Bright Dark L:s Hardness
4\7 685.714 514.286 1:1 1.000 Equalized 5L 2s
27\47 689.362 510.638 7:6 1.167
23\40 690.000 510.000 6:5 1.200
42\73 690.411 509.589 11:9 1.222
19\33 690.909 509.091 5:4 1.250
53\92 691.304 508.696 14:11 1.273
34\59 691.525 508.475 9:7 1.286
49\85 691.765 508.235 13:10 1.300
15\26 692.308 507.692 4:3 1.333 Supersoft 5L 2s
56\97 692.784 507.216 15:11 1.364
41\71 692.958 507.042 11:8 1.375
67\116 693.103 506.897 18:13 1.385
26\45 693.333 506.667 7:5 1.400 Flattone is in this region
63\109 693.578 506.422 17:12 1.417
37\64 693.750 506.250 10:7 1.429
48\83 693.976 506.024 13:9 1.444
11\19 694.737 505.263 3:2 1.500 Soft 5L 2s
51\88 695.455 504.545 14:9 1.556
40\69 695.652 504.348 11:7 1.571
69\119 695.798 504.202 19:12 1.583
29\50 696.000 504.000 8:5 1.600
76\131 696.183 503.817 21:13 1.615 Golden meantone (696.2145¢)
47\81 696.296 503.704 13:8 1.625
65\112 696.429 503.571 18:11 1.636
18\31 696.774 503.226 5:3 1.667 Semisoft 5L 2s
Meantone is in this region
61\105 697.143 502.857 17:10 1.700
43\74 697.297 502.703 12:7 1.714
68\117 697.436 502.564 19:11 1.727
25\43 697.674 502.326 7:4 1.750
57\98 697.959 502.041 16:9 1.778
32\55 698.182 501.818 9:5 1.800
39\67 698.507 501.493 11:6 1.833
7\12 700.000 500.000 2:1 2.000 Basic 5L 2s
Scales with tunings softer than this are proper
(Generators smaller than this are proper)
38\65 701.538 498.462 11:5 2.200
31\53 701.887 498.113 9:4 2.250 The generator closest to a just 3/2 for EDOs less than 200
55\94 702.128 497.872 16:7 2.286 Garibaldi / Cassandra
24\41 702.439 497.561 7:3 2.333
65\111 702.703 497.297 19:8 2.375
41\70 702.857 497.143 12:5 2.400
58\99 703.030 496.970 17:7 2.429
17\29 703.448 496.552 5:2 2.500 Semihard 5L 2s
61\104 703.846 496.154 18:7 2.571
44\75 704.000 496.000 13:5 2.600
71\121 704.132 495.868 21:8 2.625 Golden neogothic (704.0956¢)
27\46 704.348 495.652 8:3 2.667 Neogothic is in this region
64\109 704.587 495.413 19:7 2.714
37\63 704.762 495.238 11:4 2.750
47\80 705.000 495.000 14:5 2.800
10\17 705.882 494.118 3:1 3.000 Hard 5L 2s
43\73 706.849 493.151 13:4 3.250
33\56 707.143 492.857 10:3 3.333
56\95 707.368 492.632 17:5 3.400
23\39 707.692 492.308 7:2 3.500
59\100 708.000 492.000 18:5 3.600
36\61 708.197 491.803 11:3 3.667
49\83 708.434 491.566 15:4 3.750
13\22 709.091 490.909 4:1 4.000 Superhard 5L 2s
Archy is in this region
42\71 709.859 490.141 13:3 4.333
29\49 710.204 489.796 9:2 4.500
45\76 710.526 489.474 14:3 4.667
16\27 711.111 488.889 5:1 5.000
35\59 711.864 488.136 11:2 5.500
19\32 712.500 487.500 6:1 6.000
22\37 713.514 486.486 7:1 7.000
3\5 720.000 480.000 1:0 → ∞ Collapsed 5L 2s

See also

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