User:Ganaram inukshuk/5L 2s

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Revision as of 03:44, 29 June 2023 by Ganaram inukshuk (talk | contribs) (→Step ratios: Added text for ultrasoft/parasoft tunings)
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↖ 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 ¢)
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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 size around 694¢, corresponding to a step ratio within the ultrasoft to parasoft range (6:5 to 3:2).
  • Meantone, with a generator size around 696¢, corresponding to a step ratio within the hyposoft range (3:2 to 2:1).
  • Schismic, with a generator size around 702¢ (just perfect 5th, or 3/2), corresponding to a step ratio within the hypohard range (2:1 to 3:2).
  • Parapyth, with a generator size ranging between 702¢ and 705¢, corresponding to a step ratio within the quasihard to range (5:2 to 3:1).
  • Archy, with a generator size greater than 705¢, corresponding to a step ratio within the parahard to ultrahard range (greater than 3:1).

Step ratios

Ultrasoft to parasoft

Ultrasoft to parasoft tunings correspond to flattone temperaments, characterized by flattening the perfect 5th (3/2) to achieve a diatonic major 3rd that is flat of 5/4 (386¢). Examples of this include 26edo (step ratio 4:3).

Hyposoft

Main article: Meantone

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

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) Approx. JI Ratios
Steps Cents Steps Cents
Perfect 0-diadegree (unison) 0 0 0 0 1/1 (exact)
Minor 1-diadegree 2 126.3 3 116.1
Major 1-diadegree 3 189.5 5 193.5
Minor 2-diadegree 5 315.8 8 309.7
Major 2-diadegree 6 378.9 10 387.1
Perfect 3-diadegree 8 505.3 13 503.2
Augmented 3-diadegree 9 568.4 15 580.6
Diminished 4-diadegree 10 631.6 16 619.4
Perfect 4-diadegree 11 694.7 18 696.8
Minor 5-diadegree 13 821.1 21 812.9
Major 5-diadegree 14 884.2 23 890.3
Minor 6-diadegree 16 1010.5 26 1006.5
Major 6-diadegree 17 1073.7 28 1083.9
Perfect 7-diadegree (octave) 19 1200 31 1200 2/1 (exact)

Hypohard

Parahard to ultrahard

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 MOS 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 is deprecated. Please use Template: MOS tuning spectrum instead. Details:
Use of a single Comments parameter has become unmaintainable. Existing scale trees should be migrated to the new template, where comments are entered using a step ratio p/q as a parameter: 
{{MOS tuning spectrum
| 3/2 = Example comment
| 4/3 = Another example comment
}}


The parameters tuning and depth have been replaced with Scale Signature and Depth, respectively.


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
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
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
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
38\65 701.538 498.462 11:5 2.200
31\53 701.887 498.113 9:4 2.250
55\94 702.128 497.872 16:7 2.286
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
27\46 704.348 495.652 8:3 2.667
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
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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