# 5L 2s

 ↖ 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¢)
English Wikipedia has an article on:

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¢. The familiar pattern of 5 whole steps and 2 half steps, commonly written as WWHWWWH for the major scale, takes on a generalized form of LLsLLLs, where the large and small steps – denoted as L's and s's – represent whole number step sizes, thus producing different edos. These step ratios affect the sizes of the diatonic scale's intervals and correspond to different tuning systems.

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

## Name

TAMNAMS suggests the temperament-agnostic name diatonic for this scale, which commonly refers to a scale with 5 whole steps and 2 half steps. See TAMNAMS/Appendix#On_the_term_diatonic for further clarification.

## 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.

Intervals of 5L 2s
Intervals Steps subtended Range in cents Average of HE
(from HE Calc)
Min of HE
Generic[1] Specific[2] Abbrev.[3]
0-diastep Perfect 0-diastep P0ms 0 0.0¢ ~2.4654 nats ~2.4654 nats
1-diastep Minor 1-diastep m1ms s 0.0¢ to 171.4¢ ~4.6969 nats ~4.6333 nats
Major 1-diastep M1ms L 171.4¢ to 240.0¢ ~4.5872 nats ~4.5842 nats
2-diastep Minor 2-diastep m2ms L + s 240.0¢ to 342.9¢ ~4.5596 nats ~4.5379 nats
Major 2-diastep M2ms 2L 342.9¢ to 480.0¢ ~4.5540 nats ~4.4846 nats
3-diastep Perfect 3-diastep P3ms 2L + s 480.0¢ to 514.3¢ ~4.3841 nats ~4.3665 nats
Augmented 3-diastep A3ms 3L 514.3¢ to 720.0¢ ~4.6012 nats ~4.5615 nats
4-diastep Diminished 4-diastep d4ms 2L + 2s 480.0¢ to 685.7¢ ~4.6013 nats ~4.5630 nats
Perfect 4-diastep P4ms 3L + s 685.7¢ to 720.0¢ ~4.1634 nats ~4.1224 nats
5-diastep Minor 5-diastep m5ms 3L + 2s 720.0¢ to 857.1¢ ~4.5942 nats ~4.5711 nats
Major 5-diastep M5ms 4L + s 857.1¢ to 960.0¢ ~4.5145 nats ~4.4208 nats
6-diastep Minor 6-diastep m6ms 4L + 2s 960.0¢ to 1028.6¢ ~4.5760 nats ~4.5502 nats
Major 6-diastep M6ms 5L + s 1028.6¢ to 1200.0¢ ~4.6377 nats ~4.6078 nats
7-diastep Perfect 7-diastep P7ms 5L + 2s 1200.0¢ ~3.3273 nats ~3.3273 nats

1. Generic intervals are denoted solely by the number of steps they subtend.
2. Specific intervals denote whether an interval is major, minor, augmented, perfect, or diminished.
3. Abbreviations can be further shortened to 'ms' if context allows.

### Note names

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

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

## Theory

### Temperament interpretations

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)

### Ultrasoft tunings

Scale degree of 5L 2s
Scale degree 26edo (Supersoft, L:s = 4:3) 7edo (Equalized, L:s = 1:1) 33edo (L:s = 5:4) 40edo (L:s = 6:5) 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 3 138.5 1 171.4 4 145.5 5 150
Major 1-diadegree 4 184.6 1 171.4 5 181.8 6 180
Minor 2-diadegree 7 323.1 2 342.9 9 327.3 11 330
Major 2-diadegree 8 369.2 2 342.9 10 363.6 12 360
Perfect 3-diadegree 11 507.7 3 514.3 14 509.1 17 510
Augmented 3-diadegree 12 553.8 3 514.3 15 545.5 18 540
Diminished 4-diadegree 14 646.2 4 685.7 18 654.5 22 660
Perfect 4-diadegree 15 692.3 4 685.7 19 690.9 23 690
Minor 5-diadegree 18 830.8 5 857.1 23 836.4 28 840
Major 5-diadegree 19 876.9 5 857.1 24 872.7 29 870
Minor 6-diadegree 22 1015.4 6 1028.6 28 1018.2 34 1020
Major 6-diadegree 23 1061.5 6 1028.6 29 1054.5 35 1050
Perfect 7-diadegree (octave) 26 1200 7 1200 33 1200 40 1200 2/1 (exact)

### Parasoft tunings

Parasoft diatonic 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

Hyposoft diatonic 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

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 diatonic 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 diatonic 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

Parahard (3:1 to 4:1) and ultrahard (4:1 to 1:0) diatonic tunings 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.

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, a pentatonic scale, meaning 2L 3s is a subset. 5L 2s also has 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, the equalized form of both 7L 5s and 5L 7s, is also a superset of 5L 2s.

### MODMOS scales and muddles

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

## Scale tree

Scale Tree and Tuning Spectrum of 5L 2s
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
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