User:Ganaram inukshuk/5L 2s
| ↖ 4L 1s | ↑ 5L 1s | 6L 1s ↗ |
| ← 4L 2s | 5L 2s | 6L 2s → |
| ↙ 4L 3s | ↓ 5L 3s | 6L 3s ↘ |
┌╥╥╥┬╥╥┬┐ │║║║│║║││ │││││││││ └┴┴┴┴┴┴┴┘
sLLsLLL
- 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.
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
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, 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.
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User:MOS degrees is deprecated. Please use Template:MOS tunings instead. |
| 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.
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User:MOS degrees is deprecated. Please use Template:MOS tunings instead. |
| 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.
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User:MOS degrees is deprecated. Please use Template:MOS tunings instead. |
| 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¢).
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User:MOS degrees is deprecated. Please use Template:MOS tunings instead. |
| 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.
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User:MOS degrees is deprecated. Please use Template:MOS tunings instead. |
| 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.
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User:MOS degrees is deprecated. Please use Template:MOS tunings instead. |
| 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:
| 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.
| 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
- and 5L 2s Muddles
Scala files
- Meantone7 – 19edo and 31edo tunings
- Nestoria7 – 171edo tuning
- Pythagorean7 – Pythagorean tuning
- Garibaldi7 – 94edo tuning
- Cotoneum7 – 217edo tuning
- Pepperoni7 – 271edo tuning
- Supra7 – 56edo tuning
- Archy7 – 472edo tuning