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 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 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 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 large and small 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 into the form LLsLLLs, with whole-number sizes for the large steps (denoted as "L") and small steps (denoted as "s").
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.
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.
The step ratios shown above form a continuum of step ratios. The section Tuning spectrum shows how this is made, as well as a larger spectrum.
Rank-2 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¢.
Step ratio ranges
Simple step ratios
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 step ratios
- Main article: Flattone
Parasoft step ratios (between 4:3 and 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¢).
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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 step ratios
- Main article: Meantone
Hyposoft step ratios (between 3:2 and 2:1) correspond to meantone temperaments, characterized by flattened perfect 5ths (flat of 702¢) to produce diatonic major 3rds that approximate 5/4 (386¢).
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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 step ratios
- Main article: Pythagorean tuning and schismatic temperament
The range of hypohard step ratios can be divided into a minihard range (between 2:1 and 5:2) and quasihard range (between 5:2 and 3:1).
Minihard step ratios
Minihard step ratios 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 step ratios
Quasihard step ratios 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¢).
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 step ratios
- Main article: Archy
The parahard and ultrahard ranges (between 3:1 and 1:1) correspond to archy systems, with perfect 5ths that are significantly sharper than than 702¢.
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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
Tuning spectrum
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.
| 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 | |||
This process can be repeated to produce a finer, larger continuum of step ratios as shown below, with each ratio producing a different edo.Template:Scale tree