User:Ganaram inukshuk/5L 2s
- This is a test page. For the main page, see 5L 2s.
| ↖ 4L 1s | ↑ 5L 1s | 6L 1s ↗ |
| ← 4L 2s | 5L 2s | 6L 2s → |
| ↙ 4L 3s | ↓ 5L 3s | 6L 3s ↘ |
┌╥╥╥┬╥╥┬┐ │║║║│║║││ │││││││││ └┴┴┴┴┴┴┴┘
sLLsLLL
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, referring to the use of this term to refer to a scale with 5 whole steps and 2 small steps.
On the term diatonic
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
5L 2s as a moment-of-symmetry scale
The familiar interpretation of whole and half steps has step sizes of 2 and 1 steps, producing 12edo. Viewing 5L 2s as a moment-of-symmetry scale involves generalizing the step pattern of 5 whole steps and 2 whole steps, commonly written as WWHWWWH for the major scale, as being different step sizes. As such, the generalized form LLsLLLs is used, as most step ratios have step sizes that cannot be interpreted as being "whole" or "half" steps.
Substituting step sizes
Different edos are produced by substituting different 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 12, such as 24edo or 72edo, are reached by using step ratios that are not simplified, such as 4:2 for 24edo and 12:6 for 72edo. The step sizes may be called whole and half in this case.
Expanding the tuning spectrum
The tuning spectrum shown in the previous section starts with the step ratios 1:1 and 1:0 and repeatedly finding the mediants between adjacent ratios. The first few 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 | |||
Thus, reaching larger edos, such as 53edo or 311edo, is a matter of expanding the tuning spectrum until the desired step ratio is found. The section tuning spectrum contains a much larger tuning spectrum.
The step ratios 1:1 and 1:0 represent the extremes of the tuning spectrum. A step ratio where the large and small step approach being equal to each other approaches 7edo, and a step ratio where the size of the small step approaches 0 approaches 5edo.
Temperament interpretations
Tuning ranges
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 |