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 ¢.
Name
TAMNAMS suggests the name diatonic for this scale, which commonly refers 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
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 of around 4:3.
- Meantone, with a generator size around 696¢, corresponding to a step ratio of around 5:3.
- Schismic, with a generator size around 702¢ (just perfect 5th, or 3/2), corresponding to a step ratio between 2:1 and 5:2.
- Pythagorean tuning also has a generator of 702¢.
- Parapyth, with a generator size ranging between 702¢ and 705¢, corresponding to a step ratio between 5:2 and 3:1.
- Archy, with a generator size greater than 705¢, corresponding to a step ratio between 3:1 and 5:1.
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 |