5L 2s (3/1-equivalent)
| ↖ 4L 1s⟨3/1⟩ | ↑ 5L 1s⟨3/1⟩ | 6L 1s⟨3/1⟩ ↗ |
| ← 4L 2s⟨3/1⟩ | 5L 2s (3/1-equivalent) | 6L 2s⟨3/1⟩ → |
| ↙ 4L 3s⟨3/1⟩ | ↓ 5L 3s⟨3/1⟩ | 6L 3s⟨3/1⟩ ↘ |
┌╥╥╥┬╥╥┬┐ │║║║│║║││ │││││││││ └┴┴┴┴┴┴┴┘
sLLsLLL
5L 2s⟨3/1⟩, also called triatonic, is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 5 large steps and 2 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 1086.8 ¢ to 1141.2 ¢, or from 760.8 ¢ to 815.1 ¢.
Name
The name triatonic was coined by CompactStar, and is a back-formation from "diatonic" with di- being interpreted as 2 (the octave) and replaced with tri- for 3 (the tritave). It is not an official name in TAMNAMS.
Theory
As a macrodiatonic scale
It is the macrodiatonic scale with the period of a tritave. This means it is a diatonic scale, but has octaves stretched out to the size of a tritave. Other intervals are also stretched in a way that makes the unrecognizable–the diatonic fifth is now the size of a major seventh. Interestingly, 19edt, an approximation of 12edo, has a tuning of this scale, meaning it contains both a diatonic scale (which approximates 12edo's diatonic scale) and a triatonic scale.
Temperament interpretations
Although they have not been studied in detail, it is possible to construct no-twos rank-2 temperament interpretations of this scale, such as the as-of-yet unnamed b12 & b5 temperament in the 3.13.17 subgroup, in which the generator (the stretched counterpart of the fifth) is ~17/9 and a stack of 4 generators tritave-reduced (the stretched counterpart of the major third) is ~13/9. See also the page for 12edt.
Modes
The modes have step patterns which are the same as the modes of the diatonic scale.
| UDP | Cyclic order |
Step pattern |
|---|---|---|
| 6|0 | 1 | LLLsLLs |
| 5|1 | 5 | LLsLLLs |
| 4|2 | 2 | LLsLLsL |
| 3|3 | 6 | LsLLLsL |
| 2|4 | 3 | LsLLsLL |
| 1|5 | 7 | sLLLsLL |
| 0|6 | 4 | sLLsLLL |
Scale degrees
| UDP | Cyclic order |
Step pattern |
Scale degree (mosdegree) | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |||
| 6|0 | 1 | LLLsLLs | Perf. | Maj. | Maj. | Aug. | Perf. | Maj. | Maj. | Perf. |
| 5|1 | 5 | LLsLLLs | Perf. | Maj. | Maj. | Perf. | Perf. | Maj. | Maj. | Perf. |
| 4|2 | 2 | LLsLLsL | Perf. | Maj. | Maj. | Perf. | Perf. | Maj. | Min. | Perf. |
| 3|3 | 6 | LsLLLsL | Perf. | Maj. | Min. | Perf. | Perf. | Maj. | Min. | Perf. |
| 2|4 | 3 | LsLLsLL | Perf. | Maj. | Min. | Perf. | Perf. | Min. | Min. | Perf. |
| 1|5 | 7 | sLLLsLL | Perf. | Min. | Min. | Perf. | Perf. | Min. | Min. | Perf. |
| 0|6 | 4 | sLLsLLL | Perf. | Min. | Min. | Perf. | Dim. | Min. | Min. | Perf. |
Notation
Being a macrodiatonic scale, it can notated using the traditional diatonic notation, if all intervals are reinterpreted as their stretched versions (like octaves as tritaves). However, this approach involves 1-based indexing for a non-diatonic MOS which is generally discouraged. Alternatively, a generic MOS notation may be used like diamond MOS notation, which enables 0-based indexing at the cost of obscuring the connection to the standard diatonic scale.