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
It is possible to construct no-twos rank-2 temperament interpretations of this scale, although most of these do not fit neatly into the 3.5.7 subgroup used for Bohlen-Pierce. Two intervals that can serve as macrodiatonic "fifths" are ~17/9, which is just near 19edt in the soft range, and ~21/11 which is just near 17edt in the hard range.
Very soft scales (in the range between 26edt and 45edt, serving as a macro-flattone) can be interpreted in the 3.5.7.17 subgroup as Mizar, in which the generator of a flattened ~17/9 stacks twice and tritave-reduces to 25/21, which generates Sirius temperament. Scales close to basic have an interpretation in the 3.13.17 subgroup, documented as Sadalmelik in which the generator (the stretched counterpart of the fifth) is also ~17/9 and a stack of 4 generators tritave-reduced (equivalent to the major third) is ~13/9; see also the page for 12edt. Harder scales can be interpreted in Mintaka temperament in the 3.7.11 subgroup, which tempers out 1331/1323 so that the dark generator (the stretched counterpart of the fourth) is ~11/7, a stack of 2 generators (equivalent to the minor seventh) is ~27/11, and a stack of three generators (equivalent to the minor third) is ~9/7.
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.
Intervals
| Intervals | Steps subtended |
Range in cents | ||
|---|---|---|---|---|
| Generic | Specific | Abbrev. | ||
| 0-mosstep | Perfect 0-mosstep | P0ms | 0 | 0.0 ¢ |
| 1-mosstep | Minor 1-mosstep | m1ms | s | 0.0 ¢ to 271.7 ¢ |
| Major 1-mosstep | M1ms | L | 271.7 ¢ to 380.4 ¢ | |
| 2-mosstep | Minor 2-mosstep | m2ms | L + s | 380.4 ¢ to 543.4 ¢ |
| Major 2-mosstep | M2ms | 2L | 543.4 ¢ to 760.8 ¢ | |
| 3-mosstep | Perfect 3-mosstep | P3ms | 2L + s | 760.8 ¢ to 815.1 ¢ |
| Augmented 3-mosstep | A3ms | 3L | 815.1 ¢ to 1141.2 ¢ | |
| 4-mosstep | Diminished 4-mosstep | d4ms | 2L + 2s | 760.8 ¢ to 1086.8 ¢ |
| Perfect 4-mosstep | P4ms | 3L + s | 1086.8 ¢ to 1141.2 ¢ | |
| 5-mosstep | Minor 5-mosstep | m5ms | 3L + 2s | 1141.2 ¢ to 1358.5 ¢ |
| Major 5-mosstep | M5ms | 4L + s | 1358.5 ¢ to 1521.6 ¢ | |
| 6-mosstep | Minor 6-mosstep | m6ms | 4L + 2s | 1521.6 ¢ to 1630.2 ¢ |
| Major 6-mosstep | M6ms | 5L + s | 1630.2 ¢ to 1902.0 ¢ | |
| 7-mosstep | Perfect 7-mosstep | P7ms | 5L + 2s | 1902.0 ¢ |