5L 2s (3/1-equivalent): Difference between revisions
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=== Temperament interpretations === | === Temperament interpretations === | ||
It is possible to construct no-twos [[rank-2 temperament]] interpretations of this scale, but it is difficult to interpret within commonly-studied [[no-twos subgroup temperaments|no-twos subgroup]]s like the 3.5.7 [[subgroup]] used for [[Bohlen-Pierce]]. Scales close to basic have an interpretation | It is possible to construct no-twos [[rank-2 temperament]] interpretations of this scale, but it is difficult to interpret within commonly-studied [[no-twos subgroup temperaments|no-twos subgroup]]s like the 3.5.7 [[subgroup]] used for [[Bohlen-Pierce]]. Two intervals that can serve as macrodiatonic generators 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 [[no-twos subgroup temperaments#Mizar|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 [[no-twos subgroup temperaments#Sadalmelik|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== | ==Modes== | ||
Revision as of 03:40, 11 October 2024
| ↖ 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–them 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, but it is difficult to interpret within commonly-studied no-twos subgroups like the 3.5.7 subgroup used for Bohlen-Pierce. Two intervals that can serve as macrodiatonic generators 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.