4L 5s (3/1-equivalent): Difference between revisions
m Removed categories to test out infobox mos adding them |
Massive terminological update |
||
| Line 1: | Line 1: | ||
{{Infobox MOS | {{Infobox MOS | ||
| | | Other names = Lambda | ||
}} | }} | ||
{{MOS intro}} | {{MOS intro | ||
Suggested for use as | | Other Names = Lambda | ||
This is a very good temperament on the 3.5.7 | }} | ||
Suggested for use as the analog of the [[5L 2s|diatonic scale]] when playing [[Bohlen-Pierce]] is this 9-note Lambda scale, which is the 4L 5s mos with [[equave]] 3/1. This can be thought of as a mos generated by a 3.5.7-[[subgroup]] [[rank-2 temperament]] called [[BPS|BPS (Bohlen-Pierce-Stearns)]] that eliminates only the comma [[245/243]], so that (9/7)<sup>2</sup> is equated with 5/3. This is a very good temperament on the 3.5.7 subgroup, and additionally is supported by many [[edt]]'s (and even [[edo]]s!) besides [[13edt]]. | |||
Some low-numbered | Some low-numbered edos that support BPS are {{EDOs| 19, 22, 27, 41, and 46 }}, and some low-numbered edts that support it are [[9edt|9]], [[13edt|13]], [[17edt|17]], and [[30edt|30]], all of which make it possible to play BP music to some reasonable extent. These equal temperaments contain not only the Lambda "BP diatonic" scale, but, with the exception 9edt, also the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a "detempered" version of the 13edt Bohlen-Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how 19edo and 31edo do not contain 12edo as a subset, but they do contain the meantone[12] chromatic scale. | ||
When playing this temperament in some | When playing this temperament in some edo, it may be desired to [[stretched and compressed tuning|stretch/compress the tuning]] so that the tritave is pure, rather than the octave being pure - or in general, to minimize the error on the 3.5.7 subgroup while ignoring the error on 2/1. | ||
One can | One can add the octave to BPS temperament by simply creating a new mapping for 2/1. A simple way to do so is to map the 2/1 to +7 of the ~9/7 generators, minus a single tritave. This is [[sensi]] temperament, in essence treating it as a "3.5.7.2-subgroup extension" of the original 3.5.7-subgroup BPS temperament. | ||
== Modes == | == Modes == | ||
{{MOS modes}} | {{MOS modes}} | ||
== List of | == List of edts supporting the Lambda scale == | ||
Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 cents and 475.5 cents. | |||
Below is a list of | |||
{{Scale tree|depth=7|Comments=9/4:BPS is in this region}} | {{Scale tree|depth=7|Comments=9/4:BPS is in this region}} | ||
Schism, by which I<sup>[''who?'']</sup> mean, the most accurate value for 5/3 and-or 7/3 is found outside the 4L 5s MOS. | |||
Also, the way I see it, as 4edt and 9edt are comparable to 5edo and 7edo, then the "counterparts" of Blackwood and Whitewood would be found in multiples therein and would be octatonic and octadecatonic, e.g. 12edt and 27edt.{{clarify}} | |||
Revision as of 07:45, 20 May 2024
| ↖ 3L 4s⟨3/1⟩ | ↑ 4L 4s⟨3/1⟩ | 5L 4s⟨3/1⟩ ↗ |
| ← 3L 5s⟨3/1⟩ | 4L 5s (3/1-equivalent) | 5L 5s⟨3/1⟩ → |
| ↙ 3L 6s⟨3/1⟩ | ↓ 4L 6s⟨3/1⟩ | 5L 6s⟨3/1⟩ ↘ |
┌╥┬╥┬╥┬╥┬┬┐ │║│║│║│║│││ │││││││││││ └┴┴┴┴┴┴┴┴┴┘
ssLsLsLsL
4L 5s⟨3/1⟩, also called Lambda, is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 4 large steps and 5 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 422.7 ¢ to 475.5 ¢, or from 1426.5 ¢ to 1479.3 ¢. Suggested for use as the analog of the diatonic scale when playing Bohlen-Pierce is this 9-note Lambda scale, which is the 4L 5s mos with equave 3/1. This can be thought of as a mos generated by a 3.5.7-subgroup rank-2 temperament called BPS (Bohlen-Pierce-Stearns) that eliminates only the comma 245/243, so that (9/7)2 is equated with 5/3. This is a very good temperament on the 3.5.7 subgroup, and additionally is supported by many edt's (and even edos!) besides 13edt.
Some low-numbered edos that support BPS are 19, 22, 27, 41, and 46, and some low-numbered edts that support it are 9, 13, 17, and 30, all of which make it possible to play BP music to some reasonable extent. These equal temperaments contain not only the Lambda "BP diatonic" scale, but, with the exception 9edt, also the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a "detempered" version of the 13edt Bohlen-Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how 19edo and 31edo do not contain 12edo as a subset, but they do contain the meantone[12] chromatic scale.
When playing this temperament in some edo, it may be desired to stretch/compress the tuning so that the tritave is pure, rather than the octave being pure - or in general, to minimize the error on the 3.5.7 subgroup while ignoring the error on 2/1.
One can add the octave to BPS temperament by simply creating a new mapping for 2/1. A simple way to do so is to map the 2/1 to +7 of the ~9/7 generators, minus a single tritave. This is sensi temperament, in essence treating it as a "3.5.7.2-subgroup extension" of the original 3.5.7-subgroup BPS temperament.
Modes
| UDP | Cyclic order |
Step pattern |
|---|---|---|
| 8|0 | 1 | LsLsLsLss |
| 7|1 | 3 | LsLsLssLs |
| 6|2 | 5 | LsLssLsLs |
| 5|3 | 7 | LssLsLsLs |
| 4|4 | 9 | sLsLsLsLs |
| 3|5 | 2 | sLsLsLssL |
| 2|6 | 4 | sLsLssLsL |
| 1|7 | 6 | sLssLsLsL |
| 0|8 | 8 | ssLsLsLsL |
List of edts supporting the Lambda scale
Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 cents and 475.5 cents.
Schism, by which I[who?] mean, the most accurate value for 5/3 and-or 7/3 is found outside the 4L 5s MOS.
Also, the way I see it, as 4edt and 9edt are comparable to 5edo and 7edo, then the "counterparts" of Blackwood and Whitewood would be found in multiples therein and would be octatonic and octadecatonic, e.g. 12edt and 27edt.[clarification needed]