4L 5s (3/1-equivalent): Difference between revisions
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| Other Names = Lambda | | Other Names = Lambda | ||
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Suggested for use as the analog of the [[5L 2s|diatonic scale]] when playing [[ | 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 [[Bohlen–Pierce–Stearns]] that tempers out 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 edos that support BPS are {{EDOs| 19, 22, 27, 41, and 46 }}, and some low-numbered edts that support it are {{EDTs| 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 of 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. | Some low-numbered edos that support BPS are {{EDOs| 19, 22, 27, 41, and 46 }}, and some low-numbered edts that support it are {{EDTs| 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 of 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. | ||
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== Notation == | == Notation == | ||
Bohlen–Pierce theory possesses a well-established [[nonoctave]] notation system for [[EDT]]s and no-twos music, which is based on this MOS scale as generated by approximately [[7/3]], relating it to [[Bohlen–Pierce–Stearns]] temperament, where two 7/3 generators are equated to 27/5. The preferred generator for any edt is its patent val approximation of 7/3. | |||
This notation uses 9 nominals: for compatibility with [[diamond-MOS notation]], the current recommendation is to use the notes {{nowrap|J K L M N O P Q R J}} as presented in the J Cassiopeian (symmetric, sLsLsLsLs) mode, and represented by a circle of generators going as follows: {{dash|...Q♯, O♯, M♯, K♯, R, P, N, L, J, Q, O, M, K, R♭, P♭, N♭, L♭...|hair|med}} However, an alternative convention ({{w| | This notation uses 9 nominals: for compatibility with [[diamond-MOS notation]], the current recommendation is to use the notes {{nowrap|J K L M N O P Q R J}} as presented in the J Cassiopeian (symmetric, sLsLsLsLs) mode, and represented by a circle of generators going as follows: {{dash|...Q♯, O♯, M♯, K♯, R, P, N, L, J, Q, O, M, K, R♭, P♭, N♭, L♭...|hair|med}} However, an alternative convention ({{w|Bohlen–Pierce scale#Intervals and scale diagrams|as used on Wikipedia}} and certain articles of this wiki) labels them {{nowrap|C D E F G H J A B C}} in the C Andromedan (LssLsLsLs) mode, which rotates to the E symmetric mode. | ||
An extension of [[ups and downs notation]], in the obvious way, can be found at [[Lambda ups and downs notation]]. | An extension of [[ups and downs notation]], in the obvious way, can be found at [[Lambda ups and downs notation]]. | ||
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|+ style="font-size: 105%;" | 4L 5s in [[13edt]] ( | |+ style="font-size: 105%;" | 4L 5s in [[13edt]] (Bohlen–Pierce) | ||
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== List of edts supporting the Lambda scale == | == 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 equal temperaments which contain a 4L 5s scale using generators between 422.7 cents and 475.5 cents. | ||
{{Scale tree|depth=7|Comments=13/6: [[ | {{Scale tree|depth=7|Comments=13/6: [[Bohlen–Pierce–Stearns]] is in this region; 22/13: Essentially just 7/3}} | ||
Analogously to how the diatonic scale equalizes approaching [[7edo]] and its small steps collapse to 0 in [[5edo]], this scale equalizes approaching [[9edt]] and its small steps collapse in [[4edt]]; therefore, temperaments setting the 7/3 generator to precisely 7\9edt and to precisely 3\4edt are analogs of [[whitewood]] and [[blackwood]] respectively; however, unlike for the diatonic scale, the just point is not close to the center of the tuning range, but approximately 1/4 of the way between 9edt and 4edt, being closely approximated by 37\[[48edt]] and extremely closely approximated by 118\[[153edt]]. | Analogously to how the diatonic scale equalizes approaching [[7edo]] and its small steps collapse to 0 in [[5edo]], this scale equalizes approaching [[9edt]] and its small steps collapse in [[4edt]]; therefore, temperaments setting the 7/3 generator to precisely 7\9edt and to precisely 3\4edt are analogs of [[whitewood]] and [[blackwood]] respectively; however, unlike for the diatonic scale, the just point is not close to the center of the tuning range, but approximately 1/4 of the way between 9edt and 4edt, being closely approximated by 37\[[48edt]] and extremely closely approximated by 118\[[153edt]]. | ||