4L 5s (3/1-equivalent): Difference between revisions

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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 ~~[[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 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.

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 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.

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 leads to [[sensi]], in essence treating it as a "3.5.7.2-subgroup extension" of the original 3.5.7-subgroup BPS temperament.

== Modes ==

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 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&ndash;Pierce&ndash;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 [[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 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.
 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 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.
+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 leads to [[sensi]], in essence treating it as a "3.5.7.2-subgroup extension" of the original 3.5.7-subgroup BPS temperament.
 == Modes ==