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

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4L 5s{{angbr|3/1}} can be thought of as a MOS generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[Bohlen–Pierce–Stearns]] (BPS), a 3.5.7-[[subgroup]] [[rank-2 temperament]] that tempers out [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and 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 Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain 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 meantone temperaments such as {{EDOs| 19, 31, and 43 }} and EDOs approximating Pythagorean tuning {{EDOs| 41 and 53 }} contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.

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 Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain 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 meantone temperaments such as {{EDOs| 19, 31, and 43 }} and EDOs approximating Pythagorean tuning ({{EDOs| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.

When playing this scale in some EDO, it may be desired to [[stretched and compressed tuning|stretch or compress the octaves]] to make 3/1 just (or closer to just), 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.

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 L&nbsp;5s{{angbr|3/1}} can be thought of as a MOS generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[Bohlen–Pierce–Stearns]] (BPS), a 3.5.7-[[subgroup]] [[rank-2 temperament]] that tempers out [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and 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 Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain 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 meantone temperaments such as {{EDOs| 19, 31, and 43 }} and EDOs approximating Pythagorean tuning {{EDOs| 41 and 53 }} contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.
+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 Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain 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 meantone temperaments such as {{EDOs| 19, 31, and 43 }} and EDOs approximating Pythagorean tuning ({{EDOs| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset.
 When playing this scale in some EDO, it may be desired to [[stretched and compressed tuning|stretch or compress the octaves]] to make 3/1 just (or closer to just), 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.