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

}} This scale is oftsn suggested for use as [[Bohlen–Pierce]]'s analog of the [[5L 2s|diatonic scale]], which is the 4L 5s mos with tritave equivalence. 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)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 [[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 this 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 leads to [[sensi]], in essence treating it as a "3.5.7.2-subgroup" extension of BPS.

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}} 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|seen on Wikipedia}} and certain articles of this wiki) labels them {{nowrap|C D E F G H J A B}} in the C Andromedan (LssLsLsLs) mode, which rotates to the E symmetric mode.

Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 cents and 475.5 cents.

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]].