BPS: Difference between revisions

'''BPS''' (for ''Bohlen–Pierce–Stearns'') is a [[non-octave]] [[regular temperament|temperament]] in the 3.5.7 [[subgroup]] generated by a sharp [[~]][[9/7]] (or equivalently a flat ~[[7/3]]), [[tempering out]] the sensamagic comma ([[245/243]]) so that a stack of two generators represents [[5/3]] in addition to 81/49. This generates a [[mos scale]] of {{sl|4L 5s}} against the tritave, known as the Bohlen–Pierce ''Lambda'' scale. The "canonical" tuning for the generator is [[13edt|3\13edt]], representing the equal-tempered [[Bohlen–Pierce]] scale, but a range of other tunings are valid, including [[17edt|4\17edt]], [[30edt|7\30edt]], and [[43edt|10\43edt]].

As the generator of the Bohlen–Pierce scale, and the simplest decently accurate temperament of the 3.5.7 subgroup, this temperament fulfills a niche similar to [[meantone]] of the 2.3.5 subgroup, allowing for the tetrad 3:5:7:9 to serve as the theory's primary consonant tetrad.

In the [[11-limit]], [[1331/1323]] is the most convenient comma that can be tempered out, which produces [[mintra]] temperament; this splits the 9/7 generator (plus a tritave) in three and therefore functions instead as a weak extension of BPS, and a strong add-5 extension of [[mintaka]], which produces {{sl|5L 2s}} and {{sl|5L 7s}} mos scales (functioning as a macro-[[superpyth]]). Simple tunings include [[17edt]] and [[39edt]].

Another weak extension to add prime 17, known as ''[[dubhe]]'', splits the 9/7 BPS generator in half, by tempering out [[2025/2023]] and equating two of [[17/15]] to 9/7. This produces {{sl|8L 1s}} enneatonic and {{sl|9L 8s}} chromatic mos scales. Simple tunings include [[17edt]] and [[26edt]].

== Tuning spectrum ==

! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*

| [[7/3]]

| 0-comma

| [[49/45]]

| [[7/5]]

* [[DKW theory|DKW]] (3.5.7): ~3 = 1901.955, ~9/7 = 440.554

 

[[Category:Bohlen-Pierce]]

[[Category:Temperaments]]

[[Category:Tritave-equivalent temperaments]]