BPS: Difference between revisions

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-'''Bohlen-Pierce-Stearns''' ('''BPS''') is a [[regular temperament]] in the 3.5.7 [[subgroup]] where [[~]][[7/3]] is a [[generator]], and the comma [[245/243]] is [[tempering out|tempered out]],
+'''Bohlen-Pierce-Stearns''' (BPS) is a [[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, which generates a [[MOS scale]] of [[4L 5s (3/1-equivalent)|4L 5s]] against the tritave, known as the Bohlen-Pierce ''Lambda'' scale. The "canonical" tuning for the generator is [[13edt|3\13]]edt, representing the equal-tempered [[Bohlen-Pierce]] scale, but a range of other tunings are valid, including [[17edt|4\17]]edt, [[30edt|7\30]]edt, and [[43edt|10\43]]edt.
-For technical data, see [[Sensamagic clan#BPS]].
+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.
+For technical data, see ''[[Sensamagic clan#BPS]]''.
+== Extensions ==
+Several extensions of this temperament are possible to incorporate additional harmonics.
+== Interval chains ==
+<div><div style="display: inline-grid; margin-right: 25px;">
+{| class="wikitable center-1 right-2"
+|+ style="font-size: 105%;" | Basic 3.5.7 temperament
+|-
+! &#35; !! Cents&#42; !! Approximate Ratios
+|-
+| 0 || 0.0 || '''1/1'''
+|-
+| 1 || 440.6 || 9/7, 35/27
+|-
+| 2 || 881.1 || 5/3, 81/49
+|-
+| 3 || 1321.7 || 15/7, 175/81
+|-
+| 4 || 1762.2 || 25/9, 135/49
+|-
+| 5 || 300.8 || 25/21, 405/343, 875/729
+|-
+| 6 || 741.4 || 75/49, 125/81
+{{table notes|cols=3
+| In 3.5.7-subgroup [[DKW theory|DKW]] tuning
+}}
+|}
+</div>
+=== Other tunings ===
+* [[DKW theory|DKW]] (3.5.7): ~3 = 1\1, ~9/7 = 440.554
 [[Category:Temperaments]]