Editing Bohlen–Pierce scale (section)

==Bohlen–Pierce temperament==
[[Image:Bohlen-Pierce chromatic circle.png|thumb|"[[Chromatic circle]]" for the Bohlen–Pierce scale, with the third mode of the Lambda scale marked. The VII degree is mistakenly marked twice.<ref name="Cognition"/>]]

Bohlen originally expressed the BP scale in both [[just intonation]] and [[equal temperament]]. The [[Musical temperament|tempered]] form, which divides the tritave into thirteen equal steps, has become the most popular form. Each step is {{radic|3|13}} = 3<sup>{{frac|1|13}}</sup> = 1.08818… above the next, or 1200&nbsp;log<sub>2</sub>&nbsp;(3<sup>{{frac|1|13}}</sup>) = 146.3… [[cent (music)|cents]] per step. The octave is divided into a fractional number of steps. Twelve equally tempered steps per octave are used in [[equal temperament#Twelve-tone equal temperament|12-tet]]. The Bohlen–Pierce scale could be described as 8.202087-tet, because a full octave (1200 cents), divided by 146.3… cents per step, gives 8.202087 steps per octave.

Dividing the tritave into 13 equal steps tempers out, or reduces to a unison, both of the intervals 245:243 (about 14 cents, sometimes called the minor Bohlen–Pierce [[diesis]]) and 3125:3087 (about 21 cents, sometimes called the major Bohlen–Pierce diesis) in the same way that dividing the octave into 12 equal steps reduces both 81:80 ([[syntonic comma]]) and 128:125 ([[5-limit limma]]) to a unison. A [[regular temperament|7-limit linear temperament]] tempers out both of these intervals; the resulting ''Bohlen–Pierce temperament'' no longer has anything to do with tritave equivalences or non-octave scales, beyond the fact that it is well adapted to using them. A tuning of [[41 equal temperament|41 equal steps to the octave]] ({{frac|1200|41}} = 29.27 cents per step) would be quite logical for this temperament. In such a tuning, a tempered perfect twelfth (1902.4 cents, about a half cent larger than a just twelfth) is divided into 65 equal steps, resulting in a seeming paradox: Taking every fifth degree of this octave-based scale yields an excellent approximation to the non-octave-based equally tempered BP scale. Furthermore, an interval of five such steps generates (octave-based) [[Generated collection|MOS]]es (moments of symmetry) with 8, 9, or 17 notes, and the 8-note scale (comprising degrees 0, 5, 10, 15, 20, 25, 30, and 35 of the 41-equal scale) could be considered the octave-equivalent version of the Bohlen–Pierce scale.