Bohlen-Pierce scale
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The Bohlen-Pierce scale (BP scale) is a musical scale that offers an alternative to the 12-tone equal temperament typical in Western music. It was independently discovered by Heinz Bohlen, Kees van Prooijen, and also John R. Pierce. Pierce, who, with Max V. Mathews and others, published his discovery in 1984, renamed the scale the Bohlen-Pierce scale after learning of Bohlen's earlier publication.
While most scales have octave-equivalence, the BP scale instead has tritave equivalence. This means that its pitch classes are based on the interval 3:1 (tritave, or "perfect 12th" in diatonic nomenclature) rather than the 2:1 (octave). Thus the scale contains many consonant harmonies. A tritave is equivalent to a full octave plus a perfect fifth.
Though Bohlen originally expressed the BP scale in just intonation, a tempered form of the scale, which divides the tritave into 13 equal steps, has become the most popular form. In it, the chord formed by the ratio 3:5:7 serves much the same role as the 4:5:6 chord (a major triad) does in 12-tone equal temperament (3:5:7 = 1:1.66:2.33 and 4:5:6 = 2:2.5:3 = 1:1.25:1.5). This tempered BP scale can be seen as an approximation of a just intonation system based only on ratios of odd whole numbers, appropriate for timbres containing only odd harmonics.
Because the clarinet's spectrum (in the chalumeau register) consists of primarily the odd harmonics, and the instrument overblows at the twelfth rather than the octave as most other woodwind instruments do, there is a natural affinity between the clarinet and the Bohlen-Pierce scale. In early 2006 clarinet maker Stephen Fox began offering Bohlen-Pierce soprano clarinets for sale, and lower pitched instruments ("tenor" and "contra") are being developed.
Bohlen-Pierce temperament
Dividing the tritave into 13 equal steps tempers out, or reduces to a unison, both of the intervals 245/243 (sometimes called the minor Bohlen-Pierce diesis) and 3125/3072 (sometimes called the major Bohlen-Pierce diesis) in the same way that dividing the octave into 12 equal steps reduces both 81/80 and 128/125 to a unison. One can produce a 7-limit linear temperament by tempering 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. However, a tuning of 41 equal steps to the octave and a scale of 8, 9 or 17 steps would be quite logical for this temperament, and the 8 step scale could be considered the octave-equivalent version of the Bohlen-Pierce scale.
The octave is divided into a fractional number of steps. While 12 equally tempered steps are used in 12-tet, the Bohlen-Pierce scale could be described as 8.202087-tet, because a fractional number of 8.202087 steps produce an octave.