Bohlen-Pierce scale
From Wikipedia, the free encyclopedia
The Bohlen-Pierce scale (BP scale) is a musical scale that offers an alternative to the octave-repeating scales 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.
The BP scale pitch classes are based on odd integer frequency ratios. Specifically, they are based on ratios of integers whose factors are 3, 5, and 7. Thus the scale contains consonant harmonies based on the odd harmonic overtones 3/5/7/9. The fundamental role played by the 2:1 ratio (the octave) in conventional scales is instead played by the 3:1 ratio. This interval is a "perfect 12th" in diatonic nomenclature, but as this terminology is based on step sizes not used in the BP scale, it is often called by a new name, tritave, in BP contexts. In conventional scales, if a given pitch is part of the system, then all pitches one or more octaves higher or lower also are part of the system. In the BP scale, if a given pitch is present, then none of the pitches one or more octaves higher or lower are also present, but all pitches one or more tritaves higher or lower are part of the system. 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 diatonic scales (3:5:7 = 1:1.66:2.33 and 4:5:6 = 2:2.5:3 = 1:1.25:1.5).
A diatonic Bohlen-Pierce scale is constucted with the following ratios (chart shows the "Lambda" scale):
| C | D | E | F | G | H | J | A | B | C | |||||||||||
| Ratio | 1/1 | 25/21 | 9/7 | 7/5 | 5/3 | 9/5 | 15/7 | 7/3 | 25/9 | 3/1 | ||||||||||
| Step | T | s | s | T | s | T | s | T | s | |||||||||||
Though Bohlen originally expressed the BP scale in just intonation, a tempered form of the scale, which divides the tritave into thirteen equal steps, has become the most popular form. Each step is 31 / 13 = 1.08818... above the next, or about (log(31 / 13))1200 / log(2) = 146.3... cents per step. The octave is divided into a fractional number of steps. Twelve equally tempered steps per octave are used in 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.
The BP scale's use of odd integer ratios is 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 (or tritave) 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.
[edit] 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 3125/3087 (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 (1200/41 = 29.27 cents per step) 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.
[edit] External links
- Bohlen-Pierce Scale Research by Elaine Walker
- Bohlen-Pierce clarinets by Stephen Fox
- The Bohlen-Pierce Site: Web place of an alternative harmonic scale
- Kees van Prooijen's BP page
- Jyrki Alakuijala's experiments with an 8.19-tone equal temperament tuning system
| Tunings | edit | ||||
| Pythagorean · Just intonation · Harry Partch's 43-tone scale | |||||
| Regular temperaments | |||||
|
|||||
| Irregular temperaments | |||||
| Well temperament | |||||
