The Bohlen-Pierce scale is an alternative musical scale, arrived at by dividing the harmonic twelfth (a 3:1 frequency ratio) into thirteen steps. The result is an exotic sequence of tones providing numerous consonant intervals and hence the promise of extensive musical possibilities to those willing to explore non-traditional sounds.
The history, construction and previous musical application of the scale are described in detail on a website maintained by its co-inventor, Heinz Bohlen.
While the mathematics of the Bohlen-Pierce scale have been analyzed in some depth over the last thirty years or so, so far the body of music created on this basis is still small, and the field is wide open to exploration by composers.
Though both just and equal tempered Bohlen-Pierce scales have been postulated, the equal tempered version would normally be chosen for tuning instruments of fixed pitch. (As with the conventional scale, when playing instruments of variable or partially variable pitch, notes are tempered by the player in order to bring intervals and chords into tune in a particular situation.)
Each equal tempered Bohlen-Pierce scale step has a frequency ratio of the 13th root of 3, or approximately 1.088 (corresponding to roughly 146 cents on the conventional 12 tone scale).
Consonant Bohlen-Pierce intervals (see below) are mostly based on odd number ratios: 3:1, 5:3, 7:3, 7:5, 9:5, etc. Some of these intervals (3:1 = perfect twelfth, 5:3 = just major sixth, 6:5 = minor third, etc.) are familiar, while others are not heard, or at least not recognised as consonances, in conventional Western music. The octave has no special status in this tuning system.
For present purposes, 440Hz (or 442Hz in some countries) has been chosen as the central reference pitch. On this basis, the pitches of the equal tempered Bohlen-Pierce scale, and their relation to the conventional 12 tone scale, are as given below.
At present, neither a system of notation nor a method of naming the notes of the Bohlen-Pierce chromatic scale has yet been universally accepted, but suggested note names are given along with the pitch chart. In order to prevent confusion with conventional scales, the letters of the Greek alphabet are used; the thirteen notes of the scale are α (alpha), β (beta), γ (gamma), δ (delta), ε (epsilon), ζ (zeta), η (eta), θ (theta), ι (iota), κ (kappa), λ (lambda), μ (mu), ν (nu). The name α is chosen for our reference pitch, 440Hz, and for the tritaves above and below that pitch. Where necessary, in order to identify the the tritaves in which notes are located, numbers can be added to the letter names (e.g., β2).
One possible notation system, applicable to scores and parts for all instruments and voices, is also given below. (This is distinct from the pragmatic fingering-based notation systems specific to clarinets and recorders, which are described on the instruments page of this site.)
N.B.- The numbers after the note names are only needed when it is necessary to identify the tritave in which a note is located; otherwise, they can be omitted (as in conventional musical terminology).
*, **, *** the number of asterisks indicates the closeness of consonance, and hence the amount of adjustment necessary when playing (except for the tritave, which is perfect)
[ ] intervals in square brackets are not usually recognised as BP intervals, but they are available nonetheless
For other interval names, see www.xs4all.nl/~huygensf/doc/intervals.html
noted previously, methods of notation of Bohlen-Pierce music are still
evolving, but the following system is suggested for notating Bohlen-Pierce
scores and individual instrumental or vocal parts in concert pitch.
For simplicity, normal five line staves are used, with one leger line between
adjacent staves; each (chromatic) scale step is represented by moving up
or down one line or space (with no accidentals). The anticipated
total instrumental range requires three staves, with clef symbols (S =
soprano, A = alto, B = bass) chosen to prevent confusion with conventional