Posted by midiguru on October 30, 2011
Why has the Bohlen-Pierce scale been so widely embraced? (For very small values of “widely,” to be sure — but still.)
I’m not sure what the attractions of this scale may be. I’ve been trying to keep an open mind, but I have to say, it’s an exceptionally ugly scale … and you’re hearing this from a guy who writes music with 17 equal steps in the octave, a scale that probably causes tumors in mice.
The original form of BP, as I’ll call it, is a form of just intonation in which there are no octaves. All of the ratios in the scale use odd numbers. The royal seat of the octave is usurped by the “tritave,” an interval with a ratio of 3:1 — what we would call an octave and a fifth, or a twelfth.
The tritave is parceled out into 13 steps, which vary somewhat in size. These steps repeat at the interval of the tritave. (There’s also an equal-tempered version of BP, but it’s based on the tritave too.) The smallest of the steps are larger than the half-steps in our standard 12-note equal temperament, with the result that melodies in BP tend to sound rather gawky, as if their elbows were sticking out.
The major triad, in forms of just intonation that don’t avoid the octave, is defined by the ratios 4:5:6. In BP, the analogous structure uses the ratios 3:5:7. If you were to approximate this on a conventional keyboard, starting with C as the “root”, the structure would consist of C, a slightly low A (with a ratio of 5:3 to C), and, in the next octave, a rather low E-flat (a ratio of 7:3 over the C).
What I discovered today, while poking around and trying to keep an open mind, is that this structure doesn’t invert well. Because of the ease with which the ear perceives octaves, a conventional C-E-G triad can be inverted, moving the lowest note up so that it’s above the other two, in an easy, pleasant-sounding way. C-E-G becomes E-G-C and then G-C-E, and the ear has no trouble hearing that these are all permutations of the same underlying structure.
When I tried this with the BP “triad,” the results were disturbing. To invert the “triad,” we have to move the bottom note up by a twelfth, so C-A-Eb (remember, the second and third note names are only approximations) inverts to A-Eb-G and then to Eb-G-E. In the second inversion, we hear a close approximation of what in conventional tuning would be called a minor ninth. The bottom pitch is 7:3 over the original C, and the top pitch is 5:1 over the original C, making the interval ratio of the outer notes 15:7. (14:7 would be an octave.)
The BP “triad” in its root position is unsettling enough: The implied fundamental beneath C-A-Eb is the F a twelfth below the C, so we’re hearing a higher slice of the natural overtone series. Add to that the clash in the second inversion, the large size of the chromatic half-step, and the complete avoidance of octaves, and you have a scale that only a mother could love.
And yet, people are boring clarinets and building keyboards to play BP music. Oh, well. A lot of people like Philip Glass too.