Bohlen-Pierce scale
The Bohlen-Pierce scale is a musical scale that offers an alternative to the equal temperament used in western music after Johann Sebastian Bach. It was discovered independently 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 discovering Bohlen's earlier publication. Despite being a harmonic scale, instead of an octave, a ratio of 2:1, the scale uses the "tritave", 3:1. One then usually divides this tritave into 13 equal tempered intervals, instead of the usual twelve.In addition to providing an alternative equal temperament which uses a pseudo-octave, the Bohlen-Pierce scale can also be viewed as an alternative just intonation. While the Bohlen-Pierce scale is currently most often played in equal temperament for practical reasons, that scale is an approximation of a just intonation system based only on ratios of odd whole numbers, similar to the spectra of instruments such as the clarinet which consist of primarly the odd harmonics. The 13 step equal temperament is an approximation of the original just 3:5:7, used as an alternative to the just 4:5:6 (a major chord, ex. C-E-G) approximated by twelve tone equal temperment.
An interesting, or frustrating, aspect of the Bohlen-Pierce scale is that traditional music theory and even's one intuition do not seem to apply. There are few examples of analogous phenomena between the equally tempered Bohlen-Pierce scale and regular twelve tone equal temperment.
The tritave is the frequency ratio 3:1 used in the Bohlen-Pierce scale as a pseudo-octave. In traditional terms it is the interval of an octave and a fifth.
