Theoretical and experimental explorations of the Bohlen–Pierce scale

A musical scale based on the 3:5:7:9 tetrachord is described. It has clearly audible harmonic properties that are derived from the harmonies of the tetrachord. In its equal-tempered version, the nine tones of the scale are selected from a set of 13 equal steps, which have a frequency ratio equal to 31/13. The tenth note of the scale, here called the tritave, has a 1:3 frequency ratio with respect to the first note and has a role similar to that of the traditional octave. Both ‘‘major’’ and ‘‘minor’’ chords can be formed from the notes of the scale. Musicians and untrained listeners rated the consonance of all possible triads formed from a 1-tritave range of the equal-tempered chromatic scale. A wide range of consonance ratings was observed, and chords were judged as most dissonant when they had 1-step intervals. A critical bandwidth dissonance model fits the data well. The same subjects rated the consonance of a harmonized passage played in different tunings. Both groups judged the equal-tempered version as most consonant, but only the musicians were influenced by previous exposure to these sounds. Subjects also judged the similarity of chords and their inversions, both for chords from the new scale and for traditional chords. Musicians were influenced by key relationships, inversions, and chord type in their ratings of the traditional chord set but judged the new chords only by pitch height. Untrained listeners relied on pitch height for both chord sets. This suggests that the ability to abstract more complex information depends on training.