Bohlen–Pierce scale

The Bohlen–Pierce scale (BP scale) is a musical tuning and scale, first described in the 1970s, that offers an alternative to the octave-repeating scales typical in Western and other musics, specifically the equal tempered diatonic scale.

The interval 3:1 (often called by a new name, tritave) serves as the fundamental harmonic ratio, replacing the diatonic scale's 2:1 (the octave). For any pitch that is part of the BP scale, all pitches one or more tritaves higher or lower are part of the system as well, and are considered equivalent.

The BP scale divides the tritave into 13 steps, either equal tempered (the most popular form), or in a justly tuned version. Compared with octave-repeating scales, the BP scale's intervals are more consonant with certain types of acoustic spectra.

The scale was independently described by Heinz Bohlen,Kees van Prooijen and John R. Pierce. Pierce, who, with Max Mathews and others, published his discovery in 1984, renamed the Pierce 3579b scale and its chromatic variant the Bohlen–Pierce scale after learning of Bohlen's earlier publication. Bohlen had proposed the same scale based on consideration of the influence of combination tones on the Gestalt impression of intervals and chords.

The intervals between BP scale pitch classes are based on odd integer frequency ratios, in contrast with the intervals in diatonic scales, which employ both odd and even ratios found in the harmonic series. Specifically, the BP scale steps 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 ( play ). The chord formed by the ratio 3:5:7 ( play ) serves much the same role as the 4:5:6 chord (a major triad  play ) 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).

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