Bohlen–Pierce scale
The BP scale divides the tritave into 13 steps, either equal tempered (the most popular form), or in a justly tuned version.
[citation needed] The scale was independently described by Heinz Bohlen,[2] Kees van Prooijen[3] and John R. Pierce.
Bohlen had proposed the same scale based on consideration of the influence of combination tones on the Gestalt impression of intervals and chords.
[7] A study of chromatic triads formed from arbitrary combinations of the 13 tones of the chromatic scale among twelve musicians and twelve untrained listeners found semitones 0, 1, 2 to be the most dissonant chord (playⓘ), but 0, 11, 13 (playⓘ) was considered the most consonant by the trained subjects (because it sounds like an octave-dropped major triad) and 0, 7, 10 (playⓘ) was judged most consonant by the untrained subjects.
(playⓘ) This interval is a perfect twelfth in diatonic nomenclature (perfect fifth when reduced by an octave), but as this terminology is based on step sizes and functions not used in the BP scale, it is often called by a new name, tritave (playⓘ), in BP contexts, referring to its role as a pseudooctave, and using the prefix "tri-" (three) to distinguish it from the octave.
Dividing the tritave into 13 equal steps tempers out, or reduces to a unison, both of the intervals 245:243 (about 14 cents, sometimes called the minor Bohlen–Pierce diesis) and 3125:3087 (about 21 cents, sometimes called the major Bohlen–Pierce diesis) in the same way that dividing the octave into 12 equal steps reduces both 81:80 (syntonic comma) and 128:125 (5-limit limma) to a unison.
In such a tuning, a tempered perfect twelfth (1902.4 cents, about a half cent larger than a just twelfth) is divided into 65 equal steps, resulting in a seeming paradox: Taking every fifth degree of this octave-based scale yields an excellent approximation to the non-octave-based equally tempered BP scale.
Dave Benson suggests it helps to use only sounds with only odd harmonics, including clarinets or synthesized tones, but argues that because "some of the intervals sound a bit like intervals in [the more familiar] twelve-tone scale, but badly out of tune", the average listener will continually feel "that something isn't quite right", due to social conditioning.
[11] In their 1989 study of consonance judgment, both intervals of the five chords rated most consonant by trained musicians are approximately diatonic intervals, suggesting that their training influenced their selection and that similar experience with the BP scale would similarly influence their choices.
[12] Other computer composers to use the BP scale include Jon Appleton, Richard Boulanger (Solemn Song for Evening (1990)), Georg Hajdu, Juan Reyes' ppP (1999-2000),[13] Ami Radunskaya's "A Wild and Reckless Place" (1990),[14] Charles Carpenter (Frog à la Pêche (1994) & Splat),[15][16] and Elaine Walker (Stick Men (1991), Love Song, and Greater Good (2011)).
[17] David Lieberman, an associate professor of architecture at the University of Toronto, directed the development of a "Stredici", a string instrument tuned to the Bohlen–Pierce scale.
The symposium participants, which included Heinz Bohlen, Max Mathews, Clarence Barlow, Curtis Roads, David Wessel, Psyche Loui, Richard Boulanger, Georg Hajdu, Paul Erlich, Ron Sword, Julia Werntz, Larry Polansky, Manfred Stahnke, Stephen Fox, Elaine Walker, Todd Harrop, Gayle Young, Johannes Kretz, Arturo Grolimund, Kevin Foster, presented 20 papers on history and properties of the Bohlen–Pierce scale, performed more than 40 compositions in the novel system and introduced several new musical instruments.
Performers included German musicians Nora-Louise Müller and Ákos Hoffman on Bohlen–Pierce clarinets and Arturo Grolimund on Bohlen–Pierce pan flute as well as Canadian ensemble tranSpectra, and US American xenharmonic band ZIA, led by Elaine Walker.
