Bohlen–Pierce scale

The Bohlen–Pierce scale is a thirteen tone scale and musical tuning system.[2] It was first described in the 1970s. Instead of dividing the octave like a traditional scale, it spans an octave plus a perfect fifth.[3]

Background

Heinz Bohlen was curious why the octave should govern scales. After reading Paul Hindemith's explanation of tonality in The Craft of Musical Composition, Bohlen remained skeptical. He settled on combination tones as a model for a scale that would span a perfect twelfth.[4]: 5  In 1972, Bohlen developed a version of the scale in just intonation and equal temperament.[5]: 132 

Bohlen wrote about his invention in 1978.[6] That same year, software engineer Kees van Prooijen independently discovered the same scale.[4]: 15 [7] In 1984, John R. Pierce, Max Mathews, and Linda A. Roberts published their own discovery of the scale.[8][5]: 132  Pierce was the primary investigator of the project. Like Bohlen, he was also an electronic engineer by trade.[4]: 14–5 

In a 1989 book, Mathews and Pierce acknowledged Bohlen's earlier discovery of the scale and renamed it Bohlen-Pierce.[1]: 167 

Structure

Just intonation 4:5:6 chordⓘ Bohlen-Pierce 3:5:7 chordⓘ BP 3:5:7:9 chordⓘ

BP semitoneⓘ

The Bohlen-Pierce (BP) scale eschews the 2:1 ratio of the octave. The scale is governed instead by a 3:1 ratio, which yields an interval spanning an octave plus a perfect fifth. This interval is called a tritave and it totals 1901.955 cents.[3]

Instead of the traditional frequency ratio of 4:5:6 which forms a major chord, the BP scale is governed purely by odd overtones. The basic chord structure is 3:5:7. Just as the tonic, subdominant, and dominant chords can be used to derive a major scale, the BP system uses the chords in 3:5:7 ratios to form a scale with 13 steps.[8][1]: 170  Each step has a frequency ratio of 31/13.[9] There are thirteen possible BP keys. Modulation is possible through changing a single note.[1]: 169 

The 3:5:7 chord comprises a lower interval of 6 semitones and an upper interval of 4 semitones.[1]: 169  The overtones of a 3:5:7 chord coincide in similar ways as traditional major chords. The 5th partial of the first note coincides with the 3rd partial of the second. Overlapping harmonics are one of the signature features of consonance. A 1984 experiment found several BP chords were perceived as consonant by both musicians and untrained listeners.[10]

The equal tempered version of the BP scale only has 9 steps.[9] Heinz Bohlen developed his own 9-step modes of the scale which he named after Greek letters.[4]: 12–4 

Music

The Bohlen-Pierce scale may sound odd due to social conditioning.[11] Mathews and Pierce were certain that the scale was conducive to clear and memorable melodies. They were indifferent about its potential for counterpoint and functional harmony.[1]: 171–2 

Clarinet maker Stephen Fox created a family of instruments in Bohlen-Pierce tuning at the instigation of composer Georg Hajdu.[4]: iv 

Several composers have written with the Bohlen-Pierce scale:

See also

References

  1. ^ a b c d e f Mathews, Max V, and John R Pierce. Current Directions in Computer Music Research. MIT Press, 1989.
  2. ^ Pierce, John R. (2001). "Consonance and scales". In Cook, Perry R (ed.). Music, Cognition, and Computerized Sound: An Introduction to Psychoacoustics. MIT Press. p. 183. ISBN 978-0-262-53190-0.
  3. ^ a b c Benson, David J. Music: a mathematical offering. Cambridge University Press, 2007. 224–7.
  4. ^ a b c d e f Müller, Nora-Louise. The Bohlen-Pierce Clarinet: Theoretical Aspects and Contemporary Applications. Hochschule für Musik und Theater Hamburg, 2019.
  5. ^ a b Narushima, Terumi. Microtonality and the Tuning Systems of Erv Wilson. Taylor & Francis, 2017.
  6. ^ Bohlen, Heinz (1978). "13 Tonstufen in der Duodezime". Acoustica. 39 (2). Stuttgart: de:S. Hirzel Verlag: 76–86.
  7. ^ Prooijen, Kees van (1978). "A Theory of Equal-Tempered Scales". Interface. 7: 45–56. doi:10.1080/09298217808570248. Retrieved 27 November 2012.
  8. ^ a b Mathews, M.V.; Roberts, L.A.; Pierce, J.R. (1984). "Four new scales based on nonsuccessive-integer-ratio chords". J. Acoust. Soc. Am. 75, S10(A) (S1): S10. Bibcode:1984ASAJ...75...10M. doi:10.1121/1.2021272.
  9. ^ a b c Mathews, Max V, et al. "Theoretical and Experimental Explorations of the Bohlen–Pierce Scale." The Journal of the Acoustical Society of America, vol. 84, no. 4, October 1988. 1214–22.
  10. ^ Roberts, Linda A, and Max V Mathews. "Intonation Sensitivity for Traditional and Nontraditional Chords'", The Journal of the Acoustical Society of America, vol. 75, no. 3, March 1984. 952–59.
  11. ^ Benson, Dave. "Musical scales and the Baker's Dozen" (PDF). Musik og Matematik. 28/06: 16.
  12. ^ Thrall, Michael Voyne (Summer 1997). "Synthèse 96: The 26th International Festival of Electroacoustic Music". Computer Music Journal. 21 (2): 90–92 [91]. doi:10.2307/3681110. JSTOR 3681110.
  13. ^ "John Pierce (1910-2002)". Computer Music Journal. 26, No. 4 (Languages and Environments for Computer Music): 6–7. Winter 2002.
  14. ^ Discography of Microtonal CDs, Huygens-Fokker Foundation, retrieved 2016-12-13.
  15. ^ d'Escrivan, Julio (2007). Collins, Nick (ed.). The Cambridge Companion to Electronic Music. Cambridge Companions to Music. p. 229. ISBN 9780521868617.
  16. ^ "Concerts". Bohlen-Pierce-Conference.org. Retrieved 27 November 2012.
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