Bohlen-Pierce scale

The Bohlen-Pierce scale (BP scale) is a musical scale that offers an alternative to the octave-repeating scales typical in Western and other musics, specifically the diatonic scale. In addition, compared with octave-repeating scales, its intervals are more consonant with certain types of acoustic spectra. It was independently described by Heinz Bohlen [H. Bohlen, "13 Tonstufen in der Duodezeme," "Acoustica" 39, 76-86 (1978).] , Kees van Prooijen, and John Pierce. Pierce, who, with Max Mathews and others, published his discovery in 1984 [M.V. Mathews, L.A. Roberts, and J.R. Pierce, "Four new scales based on nonsuccessive-integer-ratio chords," "J. Acoust. Soc. Amer." 75, S10(A) (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 on the basis of combination tonesMax V. Mathews and John R. Pierce (1989). "The Bohlen-Pierce Scale", p.167. "Current Directions in Computer Music Research", Max V. Mathews and John R. Pierce, eds. MIT Press.] .

The intervals between BP scale pitch classes are based on odd integer frequency ratios, in contrast with the intervals in diatonic scales, which may be considered as based on 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 (Audio|BP chord just.mid|play). The chord formed by the ratio 3:5:7 (Audio|BP chord 357 just.mid|play) serves much the same role as the 4:5:6 chord (a major triad Audio|Just major triad on C.mid|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).

Chords and modulation

The perceptibility of the harmonic basis of the BP scale is suggested by 3:5:7 having a very similar pattern of intonation sensitivity to 4:5:6 (the just major chord), more similar than that of the minor chordMathews and Pierce (1989). "The Bohlen-Pierce Scale", p.165-66.] .

The 3:5:7 chord may thus be considered the major triad of the BP scale. It is approximated by an interval of six 6 equal tempered BP semitones (Audio|BP 0 1 on C.mid|play one semitone) on bottom and an interval of 4 equal tempered semitones on top (semitones: 0,6,10; Audio|BP major triad on C.mid|play). A minor triad is thus 6 semitones on top and 4 semitones on bottom (0,4,10; Audio|BP minor triad on C.mid|play). 5:7:9 is the first inversion of the the major triad (6,10,13; Audio|BP 579 or 1st inversion chord.mid|play).Mathews and Pierce (1989). "The Bohlen-Pierce Scale", p.169.]

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 0,1,2 (semitones) to be the most dissonant chord (Audio|BP 0 1 2 on C.mid|play) but 0,11,13 (Audio|BP 0 11 13 on C.mid|play) was considered the most consonant by the trained subjects and 0,7,10 (Audio|BP 0 7 10 on C.mid|play) was judged most consonant by the untrained subjects.Mathews and Pierce (1989). "The Bohlen-Pierce Scale", p.171.]

Every tone of the Pierce 3579b scale is in a major and minor triad except for tone II of the scale. There are thirteen possible keys and modulation is possible through changing a single note, in this case moving note II up one semitone causes the tonic to rise to what was note III (semitone: 3), which is considered the dominant. VIII (semitone: 10) is considered the subdominant.

Timbre and the tritave

The fundamental role played by the 2:1 ratio (the octave (Audio|Perfect octave on C.mid|play)) in conventional scales is instead played by the 3:1 ratio. 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" (Audio|Tritave on C.mid|play), in BP contexts, referring to its role as a pseudooctave, and using the prefix "tri-" (three) to distinguish it from the octave. In conventional scales, if a given pitch is part of the system, then all pitches one or more octaves higher or lower also are part of the system and, furthermore, are considered equivalent. In the BP scale, if a given pitch is present, then "none" of the pitches one or more octaves higher or lower are also present and equivalent, but "all" pitches one or more tritaves higher or lower are part of the system and considered equivalent.

The BP scale's use of odd integer ratios is appropriate for timbres containing only odd harmonics. Because the clarinet's spectrum (in the chalumeau register) consists of primarily the odd harmonics, and the instrument overblows at the twelfth (or tritave) rather than the octave as most other woodwind instruments do, there is a natural affinity between the clarinet and the Bohlen-Pierce scale. In early 2006 clarinet maker Stephen Fox began offering Bohlen-Pierce soprano clarinets for sale, and lower pitched instruments ("tenor" and "contra") are being developed.

Just BP tuning

A diatonic Bohlen-Pierce scale may be constructed with the following just ratios (chart shows the "Lambda" scale):

Music and composition

What does music using a Bohlen-Pierce scale sound like, aesthetically? 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 [Benson, Dave. "Musical scales and the Baker’s Dozen", p.16, "Musik og Matematik" 28/06.] .

Mathews and Pierce conclude that clear and memorable melodies may be composed in the BP scale, that "counterpoint sounds all right," and that, "chordal passages sound like harmony," presumably meaning progression, "but without any great tension or sense of resolution."Mathews and Pierce (1989). "The Bohlen-Pierce Scale", p.172.] In Mathews and Pierce 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.

Pieces using the Bohlen-Pierce scale include "Purity", the first movement of Curtis Roads' "Clang-Tint" ["Synthèse 96: The 26th International Festival of Electroacoustic Music", p.91. Michael Voyne Thrall. "Computer Music Journal", Vol. 21, No. 2 (Summer, 1997), pp. 90-92.] . Other computer music composers to use the BP scale include Jon Appleton, Richard Boulanger, George Hajdu, and Juan Reyes ["John Pierce (1910-2002)". "Computer Music Journal", Vol. 26, No. 4, Languages and Environments for Computer Music (Winter, 2002), pp. 6-7.] .

Other unusual tunings or scales

Other non-octave tunings investigated by Bohlen include twelve steps in the tritave, named A12 by Enrique Moreno [Moreno, Enrique Ignacio: Embedding Equal Pitch Spaces and The Question of Expanded Chromas: An Experimental Approach. Dissertation, Stanford University, Dec. 1995, pp. 12 - 22. Cited in [http://members.aol.com/bpsite/otherscales.html "Other Unusual Scales"] , "The Bohlen-Pierce Site".] and based on the 4:7:10 chord, seven steps in the octave or similar 11 steps in the tritave, and eight steps in the octave, based on 5:7:9 and of which only the just version would be used. [Bohlen, Heinz: 13 Tonstufen in der Duodezime. Acustica, vol.39 no. 2, S. Hirzel Verlag, Stuttgart, 1978, pp. 76 - 86. Cited in [http://members.aol.com/bpsite/otherscales.html "Other Unusual Scales"] , "The Bohlen-Pierce Site".] The Bohlen 833 cents scale is based on the fibonacci sequence, though was created based on combination tones, and contains a complex network of harmonic relations due to the inclusion of coinciding harmonics of stacked 833 cent intervals. For example, "step 10 turns out to be identical with the octave (1200 cents) to the base tone, at the same time featuring the Golden Ratio to step 3" [ [http://members.aol.com/hpbohlen/833cent.html "An 833 Cents Scale"] , "The Bohlen-Pierce Site".] .

An expansion of the Bohlen-Pierce tritave from 13 equal steps to 39 equal steps gives additional odd harmonics. The 13-step scale hits the odd harmonics 3/1; 5/3, 7/3; 7/5, 9/5; 9/7, and 15/7; while the 39-step scale includes all of those and many more (11/5, 13/5; 11/7, 13/7; 11/9, 13/9; 13/11, 15/11, 21/11, 25/11, 27/11; 15/13, 21/13, 25/13, 27/13, 33/13, and 35/13), while still missing almost all of the even harmonics (including 2/1; 3/2, 5/2; 4/3, 8/3; 6/5, 8/5; 9/8, 11/8, 13/8, and 15/8). The size of this scale is about 25 equal steps to a ratio slightly larger than an octave, so each of the 39 equal steps is slightly smaller than half of one of the 12 equal steps of the standard scale.Fact|date=August 2008

ources

External links

* [http://www.ziaspace.com/elaine/BP/ Bohlen-Pierce Scale Research by Elaine Walker]
* [http://www.sfoxclarinets.com/BP_sale.htm Bohlen-Pierce clarinets by Stephen Fox]
* [http://members.aol.com/bpsite/ The Bohlen-Pierce Site: Web place of an alternative harmonic scale]
* [http://www.kees.cc/music/scale13/scale13.html Kees van Prooijen's BP page]