Definitions

# 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, Kees van Prooijen, and John 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 on the basis of combination tones.

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 (). The chord formed by the ratio 3:5:7 serves much the same role as the 4:5:6 chord (a major triad ) 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 chord.

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 on bottom and an interval of 4 equal tempered semitones on top (semitones: 0,6,10; ). A minor triad is thus 6 semitones on top and 4 semitones on bottom (0,4,10; ). 5:7:9 is the first inversion of the the major triad (6,10,13; ).

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 but 0,11,13 was considered the most consonant by the trained subjects and 0,7,10 was judged most consonant by the untrained subjects.

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 ()) 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 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):

!
! Ratio
! Step
! Midi
 C D E F G H J A B C 1/1 25/21 9/7 7/5 5/3 9/5 15/7 7/3 25/9 3/1 T s s T s T s T s

A just BP scale may be constructed from four overlapping 3:5:7 chords, for example, V, II, VI, and IV, though different chords may be chosen to produce a similar scale:

(5/3) (7/5)
V   IX   III
|
III VII I
|
VI I IV
|
IV VIII II

## Bohlen-Pierce temperament

Though Bohlen originally expressed the BP scale in just intonation, a tempered form of the scale, which divides the tritave into thirteen equal steps, has become the most popular form. Each step is $3^\left\{1/13\right\} = 1.08818...$ above the next, or about $\left(log\left(3^\left\{1/13\right\} \right) \right) 1200 / log\left(2 \right) = 146.3...$ cents per step. The octave is divided into a fractional number of steps. Twelve equally tempered steps per octave are used in 12-tet. The Bohlen-Pierce scale could be described as 8.202087-tet, because a full octave (1200 cents), divided by 146.3... cents per step, gives 8.202087 steps per 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. One can produce a 7-limit linear temperament by tempering out both of these intervals; the resulting Bohlen-Pierce temperament no longer has anything to do with tritave equivalences or non-octave scales, beyond the fact that it is well adapted to using them. A tuning of 41 equal steps to the octave (1200/41 = 29.27 cents per step) would be quite logical for this temperament. 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: By taking every fifth degree of this octave-based scale one finds it contains an excellent approximation to the non-octave-based equally tempered BP scale. Furthermore, an interval of five such steps generates (octave-based) MOSes with 8, 9, or 17 notes, and the 8-note scale (comprising degrees 0, 5, 10, 15, 20, 25, 30, and 35 of the 41-equal scale) could be considered the octave-equivalent version of the Bohlen-Pierce scale.

## Intervals and scale diagrams

The following are the thirteen notes in the scale (cents rounded to nearest whole number):

Justly tuned

 Interval (cents) 133 169 133 148 154 147 134 147 154 148 133 169 133 Note name C D♭ D E F G♭ G H J♭ J A B♭ B C Note (cents) 0 133 302 435 583 737 884 1018 1165 1319 1467 1600 1769 1902

Equal-tempered

 Interval (cents) 146 146 146 146 146 146 146 146 146 146 146 146 146 Note name C D♭ D E F G♭ G H J♭ J A B♭ B C Note (cents) 0 146 293 439 585 732 878 1024 1170 1317 1463 1609 1756 1902

Steps EQ interval Cents in EQ Just intonation interval Traditional name Cents in just intonation Difference
0 $3^frac\left\{0\right\}\left\{13\right\}$ = 1.00 0.00 $begin\left\{matrix\right\} frac\left\{1\right\}\left\{1\right\} end\left\{matrix\right\}$ = 1.00 Unison 0.00 0.00
1 $3^frac\left\{1\right\}\left\{13\right\}$ = 1.09 146.30 $begin\left\{matrix\right\} frac\left\{27\right\}\left\{25\right\} end\left\{matrix\right\}$ = 1.08 Great limma 133.24 13.06
2 $3^frac\left\{2\right\}\left\{13\right\}$ = 1.18 292.61 $begin\left\{matrix\right\} frac\left\{25\right\}\left\{21\right\} end\left\{matrix\right\}$ = 1.19 Quasi-tempered minor third 301.85 -9.24
3 $3^frac\left\{3\right\}\left\{13\right\}$ = 1.29 438.91 $begin\left\{matrix\right\} frac\left\{9\right\}\left\{7\right\} end\left\{matrix\right\}$ = 1.29 Septimal major third 435.08 3.83
4 $3^frac\left\{4\right\}\left\{13\right\}$ = 1.40 585.22 $begin\left\{matrix\right\} frac\left\{7\right\}\left\{5\right\} end\left\{matrix\right\}$ = 1.4 Lesser septimal tritone 582.51 2.71
5 $3^frac\left\{5\right\}\left\{13\right\}$ = 1.53 731.52 $begin\left\{matrix\right\} frac\left\{75\right\}\left\{49\right\} end\left\{matrix\right\}$ = 1.53 BP fifth 736.93 -5.41
6 $3^frac\left\{6\right\}\left\{13\right\}$ = 1.66 877.83 $begin\left\{matrix\right\} frac\left\{5\right\}\left\{3\right\} end\left\{matrix\right\}$ = 1.67 Just major sixth 884.36 -6.53
7 $3^frac\left\{7\right\}\left\{13\right\}$ = 1.81 1024.13 $begin\left\{matrix\right\} frac\left\{9\right\}\left\{5\right\} end\left\{matrix\right\}$ = 1.8 Greater just minor seventh 1017.60 6.53
8 $3^frac\left\{8\right\}\left\{13\right\}$ = 1.97 1170.44 $begin\left\{matrix\right\} frac\left\{49\right\}\left\{25\right\} end\left\{matrix\right\}$ = 1.96 BP eighth 1165.02 5.42
9 $3^frac\left\{9\right\}\left\{13\right\}$ = 2.14 1316.74 $begin\left\{matrix\right\} frac\left\{15\right\}\left\{7\right\} end\left\{matrix\right\}$ = 2.14 Septimal minor ninth 1319.44 -2.70
10 $3^frac\left\{10\right\}\left\{13\right\}$ = 2.33 1463.05 $begin\left\{matrix\right\} frac\left\{7\right\}\left\{3\right\} end\left\{matrix\right\}$ = 2.33 Septimal minimal tenth 1466.87 -3.82
11 $3^frac\left\{11\right\}\left\{13\right\}$ = 2.53 1609.35 $begin\left\{matrix\right\} frac\left\{63\right\}\left\{25\right\} end\left\{matrix\right\}$ = 2.52 Quasi-tempered major tenth 1600.11 9.24
12 $3^frac\left\{12\right\}\left\{13\right\}$ = 2.76 1755.66 $begin\left\{matrix\right\} frac\left\{25\right\}\left\{9\right\} end\left\{matrix\right\}$ = 2.78 Classic augmented eleventh 1768.72 -13.06
13 $3^frac\left\{13\right\}\left\{13\right\}$ = 3.00 1901.96 $begin\left\{matrix\right\} frac\left\{3\right\}\left\{1\right\} end\left\{matrix\right\}$ = 3.00 Just twelfth, "Tritave" 1901.96 0.00

## 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.

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." 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. Other computer music composers to use the BP scale include Jon Appleton, Richard Boulanger, George Hajdu, and Juan Reyes.

## Other unusual tunings or scales

Other non-octave tunings investigated by Bohlen include twelve steps in the tritave, named A12 by Enrique Moreno 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. 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" .

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.