Bohlen-Pierce scale

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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. 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 scale the Bohlen-Pierce scale after learning of Bohlen's earlier publication.

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, they 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 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. 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 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).

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

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

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 31 / 13 = 1.08818... above the next, or about (log(31 / 13))1200 / log(2) = 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.

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.

[edit] Bohlen-Pierce temperament

Dividing the tritave into 13 equal steps tempers out, or reduces to a unison, both of the intervals 245/243 (sometimes called the minor Bohlen-Pierce diesis) and 3125/3087 (sometimes called the major Bohlen-Pierce diesis) in the same way that dividing the octave into 12 equal steps reduces both 81/80 and 128/125 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.

[edit] Scale diagrams

The following are the 13 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

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Musical scales by edit
# | pentatonic | hexatonic | heptatonic | octatonic | chromatic
Types | Altered | Bebop | Diatonic scale | Enharmonic | Jazz scale | Minor scale
Name | Acoustic | Blues | Bohlen-Pierce | Diatonic | Double harmonic | Half diminished | Harmonic major | Lydian dominant | Major | Major locrian | Pelog | Phrygian dominant scale | Slendro
"Ethnic" name | Arabic | Gypsy | Jewish
Modes of the diatonic scale edit
Ionian (I) | Dorian (II) | Phrygian (III)
Lydian (IV) | Mixolydian (V) | Aeolian (VI) | Locrian (VII)
Modes of the melodic minor scale edit
Melodic minor (I) | Dorian b2 (II) | Lydian Augmented (III)
Lydian Dominant (IV) | Mixolydian b13(V) | Locrian #2 (VI) | | Altered (VII)


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