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).
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.
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.
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:
V IX III
III VII I
VI I IV
IV VIII II
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.
The following are the thirteen notes in the scale (cents rounded to nearest whole number):
|Steps||EQ interval||Cents in EQ||Just intonation interval||Traditional name||Cents in just intonation||Difference|
|0||= 1.00||0.00||= 1.00||Unison||0.00||0.00|
|1||= 1.09||146.30||= 1.08||Great limma||133.24||13.06|
|2||= 1.18||292.61||= 1.19||Quasi-tempered minor third||301.85||-9.24|
|3||= 1.29||438.91||= 1.29||Septimal major third||435.08||3.83|
|4||= 1.40||585.22||= 1.4||Lesser septimal tritone||582.51||2.71|
|5||= 1.53||731.52||= 1.53||BP fifth||736.93||-5.41|
|6||= 1.66||877.83||= 1.67||Just major sixth||884.36||-6.53|
|7||= 1.81||1024.13||= 1.8||Greater just minor seventh||1017.60||6.53|
|8||= 1.97||1170.44||= 1.96||BP eighth||1165.02||5.42|
|9||= 2.14||1316.74||= 2.14||Septimal minor ninth||1319.44||-2.70|
|10||= 2.33||1463.05||= 2.33||Septimal minimal tenth||1466.87||-3.82|
|11||= 2.53||1609.35||= 2.52||Quasi-tempered major tenth||1600.11||9.24|
|12||= 2.76||1755.66||= 2.78||Classic augmented eleventh||1768.72||-13.06|
|13||= 3.00||1901.96||= 3.00||Just twelfth, "Tritave"||1901.96||0.00|
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.
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.
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