![Bohlen-Pierce scale triad on C image](https://upload.wikimedia.org/wikipedia/commons/7/7b/Bohlen-Pierce_scale_triad_on_C_image.png)
The Bohlen–Pierce scale (BP scale) is a musical
tuning
Tuning can refer to:
Common uses
* Tuning, the process of tuning a tuned amplifier or other electronic component
* Musical tuning, musical systems of tuning, and the act of tuning an instrument or voice
** Guitar tunings
** Piano tuning, adjusti ...
and
scale, first described in the 1970s, that offers an alternative to the
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
-repeating scales typical in
Western
Western may refer to:
Places
*Western, Nebraska, a village in the US
*Western, New York, a town in the US
*Western Creek, Tasmania, a locality in Australia
*Western Junction, Tasmania, a locality in Australia
*Western world, countries that id ...
and other musics,
specifically the
equal-tempered
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, w ...
diatonic scale
In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, ...
.
The interval 3:1 (often called by a new name, ''tritave'') serves as the fundamental harmonic ratio, replacing the diatonic scale's 2:1 (the octave) with a perfect twelfth (an octave higher than a perfect fifth). For any pitch that is part of the BP scale, all pitches one or more tritaves higher or lower are part of the system as well, and are considered equivalent.
The BP scale divides the tritave into 13 steps, either
equal tempered (the most popular form), or in a
justly tuned version. Compared with octave-repeating scales, the BP scale's
intervals are more
consonant
In articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract. Examples are and pronounced with the lips; and pronounced with the front of the tongue; and pronounced wit ...
with certain types of acoustic
spectra.
The scale was independently described by
,
Kees van Prooijen Kees (Cornelis) van Prooijen (born 7 August 1952) is a creator of computer art. Although it does not bear his name, he independently discovered the Bohlen-Pierce scale, a non- octave-repeating scale based on the tritave and spectra containing odd ...
and
John R. Pierce
John Robinson Pierce (March 27, 1910 – April 2, 2002), was an American engineer and author. He did extensive work concerning radio communication, microwave technology, computer music, psychoacoustics, and science fiction. Additionally to his ...
. Pierce, who, with
Max Mathews
Max Vernon Mathews (November 13, 1926 in Columbus, Nebraska, USA – April 21, 2011 in San Francisco, CA, USA) was a pioneer of computer music.
Biography
Mathews studied electrical engineering at the California Institute of Technology and the Ma ...
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 based on consideration of the influence of
s on the
Gestalt
Gestalt may refer to:
Psychology
* Gestalt psychology, a school of psychology
* Gestalt therapy, a form of psychotherapy
* Bender Visual-Motor Gestalt Test, an assessment of development disorders
* Gestalt Practice, a practice of self-exploration ...
impression of intervals and chords.
The intervals between BP scale
pitch class
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...
es are based on odd
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
ratios, in contrast with the intervals in diatonic scales, which employ 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
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
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:: and 4:5:6 = 2::3 = 1::).
Chords and modulation
3:5:7's
intonation sensitivity pattern is similar to 4:5:6's (the just major chord), more similar than that of the minor chord.
[Mathews; Pierce (1989). pp. 165–166.] This similarity suggests that our ears will also perceive 3:5:7 as harmonic.
The 3:5:7 chord may thus be considered the major triad of the BP scale. It is approximated by an interval of 6 equal-tempered BP
semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
s () on bottom and an interval of 4 equal-tempered semitones on top (semitones: 0,6,10; ). A minor triad is correspondingly 6 semitones on top and 4 semitones on bottom (0,4,10; ). 5:7:9 is the first inversion of the major triad (0,4,7; ).
[Mathews; Pierce (1989). 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 (), but 0,11,13 () was considered the most consonant by the trained subjects (because it sounds like an octave-dropped major triad) and 0,7,10 () was judged most consonant by the untrained subjects.
[Mathews; Pierce (1989). 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. Modulation is possible through changing a single note. Moving note II up one semitone causes the tonic to rise to what was note III (semitone: 3), which therefore may be considered the
dominant. VIII (semitone: 10) may be considered the
subdominant
In music, the subdominant is the fourth tonal degree () of the diatonic scale. It is so called because it is the same distance ''below'' the tonic as the dominant is ''above'' the tonicin other words, the tonic is the dominant of the subdomina ...
.
Timbre and the tritave
3:1 serves as the fundamental harmonic ratio, replacing the diatonic scale's 2:1 (the
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
). () This interval is a perfect twelfth in
diatonic
Diatonic and chromatic are terms in music theory that are most often used to characterize Scale (music), scales, and are also applied to musical instruments, Interval (music), intervals, Chord (music), chords, Musical note, notes, musical sty ...
nomenclature (
perfect fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval fro ...
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
A pseudo-octave, pseudooctave,"Interview with Max Mathews", p.21. Author(s): C. Roads and Max Mathews. Source: ''Computer Music Journal'', Vol. 4, No. 4, (Winter, 1980), pp. 15–22. Published by: The MIT Press. or paradoxical octave"The Parad ...
, 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
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
. In the BP scale, if a given pitch is present, then ''none'' of the pitches one or more octaves higher or lower are present, but ''all'' pitches one or more tritaves higher or lower are part of the system and are considered equivalent.
The BP scale's use of odd integer ratios is appropriate for timbres containing only odd harmonics. Because the
clarinet
The clarinet is a musical instrument in the woodwind family. The instrument has a nearly cylindrical bore and a flared bell, and uses a single reed to produce sound.
Clarinets comprise a family of instruments of differing sizes and pitches ...
's spectrum (in the
chalumeau
The chalumeau (; ; plural chalumeaux) is a single-reed woodwind instrument of the late baroque and early classical eras. The chalumeau is a folk instrument that is the predecessor to the modern-day clarinet. It has a cylindrical bore with e ...
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 it and the Bohlen–Pierce scale. In early 2006, clarinet maker
Stephen Fox
Sir Stephen Fox (27 March 1627 – 28 October 1716) of Farley in Wiltshire, of Redlynch Park in Somerset, of Chiswick, Middlesex and of Whitehall, was a royal administrator and courtier to King Charles II, and a politician, who rose from ...
began offering Bohlen–Pierce soprano clarinets for sale. He produced the first BP tenor clarinet (six steps below the soprano) in 2010 and the first epsilon clarinet (four steps above the soprano) in 2011. A contra clarinet (one tritave lower than the soprano) is now (2020) played by Nora Mueller, Luebeck, Germany.
Just tuning
A diatonic Bohlen–Pierce scale may be constructed with the following just ratios (chart shows the "Lambda" (λ) scale):
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:
[Mathews; Pierce (1989). p. 170.]
(5:3)(7:5)
V IX III
,
III VII I
,
VI I IV
,
IV VIII II
Bohlen–Pierce temperament
Bohlen originally expressed the BP scale in both
just intonation
In music, just intonation or pure intonation is the tuning of musical intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to ...
and
equal temperament
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...
. The
tempered form, which divides the tritave into thirteen equal steps, has become the most popular form. Each step is = 3
= 1.08818… above the next, or 1200 log
2 (3
) = 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
Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That resultin ...
. 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
In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...
) and 128:125 (
5-limit limma) to a unison. A
7-limit linear temperament tempers 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 ( = 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: Taking every fifth degree of this octave-based scale yields an excellent approximation to the non-octave-based equally tempered BP scale. Furthermore, an interval of five such steps generates (octave-based)
MOS
MOS or Mos may refer to:
Technology
* MOSFET (metal–oxide–semiconductor field-effect transistor), also known as the MOS transistor
* Mathematical Optimization Society
* Model output statistics, a weather-forecasting technique
* MOS (filmm ...
es (moments of symmetry) 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
Equal-tempered
Music and composition
![12-tet and 13-tet BP tuning intervals compared](https://upload.wikimedia.org/wikipedia/commons/8/86/12-tet_and_13-tet_BP_tuning_intervals_compared.png)
What does music using a Bohlen–Pierce scale sound like,
aesthetically
Aesthetics, or esthetics, is a branch of philosophy that deals with the nature of beauty and taste, as well as the philosophy of art (its own area of philosophy that comes out of aesthetics). It examines aesthetic values, often expressed th ...
? 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
he more familiartwelve-tone scale
The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the ...
, but badly
out of tune
''Out of Tune'' is a British children's TV sitcom which was shown on CBBC from 1996 to 1998.
It features a group of fictional children that belong to a church choir at a school and their practice sessions. However the choir is humorously bad, an ...
", the average listener will continually feel "that something isn't quite right", due to
social conditioning
Social conditioning is the sociological process of training individuals in a society to respond in a manner generally approved by the society in general and peer groups within society. The concept is stronger than that of socialization, which is t ...
.
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; Pierce (1989). p. 172.] In their 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.
Compositions using the Bohlen–Pierce scale include "Purity", the first movement of
Curtis Roads
Curtis Roads (born May 9, 1951) is an American composer, author and computer programmer. He composes electronic and electroacoustic music, specializing in granular and pulsar synthesis.
Career and music
Born in Cleveland, Ohio, Roads studied com ...
' ''Clang-Tint''. Other computer composers to use the BP scale include
Jon Appleton
Jon Howard Appleton (January 4, 1939 – January 30, 2022) was an American composer, an educator and a pioneer in electro-acoustic music. His earliest compositions in the medium, e.g. "Chef d'Oeuvre" and "Newark Airport Rock" (1967) attracted at ...
,
Richard Boulanger
Richard Charles Boulanger (born November 10, 1956) is a composer, author, and electronic musician. He is a key figure in the development of the audio programming language Csound, and is associated with computer music pioneers Max Mathews and Bar ...
(''Solemn Song for Evening'' (1990)),
Georg Hajdu
Georg Hajdu (born 21 June 1960) is a German composer of Hungarian descent. His work is dedicated to the combination of music, science and computer technology. He is noted for his opera ' and the network music performance environment Quintet.net. ...
, Juan Reyes
''ppP''(1999-2000),
Ami Radunskaya
Ami Elizabeth Radunskaya is an American mathematician and musician. She is a professor of mathematics at Pomona College, where she specializes in dynamical systems and the applications of mathematics to medicine, such as the use of cellular aut ...
's "A Wild and Reckless Place" (1990), Charles Carpenter (''Frog à la Pêche'' (1994) & ''Splat''), and
Elaine Walker (''Stick Men'' (1991), ''Love Song'', and ''Greater Good'' (2011)).
Symposium
A first Bohlen–Pierce symposium took place in Boston on March 7 to 9, 2010, produced by composer
Georg Hajdu
Georg Hajdu (born 21 June 1960) is a German composer of Hungarian descent. His work is dedicated to the combination of music, science and computer technology. He is noted for his opera ' and the network music performance environment Quintet.net. ...
(
Hochschule für Musik und Theater Hamburg
The Hochschule für Musik und Theater Hamburg is one of the larger universities of music in Germany.
It was founded 1950 as ''Staatliche Hochschule für Musik'' (Public college of music) on the base of the former private acting school of Annem ...
) and the
Boston Microtonal Society. Co-organizers were the Boston
Goethe Institut
The Goethe-Institut (, GI, en, Goethe Institute) is a non-profit German cultural association operational worldwide with 159 institutes, promoting the study of the German language abroad and encouraging international cultural exchange and ...
e, the
Berklee College of Music
Berklee College of Music is a private music college in Boston, Massachusetts. It is the largest independent college of contemporary music in the world. Known for the study of jazz and modern American music, it also offers college-level cours ...
, the Northeastern University and the
New England Conservatory
The New England Conservatory of Music (NEC) is a private music school in Boston, Massachusetts. It is the oldest independent music conservatory in the United States and among the most prestigious in the world. The conservatory is located on Hu ...
of Music. The symposium participants, which included Heinz Bohlen, Max Mathews,
Clarence Barlow
Clarence Barlow (also Klarenz, born 27 December 1945) is a composer of classical and electroacoustic works.
Career
Barlow was one of the founders of Initiative Musik und Informatik Köln. In 1988 he was the director of music at the Internatio ...
,
Curtis Roads
Curtis Roads (born May 9, 1951) is an American composer, author and computer programmer. He composes electronic and electroacoustic music, specializing in granular and pulsar synthesis.
Career and music
Born in Cleveland, Ohio, Roads studied com ...
, David Wessel, Psyche Loui, Richard Boulanger, Georg Hajdu,
Paul Erlich, Ron Sword, Julia Werntz,
Larry Polansky
Larry Polansky (born 1954) is a composer, guitarist, mandolinist, and professor emeritus at Dartmouth College and the University of California, Santa Cruz. He is a founding member and co-director of Frog Peak Music (a composers' collective) ...
, Manfred Stahnke, Stephen Fox, Elaine Walker, Todd Harrop, Gayle Young, Johannes Kretz, Arturo Grolimund, Kevin Foster, presented 20 papers on history and properties of the Bohlen–Pierce scale, performed more than 40 compositions in the novel system and introduced several new musical instruments.
Performers included German musicians Nora-Louise Müller and Ákos Hoffman on Bohlen–Pierce clarinets and Arturo Grolimund on Bohlen–Pierce pan flute as well as Canadian ensemble tranSpectra, and US American xenharmonic band ZIA, led by Elaine Walker.
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 (
7-tet
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, w ...
) 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. Additionally, the pentave can be divided into eight steps which approximates chords of the form 5:9:13:17:21:25. The
Bohlen 833 cents scale is based on the
Fibonacci sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...
, although it was created from
s, 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
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
to step 3".
Alternate scales may be specified by indicating the size of equal tempered steps, for example
Wendy Carlos
Wendy Carlos (born Walter Carlos, November 14, 1939) is an American musician and composer best known for her electronic music and film scores. Born and raised in Rhode Island, Carlos studied physics and music at Brown University before moving ...
' 78-cent
alpha scale
The α (alpha) scale is a non-octave-repeating musical scale invented by Wendy Carlos and first used on her album ''Beauty in the Beast'' (1986). It is derived from approximating just intervals using multiples of a single interval, but without r ...
and 63.8-cent
beta scale
The β (beta) scale is a non-octave-repeating musical scale invented by Wendy Carlos and first used on her album ''Beauty in the Beast'' (1986). It is derived from approximating just intervals using multiples of a single interval without, as is ...
, and Gary Morrison's 88-cent scale (13.64 steps per octave or 14 per 1232-cent stretched octave). This gives the alpha scale 15.39 steps per octave and the beta scale 18.75 steps per octave.
Expansions
39-tone equal division of the tritave
Paul Erlich proposed dividing each step of the Bohlen–Pierce into thirds so that the tritave is divided into 39 equal steps instead of 13 equal steps. The scale, which can be viewed as three evenly staggered Bohlen-Pierce scales, 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.
65-tone equal division of the tritave
Dividing each step of the Bohlen-Pierce scale into fifths (so that the tritave is divided into 65 steps) results in a very accurate octave (41 steps) and perfect fifth (24 steps), as well as approximations for other just intervals. The scale is practically identical to
41-tone equal division of the octave except that each step is slightly smaller (less than a hundredth of a cent per step).
See also
*
Double reed
A double reed is a type of reed used to produce sound in various wind instruments. In contrast with a single reed instrument, where the instrument is played by channeling air against one piece of cane which vibrates against the mouthpiece and c ...
*
Square wave
A square wave is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. In an ideal square wave, the transitions b ...
*
Stredici
*Other non-octave repeating scales:
**
Delta scale
The δ (delta) scale is a non-octave repeating musical scale. It may be regarded as the beta scale's reciprocal, since it is "as far 'down' the ( 0 3 6 9) circle from α as β is 'up'".Taruskin, Richard (1996). ''Stravinsky and the Russian Trad ...
**
Gamma scale
The γ (gamma) scale is a non-octave repeating musical scale invented by Wendy Carlos while preparing ''Beauty in the Beast'' (1986) though it does not appear on the album. It is derived from approximating just intervals using multiples of a sin ...
References
External links
*
The Bohlen–Pierce Scale Research, ''ZiaSpace.com''.
*
, Bohlen-Pierce clarinets and other instruments, ''SFoxClarinets.com''.
*
The Bohlen–Pierce Site: Web place of an alternative harmonic scale, ''Huygens-Fokker.org''.
*
, ''Kees.cc''.
* Song in Bohlen Pierce Scale:
17tppp4 Walker Love Song, ''Xenharmonic.Wikispaces.com''.
*
The Bohlen–Pierce Symposium, ''Bohlen-Pierce-Conference.org''.
*
Bohlen–Pierce Scale Symposium, Boston 2010 laylist ''YouTube.com''.
{{DEFAULTSORT:Bohlen-Pierce Scale
Just tuning and intervals
Non–octave-repeating scales
Musical temperaments