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References

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[[File:BP chord 357 just.png|thumb|right|Chord from just Bohlen–Pierce scale: C-G-A, tuned to harmonics 3, 5, and 7. "BP" above the clefs indicates Bohlen–Pierce notation. {{Audio|BP Just 357 chord.ogg|Play}}]]

The '''Bohlen–Pierce scale''' ('''BP scale''') is a musical [[Scale (music)|scale]] that offers an alternative to the [[octave]]-repeating scales typical in [[Classical music|Western]] and other musics,<ref name="Cognition">{{cite book | title = Music, Cognition, and Computerized Sound: An Introduction to Psychoacoustics | last = Pierce | first = John R. | chapter = Consonance and scales | editor-first = Perry R | editor-last = Cook | publisher = MIT Press | year = 2001 | isbn = 978-0-262-53190-0 | page = 183 | url = http://books.google.com/books?id=L04W8ADtpQ4C&pg=PA183&dq=%22Bohlen-Pierce+scale%22+13+octave&lr=&as_brr=0&as_pt=ALLTYPES&ei=2jhdSYDPMYnwkQSi1LXSAw }}</ref> specifically the [[equal temperament|equal tempered]] [[diatonic scale]]. Compared with octave-repeating scales, its [[interval (music)|interval]]s are more [[consonance and dissonance|consonant]] with certain types of acoustic [[frequency spectrum|spectra]]. It was independently described by [[Heinz Bohlen]],<ref>{{cite journal |last1=Bohlen |first1=Heinz |year=1978 |title=13 Tonstufen in der Duodezime |journal=Acoustica |volume=39| issue = 2 |pages=76–86 |location=Stuttgart |publisher=S. Hirzel Verlag |url=http://www.huygens-fokker.org/bpsite/publication0178.html |accessdate=27 November 2012}} {{de icon}}</ref> [[Kees van Prooijen]]<ref>{{cite journal |last1=Prooijen |first1=Kees van |year=1978 |title=A Theory of Equal-Tempered Scales |journal=Interface |volume=7 |pages=45–56 |url=http://www.kees.cc/tuning/interface.html |accessdate=27 November 2012}}</ref> and [[John R. Pierce]]. Pierce, who, with [[Max Mathews]] and others, published his discovery in 1984,<ref>{{cite journal |last1=Mathews |first1=M.V. |last2=Roberts |first2=L.A. |last3=Pierce |first3=J.R. |year=1984 |title=Four new scales based on nonsuccessive-integer-ratio chords |journal=[[J. Acoust. Soc. Am.]] |volume=75, S10(A) |pages= }}</ref> 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 [[combination tone]]s on the [[Gestalt psychology|Gestalt]] impression of intervals and chords.<ref name="Current Directions, p.167">{{cite book |last1= Mathews |first1=Max V. |last2=Pierce |first2=John R. |authorlink= |editor1-first=Max V. |editor1-last=Mathews |editor2-first=John R. |editor2-last=Pierce |chapter=The Bohlen–Pierce Scale |title=Current Directions in Computer Music Research |accessdate= |year=1989 |publisher=MIT Press |location= |isbn=9780262631396 |page=167 }}</ref>

The intervals between BP scale [[pitch class]]es are based on odd [[integer]] [[frequency]] ratios, in contrast with the intervals in diatonic scales, which employ both odd and even ratios found in the [[Harmonic series (music)|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|3579 Harmonic Chord.ogg|play}}). The chord formed by the ratio 3:5:7 ({{Audio|BP Just 357 chord.ogg|play}}) serves much the same role as the 4:5:6 chord (a major triad {{Audio|JI 456 chord.ogg|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==
3:5:7's [[Intonation (music)#Intonation sensitivity|intonation sensitivity]] pattern is similar to 4:5:6's (the just major chord), more similar than that of the minor chord.<ref name="Current Directions, p.165-66">{{cite book |last1=Mathews |last2=Pierce |year=1989 |pages=165–66}}</ref> 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]]s ({{Audio|BP ET half step.ogg|play one semitone}}) on bottom and an interval of 4 equal-tempered semitones on top (semitones: 0,6,10; {{Audio|BP ET 357.ogg|play}}). A minor triad is thus 6 semitones on top and 4 semitones on bottom (0,4,10; {{Audio|BP ET minor.ogg|play}}). 5:7:9 is the first inversion of the major triad (0,4,7; {{Audio|BP ET 579.ogg|play}}).<ref name="Current Directions, p.169">{{cite book |last1=Mathews |last2=Pierce |year=1989 |pages=169}}</ref>

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 012.ogg|play}}) but 0,11,13 ({{Audio|BP 0 11 13.ogg|play}}) was considered the most consonant by the trained subjects and 0,7,10 ({{Audio|BP 0 7 10.ogg|play}}) was judged most consonant by the untrained subjects.<ref name="Current Directions, p.171">{{cite book |last1=Mathews |last2=Pierce |year=1989 |pages=171}}</ref>

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 may be considered the [[dominant (music)|dominant]]. VIII (semitone: 10) may be considered the [[subdominant]].<ref name="Current Directions, p.169"/>

==Timbre and the tritave==
3:1 serves as the fundamental harmonic ratio, replacing the diatonic scale's 2:1 (the [[octave]]). ({{Audio|Octave.ogg|play}}) This interval is a perfect twelfth in [[diatonic scale|diatonic]] nomenclature ([[perfect fifth]] when reduced by an octave), but as this terminology is based on step sizes and [[diatonic function|functions]] not used in the BP scale, it is often called by a new name, '''''tritave''''' ({{Audio|Tritave.ogg|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 [[octave equivalency|equivalent]]. 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]]'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 it and the Bohlen–Pierce scale. In early 2006 clarinet maker [[Stephen Fox (clarinet maker)|Stephen Fox]] began offering Bohlen–Pierce soprano clarinets for sale, and 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, while a contra clarinet (one tritave lower than the soprano) is under development.

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

{| class="wikitable" style="text-align:center"
!
| colspan="2" | '''C'''
| colspan="2" | '''D'''
| colspan="2" | '''E'''
| colspan="2" | '''F'''
| colspan="2" | '''G'''
| colspan="2" | '''H'''
| colspan="2" | '''J'''
| colspan="2" | '''A'''
| colspan="2" | '''B'''
| colspan="2" | '''C'''
|-
! Ratio
| colspan="2" | 1/1
| colspan="2" | [[Semitone maximus|25/21]]
| colspan="2" | [[Septimal major third|9/7]]
| colspan="2" | [[Tritone|7/5]]
| colspan="2" | [[Major sixth|5/3]]
| colspan="2" | [[Minor seventh|9/5]]
| colspan="2" | [[Septimal diatonic semitone|15/7]]
| colspan="2" | [[Septimal minor third|7/3]]
| colspan="2" | [[Just chromatic semitone|25/9]]
| colspan="2" | [[Tritave|3/1]]
|-
! Cents
| colspan="2" | 0
| colspan="2" | 301.85
| colspan="2" | 435.08
| colspan="2" | 582.51
| colspan="2" | 884.36
| colspan="2" | 1017.60
| colspan="2" | 1319.44
| colspan="2" | 1466.87
| colspan="2" | 1768.72
| colspan="2" | 1901.96
|-
! Step
|
| colspan="2" | T
| colspan="2" | s
| colspan="2" | s
| colspan="2" | T
| colspan="2" | s
| colspan="2" | T
| colspan="2" | s
| colspan="2" | T
| colspan="2" | s
|
|-
! Cents
|
| colspan="2" | 301.85
| colspan="2" | [[Just minor second|133.24]]
| colspan="2" | 147.43
| colspan="2" | 301.85
| colspan="2" | 133.24
| colspan="2" | 301.84
| colspan="2" | 147.43
| colspan="2" | 301.85
| colspan="2" | 133.24
|
|-
! Midi
| colspan="2" | {{Audio|BP Just C.ogg|C}}
| colspan="2" | {{Audio|BP Just D.ogg|D}}
| colspan="2" | {{Audio|BP Just E.ogg|E}}
| colspan="2" | {{Audio|BP Just F.ogg|F}}
| colspan="2" | {{Audio|BP Just G.ogg|G}}
| colspan="2" | {{Audio|BP Just H.ogg|H}}
| colspan="2" | {{Audio|BP Just J.ogg|J}}
| colspan="2" | {{Audio|BP Just A.ogg|A}}
| colspan="2" | {{Audio|BP Just B.ogg|B}}
| colspan="2" | {{Audio|BP Just High C.ogg|C}}
|}

{{Audio|BP Just Lambda Scale.ogg|play just Bohlen–Pierce "Lambda" scale}}
{{Audio|JI diatonic scale.ogg|contrast with just major diatonic 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:<ref name="Current Directions, p.170">{{cite book |last1=Mathews |last2=Pierce |year=1989 |pages=170}}</ref>
(5/3) (7/5)
V IX III
|
III VII I
|
VI I IV
|
IV VIII II

==Bohlen–Pierce temperament==
[[Image:Bohlen-Pierce chromatic circle.png|thumb|"[[Chromatic circle]]" for the Bohlen–Pierce scale, with the third mode of the Lambda scale marked.<ref name="Cognition"/>]]

Bohlen originally expressed the BP scale in both [[just intonation]] and [[equal temperament]]. The [[Musical temperament|tempered]] form, which divides the tritave into thirteen equal steps, has become the most popular form. Each step is <math>\sqrt [13]{3} = 3^{1/13} = 1.08818...</math> above the next, or <math>1200\log_2( 3^{1/13} )= 146.3...</math> cents per step. The octave is divided into a fractional number of steps. Twelve equally tempered steps per octave are used in [[equal temperament|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. A [[regular temperament|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 temperament|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 [[cent (music)|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) [[Generated collection|MOS]]es 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'''
{| class="wikitable"
|align=center bgcolor="#ffeeee"|'''Interval (cents)'''
|
|colspan=2 align=center bgcolor="#ffeeee"|133
|colspan=2 align=center bgcolor="#ffeeee"|169
|colspan=2 align=center bgcolor="#ffeeee"|133
|colspan=2 align=center bgcolor="#ffeeee"|148
|colspan=2 align=center bgcolor="#ffeeee"|154
|colspan=2 align=center bgcolor="#ffeeee"|147
|colspan=2 align=center bgcolor="#ffeeee"|134
|colspan=2 align=center bgcolor="#ffeeee"|147
|colspan=2 align=center bgcolor="#ffeeee"|154
|colspan=2 align=center bgcolor="#ffeeee"|148
|colspan=2 align=center bgcolor="#ffeeee"|133
|colspan=2 align=center bgcolor="#ffeeee"|169
|colspan=2 align=center bgcolor="#ffeeee"|133
|
|-
|align=center bgcolor="#fffbee"|'''Note name'''
|colspan=2 align=center bgcolor="#fffbee"|C
|colspan=2 align=center bgcolor="#fffbee"|D{{music|♭}}
|colspan=2 align=center bgcolor="#fffbee"|D
|colspan=2 align=center bgcolor="#fffbee"|E
|colspan=2 align=center bgcolor="#fffbee"|F
|colspan=2 align=center bgcolor="#fffbee"|G{{music|♭}}
|colspan=2 align=center bgcolor="#fffbee"|G
|colspan=2 align=center bgcolor="#fffbee"|H
|colspan=2 align=center bgcolor="#fffbee"|J{{music|♭}}
|colspan=2 align=center bgcolor="#fffbee"|J
|colspan=2 align=center bgcolor="#fffbee"|A
|colspan=2 align=center bgcolor="#fffbee"|B{{music|♭}}
|colspan=2 align=center bgcolor="#fffbee"|B
|colspan=2 align=center bgcolor="#fffbee"|C
|-
|align=center bgcolor="#eeeeff"|'''Note (cents)'''
|colspan=2 align=center bgcolor="#eeeeff"|  0  
|colspan=2 align=center bgcolor="#eeeeff"| 133 
|colspan=2 align=center bgcolor="#eeeeff"|302
|colspan=2 align=center bgcolor="#eeeeff"|435
|colspan=2 align=center bgcolor="#eeeeff"|583
|colspan=2 align=center bgcolor="#eeeeff"|737
|colspan=2 align=center bgcolor="#eeeeff"|884
|colspan=2 align=center bgcolor="#eeeeff"|1018
|colspan=2 align=center bgcolor="#eeeeff"|1165
|colspan=2 align=center bgcolor="#eeeeff"|1319
|colspan=2 align=center bgcolor="#eeeeff"|1467
|colspan=2 align=center bgcolor="#eeeeff"|1600
|colspan=2 align=center bgcolor="#eeeeff"|1769
|colspan=2 align=center bgcolor="#eeeeff"|1902
|}

'''Equal-tempered'''
{| class="wikitable"
|align=center bgcolor="#ffeeee"|'''Interval (cents)'''
|
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|
|-
|align=center bgcolor="#fffbee"|'''Note name'''
|colspan=2 align=center bgcolor="#fffbee"|C
|colspan=2 align=center bgcolor="#fffbee"|D{{music|♭}}
|colspan=2 align=center bgcolor="#fffbee"|D
|colspan=2 align=center bgcolor="#fffbee"|E
|colspan=2 align=center bgcolor="#fffbee"|F
|colspan=2 align=center bgcolor="#fffbee"|G{{music|♭}}
|colspan=2 align=center bgcolor="#fffbee"|G
|colspan=2 align=center bgcolor="#fffbee"|H
|colspan=2 align=center bgcolor="#fffbee"|J{{music|♭}}
|colspan=2 align=center bgcolor="#fffbee"|J
|colspan=2 align=center bgcolor="#fffbee"|A
|colspan=2 align=center bgcolor="#fffbee"|B{{music|♭}}
|colspan=2 align=center bgcolor="#fffbee"|B
|colspan=2 align=center bgcolor="#fffbee"|C
|-
|align=center bgcolor="#eeeeff"|'''Note (cents)'''
|colspan=2 align=center bgcolor="#eeeeff"|  0  
|colspan=2 align=center bgcolor="#eeeeff"| 146 
|colspan=2 align=center bgcolor="#eeeeff"|293
|colspan=2 align=center bgcolor="#eeeeff"|439
|colspan=2 align=center bgcolor="#eeeeff"|585
|colspan=2 align=center bgcolor="#eeeeff"|732
|colspan=2 align=center bgcolor="#eeeeff"|878
|colspan=2 align=center bgcolor="#eeeeff"|1024
|colspan=2 align=center bgcolor="#eeeeff"|1170
|colspan=2 align=center bgcolor="#eeeeff"|1317
|colspan=2 align=center bgcolor="#eeeeff"|1463
|colspan=2 align=center bgcolor="#eeeeff"|1609
|colspan=2 align=center bgcolor="#eeeeff"|1756
|colspan=2 align=center bgcolor="#eeeeff"|1902
|}

{{Audio|BP ET lambda scale.ogg|play equal tempered Bohlen–Pierce scale}}

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

==Music and composition==
What does music using a Bohlen–Pierce scale sound like, [[aesthetics of music|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] [[chromatic scale|twelve-tone scale]], but badly [[Musical tuning#Tuning practice|out of tune]]," the average listener will continually feel "that something isn't quite right," due to [[social conditioning]].<ref>{{cite journal |last1=Benson |first1=Dave |year= |title=Musical scales and the Baker’s Dozen |journal=Musik og Matematik |volume=28/06 |pages=16}}</ref>

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 [[chord progression|progression]], "but without any great tension or sense of resolution."<ref name="Current Directions, p.172">{{cite book |last1=Mathews |last2=Pierce |year=1989 |pages=172}}</ref> 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.<ref name="Current Directions, p.171"/>

Compositions using the Bohlen–Pierce scale include "Purity", the first movement of [[Curtis Roads]]' ''Clang-Tint''.<ref>{{cite journal |last1=Thrall |first1=Michael Voyne |year=Summer, 1997 |title=Synthèse 96: The 26th International Festival of Electroacoustic Music |journal=Computer Music Journal |volume=21| issue = 2 |pages=90–92 [91]}}</ref> Other computer composers to use the BP scale include [[Jon Appleton]], Richard Boulanger (''Solemn Song for Evening'' (1990)), [[Georg Hajdu]], and Juan Reyes' ''[http://ccrma.stanford.edu/~juanig/descrips/ppPdesc.html ppP]'' (1999-2000).<ref>{{cite journal |last1= |first1= |year=Winter 2002 |title=John Pierce (1910-2002) |journal=Computer Music Journal |volume=26, No. 4 |issue=Languages and Environments for Computer Music |pages=6–7}}</ref> Also Charles Carpenter (''Frog à la Pêche'' (1994) & ''Splat'').<ref>{{cite book |last=d'Escrivan |first=Julio |editor-last=Collins |editor-first=Nick |editor-link= Nicolas Collins |title=The Cambridge Companion to Electronic Music |year=2007 |isbn=9780521868617 |page=229 }}</ref><ref>{{cite book |last=Benson |first=Dave |title=Music: A Mathematical Offering |year=2006 |isbn=9780521853873 |page=237}}</ref> As well as [[Elaine Walker (composer)|Elaine Walker]] (''Stick Men'' (1991), ''Love Song'', and ''Greater Good'' (2011)).<ref>{{cite web |url=http://bohlen-pierce-conference.org/concert-1 |title=Concerts |work=Bohlen-Pierce-Conference.org |publisher= |accessdate=27 November 2012}}</ref>

==Symposium==
A first Bohlen–Pierce symposium took place in Boston on March 7 to 9, 2010, produced by composer [[Georg Hajdu]] ([[Hochschule für Musik und Theater Hamburg]]) and the [[Boston Microtonal Society]]. Co-organizers were the Boston [[Goethe Institut]]e, the [[Berklee College of Music]], the Northeastern University and the [[New England Conservatory]] of Music. The symposium participants, which included Heinz Bohlen, Max Mathews, Clarence Barlow, [[Curtis Roads]], David Wessel, Psyche Loui, Richard Boulanger, Georg Hajdu, [[Paul Erlich]], [[Ron Sword]], Julia Werntz, Larry Polansky, 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<ref>{{cite book |last=Bohlen |year=1978 |page=footnote 26, 84}}</ref> include twelve steps in the tritave, named [[A12 scale|A12]] by Enrique Moreno <ref>{{cite web |url=http://www.huygens-fokker.org/bpsite/otherscales.html |title=Other Unusual Scales |work=The Bohlen–Pierce Site |accessdate=27 November 2012}} Cites: {{cite journal |last1=Moreno |first1=Enrique Ignacio |year=Dec. 1995 |title=Embedding Equal Pitch Spaces and The Question of Expanded Chromas: An Experimental Approach |journal=Dissertation |pages=12–22 |publisher=Stanford University}}</ref> and based on the 4:7:10 chord {{audio|A12 4 7 10 on C.mid|Play}}, seven steps in the octave ([[7-tet]]) or similar 11 steps in the tritave, and eight steps in the octave, based on 5:7:9 {{audio|5 7 9 chord on E.mid|Play}} and of which only the just version would be used.<ref>{{cite web |url=http://www.huygens-fokker.org/bpsite/otherscales.html |title=Other Unusual Scales |work=The Bohlen–Pierce Site |accessdate=27 November 2012}} Cites:{{cite journal |last1=Bohlen |year=1978 |pages=76–86}}</ref> The [[833 cents scale|Bohlen 833 cents scale]] is based on the [[Fibonacci sequence]], although it was created from [[combination tone]]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]] to step 3".<ref>{{cite web |url=http://www.huygens-fokker.org/bpsite/833cent.html |title=An 833 Cents Scale |last1=Bohlen |first1=Heinz |work=The Bohlen–Pierce Site |accessdate=27 November 2012}}</ref>

An expansion of the Bohlen–Pierce tritave from 13 equal steps to 39 equal steps, proposed by Paul Erlich, 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.<ref>{{cite web |url=http://www.huygens-fokker.org/bpsite/scales.html |title=BP Scale Structures |work=The Bohlen–Pierce Site |accessdate=27 November 2012}}</ref>

Alternate scales may be specified by indicating the size of equal tempered steps, for example [[Wendy Carlos]]' 78 cent [[alpha scale]] and 63.8 cent [[beta scale]], and Gary Morrison's 88 cent scale (13.64 steps per octave or 14 per 1232 cent stretched octave).<ref>{{cite book |last=Sethares |first=William |authorlink=William Sethares |title=Tuning, Timbre, Spectrum, Scale |year=2004 |isbn=1-85233-797-4 |page=60}}</ref> This gives the alpha scale 15.39 steps per octave and the beta scale 18.75 steps per octave.<ref>{{cite album-notes |title=Beauty in the Beast |albumlink=Beauty in the Beast |artist=Wendy Carlos |year=2000/1986 |notestitle=Liner notes |first=Wendy |last=Carlos |authorlink=Wendy Carlos |type=CD |publisher=ESD |id=81552}}</ref>

See also: [[Delta scale]], [[Gamma scale]].

==Sources==
{{reflist|33em}}

==External links==
* [http://ziaspace.com/_microtonality/BP/ Bohlen–Pierce Scale Research by Elaine Walker]
* [http://www.sfoxclarinets.com/BP_sale.htm Bohlen–Pierce clarinets by Stephen Fox]
* [http://www.huygens-fokker.org/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]
* [http://xenharmonic.wikispaces.com/file/view/17tppp4_Walker_Love_Song.mp3 song in Bohlen Pierce Scale]
* [http://bohlen-pierce-conference.org/ Bohlen–Pierce symposium]
* "[http://www.youtube.com/playlist?list=PLE4AEC5BC4B7AA4F1&feature=plcp Bohlen–Pierce Scale Symposium, Boston 2010]" [playlist], ''YouTube.com''.

{{Microtonal music}}
{{Musical tuning}}
{{Scales}}

{{DEFAULTSORT:Bohlen-Pierce Scale}}
[[Category:Just tuning and intervals]]
[[Category:Non–octave-repeating scales]]
[[Category:Musical temperaments]]

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