Editing Bohlen–Pierce scale (section)

{{Short description|Musical scale}}
[[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.{{citation needed|date=September 2015}} {{Audio|BP Just 357 chord.ogg|Play}}]]
[[File:Bohlen-Pierce scale triad on C image.png|thumb|Same chord in Ben Johnston's notation for just intonation]]

The '''Bohlen–Pierce scale''' ('''BP scale''') is a musical [[musical tuning|tuning]] and [[scale (music)|scale]], first described in the 1970s, 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 | chapter-url = https://books.google.com/books?id=L04W8ADtpQ4C&dq=%22Bohlen-Pierce+scale%22+13+octave&pg=PA183 }}</ref> specifically the [[equal temperament|equal-tempered]] [[diatonic scale]].

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 temperament|equal tempered]] (the most popular form), or in a [[Just intonation|justly tuned]] version. Compared with octave-repeating scales, the BP scale's [[interval (music)|interval]]s are more [[consonance and dissonance|consonant]] with certain types of acoustic [[frequency spectrum|spectra]].{{Citation needed|date=October 2019}}

The scale 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 |access-date=27 November 2012|language=de}}</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 |access-date=27 November 2012 |doi=10.1080/09298217808570248}}</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) |issue=S1 |pages=S10 |doi=10.1121/1.2021272 |bibcode=1984ASAJ...75...10M |doi-access=free }}</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. |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 |year=1989 |publisher=MIT Press |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:{{sfrac|1|2|3}}:{{sfrac|2|1|3}} and 4:5:6 = 2:{{sfrac|2|1|2}}:3 = 1:{{sfrac|1|1|4}}:{{sfrac|1|1|2}}).