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| The '''Bohlen-Pierce''' ('''BP''') scale is a [[nonoctave]] scale, a 13-part equal division of the perfect-twelfth ([[3/1]]) or [[Tritave]] ('''13edt'''). Each step is about 146 ¢, making it a [[macrotonal]] scale. It is closely related to the rank two temperament [[Sensamagic clan#Bohpier|bohpier]]. Bohlen-Pierce is normally thought of (if not in these terms, then in fact) as a temperament defined on the 3.5.7 [[Just intonation subgroups|subgroup]]. However, it (or at least 3.5.7-limit [[edt|13edt]]) can be extended to the 3.5.7.11/4 subgroup. This extension is controversial because of the presence of 2 in the denominator of 11/4, but the interval is present in the sense that 3^(12\13) provides an approximation to it. Chords of Bohlen-Pierce, from this extended perspective, may be found listed on the page [[chords of bohpier]]. Bohlen-Pierce was discovered independently by [[Heinz Bohlen]], [[John Pierce]], [[Kees van Prooijen]], and perhaps others, usually noticed for its good approximation of odd-number just ratios 3:5, 5:7, 3:7, etc.; but not necessarily 4:11, 5:6, 6:7, etc.
| | {{Wikipedia}} |
| | [[File:vaisvil-BP-guitar-052-crop.jpg|thumb|x00px|Chris Vaisvil's BP electric guitar. [http://chrisvaisvil.com/the-bohlen-pierce-epiphone-roadie-guitar/ Music from this guitar.]]] |
| | [[File:Sword_BP_guitars.jpg|thumb|300px]] |
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| == Theory ==
| | The '''Bohlen–Pierce scale''' ('''BP''') is a 13-tone [[macrotonal]] [[nonoctave]] [[scale]] designed to emphasize odd-number intervals and chords, such as the 3:5:7:9 tetrad. It was first described as a 7-limit [[just intonation]] scale and as an [[equal-step tuning|equal temperament]], [[13edt|13 equal divisions of the tritave]]. The [[tritave]] (3/1) usually replaces the [[octave]] in the role of the [[equave]], such that intervals a tritave apart are considered [[equivalent]]. |
| {| class="wikitable center-all"
| |
| |+Approximation of prime intervals in 13-EDT
| |
| ! colspan="2" |Prime interval
| |
| ! 2
| |
| ! 3
| |
| ! 5
| |
| ! 7
| |
| ! 11
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| ! 13
| |
| ! 17
| |
| ! 19
| |
| |-
| |
| ! rowspan="2" |Error
| |
| ! absolute ([[cent|¢]])
| |
| | -29.6
| |
| | 0.0 | |
| | -6.5 | |
| | -3.8
| |
| | -54.8
| |
| | -51.4
| |
| | +69.4
| |
| | +23.1
| |
| |-
| |
| ! [[Relative error|relative]] (%)
| |
| | -20
| |
| | 0
| |
| | -4
| |
| | -3
| |
| | -37
| |
| | -35
| |
| | +47
| |
| | +16
| |
| |-
| |
| ! colspan="2" |[[Patent val]]
| |
| | 8
| |
| | 13
| |
| | 19
| |
| | 23
| |
| | 28
| |
| | 30
| |
| | 34
| |
| | 35
| |
| |-
| |
| ! colspan="2" |[[Fifthspan]]
| |
| | -1
| |
| | 0
| |
| | -4
| |
| | +2
| |
| | +3
| |
| | +6
| |
| | -1
| |
| | +7
| |
| |}
| |
|
| |
|
| [[File:vaisvil-BP-guitar-052-crop.jpg|frame|Chris Vaisvil's BP electric guitar. [http://chrisvaisvil.com/the-bohlen-pierce-epiphone-roadie-guitar/ Music from this guitar.]] | | It is closely related to the rank two temperament [[Sensamagic clan #Bohpier|bohpier]]. It is normally thought of (if not in these terms, then in fact) as a temperament defined on the 3.5.7 [[Just intonation subgroup|subgroup]]. However, it can be extended to the 3.5.7.11/4 subgroup, especially when considering 13edt instead of the JI version. This extension is controversial because of the presence of 2 in the denominator of 11/4, but the interval is present in the sense that 3^(12\13) provides an approximation to it. Chords of Bohlen–Pierce, from this extended perspective, may be found listed on the page [[chords of bohpier]]. |
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| [[File:Sword_BP_guitars.jpg|thumb|500px]] | | Bohlen–Pierce was discovered independently by [[Heinz Bohlen]], [[John Pierce]], [[Kees van Prooijen]], and perhaps others, usually noticed for its good approximation of odd-number just ratios 3:5, 5:7, 3:7, etc.; but not necessarily 4:11, 5:6, 6:7, etc. |
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| * [[Intervals of BP]]
| | == Theory == |
| * [[Relationship between Bohlen-Pierce and octave-ful temperaments]] | | * [[Relationship between Bohlen–Pierce and octave-ful temperaments]] |
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| == Variations ==
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| === Lambda scale === | | === Lambda scale === |
| {{main|4L 5s (tritave-equivalent)}} | | {{Main|4L 5s (3/1-equivalent)}} |
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| === Triple Bohlen-Pierce === | | == Intervals == |
| Proposed by [[Paul Erlich]], is the [[Triple BP|Triple Bohlen-Pierce Scale]], or 39th root of 3. It approximates additional odd harmonics and can be used in a variety of ways, for both just intonation chords and harmonies, as standard Bohlen-Pierce scale interlocking three times with calm sounding quarter-tones, and for various JI modulations. | | {{Main|Intervals of BP}} |
| | |
| | == Variations == |
| | === Triple Bohlen–Pierce === |
| | Proposed by [[Paul Erlich]] is [[39edt]], also known as the ''Triple Bohlen–Pierce scale''. It adds to 13edt accurate approximations to the 11th and 13th harmonics ([[11/9]] and [[13/9]]) and can be used in a variety of ways, for both just intonation chords and harmonies, as standard Bohlen–Pierce scale interlocking three times with calm-sounding quarter-tones, and for various JI modulations. |
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| == Publications == | | == Regular temperament properties == |
| * [https://ronsword.bigcartel.com/product/creative-applications-for-polyscales-and-scale-sequences ''Bohlen Pierce Scales for Guitar''] by [[Ron Sword]], IAAA Press, UK-USA. First Ed: May 2009.
| | {{Main| 13edt#Regular temperament properties }} |
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| == Instruments == | | == Instruments == |
| * [http://www.swordguitars.com/Sword_Bohlen_Pierce_acoustic.JPG Bohlen Pierce guitar] | | * [http://www.huygens-fokker.org/bpsite/instruments.html#anchor39641 Bohlen Pierce guitar] |
| * [[Microtonal clarinet #BPClarinets|Clarinets]] | | * [[Clarinet#Bohlen-Pierce clarinets|Clarinets]] |
| * Metallophone | | * Metallophone |
| * Electronic Organ | | * Electronic Organ |
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| * [http://bp.b0b.com/2012/11/sierra-pedal-steel/ Pedal Steel Guitar] | | * [http://bp.b0b.com/2012/11/sierra-pedal-steel/ Pedal Steel Guitar] |
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| == Links == | | == Music == |
| * [[Wikipedia: Bohlen-Pierce scale]]
| | {{Main| Bohlen–Pierce scale/Music }} |
| * [http://www.huygens-fokker.org/bpsite/ Bohlen Pierce website]
| | {{Catrel|Bohlen-Pierce tracks}} |
| * [http://ziaspace.com/_academic/BP/ Bohlen-Pierce Scale Research] by [[Elaine Walker]]
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| == Compositions ==
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| * [https://soundcloud.com/overtoneshock/a-mean-little-voice-bohlen-pierce-or-13ed3 A Mean Little Voice] by Stephen Weigel
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| * [http://chrisvaisvil.com/ask-for-it-over-the-ridge/ Ask For It] by [[Chris Vaisvil]]
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| * Links to available music written in BP at [http://www.huygens-fokker.org/bpsite/references2.html#anchor81501 above website].
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| * [http://www.seraph.it/XenoTunes4_files/Bohl-en%20Roll.mp3 Bohl-en Roll] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/004e4fe53f09c06fd5e709a08d53d0d5-4.html blog entry])
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| * [http://www.jeanpierrepoulin.com/mp3/BPguitares.mp3 Bohlen-Pierce electric guitar improvisation] by [[Jean-Pierre Poulin]]
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| * [http://www.ronsword.com/sounds/BP_stretched_chroma.mp3 Bohlen-Pierce "Stretched Chroma" Acoustic Improvisation] by [[Ron Sword]]
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| * [http://www.kunstmusik.com/mp3/reminiscences.mp3 Reminiscences] by [http://www.kunstmusik.com/2010/03/08/reminiscences/ Steven Yi]
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| * [http://www.jeanpierrepoulin.com/mp3/Roll%27n%27Peace.mp3 Roll'n'Peace] by [[Jean-Pierre Poulin]]
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| * [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/01%20-%201.%20Bohlen-Pierce.mp3 Comets Over Flatland 1] by [[Randy Winchester]]
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| * [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/02%20-%202.%20Bohlen-Pierce.mp3 Comets Over Flatland 2] by [[Randy Winchester]]
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| * [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/03%20-%203.%20Bohlen-Pierce.mp3 Comets Over Flatland 3] by [[Randy Winchester]]
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| * [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/04%20-%204.%20Bohlen-Pierce.mp3 Comets Over Flatland 4] by [[Randy Winchester]]
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| * [http://micro.soonlabel.com/bohlen-pierce/daily20110915_BP_strings-Bohlen-Pierce-Island.mp3 Bohlen-Pierce Island audio] by [[Chris Vaisvil]]
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| * [http://micro.soonlabel.com/bohlen-pierce/mesonic_atom_13th_root_of_3_BP_apx_X1.mp3 Mesonic Atom] by [[Chris Vaisvil]]
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| * [http://micro.soonlabel.com/MOTU/Vaisvil-MOTU-8a-bending-the-rules-in-bohlen-pierce.mp3 Bending the Rules] by [[Chris Vaisvil]]
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| * [http://micro.soonlabel.com/gene_ward_smith/Others/Hansen/Bohlen-Pierce%20Canon.mp3 Bohlen-Pierce Canon] by [http://www.youtube.com/watch?v=sd1b9Lh8iFA Kjell Hansen].
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| * [http://chrisvaisvil.com/bohlens-pierced-waltz/ Bohlen's Pierced Waltz] by [[Chris Vaisvil]]
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| * [http://chrisvaisvil.com/the-complex-plane-bp-guitar/ The Complex Plane] by [[Chris Vaisvil]]
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| * [https://www.freewebs.com/ralphjarzombek/120420.mp3 120420] by Ralph Jarzombek
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| == See also == | | == See also == |
| * [[Catalog of 3.5.7 subgroup rank two temperaments]] | | * [[Catalog of 3.5.7 subgroup rank two temperaments]] |
| * [[No-twos 31-limit]] | | * [[Lumatone mapping for Bohlen–Pierce]] |
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| | == Further reading == |
| | * [[Sword, Ron]]. [https://ronsword.bigcartel.com/product/creative-applications-for-polyscales-and-scale-sequences ''Bohlen-Pierce Scales for Guitar: Includes Paul Erlich's Triple BP Scales. Notation methods, Chord-Scales, Melody Chords, Polyscales, and new Scales for the Bohlen-Pierce tuning'']. 2009. ([http://www.metatonalmusic.com/books.html Metatonal Music link]) |
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| | == External links == |
| | * [http://www.huygens-fokker.org/bpsite/ The Bohlen-Pierce Site] |
| | * [http://ziaspace.com/_academic/BP/ Bohlen-Pierce Scale Research] by [[Elaine Walker]] |
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| [[Category:Bohlen-pierce]] | | [[Category:Bohlen–Pierce| ]] <!-- main article --> |
| [[Category:Edonoi]] | | [[Category:13-tone scales]] |
| [[Category:Edt]]
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| [[Category:Equal]]
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| [[Category:Guitar]] | | [[Category:Guitar]] |
| [[Category:Listen]]
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| [[Category:Macrotonal]] | | [[Category:Macrotonal]] |
| [[Category:Scale]]
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| [[Category:Theory]]
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