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| 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 [[Equivalence|equivalent]]. | | 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]]. |
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| 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]]. | | 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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| 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.
| | 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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| == Theory == | | == Theory == |
| * [[Relationship between Bohlen-Pierce and octave-ful temperaments]] | | * [[Relationship between Bohlen–Pierce and octave-ful temperaments]] |
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| | === Lambda scale === |
| | {{Main|4L 5s (3/1-equivalent)}} |
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| == Intervals == | | == Intervals == |
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| == Variations == | | == Variations == |
| === Lambda scale ===
| | === Triple Bohlen–Pierce === |
| {{Main|4L 5s (tritave-equivalent)}}
| | 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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| === Triple Bohlen-Pierce === | |
| 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. | |
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| == Regular temperament properties == | | == Regular temperament properties == |
| {| class="wikitable center-4 center-5 center-6" | | {{Main| 13edt#Regular temperament properties }} |
| ! rowspan="2" | Subgroup
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| ! rowspan="2" | [[Comma list]]
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| ! rowspan="2" | [[Mapping]]
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| ! rowspan="2" | Optimal<br>Equave stretch (¢)
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| ! colspan="2" | Tuning error
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| |-
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| ! [[TE error|Absolute]] (¢)
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| ! [[TE simple badness|Relative]] (%)
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| |-
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| | 3.5.7
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| | 245/243, 3125/3087
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| | [{{val| 13 19 23 }}] (b13)
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| | +1.393
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| | 1.150
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| | 0.79
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| |} | |
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| === Rank-2 temperaments ===
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| {| class="wikitable center-all right-3 left-5"
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| |+Table of rank-2 temperaments by generator
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| ! Periods<br>per tritave
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| ! Generator<br>(reduced)
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| ! Cents<br>(reduced)
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| ! Associated<br>ratio
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| ! Temperament
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| |-
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| | 1
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| | 1\13
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| | 146.30
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| | 49/45
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| | [[Procyon]]
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| |-
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| | 1
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| | 2\13
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| | 292.61
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| | 25/21
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| | [[Sirius]]
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| |-
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| | 1
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| | 3\13
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| | 438.91
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| | 9/7
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| | [[BPS]]
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| |-
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| | 1
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| | 4\13
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| | 585.22
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| | 7/5
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| | [[Canopus]]
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| |-
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| |1
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| |5\13
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| |731.63
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| |75/49
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| |-
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| | 1
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| | 6\13
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| | 877.83
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| | 5/3
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| | [[Arcturus]]
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| |}
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| == Instruments == | | == Instruments == |
| * [http://www.huygens-fokker.org/bpsite/instruments.html#anchor39641 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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| == Music == | | == Music == |
| {{See also|:Category:Bohlen-Pierce tracks}} | | {{Main| Bohlen–Pierce scale/Music }} |
| '''[[Bazil Müzik]]'''
| | {{Catrel|Bohlen-Pierce tracks}} |
| * [https://www.youtube.com/watch?v=In2nYr9o3Sk Bohlen-Pierce Atonal Orchestra]
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| '''[[Kjell Hansen]]'''
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| * [http://micro.soonlabel.com/gene_ward_smith/Others/Hansen/Bohlen-Pierce%20Canon.mp3 Bohlen-Pierce Canon]
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| '''[[Ralph Jarzombek]]'''
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| * [https://www.freewebs.com/ralphjarzombek/120420.mp3 120420]
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| * [https://ralphjarzombek.webs.com/032323.mp3 032323]
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| '''[[Jean-Pierre Poulin]]'''
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| * [http://www.jeanpierrepoulin.com/mp3/BPguitares.mp3 Bohlen-Pierce electric guitar improvisation]
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| * [http://www.jeanpierrepoulin.com/mp3/Roll%27n%27Peace.mp3 Roll'n'Peace]
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| '''[[Carlo Serafini]]'''
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| * [http://www.seraph.it/XenoTunes4_files/Bohl-en%20Roll.mp3 Bohl-en Roll] ([http://www.seraph.it/blog_files/004e4fe53f09c06fd5e709a08d53d0d5-4.html blog entry])
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| '''[[Ron Sword]]'''
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| * [http://www.ronsword.com/sounds/BP_stretched_chroma.mp3 Bohlen-Pierce "Stretched Chroma" Acoustic Improvisation]
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| '''[[Chris Vaisvil]]'''
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| * [http://chrisvaisvil.com/ask-for-it-over-the-ridge/ Ask For It]
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| * [http://micro.soonlabel.com/bohlen-pierce/daily20110915_BP_strings-Bohlen-Pierce-Island.mp3 Bohlen-Pierce Island audio]
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| * [http://micro.soonlabel.com/bohlen-pierce/mesonic_atom_13th_root_of_3_BP_apx_X1.mp3 Mesonic Atom]
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| * [http://micro.soonlabel.com/MOTU/Vaisvil-MOTU-8a-bending-the-rules-in-bohlen-pierce.mp3 Bending the Rules]
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| * [http://chrisvaisvil.com/bohlens-pierced-waltz/ Bohlen's Pierced Waltz]
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| * [http://chrisvaisvil.com/the-complex-plane-bp-guitar/ The Complex Plane]
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| '''[[Stephen Weigel]]'''
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| * [https://soundcloud.com/overtoneshock/a-mean-little-voice-bohlen-pierce-or-13ed3 A Mean Little Voice]
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| '''[[Randy Winchester]]'''
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| * [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/01%20-%201.%20Bohlen-Pierce.mp3 Comets Over Flatland 1]
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| * [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/02%20-%202.%20Bohlen-Pierce.mp3 Comets Over Flatland 2]
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| * [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/03%20-%203.%20Bohlen-Pierce.mp3 Comets Over Flatland 3]
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| * [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/04%20-%204.%20Bohlen-Pierce.mp3 Comets Over Flatland 4]
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| '''[[Steven Yi]]'''
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| * [http://www.kunstmusik.com/mp3/reminiscences.mp3 Reminiscences]
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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]] |
| * [[Lumatone mapping for Bohlen-Pierce]] | |
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| == Further reading == | | == Further reading == |
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| * [http://ziaspace.com/_academic/BP/ Bohlen-Pierce Scale Research] by [[Elaine Walker]] | | * [http://ziaspace.com/_academic/BP/ Bohlen-Pierce Scale Research] by [[Elaine Walker]] |
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| [[Category:Bohlen-Pierce| ]] <!-- main article --> | | [[Category:Bohlen–Pierce| ]] <!-- main article --> |
| [[Category:13-tone scales]] | | [[Category:13-tone scales]] |
| [[Category:Edonoi]]
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| [[Category:Edt]]
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| [[Category:Guitar]] | | [[Category:Guitar]] |
| [[Category:Listen]]
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| [[Category:Macrotonal]] | | [[Category:Macrotonal]] |