Bohlen–Pierce scale: Difference between revisions
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The ''' | 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]]. | ||
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 | 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]]. | ||
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. | |||
== Theory == | == Theory == | ||
* [[Relationship between | * [[Relationship between Bohlen–Pierce and octave-ful temperaments]] | ||
=== Lambda scale === | === Lambda scale === | ||
{{Main|4L 5s ( | {{Main|4L 5s (3/1-equivalent)}} | ||
== Intervals == | == Intervals == | ||
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== Variations == | == Variations == | ||
=== Triple Bohlen–Pierce === | |||
=== Triple | 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. | ||
Proposed by [[Paul Erlich]] | |||
== Regular temperament properties == | == Regular temperament properties == | ||
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== Music == | == Music == | ||
{{Main| | {{Main| Bohlen–Pierce scale/Music }} | ||
{{Catrel|Bohlen-Pierce tracks}} | {{Catrel|Bohlen-Pierce tracks}} | ||
== See also == | == See also == | ||
* [[Catalog of 3.5.7 subgroup rank two temperaments]] | * [[Catalog of 3.5.7 subgroup rank two temperaments]] | ||
* [[Lumatone mapping for | * [[Lumatone mapping for Bohlen–Pierce]] | ||
== 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]] | ||
[[Category: | [[Category:Bohlen–Pierce| ]] <!-- main article --> | ||
[[Category:13-tone scales]] | [[Category:13-tone scales]] | ||
[[Category:Guitar]] | [[Category:Guitar]] | ||
[[Category:Macrotonal]] | [[Category:Macrotonal]] | ||