Bohlen–Pierce scale: Difference between revisions

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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.
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.
==Theory==
{| class="wikitable center-all"
|+Approximation of prime intervals in 13-EDT
! colspan="2" |Prime interval
!2
!3
!5
!7
!11
!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.]]
[[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.]]
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[[File:Sword_BP_guitars.jpg|thumb|500px]]
[[File:Sword_BP_guitars.jpg|thumb|500px]]


== Lambda scale ==
{{main|4L 5s (tritave-equivalent)}}
=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.
=Theory=
[http://www.huygens-fokker.org/bpsite/ Bohlen Pierce website]
[http://www.huygens-fokker.org/bpsite/ Bohlen Pierce website]


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[[Relationship between Bohlen-Pierce and octave-ful temperaments]]
[[Relationship between Bohlen-Pierce and octave-ful temperaments]]


=Physical instruments tuned to the BP scale=
==Lambda scale==
{{main|4L 5s (tritave-equivalent)}}
 
==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.
 
==Physical instruments tuned to the BP scale==
[http://www.swordguitars.com/Sword_Bohlen_Pierce_acoustic.JPG Bohlen Pierce guitar]
[http://www.swordguitars.com/Sword_Bohlen_Pierce_acoustic.JPG Bohlen Pierce guitar]


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[https://www.freewebs.com/ralphjarzombek/120420.mp3 120420] by Ralph Jarzombek
[https://www.freewebs.com/ralphjarzombek/120420.mp3 120420] by Ralph Jarzombek




==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]]
*[[No-twos 31-limit]]


[[Category:Bohlen-pierce]]
[[Category:Bohlen-pierce]]

Revision as of 04:14, 15 July 2021

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 bohpier. Bohlen-Pierce is normally thought of (if not in these terms, then in fact) as a temperament defined on the 3.5.7 subgroup. However, it (or at least 3.5.7-limit 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.

Theory

Approximation of prime intervals in 13-EDT
Prime interval 2 3 5 7 11 13 17 19
Error absolute (¢) -29.6 0.0 -6.5 -3.8 -54.8 -51.4 +69.4 +23.1
relative (%) -20 0 -4 -3 -37 -35 +47 +16
Patent val 8 13 19 23 28 30 34 35
Fifthspan -1 0 -4 +2 +3 +6 -1 +7
Chris Vaisvil's BP electric guitar. [http://chrisvaisvil.com/the-bohlen-pierce-epiphone-roadie-guitar/ Music from this guitar.

Bohlen Pierce website

Wikipedia Bohlen-Pierce scale

Bohlen-Pierce Scale Research by Elaine Walker

Sword, Ronald. "Bohlen Pierce Scales for Guitar" IAAA Press, UK-USA. First Ed: May 2009.

Intervals of BP

Relationship between Bohlen-Pierce and octave-ful temperaments

Lambda scale

Main article: 4L 5s (tritave-equivalent)

Triple Bohlen-Pierce

Proposed by Paul Erlich, is the 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.

Physical instruments tuned to the BP scale

Bohlen Pierce guitar

Clarinets

Metallophone

Electronic Organ

Stredici

Kalimba (Mbira)

Pedal Steel Guitar

Compositions

A Mean Little Voice by Stephen Weigel

Ask For It by Chris Vaisvil

Links to available music written in BP at above website.


Bohl-en Roll by Carlo Serafini (blog entry)

Bohlen-Pierce electric guitar improvisation by Jean-Pierre Poulin

Bohlen-Pierce "Stretched Chroma" Acoustic Improvisation by Ron Sword

Reminiscences by Steven Yi

Roll'n'Peace by Jean-Pierre Poulin

Comets Over Flatland 1 by Randy Winchester

Comets Over Flatland 2 by Randy Winchester

Comets Over Flatland 3 by Randy Winchester

Comets Over Flatland 4 by Randy Winchester

Bohlen-Pierce Island audio by Chris Vaisvil

Mesonic Atom by Chris Vaisvil

Bending the Rules by Chris Vaisvil

Bohlen-Pierce Canon by Kjell Hansen.

Bohlen's Pierced Waltz by Chris Vaisvil

The Complex Plane by Chris Vaisvil

120420 by Ralph Jarzombek


See also

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