Bohlen–Pierce scale

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

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  Chris Vaisvil's BP electric guitar. [http://chrisvaisvil.com/the-bohlen-pierce-epiphone-roadie-guitar/ Music from this guitar.

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• 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

• Catalog of 3.5.7 subgroup rank two temperaments
• No-twos 31-limit