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.

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

Theory

• Relationship between Bohlen-Pierce and octave-ful temperaments

Intervals

• Intervals of BP (equal version)

Variations

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.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal Equave stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
3.5.7	245/243, 3125/3087	[⟨13 19 23]] (b13)	+1.393	1.150	0.79

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per equave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperament
1	1\13	146.30	49/45	Procyon
2	2\13	292.61	25/21	Sirius
1	3\13	438.91	9/8	BPS
1	4\13	585.22	7/5	Canopus
1	6\13	877.83	5/3	Arcturus

Publications

• Bohlen Pierce Scales for Guitar by Ron Sword, IAAA Press, UK-USA. First Ed: May 2009.

Instruments

• Bohlen Pierce guitar
• Clarinets
• Metallophone
• Electronic Organ
• Stredici
• Kalimba (Mbira)
• Pedal Steel Guitar

Links

• Wikipedia: Bohlen-Pierce scale
• The Bohlen-Pierce Site
• Bohlen-Pierce Scale Research by Elaine Walker

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