Bohlen–Pierce scale

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Bohlen–Pierce scale

Chris Vaisvil's BP electric guitar. Music from this guitar.

The Bohlen-Pierce (BP) scale 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 temperament, 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 bohpier. It is normally thought of (if not in these terms, then in fact) as a temperament defined on the 3.5.7 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

• Relationship between Bohlen-Pierce and octave-ful temperaments

Intervals

Main article: Intervals of BP

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 tritave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperament
1	1\13	146.30	49/45	Procyon
1	2\13	292.61	25/21	Sirius
1	3\13	438.91	9/7	BPS
1	4\13	585.22	7/5	Canopus
1	5\13	731.63	75/49
1	6\13	877.83	5/3	Arcturus

Instruments

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

Music

See also: Category:Bohlen-Pierce tracks

Kjell Hansen

• Bohlen-Pierce Canon

Ralph Jarzombek

• 120420
• 032323

Jean-Pierre Poulin

• Bohlen-Pierce electric guitar improvisation
• Roll'n'Peace

Carlo Serafini

• Bohl-en Roll (blog entry)

Ron Sword

• Bohlen-Pierce "Stretched Chroma" Acoustic Improvisation

Chris Vaisvil

• Ask For It
• Bohlen-Pierce Island audio
• Mesonic Atom
• Bending the Rules
• Bohlen's Pierced Waltz
• The Complex Plane

Stephen Weigel

• A Mean Little Voice

Randy Winchester

• Comets Over Flatland 1
• Comets Over Flatland 2
• Comets Over Flatland 3
• Comets Over Flatland 4

Steven Yi

• Reminiscences

See also

• Catalog of 3.5.7 subgroup rank two temperaments
• No-twos 31-limit
• Lumatone mapping for Bohlen-Pierce

Further reading

• Sword, Ron. Bohlen-Pierce Scales for Guitar: Includes Paul Erlich's Triple BP Scales. Notation methods, Chord-Scales, Melody Chords, Polyscales, and new Scales for the Bohlen-Pierce tuning. 2009. (Metatonal Music link)

External links

• The Bohlen-Pierce Site
• Bohlen-Pierce Scale Research by Elaine Walker