Bohlen–Pierce scale
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
Intervals
Variations
Lambda scale
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
| 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
- 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
- 032323 by Ralph Jarzombek