# Bohlen-Pierce scale

The **Bohlen-Pierce scale** (**BP**) 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

*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

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]]

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

## See also

## 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)