Bohlen-Pierce Scale
The Bohlen-Pierce (BP) scale is a unique tuning system that uses the tritave (3:1) instead of the octave (2:1) as its primary interval. It creates an otherworldly, distinctive sound unlike traditional Western music.
Overview
| Property | Value |
|---|---|
| Interval of equivalence | Tritave (3:1) |
| Steps per tritave | 13 |
| Step ratio | 3^(1/13) ≈ 1.08818 |
What Makes BP Different?
The Tritave
Traditional music treats the octave (2:1) as the interval of "equivalence" - notes an octave apart are considered the "same" note.
Bohlen-Pierce uses the tritave (3:1) instead:
- Notes a tritave apart are considered equivalent
- The 3:1 ratio comes from the third harmonic
- This creates a fundamentally different harmonic experience
No Octaves!
BP explicitly avoids the 2:1 ratio:
- There are no octave equivalents
- The sense of "returning home" at the octave is absent
- This creates an alien, floating quality
Expression Syntax
Single BP Step
// One BP step
new Fraction(3).pow(new Fraction(1, 13))Multiple Steps
// BP "fifth" (6 steps)
new Fraction(3).pow(new Fraction(6, 13))
// Full tritave (13 steps)
new Fraction(3).pow(new Fraction(13, 13)) // = 3Applying to BaseNote
// Note at 3 steps above BaseNote
module.baseNote.getVariable('frequency').mul(
new Fraction(3).pow(new Fraction(3, 13))
)The BP-13 Scale
| Step | Approximate Ratio | Cents |
|---|---|---|
| 0 | 1/1 | 0 |
| 1 | 27/25 | 146 |
| 2 | 25/21 | 293 |
| 3 | 9/7 | 435 |
| 4 | 7/5 | 583 |
| 5 | 75/49 | 731 |
| 6 | 5/3 | 884 |
| 7 | 9/5 | 1018 |
| 8 | 49/25 | 1165 |
| 9 | 15/7 | 1319 |
| 10 | 7/3 | 1467 |
| 11 | 63/25 | 1600 |
| 12 | 25/9 | 1755 |
| 13 | 3/1 | 1902 |
Using the BP-13 Module
- Open the Module Bar
- Find Melodies category
- Drag BP-13 onto the workspace
Listen carefully - you'll hear the tritave "closure" at step 13, not an octave!
BP Intervals
The BP scale has its own interval vocabulary:
| BP Interval | Steps | Ratio Approximation |
|---|---|---|
| BP minor second | 1 | 27/25 |
| BP major second | 2 | 25/21 |
| BP minor third | 3 | 9/7 |
| BP major third | 4 | 7/5 |
| BP fourth | 5 | 75/49 |
| BP fifth | 6 | 5/3 |
| BP sixth | 7 | 9/5 |
| BP seventh | 8 | 49/25 |
| BP eighth | 9 | 15/7 |
| BP ninth | 10 | 7/3 |
| BP tenth | 11 | 63/25 |
| BP eleventh | 12 | 25/9 |
| Tritave | 13 | 3/1 |
BP Triads
Traditional triads don't work in BP (they use 2:1 relationships). Instead:
BP Major Triad
Steps: 0, 4, 9 (ratios approximately 1:7/5:15/7)
root.frequency = baseNote.frequency
third.frequency = baseNote.frequency.mul(new Fraction(3).pow(new Fraction(4, 13)))
fifth.frequency = baseNote.frequency.mul(new Fraction(3).pow(new Fraction(9, 13)))BP Minor Triad
Steps: 0, 3, 9 (ratios approximately 1:9/7:15/7)
root.frequency = baseNote.frequency
third.frequency = baseNote.frequency.mul(new Fraction(3).pow(new Fraction(3, 13)))
fifth.frequency = baseNote.frequency.mul(new Fraction(3).pow(new Fraction(9, 13)))Why Use Bohlen-Pierce?
Unique Sound
BP creates sounds impossible in traditional music:
- No sense of octave return
- Different consonance/dissonance relationships
- Alien, otherworldly quality
Odd Harmonics
BP emphasizes odd-numbered harmonics (3, 5, 7, 9...):
- Clarinet-like timbres (which naturally emphasize odd harmonics) work well
- Square waves sound particularly at home
Theoretical Interest
BP explores what music could sound like in an alternate universe with different acoustical foundations.
Challenges
Unfamiliar
Everything you know about Western harmony doesn't apply directly.
Emotional Ambiguity
Without familiar major/minor distinctions, emotional content is less predictable.
Limited Repertoire
Very little music exists in BP. You're exploring new territory!
Instruments for BP
- Clarinets: Natural affinity for odd harmonics
- Synthesizers: RMT Compose is perfect for BP exploration
- Custom-built instruments: Some instruments have been built specifically for BP
Tips
- Listen without expectations - Don't look for octaves or traditional intervals
- Start with the BP-13 module - Hear the complete scale first
- Try BP triads - They're consonant in their own way
- Use odd-harmonic timbres - Square waves, clarinets
- Embrace the strangeness - That's the point!
Example: BP Scale
// Build a 13-note BP scale
note1.frequency = baseNote.frequency
note2.frequency = note1.frequency.mul(new Fraction(3).pow(new Fraction(1, 13)))
note3.frequency = note2.frequency.mul(new Fraction(3).pow(new Fraction(1, 13)))
// ... continue for all 13 notes
note14.frequency = note13.frequency.mul(new Fraction(3).pow(new Fraction(1, 13)))
// note14 = 3 × baseNote (tritave)Next Steps
- Create your own system with Custom TET
- Return to Pure Ratios for comparison
- Explore 12-TET to appreciate what's different