Bohlen-Pierce
Bohlen–Pierce throws away the octave. Instead of dividing 2/1, it divides the tritave — the ratio 3/1 — into thirteen equal steps. Notes a tritave apart are treated as the "same" note, the way notes an octave apart are in every other system you know.
The result sounds genuinely foreign, and it is not a gimmick: BP is built on odd harmonics, and it has consonances of its own.
| Property | Value |
|---|---|
| Interval of equivalence | tritave, 3/1 |
| Steps per tritave | 13 |
| Step ratio | 3 ^ (1/13) ≈ 1.088182 |
| Step size | 146.30 cents |
| Tritave | 1901.96 cents, exact |
Why the tritave
Take a clarinet, or a square wave. Its spectrum has only odd harmonics: 1, 3, 5, 7, 9… The even ones are missing, and so the 2/1 octave — which is where harmonic 2 lives — has nothing to reinforce it.
Build a scale for that spectrum and the octave stops being special. The strongest simple ratio available is 3/1, and the consonances that fall out are ratios of odd numbers: 3, 5, 7, 9. Bohlen–Pierce is what you get when you take that seriously and divide 3/1 into thirteen equal parts.
Thirteen is not arbitrary: it is the division that best approximates the just BP intervals (see the table below), the same way twelve best approximates 5-limit just intonation for the octave.
The scale
The shipped module is equal-tempered BP: every step is exactly 3 ^ (1/13) = 146.30 cents. Just BP — the ratio-based version the temperament approximates — is a different set of numbers. Both are given here, side by side, because they are easy to confuse.
| Step | Equal-tempered cents | Nearest just BP ratio | Just cents | ET error |
|---|---|---|---|---|
| 0 | 0 | 1/1 | 0 | 0 |
| 1 | 146.30 | 27/25 | 133.2 | +13.1¢ |
| 2 | 292.61 | 25/21 | 301.8 | −9.2¢ |
| 3 | 438.91 | 9/7 | 435.1 | +3.8¢ |
| 4 | 585.22 | 7/5 | 582.5 | +2.7¢ |
| 5 | 731.52 | 75/49 | 736.9 | −5.4¢ |
| 6 | 877.83 | 5/3 | 884.4 | −6.5¢ |
| 7 | 1024.13 | 9/5 | 1017.6 | +6.5¢ |
| 8 | 1170.43 | 49/25 | 1165.0 | +5.4¢ |
| 9 | 1316.74 | 15/7 | 1319.4 | −2.7¢ |
| 10 | 1463.04 | 7/3 | 1466.9 | −3.8¢ |
| 11 | 1609.35 | 63/25 | 1600.1 | +9.2¢ |
| 12 | 1755.65 | 25/9 | 1768.7 | −13.1¢ |
| 13 | 1901.96 | 3/1 | 1902.0 | 0 |
The best-tuned degrees are 3, 4, 9 and 10 — and those are exactly the ones the BP chords are built from.
BP chords
Traditional triads do not transfer: they are stacks of 5-limit ratios that assume an octave. BP has its own.
The BP major chord — 3:5:7
The characteristic Bohlen–Pierce sonority. Take the odd harmonics 3, 5 and 7 and build a chord on them. Over a root, the tones are 1/1, 5/3 and 7/3.
In equal-tempered BP those land on steps 0, 6 and 10:
base.f # 3
base.f * 3 ^ (6/13) # 5 (877.8¢ — just 5/3 is 884.4¢)
base.f * 3 ^ (10/13) # 7 (1463.0¢ — just 7/3 is 1466.9¢)Both tones land within 7 cents of pure. This chord locks the way a just major triad does — it is the reason BP works as music rather than as noise.
The BP minor chord — 5:7:9
The same three odd harmonics with the root moved to 5. Over the root, the tones are 1/1, 7/5 and 9/5, which fall on steps 0, 4 and 7:
base.f # 5
base.f * 3 ^ (4/13) # 7 (585.2¢ — just 7/5 is 582.5¢)
base.f * 3 ^ (7/13) # 9 (1024.1¢ — just 9/5 is 1017.6¢)Legacy JavaScript syntax
// BP major chord (3:5:7)
module.baseNote.getVariable('frequency')
module.baseNote.getVariable('frequency').mul(new Fraction(3).pow(new Fraction(6, 13)))
module.baseNote.getVariable('frequency').mul(new Fraction(3).pow(new Fraction(10, 13)))Hear the just version too
The library ships both chords as pure ratios. Base-3 chord (3:5:7:9) and Base-5 chord (5:7:9:11) are in the Chords section. Drop one next to a tempered BP chord and compare — the tempered one is a few cents off and beats very slowly.
Writing BP expressions
3 ^ (1/13) # one BP step
3 ^ (6/13) # six steps
3 ^ (13/13) # a full tritave = 3
base.f * 3 ^ (4/13) # four steps above the BaseNote
[1].f * 3 ^ (1/13) # one step above note 1The base is 3, not 2. That is the only structural difference from the TET pages — everything else about the syntax is the same.
RMT keeps base-2 and base-3 powers as separate terms rather than merging them into one number. That is why you can mix them in a single scale (see Mixed-Base, below) and why 3^(1/13) stays symbolic across saves.
The shipped module
The library's Scale Systems section ships a Bohlen–Pierce module (file scale-systems/BP-13.json; the tile reads Bohlen–Pierce).
| Notes | 14 — thirteen steps plus the note you start on |
| Base frequency | 440 Hz |
| Tempo | 80 |
| Beats per measure | 4 |
| Instrument | sine-wave |
| Note duration | beat(base) * (1/2) |
| Step | [n].f * 3 ^ (1/13) |
Note 1 is base.f; every later note is the previous one times 3 ^ (1/13). Note 14 is therefore exactly three times note 1 — the tritave, not an octave. Listen for it: the scale closes somewhere your ear does not expect.
To load it, drag the tile from Scale Systems (or search bohlen, tritave or base-3 with the library magnifier) onto a note or onto the BaseNote circle — a drop on empty canvas is refused. The full loading workflow is on Equal Temperament.
All thirteen steps after the first show a ≈ prefix and cross-hatching, because 3^(k/13) is irrational.
Timbre matters more here than anywhere else
BP is a scale designed for odd-harmonic spectra. Play it on a timbre with strong even harmonics and the consonances stop working — the chords will beat against overtones that the scale has no notes for.
Use an odd-harmonic voice. Of the built-in instruments, square-wave is the one to reach for: a square wave's spectrum is odd harmonics only, which is precisely what BP was designed around. sine-wave (the module's default) is also safe, because a sine has no harmonics at all to clash with.
The full instrument list is sine-wave, square-wave, sawtooth-wave, triangle-wave, organ, vibraphone, fm-epiano, piano, violin. Set the default in Settings → Audio → Default instrument, or set instrument on a single note.
Sawtooth and the sampled voices will fight the scale
sawtooth-wave has all harmonics, even ones included, and the sampled piano and violin are recordings of instruments with full harmonic spectra. They will play BP pitches accurately — the samplers pitch-shift by frequency / rootHz, so the frequencies are correct — but the harmonic clash is real, and BP chords will not lock on them.
Neighbours in the same section
Two other modules sit alongside BP in Scale Systems, and both are natural next stops from here.
Tesla is an 81-note scale of pure odd-harmonic ratios on a base-3 frame — the just expression of the same odd-harmonic idea BP tempers. Because every step is rational, no Tesla note is hatched or marked with ≈.
Mixed-Base is a 12-note experiment that alternates step bases in a single line — 2^(1/12), then 3^(1/13), with a 5^(1/7) partway through. It only works because the app keeps 2-, 3- and 5-based powers as independent terms.
Both are described in full on Equal Temperament.
What to expect
No octave return. The sense of arriving home an octave up is simply absent. This is disorienting for about a minute, and then it is not.
Consonance still exists. 3:5:7 is a stable, restful chord. It is not a major triad, but it does the same job.
Major and minor stop meaning what they meant. BP's "major" and "minor" chords are two rotations of the same odd-harmonic set. They do not carry the emotional loading of their octave-world namesakes, and you should not expect them to.
There is almost no repertoire. You are not learning a tradition. You are starting one.
Next steps
- 31-TET — septimal harmony inside the octave
- Custom TET — divide any interval you like
- Pure Ratios — the odd-harmonic chords in just intonation
- 12-TET — to hear what BP is not