Every Interval You've Heard Is a Compromise
Play a major third on a piano. Now play the same two notes as harmonics on a guitar string — touch the string at 1/5 of its length. The guitar's third sounds different. Cleaner. More locked in.
The piano's major third is 400 cents (four equal semitones). The guitar harmonic is 386.31 cents — a ratio of exactly 5:4 between the two frequencies. That 13.69 cent difference is the gap between equal temperament and just intonation, and your ear can absolutely tell.
What Are Pure Ratios?
When a string vibrates, it doesn't just produce one frequency. It produces a harmonic series: the fundamental, then 2x the fundamental, 3x, 4x, 5x, and so on. These are called partials or overtones.
Musical intervals map to these ratios:
- Octave: 2:1 (second harmonic)
- Perfect fifth: 3:2 (third harmonic, brought down an octave)
- Perfect fourth: 4:3
- Major third: 5:4 (fifth harmonic, brought down two octaves)
- Minor third: 6:5
- Harmonic seventh: 7:4 (seventh harmonic — the "blue note")
When two notes are in a pure ratio, their partials align exactly. The waveforms reinforce instead of interfering. This is why pure intervals sound "locked" — there's no beating, no wavering, just a single fused sound.
Why Don't We Use Pure Ratios Everywhere?
Because they don't stack consistently. Three pure major thirds (5:4 × 5:4 × 5:4 = 125:64) should equal an octave (2:1 = 128:64), but they don't. The gap — about 41 cents — is called the diesis. Similar gaps (commas) appear everywhere in JI. You can't build a 12-note scale where every interval is pure.
Equal temperament solves this by making every semitone exactly the same size (100 cents). Nothing is pure, but nothing is catastrophically bad either. It's the musical equivalent of a font where every letter is slightly ugly but they all fit on the same grid.
Dynamic Harmony: Having It Both Ways
What if you didn't need a fixed scale at all? What if every note's tuning was determined by its relationship to the notes around it?
That's what Arbit does. Each note is linked to another by a pure ratio. The same pitch class can have different frequencies in different contexts. No fixed scale means no compromises — every interval can be pure, and modulation is free.