Every Note Is a Chord
Pluck a guitar string tuned to A (110 Hz). You hear a single pitch — but the string is vibrating at many frequencies simultaneously. 110 Hz, 220 Hz, 330 Hz, 440 Hz, 550 Hz, and on upward. These are the harmonics, and they define everything about how that note sounds.
The first harmonic (110 Hz) is the fundamental — it determines the pitch you perceive. The second harmonic (220 Hz) is an octave above. The third (330 Hz) is an octave and a fifth. The fourth (440 Hz) is two octaves. The fifth (550 Hz) is two octaves and a major third.
This pattern — 1, 2, 3, 4, 5, 6, 7, 8... times the fundamental — is the harmonic series. It's not a human invention. It's a consequence of physics: any vibrating body with fixed endpoints produces these frequencies.
Where Intervals Come From
The harmonic series is the origin of every interval in music. Reduce any harmonic to its simplest ratio against the fundamental:
- 2nd harmonic → 2:1 = octave
- 3rd harmonic → 3:2 = perfect fifth (brought down an octave)
- 4th harmonic → 4:3 = perfect fourth
- 5th harmonic → 5:4 = major third (brought down two octaves)
- 6th harmonic → 6:5 = minor third
- 7th harmonic → 7:4 = harmonic seventh
- 8th harmonic → another octave
Notice something: the intervals get smaller as you go higher. The first few harmonics produce the consonances that dominate Western harmony — octaves, fifths, thirds. Higher harmonics produce increasingly exotic intervals. The 11th harmonic is an undecimal tritone (~551 cents), neither augmented fourth nor diminished fifth. The 13th harmonic sits between a minor sixth and a major sixth. These are intervals that 12-tone equal temperament can't represent at all.
Why Timbre Is Harmony
A clarinet and a violin playing the same note sound different because they emphasize different harmonics. The clarinet suppresses even harmonics (2nd, 4th, 6th...) — its bore only supports odd-numbered modes. The violin produces a rich spread of all harmonics, with the balance shifting depending on bow speed and pressure.
This means timbre is harmony. When you hear a rich, full tone, you're hearing a chord — the note's own partials reinforcing specific interval relationships. When two instruments blend well, it's because their harmonic spectra overlap. When they clash, it's because the partials interfere destructively.
As a composer, understanding this changes how you think about orchestration. Doubling a melody at the major third doesn't just add a harmony note — it reinforces the 5th harmonic that already exists in the timbre of the original voice.
The Harmonic Series and Consonance
Why do some intervals sound consonant and others dissonant? The simplest explanation: consonance is harmonic alignment. Two notes in a 3:2 ratio share many partials. The 2nd harmonic of the upper note equals the 3rd harmonic of the lower. The 4th of the upper equals the 6th of the lower. These shared frequencies fuse instead of beating.
Two notes in an irrational ratio — like the equal-tempered tritone (600 cents, a ratio of 2^(6/12)) — share no partials at all. Every harmonic of one collides with the harmonics of the other, producing the rough beating we call dissonance.
This is why just intonation sounds different from equal temperament. A pure major third (5:4 = 386 cents) shares partials precisely. An equal-tempered major third (400 cents) is close but not exact — the partials almost align, creating a subtle wavering called beating. On a sustained organ chord, you can hear this clearly: the just version fuses into a single resonant sound, while the tempered version shimmers with interference.
Beyond the 5-Limit
Traditional Western harmony uses intervals from the first five harmonics — ratios involving only the numbers 2, 3, and 5. This is called 5-limit just intonation. But the harmonic series keeps going.
The 7th harmonic gives the harmonic seventh (7:4 = 969 cents) — 31 cents flatter than the equal-tempered minor seventh. This is the "barbershop seventh," the blue note in blues singing, the natural resting point for dominant seventh chords. It sounds completely different from the equal-tempered version: warm, stable, resolved.
The 11th harmonic produces a neutral interval (~551 cents) that splits the tritone into two unequal parts. The 13th harmonic gives a neutral sixth. These intervals appear in traditional music from the Middle East, Central Asia, and parts of Africa — tuning systems that evolved independently of the Western 12-note framework.
Exploring higher harmonics opens compositional territory that equal temperament literally cannot reach.
Composing With the Harmonic Series
Most DAWs lock you into 12-TET — the piano roll is the grid, and the grid is the tuning. But what if you could access any harmonic directly?
Arbit is a piano roll where every note's frequency is defined by its harmonic relationship to other notes. Link a note to another at the 7th harmonic and hear the barbershop seventh instantly. Link at the 11th and hear the undecimal tritone — an interval your equal-tempered keyboard has no key for. Chain links into hierarchies where moving one anchor note cascades pure tuning through an entire chord progression.
The harmonic series isn't just theory — it's the raw material of all harmony. Arbit lets you compose with it directly, from the 1st harmonic to the 32nd and beyond.