How this instrument works
When a string, air column, or any resonant body vibrates, it doesn't just produce one pure tone: it produces a whole family of frequencies stacked on top of the lowest one, called the fundamental. Those additional frequencies, the harmonics, sit at exact whole-number multiples of the fundamental — the 2nd harmonic at twice the fundamental frequency, the 3rd at three times, the 4th at four times, and so on, extending upward without limit.
This whole-number relationship isn't an approximation; it falls directly out of the physics of how a fixed-length vibrating string or air column resonates. It's also the acoustic basis for a huge amount of music theory: the 2nd harmonic is exactly one octave above the fundamental, the 3rd harmonic is an octave and a perfect fifth above it, and the relative loudness of each harmonic in a sound is a large part of what makes one instrument's tone distinguishable from another's.
Guitarists exploit this directly with natural harmonics — lightly touching a string at points that divide it into halves, thirds, or quarters (roughly the 12th, 7th, and 5th frets) isolates the 2nd, 3rd, and 4th harmonics as clear, bell-like tones. This instrument computes the exact frequency of any harmonic number above any fundamental.
- Enter Fundamental frequency (Hz) — the base pitch's frequency.
- Enter Harmonic number (n) — a whole number, 1 or greater, for which overtone you want.
- Read Frequency of the nth harmonic (Hz) directly beneath the inputs.
- Harmonic number 1 always returns the fundamental itself, unchanged.
- Both the fundamental frequency and harmonic number must be positive; harmonic number must be at least 1.
Worked example — 3rd harmonic of a 110 Hz fundamental
Enter Fundamental frequency = 110 Hz (A2, a guitar's open A string) and Harmonic number = 3. That's 110 x 3 = 330.0 Hz. Frequency of the nth harmonic reads 330.0.
The 3rd harmonic is always an octave and a perfect fifth above the fundamental — here, 330 Hz sits a twelfth above the open 110 Hz A string, the interval a guitarist hears when lightly touching that string at the 7th fret to sound its natural harmonic.
Questions
What's the difference between harmonic number 1 and the fundamental?
They're the same thing. Harmonic number 1 is defined as the fundamental frequency itself — multiplying by 1 leaves it unchanged. The harmonic series starts counting from the fundamental as the 1st harmonic, then builds upward from there at each successive whole-number multiple.
Why are harmonics always exact whole-number multiples of the fundamental?
It comes from the physics of standing waves on a fixed-length string or in an air column: only wave patterns that fit a whole number of half-wavelengths into that fixed length can resonate stably. Each of those stable patterns corresponds to a frequency that's an exact integer multiple of the lowest (fundamental) resonant frequency.
How do harmonics affect an instrument's timbre?
The specific mix of harmonic strengths present in a tone — which ones are loud, which are weak or missing — is a major part of what makes a violin sound different from a trumpet playing the same fundamental pitch. Timbre isn't just the fundamental frequency; it's the whole harmonic 'fingerprint' layered on top of it.
Where do natural harmonics show up on a guitar fretboard?
Lightly touching (without pressing down) a string at points that divide its length into equal fractions isolates specific harmonics: the halfway point (12th fret) gives the 2nd harmonic, one-third of the way along (7th fret) gives the 3rd harmonic, and one-quarter along (5th fret) gives the 4th harmonic, each ringing out as a clear, bell-like tone.
Is 'harmonic series' the same thing as 'overtone series'?
They describe the same sequence of frequencies but count differently: the harmonic series numbers the fundamental itself as the 1st harmonic, while the overtone series only counts the tones above the fundamental, so the 1st overtone is the 2nd harmonic, the 2nd overtone is the 3rd harmonic, and so on — an offset of one in the numbering.