Harmonic Series Calculator
Calculate all harmonics (overtones) for any fundamental frequency. See the frequency of each partial, its nearest musical note, cents deviation, and an interactive harmonic spectrum display.
Fundamental Frequency
Harmonic Spectrum
Harmonic Series Table
| n | Frequency (Hz) | Nearest Note | Cents ± | Interval from f₁ | Multiplier |
|---|
Understanding the Harmonic Series
The harmonic series is a sequence of harmonic frequencies that are integer multiples of the fundamental frequency (f₁): f₁, 2f₁, 3f₁, 4f₁, ... Each frequency is called a harmonic or partial. The fundamental (1st harmonic) is what we perceive as the pitch; the higher harmonics (overtones) determine the timbre or tone quality of the sound. This calculator lets you find the complete set of musical harmonics for any root pitch, making it a practical overtone reference for composers, instrument builders, and synthesis designers.
Musical Instruments and Harmonics
- String instruments — a vibrating string vibrates simultaneously at its fundamental and all integer harmonics. The relative amplitudes of harmonics give each instrument its characteristic timbre. Real strings exhibit slight inharmonicity (stiffness causes upper partials to stretch slightly sharp), which is most audible on piano bass strings.
- Wind instruments — open-bore instruments (flute, trumpet) support all harmonics. Closed-bore instruments (clarinet) support only odd harmonics (1st, 3rd, 5th...), giving a hollow tone quality. This odd-harmonic restriction is audible in the clarinet's distinctive timbre compared to the oboe, which supports a full harmonic series.
- Singing voice — the vocal tract shapes the harmonic content by selectively amplifying certain formant frequencies. Vowel sounds differ in which harmonics are emphasized, a process central to spectral analysis of speech and singing. Throat singers (overtone singers) can consciously reinforce individual harmonics to hear them as separate pitches.
- Electronic synthesis — additive synthesis builds complex timbres by summing sine waves at harmonic frequencies with controlled amplitudes. You can explore a reference pitch for additive synthesis experiments using the online tone generator.
Harmonic Series and Equal Temperament
The natural harmonic series doesn't align perfectly with 12-tone equal temperament (12-TET). The 7th harmonic (7f₁) is about 31 cents flat compared to the nearest 12-TET note. The 11th harmonic falls almost exactly between two notes. This "harmonic dissonance" is why just intonation and equal temperament represent different compromises between physical reality and musical practicality. Many microtonal and alternative tuning systems — such as 31-TET or 53-TET — are designed to accommodate more of the natural harmonic series without the compromises of standard 12-TET. The calculator's "Cents ±" column makes it straightforward to see exactly how far each partial deviates from the nearest equal-tempered pitch, a useful reference when exploring beat frequencies caused by those tuning discrepancies.
Why Is Piano Tuning Stretched? Inharmonicity and the Railsback Curve
Real piano strings are stiff, not perfectly flexible, so their vibrating modes fall slightly sharp of exact integer harmonics — a property called inharmonicity. The effect is largest on the thick, short strings at the bass and treble extremes of the keyboard. Because the overtones of each string are stretched sharp of the ideal harmonic series, tuning octaves to "zero beats" forces the tuner to raise treble pitches and lower bass pitches relative to equal temperament. The resulting S-shaped deviation from 12-TET is known as the Railsback curve (or stretch tuning). On a well-tuned concert grand, the highest notes may sit 15–20 cents sharp of equal temperament and the lowest notes similarly flat. You can explore the practical implications with the piano stretch tuning calculator.
Stretch tuning is therefore not an error — it is the direct acoustic consequence of inharmonicity. A piano tuned to flat equal temperament would have octaves that beat audibly, because the 2nd partial of the lower note would clash with the (stretched) fundamental of the upper note. When checking individual string pitches with a chromatic tuner, expect treble notes to read sharp and bass notes to read flat compared to equal temperament; that is correct for a stretched tuning.
Fourier Analysis and the Harmonic Series
The harmonic series is the practical backbone of Fourier analysis applied to audio: any periodic sound wave can be decomposed into sine waves at the fundamental frequency and its integer multiples, each with its own amplitude and phase. This Fourier decomposition is what spectrum analyzers and standing wave models rely on. The relative amplitudes of the harmonics — the Fourier coefficients — uniquely define the waveform shape: a square wave contains only odd harmonics with amplitudes that fall as 1/n, a sawtooth contains all harmonics falling as 1/n, and a triangle wave contains odd harmonics falling as 1/n².