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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

Hz
20 Hz1 kHz
n
Hz
Quick Presets

Harmonic Spectrum

f₁ = 110.0 Hz A2

Harmonic Series Table

n Frequency (Hz) Nearest Note Cents ± Interval from f₁ Multiplier

Understanding the Harmonic Series

The harmonic series is a sequence of 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.

Harmonic Formula
fₙ = n × f₁
n = harmonic number (1, 2, 3...), f₁ = fundamental frequency. The 2nd harmonic (n=2) is exactly one octave above the fundamental (2× frequency).
Interval Pattern in Harmonics
Octave (2:1), 5th (3:2), 4th (4:3)...
The gaps between successive harmonics follow the natural overtone series: octave, perfect 5th, perfect 4th, major 3rd... This is why these intervals sound consonant — they share harmonic content.

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.
  • 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.
  • Singing voice — the vocal tract shapes the harmonic content by selectively amplifying certain formant frequencies. Vowel sounds differ in which harmonics are emphasized.
  • Electronic synthesis — additive synthesis builds complex timbres by summing sine waves at harmonic frequencies with controlled amplitudes.

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.

Frequently Asked Questions

What is the difference between harmonics and overtones?
Harmonics are numbered from 1 (including the fundamental): 1st harmonic = fundamental, 2nd harmonic = first overtone, 3rd harmonic = second overtone, etc. Overtones are harmonics above the fundamental, so the numbering is offset by one. In physics and acoustics, "harmonic" and "partial" are more precise terms; "overtone" is common in music contexts.
Why do harmonics form musical intervals?
The 2nd harmonic (2:1) is an octave. The 3rd harmonic relative to the 2nd (3:2) is a perfect 5th. The 4th relative to the 3rd (4:3) is a perfect 4th. These ratios of small integers correspond to the most consonant musical intervals — a relationship discovered by Pythagoras. Consonance arises because these intervals produce minimal beating between shared harmonics.
What is the 7th harmonic and why is it "out of tune"?
The 7th harmonic of A2 (110 Hz) is 770 Hz. The nearest 12-TET note is G5 (783.99 Hz) or F#5 (739.99 Hz). The 7th harmonic at 770 Hz is about 31 cents flat from G5 — noticeably out of tune with equal temperament. This is why horn players and singers naturally use a slightly "bent" 7th — they're following the natural harmonic series rather than equal temperament.
How does timbre relate to harmonics?
Timbre is the quality that distinguishes two instruments playing the same note. It is determined by the relative amplitudes of the harmonics present. A flute has a pure tone with mostly fundamental and few upper harmonics. A violin has rich upper harmonics. A clarinet emphasizes odd harmonics. Synthesizers can replicate any timbre by controlling harmonic amplitude envelopes.
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