Calculate overtones from any fundamental frequency — with spectrum visualisation, audio synthesis, and the physics of why instruments sound different.
HARMONIC SPECTRUM ANALYSER
Click any bar to inspect that harmonic's standing wave mode.
Enter 20–2000 Hz. Harmonics extend to 20 kHz.
Showing harmonics 1 through 8 (capped at 20 kHz)
Audio Preview
Or click ▶ in the table to hear any individual harmonic.
| H# | Frequency | Note | ¢ ET | Interval | Play |
|---|---|---|---|---|---|
| 1 | 440.00 Hz | A4 | 0.0 | Fundamental | |
| 2 | 880.00 Hz | A5 | 0.0 | Octave | |
| 3 | 1320.00 Hz | E6 | +2.0 | Perfect 5th | |
| 4 | 1760.00 Hz | A6 | 0.0 | 2nd Octave | |
| 5 | 2200.00 Hz | C♯/D♭7 | -13.7 | Major 3rd | |
| 6 | 2640.00 Hz | E7 | +2.0 | Perfect 5th | |
| 7 | 3080.00 Hz | G7 | -31.2 | Harm. 7th ♮ | |
| 8 | 3520.00 Hz | A7 | 0.0 | 3rd Octave |
¢ ET = cents deviation from equal temperament. Harmonics 7, 11, 13, 14 deviate noticeably — they exist "between the keys."
Each instrument emphasises different harmonics. The dots below show which harmonics are prominent for each instrument at a given fundamental. Click a row to see those harmonics in the spectrum above.
| Instrument | Character | Active Harmonics (dots = strong) | Physics note |
|---|---|---|---|
| 🪈 Flute | Warm & pure | H1–H16 | Cylindrical bore + soft embouchure — near-pure sine. Fundamental dominates; above H3 almost absent. |
| 🎵 Clarinet | Hollow & reedy | H1–H16 | Cylindrical closed-end tube: even harmonics suppressed. Odd series (1, 3, 5, 7) gives its distinctive hollow quality. |
| 🎻 Violin | Rich & singing | H1–H16 | Bowing excites a rich spectrum. H2–H6 give warmth; formant resonances in the body amplify certain regions. |
| 🎺 Trumpet | Bright & cutting | H1–H16 | Conical bore + brass bell amplifies high harmonics. Strong H4–H8 creates the brilliant, piercing timbre. |
| 🎸 Bass guitar | Warm & round | H1–H16 | Heavy string mass emphasises the fundamental. H2 and H3 add warmth. Higher harmonics roll off quickly. |
| 🎹 Piano | Complex & inharmonic | H1–H16 | Stiff steel strings deviate slightly from perfect harmonic ratios (inharmonicity), giving each note a characteristic shimmer. |
The harmonic series produces just intervals — perfect whole-number ratios. Modern tuning uses equal temperament (12 equal semitones per octave). These systems almost agree, but harmonics 7, 11, 13 and 14 fall noticeably "between the keys." The cents column in the table above shows the gap. Slide players, singers, and fretless strings naturally gravitate toward just intonation.
H5 — Major 3rd is 13.7¢ flat of equal temperament C♯/D♭7
H7 — Harm. 7th ♮ is 31.2¢ flat of equal temperament G7
A Simple Example: Why Does a Flute Sound Different from a Clarinet?
Both instruments can play the same note — say, A4 at 440 Hz — yet sound completely distinct. The harmonic series explains why. Try this:
1️⃣ Click the A4 preset — fundamental set to 440 Hz, harmonics 1–8 visible.
2️⃣ Hit ▶ All Harmonics to hear all 8 harmonics as additive synthesis. Notice the rich, buzzy timbre.
3️⃣ Now click ▶ on H1 (440 Hz alone). That is the flute — almost pure, only a little air noise.
4️⃣ Click ▶ on H3 (1320 Hz) then H5 (2200 Hz) in sequence. A clarinet's barrel-shaped bore suppresses even harmonics; these odd ones are its signature voice.
5️⃣ Look at the Timbre Reference table — the Clarinet row shows only odd harmonics (H1, H3, H5, H7) lit up.
6️⃣ Click H7 in the spectrum — notice its bar is orange because it falls 31 cents flat of an equal-tempered G (the natural 7th of A). This harmonic lives "between the keys."
Pro tip: For a blog post, try setting the fundamental to C3 (130.81 Hz) with 12 harmonics — you get the entire C major triad and then some within the first five harmonics, exactly as Pythagoras found when stretching strings 2,500 years ago.
Data Source: Helmholtz, On the Sensations of Tone (1863); ISO 16 standard pitch; equal temperament mathematics — all public domain • Public domain • Solo-developed with AI
The Secret Inside Every Sound: Strike a piano key labeled "A" and you do not hear a single pure frequency — you hear a whole cascade of frequencies vibrating simultaneously. The string vibrates as a whole (producing A4 at 440 Hz), but also in halves (880 Hz), thirds (1320 Hz), quarters (1760 Hz), and so on. Each of these "partial vibrations" is a harmonic overtone, and together they form the harmonic series. In 1863, Hermann von Helmholtz published On the Sensations of Tone, the landmark work that explained how the balance of these overtones — which ones are strong, which are weak — is the very definition of an instrument's timbre: why a violin and a flute playing the exact same note still sound completely different.
Why Simple Ratios Sound Good: The frequencies of the harmonic series are always whole-number multiples of the fundamental: 1×, 2×, 3×, 4×... This is not a musical convention but a physical law. A string fixed at both ends can only sustain vibrations where an exact integer number of half-wavelengths fit between the endpoints. These are the "standing wave modes." The elegant consequence is that the musical intervals we perceive as most consonant — the octave (2:1), the perfect fifth (3:2), the major third (5:4) — arise directly from these simple integer ratios. Pythagoras discovered this around 530 BC by comparing lengths of vibrating strings, but he had no idea he was also describing the physics of every sound ever made.
Just Intonation vs. Equal Temperament: Here's the fascinating wrinkle: the harmonic series gives us "just" intervals with perfect whole-number ratios, but modern pianos and guitars use "equal temperament," where the octave is divided into 12 exactly equal semitones. These systems almost agree — but not quite. The 7th harmonic sits 31 cents flat of a well-tempered minor 7th (B♭ if the fundamental is C; G if the fundamental is A). The 11th harmonic is 49 cents flat of an equal-tempered tritone (augmented 4th). Instruments like the trombone (continuous slide) and human voice naturally gravitate toward just intonation — which is why a skilled singer or brass quartet can achieve a purity of harmony that a piano simply cannot.
Additive Synthesis and the Birth of Electronic Music: In the 1960s, engineers at Bell Labs and Robert Moog realised you could reverse-engineer timbre: start with pure sine waves and add them together in the right ratios to build any sound. This is additive synthesis — the foundation of all electronic music. The "Play All Harmonics" button in this tool does exactly that, letting you hear the raw sum of sine waves and understand why a set of pure tones can sound like something rich and musical. Modern wavetable synthesizers and digital audio workstations still use variants of this principle every day.
🐾 From the Lab Cat's Resonance & Purr-quency Division: I have Conducted extensive research into biological harmonic generation, specifically the Purr. My purr fundamental sits at approximately 25–50 Hz — firmly in the sub-bass register. However, the therapeutic effects come from harmonics 2 through 8, which span 50–400 Hz — the precise range shown to stimulate bone density and tissue repair. In other words, my purring is not emotional. It is a precision-calibrated additive synthesis engine delivering targeted physiological intervention. You are welcome.
Meanwhile, my meow peaks near H3 of 200 Hz (≈ 600 Hz, D♯5/E♭5), which I have optimised — through decades of selective vocal evolution — to match the frequency range of maximum human guilt response. The harmonic series is not merely physics. It is leverage. 🎵