Harmonic Series

Build sounds from sine waves: weight the harmonics freely or load sine, sawtooth, square and triangle – with waveform, keyboard and distortion figures.

Explanation Tasks

Drag a bar to change the amplitude; below zero means phase 180°. Double-click sets a harmonic to 0. Pointing at a bar highlights the harmonic in the waveform and on the keyboard.

Waveform (two periods)

Harmonics on the keyboard (deviation from equal temperament in cents)

Playable like a MIDI keyboard: pressing a key sets the fundamental, dragging across the keys glides. If the sound is off, it plays only while the key is held; if it is on, it stays at the new pitch.

Adds the harmonics one after another, from the fundamental up to the selected number. Use ◀︎ and ▶︎ (or ← and →) to go step by step. Solid bars are heard, faint bars with a dashed handle are (still) silent – as are harmonics above 20 kHz.

Harmonics

Load waveform

Number of harmonics N Fundamental f₀

Bar scale

linear dB

Show single harmonics

off on

Waveform
Highest harmonic
Distortion (THD)
Crest factor

Keyboard: Space = play/stop build-up, ← / → = remove/add harmonic, F = presentation mode, M = sound on/off

What is happening?

One tone from many sine waves

A plucked guitar string sounds like a single tone, yet it vibrates at the same time as a whole, in two halves, in three thirds and so on. Each pattern produces a sine tone, and their frequencies are integer multiples of the fundamental frequency f₀: f₀, 2·f₀, 3·f₀ … This is the harmonic series. Hearing fuses these harmonics into one sound with the pitch f₀ – how strong the individual harmonics are is what you hear as timbre.

This holds for every periodic sound (Fourier series): x(t) = Σ aₙ · sin(2π · n · f₀ · t), n = 1 … N. Here x(t) is the signal value at time t, n the number of the harmonic, aₙ its amplitude and N the number of harmonics; Σ adds up all harmonics. This is exactly the sum the module computes. In the top diagram “Amplitudes of the harmonics aₙ” each bar is one aₙ; below it, “Waveform (two periods)” shows the sum as an orange curve. Mind the counting: the 1st harmonic is the fundamental, the 2nd harmonic is the 1st overtone.

Try it: click “Play build-up”. The harmonics are added one after another, from the fundamental up to the selected “Number of harmonics N”. Solid bars are heard, faint bars with a dashed handle are still silent. Use ◀ and ▶ (or ← and →) to go step by step.

Timbre: how strong each harmonic is

The buttons under “Load waveform” set the textbook amplitudes. The sine has only the fundamental. The sawtooth contains all harmonics with magnitude 1/n (with alternating sign), the square wave only the odd ones, also with 1/n. The triangle has the same odd harmonics but with 1/n², and every second one is inverted.

Why does a 1/n spectrum fall by 6 dB per octave? An octave higher means twice the harmonic number, so the amplitude halves: 20 · log₁₀(1/2) ≈ −6 dB. With 1/n² it drops to a quarter, i.e. −12 dB per octave. Set the “Bar scale” to dB and you see it directly: in the square wave the 3rd harmonic lies at −9.5 dB and the 9th at −19.1 dB; in the triangle the 3rd is already at −19.1 dB. In this view the zero line stands for −60 dB and below.

That is why the sawtooth sounds bright and sharp, the square wave hollow and the triangle soft, almost flute-like. And that is why the triangle looks almost perfect with only a few harmonics: the dashed “ideal waveform” shows what the sum approaches with infinitely many harmonics.

Listen: switch the sound on with the speaker icon and change between the waveforms. The module keeps the RMS value constant, so you compare timbres, not levels.

The harmonic series as a scale

Because each harmonic has a fixed frequency ratio to the fundamental, the harmonics always form the same sequence of intervals: octave (2), a fifth above that (3), double octave (4), major third (5), fifth (6) … Harmonics 4, 5 and 6 together form a major triad. Intervals are measured in cents: the nth harmonic lies c = 1200 · log₂(n) cents above the fundamental; 100 cents are one equal-tempered semitone.

Worked example with f₀ = 110 Hz (A2): the 3rd harmonic lies at 330 Hz, 1200 · log₂(3) ≈ 1902 cents – an octave plus a fifth (1900 cents) and only 2 cents above, so practically the key E4. The 5th harmonic (550 Hz) has 1200 · log₂(5) ≈ 2786 cents and so lies 14 cents below the equal-tempered major third C♯5. Even more clearly beside the keys are the 7th harmonic (31 cents flat), the 11th (49 cents flat, almost exactly between two keys) and the 13th (41 cents sharp).

Try it: point at a bar. The header line gives the harmonic number, frequency, note with cents deviation, interval and amplitude. On the keyboard below, each harmonic is a line with a dot; if it lies 15 cents or more beside the nearest key, it appears in teal with its deviation. The keyboard is playable: pressing a key sets f₀, dragging across the keys glides. A MIDI keyboard is switched on in the settings menu (gear icon).

Listen: set the fundamental to 0 with a double-click. The pitch remains, because hearing recognises the common spacing f₀ of the remaining harmonics (“missing fundamental”). That is why you can hear bass even on a phone speaker.

Phase: same spectrum, different wave

A bar below zero means that this harmonic contributes with inverted phase (180°). The magnitude spectrum does not change, and neither does the RMS value, because it depends only on the magnitudes: RMS = √(Σ aₙ² / 2). The waveform, however, can look completely different – and with it the peak value.

Try it: load the square wave and drag the 3rd harmonic from +0.33 to −0.33. The flat plateau turns into a peak in the middle. The peak value rises from about 0.93 to 1.45, i.e. by more than half, and the “Crest factor” from 1.5 dB to 5.4 dB, while “Distortion (THD)” stays the same.

Listen: with the sound on, the inverted square wave sounds hardly any different. A common misconception follows from this: “What you cannot hear does not matter.” For levels, headroom and limiters, however, the peak value counts, and phase helps to determine it. A phase-rotating EQ or all-pass can therefore raise the peak levels in the master by a few dB without visibly changing the frequency response – measure the peaks again afterwards, ideally with a true-peak meter.

Distortion and crest factor

Total harmonic distortion (THD) compares all overtones together with the fundamental: THD = √(a₂² + a₃² + …) / |a₁|, where |a₁| is the magnitude of the fundamental's amplitude, i.e. its value without sign. It depends only on the magnitudes. Example: a sine with an additional 0.1 at the 3rd harmonic has THD = 0.1/1 = 10 % = −20 dB. Without a fundamental, THD is undefined and the readout shows “—”. For comparison: the square wave with 32 harmonics has 46.7 %, the triangle 12.1 %. A good studio amplifier stays below 0.01 %, a tape machine at its maximum level (MOL) at 3 %.

When an amplifier distorts a sine, exactly such additional harmonics appear, and their pattern reveals the cause. Odd harmonics keep the positive and negative half-waves mirror images of each other – this is how symmetric clipping acts. Even harmonics make the half-waves different – this is how an asymmetric transfer curve acts. Try it: load the sine and drag the 3rd harmonic to +0.1 – the tops become flatter (peak 0.9). Set it back to 0 and drag the 2nd to +0.1: the half-waves lean differently, and THD is again 10 %.

The crest factor is the ratio of peak value to RMS value. For a sine it is √2 ≈ 1.41 (3.0 dB), for an ideal square wave 1 (0 dB) – in the module, with 32 harmonics, 1.19 (1.5 dB) because of the overshoots. Sawtooth and triangle ideally have √3 ≈ 1.73 (4.8 dB). Music lies far above that: unprocessed drums easily reach 20 dB, a heavily limited master often less than 10 dB. That is how much headroom a signal needs above its RMS level.

Gibbs phenomenon: the overshoots remain

With few harmonics, overshoots remain at the jumps of the square and sawtooth waves: the Gibbs phenomenon. Try it: load the square wave, set N to 1 and increase it step by step. From N = 5 on (harmonics 1, 3, 5) the shape is recognisable. With more harmonics the overshoots become narrower but not smaller: even with 31 harmonics the peak still lies about 18 % above the plateau (0.93 instead of π/4 ≈ 0.79), i.e. about 9 % of the jump.

A finite sum of sine waves is smooth and can only approximate a vertical jump. Common misconception: “With enough harmonics the overshoot disappears.” As N grows, the ripple only contracts towards the jump – only infinitely many harmonics give a perfect square wave.

Leaving out the harmonics above a limit is exactly what a low-pass does. The readout “Highest harmonic” (N · f₀) is the cut-off frequency here. A practical example: a 1 kHz square wave through a 20 kHz filter keeps only the odd harmonics up to 19 kHz – on an oscilloscope you then see exactly these overshoots. Gently, as in the spring–mass model, the corners become round; steeply, as in the digital filter, overshoot and pre-ringing appear.

Limits of the model

The model works with a strictly periodic, steady sound of at most 32 harmonics. Real instruments have attack transients, their spectrum changes over time, and with stiff strings (on the piano, for example) the partials lie slightly above the integer multiples. The module does not reproduce this.

Phase has only two values here: 0° (bar above zero) or 180° (bar below zero); all harmonics are sine waves that start together at zero. Harmonics above 20 kHz are drawn faintly and are silent. The sound is normalised to the RMS value so that the volume stays the same when harmonics are added or inverted; a limiter catches waveforms with a high crest factor.

The overshoots at jumps are not a calculation error but the Gibbs phenomenon (about 9 % of the jump for the square wave) – just as behind a low-pass that removes the high harmonics.

Tasks

Practice questions

Questions from studio, stage and mastering – to think about, discuss and check with the module.

1. Why can you still hear the pitch of a bass line on a phone speaker that radiates hardly anything below about 200 Hz?

Show answer

The overtones of a bass note are spaced f₀ apart (2·f₀, 3·f₀, 4·f₀ …). Hearing recognises this pattern and perceives the fundamental pitch even when the fundamental itself is missing (residue pitch or “missing fundamental”). Bass enhancement plug-ins for small speakers deliberately generate overtones to exploit this effect.
2. A guitar is perfectly tuned with a tuner. Still, chords with a major third beat slightly, while fifths sound calm. Why?

Show answer

The equal-tempered third is 14 cents wider than the pure third of the harmonic series (5th harmonic: −14 ct), the fifth only 2 cents narrower. For a third C4–E4, the 5th harmonic of C and the 4th harmonic of E are therefore about 10 Hz apart – heard as beating. With the fifth the overtones almost coincide. Some musicians therefore tune slightly differently for certain keys.
3. An EQ or all-pass barely changes the frequency response, yet the peak levels in the master rise by a few dB and the limiter works harder. How is that possible?

Show answer

The peak value depends not only on the magnitudes of the harmonics but also on their phases. In the module: inverting only the 3rd harmonic of the square wave changes no magnitude and no RMS value, but raises the peak by more than half and the crest factor from 1.5 to 5.4 dB. Phase-rotating processing can therefore cost headroom. Measure peak levels again after every processing step – ideally with a true-peak meter.

Tasks

Predict first, then try it out, then open the solution. “Load starting point” sets the matching initial values.

1. Build a square wave

Only the fundamental is active. Increase the number of harmonics N step by step up to 32, or play the build-up. When do you recognise the square wave? What happens to the overshoots at the edges?

Load starting point

Show solution

From N = 5 on, i.e. with harmonics 1, 3 and 5, the square shape is clearly recognisable. With more harmonics the overshoots at the edges become narrower but not smaller: even with 31 harmonics the peak lies about 18 % above the plateau, i.e. about 9 % of the jump. This is the Gibbs phenomenon. Only infinitely many harmonics give a perfect square wave.
2. Square or triangle?

Load square and triangle one after the other. Which harmonics are present in each? How do the two spectra differ although the same harmonics occur?

Load starting point

Show solution

Both contain only the odd harmonics (1, 3, 5 …). In the square wave they fall with 1/n (−6 dB per octave), in the triangle with 1/n² (−12 dB per octave), and every second one is inverted. Already the 3rd harmonic of the triangle is only 1/9 of the fundamental (1/3 for the square). That is why the triangle is almost perfect with only a few harmonics and sounds softer.
3. Learning the harmonic series

f₀ = 110 Hz is A2. Point at the bars one after another. Which harmonics are an A again? Which harmonic is the first to bring a new pitch? Which three lie most clearly beside a piano key?

Load starting point

Show solution

Harmonics 1, 2, 4, 8 and 16 are A again – the powers of two, each an octave higher. The 3rd harmonic (330 Hz) is the first new pitch, E, a fifth above the octave. Most clearly beside a key are the 7th (G, 31 cents flat), the 11th (D♯, 49 cents flat) and the 13th (F, 41 cents sharp). The 14th is the octave of the 7th and deviates by the same amount.
4. Same spectrum, different wave

In the square wave, drag the 3rd harmonic from +0.33 to −0.33. What changes in the magnitudes of the harmonics, what in the waveform? Watch THD and crest factor.

Load starting point

Show solution

The magnitudes of all harmonics stay the same, and so do RMS value and THD. The waveform changes a lot, though: instead of a flat plateau there is a peak in the middle; the peak value rises by more than half. The crest factor rises from about 1.5 dB to 5.4 dB. At the same loudness this signal needs considerably more headroom.
5. Distortion

The sine is pure. Drag the 3rd harmonic to +0.1. How large is the THD? How does the shape change? Set the 3rd harmonic back to 0 and drag the 2nd to +0.1 instead.

Load starting point

Show solution

Both times the THD is 10 % (−20 dB), because it depends only on the magnitudes. With the 3rd harmonic the tops become flatter (peak 0.9 instead of 1), similar to symmetric clipping; the half-waves remain mirror images of each other. With the 2nd harmonic the positive and negative half-waves become different: they lean in different directions. Even harmonics arise from asymmetric distortion, odd ones from symmetric distortion.

Settings

Sound

Level calibration (dB SPL)

MIDI keyboard

Harmonic Series