ETM Learn · 02 / 08
Add sines, discover timbre and learn when a partial really is a harmonic.
The previous lesson used one sine. Now we add several to build a richer sound. Each sine is a component, or partial. Choose a 110 Hz fundamental: components at 220, 330 and 440 Hz are integer multiples, called harmonics. The fundamental is 1; 220 Hz is harmonic 2, also the first overtone in this series.
The waveform shows the sum changing over time. The spectrum shows which frequencies participate and their amplitudes. Timbre also depends on how a sound starts, decays and changes. Not every sound has components at exact harmonic frequencies.
Dotted terms offer a definition on hover, keyboard focus or tap. Click to keep it open; Escape or an outside click closes it. All definitions are also available here.
A component at a positive integer multiple of the fundamental frequency. The fundamental itself is harmonic 1.
An algorithm for the discrete Fourier transform: obtains spectral information from a block of samples. Block length and window affect the display.
Half the sample rate. In this synthesis, components are kept below this frequency to avoid frequency folding.
Begin with one partial. Add components gradually, then change waveform while keeping the fundamental fixed.
Predict → change one control → compare → explain the result.
Compare sawtooth, square and triangle with the same partial count. Predict brightness before listening. More partials do not automatically make a better sound.
An ideal flexible string fixed at both ends must have zero displacement at both boundaries. Real stiffness and boundary conditions can shift modal frequencies away from integer multiples.
Change interval, partial count and register separately. Roughness is a feature of this model, not a universal musical consonance score.
Compare a loop’s waveform and spectrum. A finite window estimates the spectrum during that window; a broad peak need not be a separate stable sine.
A partial is at 330 Hz and the fundamental is 110 Hz. Which harmonic?
330/110 = 3: harmonic 3 and, in this series, overtone 2.For a periodic signal, , plus DC if its mean is nonzero. Ideal sawtooth amplitudes decay as 1/n; a symmetric square has odd harmonics decaying as 1/n; a symmetric triangle has odd harmonics decaying as 1/n² with alternating signs in a suitable sine representation. Phase and sign are part of the recipe.
Near discontinuities, a truncated Fourier series exhibits Gibbs ringing. More terms narrow the region but do not eliminate the limiting relative overshoot. Sampled synthesis must keep components below (fs/2) or filter them. An ideal discontinuity cannot be reproduced exactly at a finite sample rate.
Beating and roughness depend on frequency separation, level and spectrum. Musical consonance also involves context and learning. Equal-tempered intervals generally approximate rather than exactly equal simple frequency ratios.
fₙ = n · f₀
Multiply fundamental frequency f₀ by harmonic number n. For a 110 Hz fundamental, harmonic 3 is 330 Hz.
x(t) = Σ Aₙ cos(2πnf₀t + φₙ)
For each component choose amplitude Aₙ, frequency n times f₀ and phase φₙ. Calculate its value at time t and add the components; Σ means sum. Add a constant component if the mean is nonzero.
Aₙ ∝ 1/n; Aₙ ∝ 1/n²
The symbol ∝ means proportional to. With 1/n decay, a harmonic with three times the index has one-third the amplitude. With 1/n² decay it has one-ninth. Symmetric square and triangle waves use odd harmonics only; appropriate phases or signs are also needed.
Use 20 log₁₀(1/9), because the ratio is amplitude.
20 log₁₀(1/9) = −19.0849 dBc. The corresponding square-wave harmonic is 1/3, or −9.5424 dBc. Equal harmonic numbers need not carry equal energy.
Solve n × 3000 < 24000. The word strictly excludes the boundary.
n < 8, so harmonics 1–7. Harmonic 8 lands at Nyquist, where sampled sine amplitude depends on phase and arbitrary phase cannot be represented. Harmonic 9 at 27 kHz folds to 21 kHz if generated without anti-alias filtering.
Further reading: Julius O. Smith: Mathematics of the DFT