Interactive Fourier Harmonic Synthesizer Simulator

FOURIER HARMONIC SYNTHESIZER

Adjust Harmonic Amplitudes (1-12)

TIME DOMAIN (COMPOSITE WAVE)

FREQUENCY DOMAIN (PARTIALS)

y(t) = ∑ A_n sin(2π n f t)

The Fourier Series Theorem

The simulator implements the Fourier Series, which states that any periodic function can be represented as a summation of simple oscillating functions (sines and cosines).

y(t) = ∑_n=1^N A_n sin(2π n f₀ t)

Deconstructing the Equation

• 1. The Fundamental (f₀)

  This is the first harmonic (H1). It defines the perceived pitch of the sound. All other frequencies are integer multiples of this frequency.

• 2. Harmonic Multiples (n)

  The variable n represents the harmonic number. If the fundamental is 200Hz, then n=2 is 400Hz, n=3 is 600Hz, and so on.

• 3. Amplitude Coefficients (A_n)

  This is what you control with the sliders. By changing the "weight" of each harmonic, you change the Timbre (tone color) of the wave.

Classical Waveform Recipes

By following specific mathematical "recipes" for the amplitudes, we can reconstruct standard geometric waves:

Square Wave: Sum of ODD harmonics only, with amplitude 1/n.
Recipe: H1=1.0, H2=0, H3=0.33, H4=0, H5=0.20...

Sawtooth Wave: Sum of ALL harmonics, with amplitude 1/n.
Recipe: H1=1.0, H2=0.5, H3=0.33, H4=0.25, H5=0.20...

Note: Gibbs Phenomenon
In the simulator, you will notice that even with all odd harmonics active, the "Square" wave has ripples at the corners. This is because a true square wave requires an infinite number of harmonics. Since we only use 12, we see the mathematical "approximation error" known as the Gibbs Phenomenon.