FOURIER HARMONIC SYNTHESIZER
Adjust Harmonic Amplitudes (1-12)
TIME DOMAIN (COMPOSITE WAVE)
FREQUENCY DOMAIN (PARTIALS)
y(t) = ∑ An 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=1N An sin(2π n f0 t)
Deconstructing the Equation
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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.
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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.
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3. Amplitude Coefficients (An)
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...
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...
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.
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.