Devansh Mittal
Devansh Mittal
Physics Educator and Mentor • Lead Developer
Quality of Sound (Timbre) & Fourier Harmonic Synthesis
Fundamental Pitch:
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1. How to Use this Interactive Simulation User Guide & Experimentation
  1. Select the Fundamental Pitch: Use the dropdown menu in the top-right corner to select a musical note: do (262 Hz), re (294 Hz), mi (329 Hz), fa (349 Hz), or so (392 Hz). This sets the fundamental base frequency (f0).
  2. Audition Isolated Pure Harmonics: Click the blue "Hear" button inside any of the 10 harmonic boxes on the left to listen to that isolated pure sine wave overtone (n × f0). Notice how the pitch increases with higher harmonic order numbers.
  3. Adjust Overtone Amplitudes: Drag the green slider thumbs in harmonics 2 through 10 to adjust their relative strength from 0% to 50% relative to the fundamental tone.
  4. Observe Waveform Synthesis & Relative Spectrum: Notice how changing each individual harmonic slider immediately alters the red Resultant Waveform and updates the black Relative Amplitudes bar chart on the right in real time.
  5. Hear the Synthesized Composite Timbre: Click the "Hear Resultant Wave" button in the top right to play the composite sound wave synthesized live in your browser. Experiment with different harmonic balances to hear how tone color changes from mellow to bright or buzzy.
2. Physics Theory: Timbre & Fourier Superposition Theory & Acoustics
Fourier Harmonic Superposition Formula:   y(t) = ∑n=110 An sin(2π n f0 t) = A1 sin(2π f0 t) + A2 sin(2π (2f0) t) + … + A10 sin(2π (10f0) t)

• What is Timbre (Tone Quality)? Timbre is the distinct sonic characteristic or tone color that allows human ears to distinguish between different musical instruments (such as a flute, violin, trumpet, or piano) or human voices, even when they play the exact same musical note at the same pitch (same fundamental frequency f0) and equal loudness.

• The Fourier Theorem: According to Fourier's theorem, any complex periodic sound wave is mathematically equivalent to the linear sum of pure sinusoidal waves comprising a fundamental frequency (f0) and a sequence of integer multiple overtones or harmonics (2f0, 3f0, 4f0, and so on). While the fundamental frequency determines the perceived pitch, the relative strengths (amplitudes A1, A2, A3, ..., A10) of the overtones uniquely determine the timbre and the resulting composite wave shape.

• Acoustic Spectrum in Musical Instruments: A flute produces a nearly pure tone dominated almost entirely by the fundamental frequency (n = 1), yielding a smooth sinusoidal wave. In contrast, a violin or trumpet generates rich, prominent higher harmonics (n = 2 through 10), producing complex, sharp resultant waveforms with bright, resonant acoustic textures.