Harmonic Decay Models
A quantitative study of musical timbre via time-resolved Fourier analysis.
The problem
Instruments playing the same pitch often sound distinct despite sharing an identical fundamental frequency. These differences are not captured by pitch alone, but emerge from the distribution and temporal evolution of harmonic energy.
Mathematical model
f(t) = Σₙ aₙ(t) cos(2π fₙ t + φₙ)
- initial harmonic amplitudes aₙ(0)
- exponential decay rates λₙ
- deviations from ideal harmonicity
- time-distributed spectral energy
Methodology
- Controlled acoustic recordings
- Fixed room, microphone, and gain
- Short-Time Fourier Transform (STFT)
- Harmonic tracking and parameter estimation
- Fourier reconstruction via truncation
What the analysis reveals
Harmonic decay patterns differ systematically between instruments and techniques. Truncating Fourier representations removes low-energy components that remain perceptually significant, revealing a gap between mathematical convergence and auditory fidelity.
What Fourier truncation loses
∥f − fₙ∥² = Σ |cₖ|² , |k| > N
Small L² error does not imply perceptual equivalence.
Artifacts
GitHub repository
Paper and audio appendix forthcoming