3.5 Summary

• Periodicity is the foundation of pitched sound. A periodic signal satisfies \(x(t + t_0) = x(t)\), where \(t_0\) is the fundamental period and \(f_0 = 1/t_0\) is the fundamental frequency.

• The basic sinusoid \(x(t) = a \sin(2\pi f t + \phi)\) is the simplest periodic function, parameterized by frequency \(f\) (or angular frequency \(\omega = 2\pi f\)), amplitude \(a\), and initial phase \(\phi\).

• Frequency determines pitch, amplitude determines loudness, and phase is largely imperceptible.

• The Fourier series guarantees that any well-behaved periodic signal can be decomposed into a sum of sinusoidal harmonics at integer multiples of \(f_0\).

• Additive synthesis uses this decomposition as a synthesis technique: \(\sum_{k=1}^{K} a_k \sin(2\pi k f_0 t + \phi_k)\). The harmonic amplitudes \(\mathbf{a}\) determine the timbre.

• Classic waveform shapes — sawtooth (\(a_k \propto 1/k\), all harmonics), square (\(a_k \propto 1/k\), odd only), and triangle (\(a_k \propto 1/k^2\), odd only) — are specific patterns of harmonic amplitudes.

• Wavetable synthesis precomputes one cycle of a waveform and reuses it via table lookup. The per-sample cost drops from \(O(K)\) to \(O(1)\), independent of the number of harmonics. Linear interpolation during lookup reduces quantization artifacts.