A periodic grain envelope creates AM sidebands at 1/grain-period intervals

When grains repeat periodically (as in synchronous granular synthesis), the periodic amplitude envelope acts as an amplitude modulator. For each sinusoidal component in the carrier waveform, this periodic envelope adds sidebands to the spectrum. The sidebands are spaced from the carrier frequency at integer multiples of 1/T, where T is the period of the envelope repetition (the inverse of the grain density). This creates a harmonic series of sidebands around each waveform component. In practice, the grain density directly determines the spectral bandwidth of sidebands, and grain duration affects sideband width via the AM spectrum of the individual envelope shape.

Examples

A sine grain at 1000 Hz, repeated every 10ms (100 Hz density): sidebands appear at 900, 1100, 800, 1200 Hz etc. (100 Hz spacing). Increase density to 200 Hz: sidebands at 200 Hz spacing.

Assessment

In synchronous granular synthesis with a carrier frequency of 500 Hz and grain density of 50 grains/sec, at what frequencies do the first-order AM sidebands appear? How does increasing the density affect the sideband spacing?