Noise-based fBm fills the frequency spectrum with far fewer octaves than sine-wave additive synthesis

An fBm can also be synthesised additively with sine waves (an inverse FFT), but this is costly: each sine wave contributes energy at a single frequency, so filling a broadband spectrum needs a huge number of terms. Value noise and gradient noise instead have wide spectral density — a single noise ‘wave’ already covers a long range of frequencies with most energy in the low end. Summing a handful of rescaled noise octaves therefore fills the spectrum rapidly, which is why geometric (×2) octave stacking with only ~10-24 iterations produces good terrain. Sine-based fBm’s only advantages are raw hardware speed and exact periodicity (useful for tiling/ocean textures), at the cost of poor, repeating landscapes.

Examples

iq’s spectral-density plots: a single sine covers one frequency line, whereas value/gradient noise plots are broad. This is why ~24 noise octaves can cover planet-Earth scale down to 2-metre detail, while linearly-spaced sine frequencies would need orders of magnitude more terms.

Assessment

Explain why an IFFT (sine-based) fBm needs many more terms than a noise-based fBm to cover the same spectrum. Give one situation where sine-based fBm is still preferable.