Then, during resysnthesis, these spectral descriptions are manipulated and combined in real-time by a performer to create fades, morphs, filtering, etc with very fine control over timbre. The key novel idea in our sythesizer was the recasting of the recursion equation for the sine computation, keeping error outside the bounds of auditory perception while minimizing the number of additional operations required to implement the modified equation on our hardware. Here is a ten second tenor voice resynthesis with ten partials and 128 samples per frame. Audibly the waveforms are indistinguishable, but physically they differ. A square wave with 500 partials spanning from 20 to 19980Hz at 44.1KHz sampling rate.
Square waves are nice visual examples because the error is easily estimated.
The most widely used digital sinusoidal oscillator structure employs a first order recursion to accumulate and wrap a phasor, followed by a sinusoidal functional evaluation - usually a table lookup. Second order recursions are an attractive alternative because the sinusoid is computed within the structure, avoiding the need for a large table or specialized hardware function evaluator. Unfortunately, numerous theoretical and practical challenges have to be overcome to use second order structures. We were motivated to overcome these because the next generation of microprocessors are hostile to efficient implementations of first order oscillators. Although primary cache memory may be large enough to hold a sine table, no locality of reference is observed for arbitrary frequencies.