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Protocol for synthesis of complex harmonic waveforms

Additive synthesis is the practice of creating a pitched complex waveform by summing sine waves that are harmonically related to a single root frequency, called the fundamental frequency. Additive synthesis of a musical A, for instance, would involve summing one sine at 440 Hz with another sine at 880 Hz, and another at 1320 Hz (3 * 440), etc.

The relative amplitudes at which these frequencies are summed define the timbre of the sound. There is a distinction to be made here between the even and odd harmonics of the wave. The even harmonics are those of which the frequency is an even multiple of the base frequency (for example 880 Hz, 1760 Hz, 2640 Hz, etc.), the odd frequencies are those that are not even, including the fundamental itself.

Further, we can distinguish between lower and upper harmonics. There is no conclusive definition as to what is considered as 'lower' and 'upper', but lower harmonics are roughly those below seven times the fundamental frequency.

An emphasis on the even lower harmonics gives the sound a 'warm' quality. Increased energy in the higher even harmonics gives the sound clearness and definition. Odd harmonics give the sound a hollow character in the lower range and a bright, agressive character in the higher range.

In practice, additive synthesis is difficult to accomplish in real time applications like synthesizers because it requires either too much expensive electronic components (oscillators) or, in a digital system, too much computational overhead. An average complex wavevorm can have up to 50 harmonics, so a synthesizer that wants to implement this protocol would either have 50 oscillators or enough DSP power to compute 50 sine values and make a weighted sum of them for every sample. At CD quality audio that would amount to 44100 samples per second.

The Kawai K5 is a rare example of a synthesizer that does utilize additive synthesis. The Casio FZ-1 sampler allows non real time additive synthesis.

Thanks to Ashley Pomeroy for the K5 and FZ-1 information

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