Wavetable Synthesis

The most common computer implementation of the sinusoidal oscillator is not through direct calculation of values of the cosine or sine function, but, rather, through the use of a stored table containing values of one period of a sinusoidal waveform. A sinusoid at a given frequency may then be generated by reading through the table, circularly, at an appropriate rate. If the table contains $N$ values, and the sample rate is $f_{s}$, then the generation of a sinusoid at frequency $f_{0}$ will require a jump of $f_{s}/f_{0}N$ values in the table over each sample period, using interpolation of some form. Clearly, the quality of the output will depend on the number of values stored in the table, as well as on the type of interpolation employed. (Linear interpolation is frequently employed, but other more accurate methods, built around allpass filter approximations to fractional delays, are possible, and of special interest in physical modeling applications [134].)

Figure 1.4: Wavetable synthesis. A buffer, filled with values, is read through at intervals of $1/f_{s}$ s, where $f_{s}$ is the sample rate. Interpolation is employed.

It should be clear that one can store values of an arbitrary waveform in the table, not merely those corresponding to a sinusoid. See Figure 1.4. Reading through such a table at a fixed rate will generate a quasi-periodic waveform with, in general, a full harmonic spectrum, all at the price of a single table read and interpolation operation per sample period--in other words, it is no more expensive, in terms of computer arithmetic, than a single oscillator, though memory requirements are obviously greater. As will be seen shortly, there is an extremely fruitful physical interpretation of wavetable synthesis, namely the digital waveguide, which revolutionized physical modeling sound synthesis through the same efficiency gains. See §1.2.3. Various other variants of wavetable synthesis have seen use, in particular wavetable stacking, involving multiple wavetables, the outputs of which are combined using crossfading techniques [174].

Tables of data are also associated with so-called sampling synthesis techniques, as a de facto means of data reduction. Many musical sounds consist of a short attack, followed by a steady pitched tone. Such a sound may be efficiently reproduced through storage of only the attack and a single period of the pitched part of the waveform, which is stored in a wavetable and looped [219].