Wavetable synthesis
Wavetable synthesis is used in digital musical instruments (synthesizers) to produce natural tone-like sounds. The sound of a existing instrument (a single note) is sampled and parsed into a sequence of wavetables having one period or cycle in each wavetable. Upon playback, these wavetables are used for sample "lookup" in the same manner as in a Numerically-controlled oscillator (or a Digitally controlled oscillator) but in wavetable synthesis, the output waveform is not normally static and evolves slowly in time as one wavetable is mixed with another, creating a changing waveform. Looping occurs when the wavetable evolution is halted, slowed, or reversed in time.
Different wavetables can be used for the attack phase (at the beginning of the sound, as the volume increases) and release phase (at the end of the sound, as the volume decreases) of the sound. These are normally very difficult to synthesize with other synthesis techniques, but because these are stored as samples very realistic sounds can be produced with little processing power. The Roland Corporation's series of "Linear Additive" synthesizers such as the D-50 made use of a combination of digitally sampled attack phases.
Because single samples are somewhat limited for synthesis of new sounds, modern wavetable synthesizers can combine multiple samples or even change the sound with filters.
Because of the low processing power required early synthesizers imitated filters and other expensive synthesis methods by rapidly playing successive wavetables in sequence. If each waveform is a little duller (or brighter) than the previous, a filter effect can be imitated.
Special effects can be achieved by selecting a wavetable at random or in a special pattern from the table on a tempo-tick from a sequencer. But the effects are quite strong and don't sound very natural. Of course this can be used on purpose to give the sound a special "edge".
Palm's wavetable systems
The German synthesizer designer Wolfgang Palm began experimenting with wavetable synthesis in the late 1970s, with his research realized in PPG's line of synthesizers such as the Wavecomputer, Waveterm, and Wave. Palm's implementation of wavetable synthesis employed an array containing 64 pointers to individual (symmetrical) single-cycle waves stored within the instrument. Usually, only a few pointers to these waves were actually used, spread throughout the breadth of the wavetable. The distinguishing feature of the PPG Wave series was that it would interpolate the remaining waves in between the defined pointers, so that changing the position within the table would result in a smooth, unique "morphing" effect between the waves. It should be stressed that Palm's wavetable scheme's strength was in its generation of harsh digital sounds and bell-like timbres, not the emulation of acoustic instruments. It was possible to sample a complex sound into a wavetable by way of the Waveterm device, but the results were invariably artificial, and usually not as interesting as the powerful and bizarre sounds resulting from competent exploitation of the synthesizer's capabilities.
After the demise of the PPG company, Waldorf Music adapted Palm's wavetable oscillator design into their wildly successful Microwave synthesizer module (1988). This design was extrapolated into the Waldorf Wave in the early 1990s, a very large and expensive instrument offering facilities for resynthesis and user-friendly wave and wavetable construction. An all-digital revision of the Microwave soon followed, with a complete line of wavetable synthesizers remaining in production until 2003. Aside from a few improvements designed to eliminate audible aliasing and quantization (signal processing) errors, this wavetable oscillator scheme had not significantly changed since the first days of the PPG wave series.
Relation to sample playback
A common misdefinition of the term "wavetable synthesis" is that of any one of many sample based playback devices in popular soundcards and sampling keyboards. The term "sample playback" was deemed inadequate, from a marketing perspective, and the term "wavetable synthesis" was borrowed for use for these soundcards and keyboards. The difference between straight-through sample playback and wavetable synthesis as defined here is that the wave is not kept as a single PCM recording, but as a series of single-cycle waves (or, on later equipment like the Roland D-50, very short PCM samples) that can be played in any order or even omitted.
Early sampling keyboards and PC soundcards usually did not work in this mode; before about 1990, limitations in sample memory and computational power meant that samplers and PC trackers could only handle 1 relatively long PCM sample per instrument. Since General MIDI didn't call for anything more sophisticated, and crossfading 3 or more samples per note in tracker-style software was still prohibitively slow at the time, most manufacturers (with the notable exception of Roland's MT-32) decided to stick to this simple method even when better hardware was available. More modern sound cards like the Sound Blaster Live! have multiple independent DCOs that are completely programmable, making Palm-style wavetable effects possible; also, modern PCs have improved in speed and capabilities to the point that using more than one sample per note in a software synthesizer is much easier, though rarely done outside of professional audio software.
The description of wavetable synthesis above is the most original definition and it should be noted (shown in the reference below) that wavetable synthesis is equivalent to additive synthesis in the case that all partials or overtones are harmonic (that is all overtones are at frequencies that are an integer multiple of a fundamental frequency of the tone).