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This paper describes a paradigm called procedural concatenation for defining functions of time. In this paradigm, a function is represented by a computational process that concatenates function primitives by calling them in the desired order. Procedural concatenation has proven useful for specifying volume, tempo, and other time-varying musical parameters. We are concerned here with programming systems for the real-time control of synthesizers. Such systems can be divided into two classes: functionoriented and event-oriented. Function-oriented systems produce periodically sampled synthesizer control signals, such as amplitude and timbral envelopes. Examples include the venerable GROOVE system (Mathews and Moore 1970), ARCTIC (Dannenberg 1984), and FORMES (Rodet and Cointe 1984). Event-oriented systems generate discrete commands (e.g., note-start and note-end) at irregular intervals. Continuous control signals (such as amplitude envelopes) are usually generated internally by the synthesizer. Examples include POD (Truax 1977), MOXIE (Collinge 1984), and FORMULA (Anderson and Kuivila 1986). The function-oriented approach is the more general of the two, but is difficult to implement with acceptable real-time performance on inexpensive hardware. This, together with the widespread use of event-oriented MIDI synthesizers, has made the event-oriented approach more prevalent. Musical phenomena such as volume and tempo fluctuation can be viewed as functions of continuous time. However, in an event-oriented system it may suffice to sample these functions only at the discrete times of note initiations or other control
Anderson et al. (Sun,) studied this question.