Reflection: Theoretical and practical fundamentals for computed sound We have seen in Section 1.4.2 how the introduction of computed sound, over the

decades, resulted in the development of a wide range of new sound-generating processes, including forms of direct wave table manipulation, parametrized waveform synthesis, sampling, spectrum-based operations, amplitude-based signal processing, granular synthesis, physical modeling, and sequencing. In Section 1.4.3, we have discussed how these became widely used for producing musical sound. Clearly, computed sound has become important to the instrumental control of musical sound, in general. But what are the reasons underlying its newness, wide variety, and wide use?

First of all, reasons can be found in the basic model of computed sound. The basic setup of electronic digital computer, wave table and electric loudspeaker described in Section 1.4.1 can be abstracted to a basic model illustrated in Figure 1.11a. Here, in the middle is the electric loudspeaker, as a sound-producing transducer. This transducer has a state, which can be defined by the current values for a set of measurable physical properties describing what of the transducer will vary over time. On the one hand, human sensory perception is exposed to this transducer state, as it changes over time, resulting in perceptual phenomena being induced (in this case, in sounds being heard). On the other hand, the transducer state has been made to causally depend on part of the state of an automaton (in this case, on the wave table of an electronic digital computer). This part of the automaton's state also is subject to the computations it can perform. Crucially, these are Turing-complete.

Turing Machines are the well-known models of automata that can be described by a tuple (Γ, Σ, □, Q, q0, F, δ) (see for example [Linz 1997]). Here, Γ is an alphabet for

symbols read from and written to a hypothetical infinite tape, with Γ including a subset Σ for input placed initially on the tape, and also a separate blank symbol □. As it operates on the infinite tape, the automaton has an internal symbolic state taken from the set Q, initially q0. If this internal state at some point changes into one of the final

states contained in the subset F, execution within the automaton halts. This point may or may not be reached, depending on the initial input and the transition function δ, which describes how reading a symbol from the tape automatically results in a new internal state, a new replacement symbol written to the tape, and left/right movement to the next tape symbol to be processed. According to the famous Church-Turing conjecture, any algorithmic method for computing numbers can be modeled and executed in this way. A special case, related to this, is the Universal Turing Machine, which accepts the description of any Turing machine with any input as its input, and then models the execution of an exact simulation. In this way, a universal model of computation is provided [Turing 1936]. The computational steps of an actual,

practically implemented automaton are said to be Turing-complete if and only if they have a similar capability for simulating any possible Turing machine.

In the context of computed sound, this is important not only for the computation of numbers in a narrow sense, but also for the related ability to simulate any machine that can be characterized by automatic causal transitions between a set of possible discrete states. Many different machines of such a general type may be developed, given a specific type of transducer, and causal dependence for controlling its state, in order to produce heard sound in various specific ways. Where the computed sound model is followed, given sufficient memory and computational speed, the Turing-complete automaton can be used to effectively implement any such conceivable machine. This is what fundamentally underlies the wide variety of sound-generating processes developed after the introduction of computed sound.

Figure 1.11a A basic model of computed sound. Sound becomes heard as human

perception is exposed to changes over time in the state of a sound-producing transducer. The transducer state causally depends on part of the state of a Turing- complete automaton.

Second, more reasons for the wide variety, and also the wide use of computed sound can be found in the existence of increasingly powerful implementations of the computed sound model. These enabled actual development and use. Given some practically implemented Turing-complete automaton, the causal relationship between part of its state and the transducer state usually will be well-known (as for the initial combination of electric loudspeaker and electronic digital computer discussed in Section 1.4.1). But what heard sound this relationship ultimately can result in is subject to discovery, and may be specific to the implementation used. On the one hand, the transducing technologies used for this were made to be increasingly powerful, in the sense of transducer state changes over time being capable of inducing a widening range of perceptual phenomena. (This was often tied to a general goal of increased realism in sound reproduction, driven by demand from basic applications such as the playback of musical sound, film soundtracks, and various forms of voice communication.) On the other hand, the computing technologies used also were made to be increasingly powerful, in the sense of practically allowing the execution of algorithms of increasing

complexity. The flawless execution, with infinite memory, in unspecified time of the Turing machine model was matched by the increasing reliability, larger finite memories and faster processing speeds of actual machines. These developments, already mentioned in Section 1.4.3, all resulted from ongoing miniaturization of reliable interconnected electronic switches, strongly driven by the commercial value of the resulting technology. This has been described by Moore's Law, the famous heuristic for integrated electronic circuits, which predicted the sustained and exponential increase, over the years, in the number of transistors per unit area [Moore 1965].

Third, more reasons for the wide variety and wide use of computed sound can be found in the increasing availability of implementations of the computed sound model. This enabled the use of computed sound by many people, and thereby accelerated its development. One way of characterizing this increasing availability could be to trace the presence of component technologies in affordable consumer electronics products. For transducing technologies, this would for example include the appearance of home stereo systems. For computing technologies, it would include the later introduction of home- and personal computers. In general, the increasing availability of computing technologies was possible because of the shrinking physical sizes and costs of increasingly user-friendly devices, already discussed in Section 1.4.3. The underlying reason for this, again, was the ongoing miniaturization described by Moore's Law. In this way, the existence and wide availability of powerful computing technology were closely intertwined, with existence depending on miniaturization – only possible because of the financial rewards of mass production and mass use. In the development of transducing technologies, e.g. for personal audio, expected mass use was similarly present as a fundamental factor.

Concluding, fundamental reasons for the newness, wide variety, and wide use of computed sound can be summarized as the development of cheaply mass-producible, powerful implementations of the computed sound model.