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What is a Model?

Model-building is a fundamental human activity. For our purposes, a model can be defined as any form of computation that predicts the behavior of a physical object or phenomenon based on its initial state and any ``input'' forces. Our first successful models occurred in our heads [185]. By mentally predicting future effects of seasonal climate change, for example, we were able to plan for winter food supplies. Current theories of human evolution postulate that increased brain size came about as an adaptation to larger scale climate changes, including centuries of drought. Mental simulation was later extended by mathematical simulation, enabling, for example, accurate prediction of astronomical events and physical phenomena at all scales. A mathematical physical theory, such as Newton's laws of motion, gives us a mathematical model that readily predicts the outcome of any physical experiment in the real world for which the theory is applicable. More recently, our brains have been supplemented even further by computer simulation, enabling formerly intractable problems to be simulated as well, thereby further extending our ability to predict. For example, it is now possible to predict even the weather some days into the future with a useful degree of accuracy.

The Basic Science Loop

The basic ``science loop'' consists of the following steps:

  1. Observations
  2. Model construction
  3. Comparison of simulated observations to real-world observations
  4. Model refinement
  5. Go to step 3
A successful model will accurately predict the outcome of any experiment given the same input conditions and initial state.

Models for Music and Audio

For virtual musical instruments and audio effects, the model replaces the real thing. Therefore, the model's prediction from its inputs (controls and possible input sounds), is simply the desired output sound. Since a sound is a special case of a signal, we can say that all models of interest in this book are signal models.

Also in the musical case, we often desire that our models run in real time. That is, we may supply model inputs from a live human performer, and the task of the model becomes rendering high-quality sound based on performance gestures. Real-time computational forms are ubiquitous in the field of digital signal processing, and in large measure, this book is about signal processing methods most applicable to real-time computational physical models.

Physical models are desirable when many dimensions of expressive control are needed. This is generally best appreciated by performing musicians. Audience members only hear the final music played, so they are not in a position to judge the quality of multidimensional control, as skilled musicians utilize only a narrow control subspace in any given performance.2.1

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Overview of Model Types
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But How Does It Sound?