... article¹

This is the author's final draft version of an article submitted to the Journal of New Music Research. The published version should appear early in 2005.

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... way.²

For purposes of this article, ``virtualization'' refers to the process of creating real-time, performable synthesis algorithms which are competitive with natural physical instruments such as pianos, guitars, clarinets, and so on.

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... conditions.³

Sound examples: http://ccrma.stanford.edu/~jos/waveguide/Sound_Examples.html

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... standard.⁴

One exception is the Yamaha VL series of synthesizers [124], which provided a separate high-speed, low-latency input for the breath controller.

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...Morse,MorseAndIngard,Pierce,Vallette95:⁵

This section follows portions of [162, Appendix B]. Figures in this article not otherwise attributed are reprinted from [162].

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...SmithDWMMI.⁶

http://ccrma.stanford.edu/~jos/waveguide/Force_Waves.html

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...⁷

A GIF89A animation of Fig. 6 is available on-line at http://ccrma.stanford.edu/~jos/swgt/movet.html .

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... increases.⁸

Note that the force goes to infinity for the wrong reason in this model. The basic approximation of force as tension times slope is only valid for slope magnitudes much less than unity, as any derivation of the ideal string wave equation shows [162, Appendix B]. Compensating for this approximation would yield a maximum force equal to the string tension. However, that force would go to infinity in a model incorporating tension modulation (which is neglected here), because the string is being stretched to infinite length. Of course, in a truly complete model, the string should break at some point.

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...⁹

http://ccrma.stanford.edu/~jos/waveguide/Comb_Filters.html

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....¹⁰

If this were instead a velocity-wave simulation, we could say we were simulating magnetic pick-ups on an electric-guitar string.

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...SmithDWMMI.¹¹

http://ccrma.stanford.edu/~jos/waveguide/Digital_Waveguide_Theory.html

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...JOSFP¹²

http://ccrma.stanford.edu/~jos/filters/Modal_Representation.html

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...SmithDWMMI,¹³

http://ccrma.stanford.edu/~jos/waveguide/Sound_Examples.html

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... amplitudes.¹⁴

Another measurable effect in nonlinear musical string vibration is combination tones [34].

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...Askenfelt90¹⁵

http://www.speech.kth.se/music/5_lectures/conklin/longitudinal.html

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... interactions.¹⁶

While both the digital waveguide and second-order finite difference models are exact simulations in the case of bandlimited ideal strings [159], they become different models in the presence of nonlinear interactions.

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...JOSSVDCP,SmithDWMMI,SVDJOSCP.¹⁷

http://ccrma.stanford.edu/~jos/waveguide/Commuted_Piano_Synthesis.html

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...SmithDWMMI.¹⁸

http://ccrma.stanford.edu/~jos/waveguide/Sound_Examples.html

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... Kit¹⁹

http://ccrma.stanford.edu/CCRMA/Software/STK/ -- version 4.1.2 was current when this article was written.

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... VL70-m.²⁰

http://www.patchmanmusic.com/turbovl.html

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...Putland.²¹

For velocity waves, the flare may be hyperbolic [23].

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