This thesis summarizes several contributions to the areas of musical signal processing and virtual-analog musical sound synthesis. In virtual analog music synthesis, the waveform oscillators and the audio filters are usually required to be smoothly modulated, often to audio rates. As such, efficient synthesis requires the design of efficiently variable algorithms for these synthesis blocks.; The design of efficiently variable filters requires the understanding of how they vary with their parameters. The most popular virtual-analog filters are of a class (pole-variable filters) which vary primarily via the motion of their poles. The Root Locus Method is proposed as a framework for understanding the variation of such filters, and several filters are analyzed as a demonstration, including popular analog filter types (the State-Variable Filter and the four-pole lowpass filter designed by Robert Moog). Root locus analysis also suggests directions for designing filters, or may be used directly in designs. This informs a series of experiments in creating digital filters in the style of Moog's lowpass filter, culminating in a few new designs.; The design of efficiently variable waveform generators is approached with the goal of reducing abasing artifacts in the generated waves. Popular waveforms in virtual analog synthesis are rich in harmonics, and their analog counterparts have extremely wide bandwidths, such that care must be taken when generating digital versions. A methodology is proposed whereby a useful subset of the desired waveforms can be derived from a Bandlimited Impulse Train (BLIT) via linear operations, which will preserve any bandlimiting which may exist in the BLIT. Further, methods for implementing BLITs are explored and contrasted, including a method introduced by the author.; Finally, methods for rendering visualizations of root loci are reviewed; as such visualizations are necessary in the use of the method as an analysis tool. A recent advance in implicit-function rendering was found to be of particular use, and its application to locus rendering allowed the creation of programs for interactive exploration of root loci and their behaviors, which assist in the goal of building user intuition into variable-filter behavior.
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