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Kendall, GS; Haworth, C; Cádiz, RF (2014)
Publisher: Massachusetts Institute of Technology Press
Languages: English
Types: Article
This article describes methods of sound synthesis based on auditory distortion products, often called combination tones. In 1856, Helmholtz was the first to identify sum and difference tones as products of auditory distortion. Today this phenomenon is well studied in the context of otoacoustic emissions, and the “distortion” is understood as a product of what is termed the cochlear amplifier. These tones have had a rich history in the music of improvisers and drone artists. Until now, the use of distortion tones in technological music has largely been rudimentary and dependent on very high amplitudes in order for the distortion products to be heard by audiences. Discussed here are synthesis methods to render these tones more easily audible and lend them the dynamic properties of traditional acoustic sound, thus making auditory distortion a practical domain for sound synthesis. An adaptation of single-sideband synthesis is particularly effective for capturing the dynamic properties of audio inputs in real time. Also presented is an analytic solution for matching up to four harmonics of a target spectrum. Most interestingly, the spatial imagery produced by these techniques is very distinctive, and over loudspeakers the normal assumptions of spatial hearing do not apply. Audio examples are provided that illustrate the discussion.
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    • Expressing this equation as a polynomial in A2 we obtain: of order N−2. For a large N, this is not only computationally challenging, it is indeed unsolvable if N > 6. Abel's impossibility theorem states that, in general, (22) polynomial equations higher than fourth degree are incapable of algebraic solutions in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions operating on the coefficients (Cheney and Kincaid 2009, pp. 705). This does not mean that high-degree polynomials are not solvable, because the fundamental theory of algebra guarantees that at least one complex solution exists. What this really means is that the solutions cannot be always expressed in radicals. Therefore, as seeking an algebraic expression for any N is impractical; if we want to specify s(t) with more than four harmonics by calculating the coefficients of x(t), the only way of doing that is by numerical methods such as the Newton-Rhapson, Laguerre, or the Lin-Bairstrow algorithm (Rosloniec (23) 2008, pp. 29-47). This is the main reason why a numerical rather than an algebraic solution is the correct approach for this problem when a high number of harmonics in the target signal is desired.
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