Analysis, Synthesis, and Perception of Musical Sounds: The by James Beauchamp

By James Beauchamp

This e-book includes a entire and exact mathematical therapy of the sounds of tune with an emphasis on musical timbre. The e-book spans the diversity from instructional creation to complex learn and alertness to speculative evaluation of its a variety of suggestions. the entire individuals use a generalized additive sine wave version for describing musical timbre which provides a conceptual cohesion, yet is of adequate software to be tailored to many alternative tasks.

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Extra resources for Analysis, Synthesis, and Perception of Musical Sounds: The Sound of Music (Modern Acoustics and Signal Processing)

Example text

28c) so that Eq. 27) becomes 8 θk (0). 29b) H where i is the frame number, running from i = 0 to i = I − 1 (there are I frames). This results in I equations with I + 2 unknowns. 29d) H and a similar equation obtains for i = I − 1. Therefore, there are now I equations with I unknowns, and these can be readily solved by a process of elimination. 8 f k1 = Analysis and Synthesis of Musical Instrument Sounds 23 The same process can be applied to interpolation of the amplitude values. The quadratic interpolation method works very well.

Going to lower frequencies by a certain factor compromises the time resolution by the inverse factor. Because the detected peaks must be at least three window bins apart, for each frame i the maximum number of peaks K i that can be detected in a spectrum is N /6. Because the frequency resolution is f b = f s /N , the maximum number of peaks (or partials) is equal to f s /(6 f b ). For example, if a minimum peak separation of 40 Hz were required at a sample rate of 44,100 Hz, a power-of-2 value of N must first be chosen to yield a value at least that small.

19a) is in the correct form for the DFT. However, with P > 1, Eq. 19a) indicates analysis only at frequencies P/N , 16 James W. Beauchamp 2P/N , 3P/N , . . , whereas the DFT is defined for all frequencies 0, 1/N , 2/N , 3/N , . . This problem is solved by taking the FFT for all frequencies (replacing kP by k and letting k = 0, 1, 2, . , k = P, 2P, 3P, . ). 5 f a are first computed, and then the odd-numbered components are thrown away while keeping the even-numbered ones, which correspond to harmonics of f a .

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