By James Beauchamp
This booklet incorporates a entire and exact mathematical remedy of the sounds of song with an emphasis on musical timbre. The publication spans the diversity from instructional advent to complicated examine and alertness to speculative evaluation of its a number of suggestions. the entire members use a generalized additive sine wave version for describing musical timbre which supplies a conceptual cohesion, yet is of adequate software to be tailored to many alternative tasks.
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Additional info for Analysis, Synthesis, and Perception of Musical Sounds: The Sound of Music
The method programmed by Maher (1989) for use in the SNDAN analysis/synthesis package (Beauchamp, 1993) convolves a Hamming-windowed sinc function with the input signal, and the upsampled result is linearly interpolated. For the DFT requirement, the substitution m ← m + n − N /2 is made in Eq. 17c), resulting in fs = c˜k (n) = e jπ k P(1−2n/N ) P N N −1 w (N /2 − m)s(m + n − N /2)e− j2π k Pm/N . 19a) m=0 With P = 1, the summation of Eq. 19a) is in the correct form for the DFT. However, with P > 1, Eq.
Another distinct method uses the cepstrum, which seeks to determine the periodicity of the DFT magnitude by taking its log and then applying a second FFT. This has been used extensively in speech applications (Noll, 1967) but seldom for music [for an exception, see Chen (2001)]. All of these methods rely on the selection of a minimum or maximum of a function. Determining which of several maxima or minima correspond to the correct fundamental frequency turns out to be the biggest problem in making these methods reliable.
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.
Analysis, Synthesis, and Perception of Musical Sounds: The Sound of Music by James Beauchamp