By James Beauchamp
This publication includes a whole and exact mathematical therapy of the sounds of track with an emphasis on musical timbre. The ebook spans the variety from instructional advent to complicated study and alertness to speculative overview of its a number of ideas. the entire individuals use a generalized additive sine wave version for describing musical timbre which provides a conceptual solidarity, yet is of adequate application to be tailored to many various tasks.
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Extra info for Analysis, Synthesis, and Perception of Musical Sounds: The Sound of Music
10. Fundamental frequency f 0 vs time for the G3 tenor voice sound. Note that the forms of the two error terms are the same and that ρ is a factor used to weight the relative importance of the two terms in calculating the total error. , f n = | f n — f k |. 0. For the first summation, ak←n refers to the normalized amplitude of the peak component k that is closest to the harmonic n. , f k = | f n — f k |. 33, respectively. For each frame of the analysis, the fundamental frequency f o is varied over a designated frequency range and the “true” fundamental frequency is deemed to be the one that yields the lowest value of Errtotal .
The first zero occurs when the argument of the sin is π , and this leads to f o = 1 + (α/π )2 /T . For f > f o the function approximately follows a sinc function: α α sinc α (π T f /α)2 − 1 ≈ sinc (π T f ) . 13c) W( f ) = sinh(α) sinh(α) So for a given window width T , the first zero frequency and the amount of stopband rejection depends on the value of the parameter α, and there is a trade-off between the two. 441 forces f o = 2/T . The minimum stop-band attenuation is approximately 40 dB, which is comparable to the Hamming.
It works fine on single musical instrument tones. 4 Piecewise Cubic Interpolation of Phases. In this case, it is assumed that both the phases and the frequencies are known at the beginning and end of each frame. By asserting these, continuity of frequency and phase can be guaranteed, and unlike the quadratic case, each frame segment is computed independently (McAulay and Quatieri, 1986). To begin, the equation for phase is postulated in terms of a cubic function of n: θk (n) = θa + γ (n − H i) + α(n − H i)2 + β(n − H i)3 , H i ≤ n ≤ H (i + 1).