Additional material can be found at http://www.musimathics.com.

benefit of all of us. loy79081_fm.fm Page xii Tuesday, February 27, 2007 11:10 AM xii Foreword The two volumes of Musimathics are a kind of instantiation of the process of learning that had such a powerful facilitating effect on the work at CCRMA in those years: explanations presented with wit and in great detail but here logically ordered and ever available, not just once! This second volume of Musimathics is comprehensive. Loy focuses on the digital domain, from elemental binary numbers

floor below the helix). a) Sine W ave b) s(t) = e iωt Im{s(t)} Imaginary 0 z x Real y Time t t Re{s(t)} Cosin e Wav e Figure 2.26 Projection of a complex signal. loy79081_ch02.fm Page 87 Tuesday, March 13, 2007 1:35 PM Musical Signals 87 If the equation for the helix is s ( t ) = e i ω t, then the cosine projection is just the real part, denoted Re { s ( t ) }, and the sine projection is just the imaginary part, denoted Im { s ( t ) }. The value of the helix when t = 0 in

signal. The DST only detects odd loy79081_ch03.fm Page 120 Monday, February 26, 2007 8:28 PM 120 Chapter 3 1 2 1 1 2 0 – 1 2 –1 – 1 2 0 0 1 2 3 4 5 6 7 Figure 3.10 New test signal, offset in phase. components of the spectrum, and the DCT only detects even components. Since the DFT combines the DST and DCT, it shows the entire spectrum. 3.2.6 DFT of Arbitrary Phase Signals In the preceding section, I developed the use of the DST and DCT to detect the even (cosine) and odd

( N 1 ) ⁄ N , which is one step short of a complete clockwise rotation. When k = 2, the phasor is e i2 ω n ⁄ N . It will spin clockwise twice as n goes from 0 to N 1. When k = 3, the phasor spins three times, and so on. Q Summarizing the roles of k and n , k selects the frequency on which to operate and determines where in the output function X to place the result. Q n steps through the samples of the input function x and also determines the phase value of the phasor for each calculation.

measurement precision (if possible) or increasing the sampling rate (if possible) or both. In terms of figure 1.9, doubling the precision would halve the height of each row; doubling the sampling rate would halve the width of each column. Doing either would increase the amount of information to be stored or transmitted: doubling the precision doubles the number of quanta per unit of magnitude, and doubling the sampling rate doubles the number of samples per unit of time. Reducing the area of the