Musimathics (The Mathematical Foundations of Music, Volume by Gareth Loy
By Gareth Loy
Gareth Loy is going deeper into the maths of song and sound, starting with electronic audio, sampling, and binary numbers, in addition to complicated numbers and the way they simplify illustration of musical signals.
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Additional info for Musimathics (The Mathematical Foundations of Music, Volume 2)
Example text
For example, consider swing rhythm. 27a shows the way eighth notes in a measure of 4/4 time are interpreted using even quantization. The written duration of each eighth note is identical to the performed duration, as shown by the proportions of the boxes below the notes. fm Page 33 Monday, February 26, 2007 4:01 PM Digital Signals and Sampling Even Quantization b) Quarter Note Performed March time a) 33 2d Eighth 1st Eighth Written Swing Quantization d) Quarter Note Performed Swing it! 27 Straight time and swing time.
28c is nonlinear because doubling the input will not double the output since the limit A max has already been exceeded and the output is clipped. Suppose x 1 and x 2 are input signals, y 1 and y 2 are output signals, and a and b are scaling terms. If a system behaves such that ax 1 ∝ y 1 and bx 2 ∝ y 2, then it is linear if it preserves ax 1 + bx 2 ∝ y 1 + y 2 for all possible values of a and b . In other words, if input x 1 produces output y 1, and input x 2 produces output y 2 , then any weighted combination of x 1 and x 2 will result in a proportional combination of outputs y 1 and y 2 if the system is linear.
6 kilobits/s. 2 kilobits/s. If we were to store 1 second of CD-quality audio on a computer disk, how much memory would be required? Computer disks are conventionally measured in megabytes (MB). A byte is 8 bits of data, and so a megabyte is . . well, there is more than one definition for this basic unit of storage. It is defined either as 220 = 1,048,576 bytes or as106 = 1,000,000 bytes. The first definition derives from the common computer practice of counting bytes in powers of 2 because computers universally use binary arithmetic.