# lab2 - 1 ENGIN 100 Music Signal Processing LAB#2...

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1 ENGIN 100: Music Signal Processing LAB #2: Frequencies of Musical Tones Professor Andrew E. Yagle Dept. of EECS, The University of Michigan, Ann Arbor, MI 48109-2122 I. ABSTRACT This lab begins the study of music by determining the frequencies of musical notes. The goals of this lab are: (1) To be able to use a simple signal processing algorithm for computing frequencies of sampled sinusoids; (2) To use this algorithm to compute the frequencies of musical tones in a pure tonal version of “The Victors”; (3) To use simple data visualization to determine the relations between these frequencies. The result will be the 12-tone chromatic scale used for most Western music. You will need to know these frequencies to build simple music synthesizers and transcribers in Projects #1 and #3. II. BACKGROUND We now make a Frst stab at analyzing musical signals. It makes sense to start with the simplest musical signals: pure tones. Even an untrained ear can sense that these are the simplest signals. We will analyze a basic tonal version of “The Victors” (if you don’t know what “The Victors” is, your admission to the University of Michigan may be rescinded!). We will quickly discover that these are sinusoids of di±erent frequencies. We then need a way of computing the frequencies of sinusoids from their samples. Then we need a way of interpreting these frequencies: why them? What are the relations between them? How can we compute them from the tonal signals? III. FREQUENCY COMPUTATION We will make extensive use of the cosine addition formulae cos( x + y )=cos( x )cos( y ) sin( x )sin( y ) , (1) cos( x y x y )+sin( x y ) . (2) Adding and subtracting these gives 2cos( x y x + y )+cos( x y ) . (3) This formula will yield two di±erent methods for determining frequency.

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2 A. Tuning Fork Method This is best illustrated using an example. Let x =2 π 441 t and y πt in (3). This gives cos(2 π 440 t )+cos(2 π 442 t )=2cos(2 )cos(2 π 441 t )( 4 ) Generalizing, this shows that the sum of two sinusoids with the same amplitude and with approximately equal frequencies f and f+∆ is equal to a sinusoid having frequency midway between these frequencies with amplitude that varies sinusoidally with frequency ∆. You heard an example of this in Lab #1.
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lab2 - 1 ENGIN 100 Music Signal Processing LAB#2...

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