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 481092122
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 12tone 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admission to the University of Michigan may be rescinded!).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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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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 Spring '10
 123
 Frequency, Cos, The Victors

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