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Chapter 8 (Hall)
Sound Spectra
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Introduction
Question: When you hear the music “Danny Boy”, what lets you distinguish between a trumpet and a flute?
Answer: Each periodic waveform has its corresponding spectrum, which determines the timbre, or tone quality of the sound.
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Waveforms and spectra of a flute and a trumpet
Flute C Note Trumpet C Note
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Outline
The harmonic series Prototype steady tones Periodic waves and Fourier spectra
Fourier spectrum Fourier components Fourier synthesis Fourier analysis
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The harmonic series An example of a harmonic series: f1 = 110 Hz, f2
= 220 Hz, f3 = 330 Hz, … f10 = 1100 Hz,…so on.
Harmonic series: A Harmonic series contains a group of frequencies that are based on a single frequency, f1, which is called the fundamental frequency. The frequencies of the other members are simple multiples of the fundamental.
fn = nf1, n = 1, 2, 3,… f1: the fundamental frequency; f2: the 2nd harmonic; f3:
the 3rd harmonic, … and so on.
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Prototype of periodic steady tones
(a) Sine wave (b) Square wave (c-d) Pulse wave (e) Triangular wave (f-h) Saw-tooth wave
What is the simplest of all wave forms?
Answer: Sine waves. They are the “building blocks” for other more complex wave forms.
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Two things to show
(1) Take simple periodic sine waves and put them together to form a more complex wave.
(2) Take a complex periodic wave and break it down into simple sine wave components.
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f = f1= 110 Hz
Combination of sine waves
+
f2=220 Hz
f1=110 Hz
T Any set of sine waves whose frequencies belong to a harmonic series will combine to make a periodic complex wave, whose repetition frequency is that of the series fundamental.
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Combination of sine waves (cont.)
In general, for a set of sine waves whose frequencies do not belong to a harmonic series, the combined wave will be non-periodic.
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Breaking a periodic complex wave
Any periodic waveform of period T may be built from a set of sine waves whose frequencies form a harmonic series with fundamental f1 = 1/T. Each sine wave must have just the right amplitude and relative phase, and those can be determined from the shape of the complex waveform.
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Recipe for building a square wave
…
After 200 selected sine waves added together
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Fourier spectrum Fourier spectrum: The
recipe of sine wave amplitudes involved in a complex wave.
Fourier components: Each sine wave ingredient is called a Fourier component.
Fourier synthesis: Putting sine waves together to make complex waves.
Fourier analysis: Taking complex waves apart into their sine wave components.
Fourier spectrum of a square wave
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Homework
Ch. 8 (Hall), P. 146, Exercises: #1, 2.