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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.