Complex Waveforms

We have established a relationship between the measureable quantity of sound waves called amplitude and the qualitative attribute of sound perception called loudness. Our task now is to understand differences in tones associated with pitch and timbre in terms of measureable things about sound waves.

Periodic Sounds

We have already discussed the relationship between the sound quality called pitch and the measured frequency of a sound wave. So far, we talked only about simple sound wave patterns, shaped like a sine wave. Below, is a graph of the changes in air pressure versus time for a pure tone sound.

Rossing discusses many subtleties about our perception of sounds in chapter 7. In particular, he talks about the keen ability our auditory system has to distinguish two pure tones that are very close in frequency. Rossing defines this as the "just noticeable difference (jdn)" in pitch. It corresponds to 1-2% of the critical bandwidth. For sound wave frequencies less than 1 kHz, the critical bandwidth is ~100 Hz, meaning we can perceive the difference between tones differing by only 1 Hz! Be sure to read through the beginning of chapter 7 for further information.

Pure tones are simple to understand -- they are just sine waves. Pure tones don't represent the full richness of sounds that we hear. Two sounds can have the same pitch but differ in other aspects, qualitatively attributed to timbre. Our job now is to understand these differences.

Complex sound waves can still have a periodicity, T (the time for a complete cycle; the sound wave then being many repeating cycles), and hence have a dominant frequency, f = 1 / T, that we perceive as the pitch of the sound. Since the pitch of a tone is primarily determined from its frequency, and hence the time it takes for the wave to complete one cycle, pure tones and complex tones with the same period have roughly the same pitch. Below are shown three different complex waveforms that all have the same period and amplitude of the pure-tone waveform shown above.

How can we quantify the differences between these sound waves? We'll quantify these differences first, and then talk about qualitative differences in how we perceive the sounds from these different waves.

Harmonic Series

We will be able to show that any complex periodic sound wave can be represented as a superposition of pure tones of different frequencies, forming a harmonic series. Before we get there, lets first remind ourselves about the harmonic series.

Pure tones of frequencies

f, 2f, 3f, 4f, ...

form a harmonic series. For example, the sequence of frequencies

200 Hz, 400 Hz, 600 Hz, 800 Hz, ...

is a harmonic series with its fundamental equal to 200 Hz. A different series of frequencies is

330 Hz, 440 Hz, 550 Hz, 660 Hz, ...

It is also a harmonic series; but in this case, the fundamental is 110 Hz, and the fundamental and first and second overtones are missing from the series.

The wave periods of pure tones within the harmonic series also forms a series. Remembering that T = 1 / f, the wave periods in a harmonic series are

T, T/2 , T/3 , T/4 , ...

Things to notice in this series are By looking at this series, it shouldn't be too surprising that a wave formed from the addition of waves from the harmonic series will have the frequency of the fundamental in the series.

What is a superposition of harmonics?

A superposition of waves is just the sum of the two waves. For sound, this amounts to adding the differences in pressure at a given position associated with each wave at every instant of time. Mathematically, this is easy to express. Let's consider the addition of the first and second harmonics

We'll go through this addition in detail on Wednesday.

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Last updated: 30 Oct 1999
Comments: bland@indiana.edu