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Chapter 1: The Digital Representation of Sound,
Whats the difference between a tuba and a flute (or, more accurately, the sounds of each)? How do we tell the difference between two people singing the same song, when theyre singing exactly the same notes? Why do some guitars "sound" better than others (besides the fact that theyre older or cost more or have Eric Claptons autograph on them)? What is it that makes things "sound" like themselves?
Its not necessarily the pitch of the sound (how high or low it is)if everyone in your family sang the same note, you could almost surely tell who was who, even with your eyes closed. Its also not just the loudnessyour voice is still your voice whether you talk softly or scream at the top of your lungs. So whats left? The answer is found in a somewhat mysterious and elusive thing we call, for lack of a better word, "timbre," and thats what this section is all about.
"Timbre" (pronounced "tam-ber") is a kind of sloppy word, inherited from previous eras, that lumps together lots of things that we dont fully understand. Some think we should abandon the word and concept entirely! But its one of those words that gets used a lot, even if it doesnt make much sense, so well use it here toowere sort of stuck with it for the time being.
What Makes Up Timbre?
Timbre can be roughly defined as those qualities of a sound that arent just frequency or amplitude. These qualities might include:
Envelope and spectra are very complicated concepts, encompassing a lot of subcategories. For example, spectral features are very important, different ways that the spectral aggregates are organized statistically in terms of shape and form (e.g., the relative "noisiness" of a sound is a result, in large part, of its spectral relationships). Many facets of envelope (onset time, harmonic decay, spectral evolution, steady-state modulations, etc.) are not easily explained by just looking at the envelope of a sound. Researchers spend a great deal of time on very specific aspects of these ideas, and its an exciting and interesting area for computer musicians to research.
Figure 1.18 shows a simplified picture of the envelope of a trumpet tone.
Its helpful here to bring another descriptive term into our vocabulary: spectrum. Spectrum is defined by a waveforms distribution of energy at certain frequencies. The combination of spectra (plural of spectrum) and envelope helps us to define the "color" of a sound. Timbre is difficult to talk about, because its hard to measure something subjective like the "quality" of a sound. This concept gives music theorists, computer musicians, and psychoacousticians a lot of trouble. However, computers have helped us make great progress in the exploration and understanding of the various components of whats traditionally been called "timbre."
Basic Elements of Sound
As weve shown, the average piece of music can be a pretty complicated function. Nevertheless, its possible to think of it as a combination of much simpler sounds (and hence simpler functions)even simpler than individual instruments. The basic atoms of sound, the sinusoids (sine waves) we talked about in the previous sections, are sometimes called pure tones, like those produced when a tuning fork vibrates. We use the tuning fork to talk about these tones because it is one of the simplest physical vibrating systems.
Although you might think that a discussion of tuning forks belongs more in a discussion of frequency, were going to use them to introduce the notion of sinusoids: Fourier components of a sound.
When you hit the tines of a tuning fork, it vibrates and emits a very pure note or tone. Tuning forks are able to vibrate at very precise frequencies. The frequency of a tuning fork is the number of times the tip goes back and forth in a second. And this number wont change, no matter how hard you hit that fork. As we mentioned, the human ear is capable of hearing sounds that vibrate all the way from 20 times a second to 20,000 times a second. Low-frequency sounds are like bass notes, and high-frequency sounds are like treble notes. (Low frequency means that the tines vibrate slowly, and high frequency means that they vibrate quickly.)
When you whack the tines of a tuning fork, the fork vibrates. The number of times the tines go back and forth in one second determines the frequency of a particular tuning fork.
Click Soundfile 1.18. Youll hear composer Warren Burts piece for tuning forks, "Improvisation in Two Ancient Greek Modes."
Now, why do tuning fork functions have their simple, sinusoidal shape? Think about how the tip of the tuning fork is moving over time. We see that it is moving back and forth, from its greatest displacement in one direction all the way back to just about the same displacement in the opposite direction.
Imagine that you are sitting on the end of the tine (hold on tight!). When you move to the left, that will be a negative displacement; and when you move to the right, that will be a positive displacement. Once again, as time progresses we can graph the function that at each moment in time outputs your position. Your back-and-forth motion yields the functions many of you remember from trigonometry: sines and cosines.
Any sound can be represented as a combination of different amounts of these sines and cosines of varying frequencies. The mathematical topic that explains sounds and other wave phenomena is called Fourier analysis, named after its discoverer, the great 18th century mathematician Jean Baptiste Joseph Fourier.
Figure 1.22 shows the relative amplitudes of sinusoidal components of simple waveforms.
For example, Figure 1.22(a) indicates that a sawtooth wave can be made by addition in the following way: one part of a sine wave at the fundamental frequency (say, 1 Hz), then half as much of a sine wave at 2 Hz, and a third as much at 3 Hz, and so on, infinitely.
In Section 4.2, well talk about using the Fourier technique in synthesizing sound, called additive synthesis. If you want to jump ahead a bit, try the applet in Section 4.2, that lets you build simple waveforms from sinusoidal components. Notice that when you try to build a square wave, there are little ripples on the edges of the square. This is called Gibbs ringing, and it has to do with the fact that the sum of any finite number of these decreasing amounts of sine waves of increasing frequency is never exactly a square wave.
What the charts in Figure 1.22 mean is that if you add up all those sinusoids whose frequencies are integer multiples of the fundamental frequency of the sound and whose amplitudes are described in the charts by the heights of the bars, you’ll get the sawtooth and square waves.
This is what Fourier analysis is all about: every periodic waveform (which is the same, more or less, as saying every pitched sound) can be expressed as a sum of sines whose frequencies are integer multiples of the fundamental and whose amplitudes are unknown. The sawtooth and square wave charts in Figure 1.22 are called spectral histograms (they dont show any evolution over time, since these waveforms are periodic).
These sine waves are sometimes referred to as the spectral components, partials, overtones, or harmonics of a sound, and they are what was thought to be primarily responsible for our sense of timbre. So when we refer to the tenth partial of a timbre, we mean a sinusoid at 10 times the frequency of the sounds fundamental frequency (but we dont know its amplitude).
The sounds in some of the following soundfiles are conventional instruments with their attacks lopped off, so that we can hear each instrument as a different periodic waveform and listen to each instruments different spectral configurations. Notice that, strangely enough, the clarinet (whose sound wave is a lot like a sawtooth wave) and the flute, without their attacks, are not all that different (in the grand scheme of things).
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