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Chapter 5: The Transformation of Sound by Computer
Section 5.4: Introduction to Spectral Manipulation
There are two different approaches to manipulating the frequency content of sounds: filtering, and a combination of spectral analysis and resynthesis. Filtering techniques, at least classically (before the FFT became commonly used by most computer musicians), attempted to describe spectral change by designing time-domain operations. More recently, a great deal of work in filter design has taken place directly in the spectral domain.
Spectral techniques allow us to represent and manipulate signals directly in the frequency domain, often providing a much more intuitive and user-friendly way to work with sound. Fourier analysis (especially the FFT) is the key to many current spectral manipulation techniques.
Perhaps the most commonly used implementation of Fourier analysis in computer music is a technique called the phase vocoder. What is called the phase vocoder actually comprises a number of techniques for taking a time-domain signal, representing it as a series of amplitudes, phases, and frequencies, and manipulating this information and returning it to the time domain. (Remember, Fourier analysis is the process of turning the list of samples of our music function into a list of Fourier coefficients, which are complex numbers that have phase and amplitude, and each corresponds to a frequency.)
Two of the most important ways that musicians have used the phase vocoder technique are to use a sounds Fourier representation to manipulate its length without changing its pitch and, conversely, to change its pitch without affecting its length. This is called time stretching and pitch shifting.
Why should this even be difficult? Well, consider trying it in the time domain: play back, say, a 33 1/3 RPM record at 45 RPMs. What happens? You play the record faster, the needle moves through the grooves at a higher rate, and the sound is higher pitched (often called the "chipmunk" effect, possibly after the famous 1960s novelty records featuring Alvin and his friends). The sound is also much shorter: in this case, pitch is directly related to frequencytheyre both controlled by the same mechanism. A creative and virtuosic use of this technique is scratching as practiced by hip-hop, rap, and dance DJs.
The Pitch/Speed Relationship in the Digital World
Now think of altering the speed of a digital signal. To play it back
faster, you might raise the sampling rate, reading through the samples
for playback more quickly. Remember that sometimes we refer to the sampling
rate as the rate at which we stored (sampled) the sounds, but it also
can refer to the kind of internal clock that the computer uses with reference
to a sound (for playback and other calculations). We can vary that rate,
for example playing back a sound sampled at 22.05 kHz at 44.1 kHz. With
more samples (read) per second, the sound gets shorter. Since frequency
is closely related to sampling rate, the sound also changes pitch.
Using the Phase Vocoder
Using the phase vocoder, we can realize Steve Reichs piece (see Xtra bit 5.1), and a great many others. The phase vocoder allows us independent control over the time and the pitch of a sound.
How does this work? Actually, in two different ways: by changing the speed and changing the pitch.
To change the speed, or length, of a sound without changing its pitch, we need to know something about what is called windowing. Remember that when doing an FFT on a sound, we use what are called framestime-delimited segments of sound. Over each frame we impose a window: an amplitude envelope that allows us to cross-fade one frame into another, avoiding problems that occur at the boundaries of the two frames.
What are these problems? Well, remember that when we take an FFT of some portion of the sound, that FFT, by definition, assumes that were analyzing a periodic, infinitely repeating signal. Otherwise, it wouldnt be Fourier analyzable. But if we just chop up the sound into FFT-frames, the points at which we do the chopping will be hard-edged, and well in effect be assuming that our periodic signal has nasty edges on both ends (which will typically show up as strong high frequencies). So to get around this, we attenuate the beginning and ending of our frame with a window, smoothing out the assumed periodical signal. Typically, these windows overlap at a certain rate (1/8, 1/4, 1/2 overlap), creating even smoother transitions between one FFT frame and another.
By changing the length of the overlap when we resynthesize the signal, we can change the speed of the sound without affecting its frequency content (that is, the FFT information will remain the same, itll just be resynthesized at a "larger" frame size). Thats how the phase vocoder typically changes the length of a sound.
What about changing the pitch? Well, its easy to see that with an FFT we get a set of amplitudes that correspond to a given set of frequencies. But its clear that if, for example, we have very strong amplitudes at 100 Hz, 200 Hz, 300 Hz, 400 Hz, and so on, we will perceive a strong pitch at 100 Hz. What if we just take the amplitudes at all frequencies and move them "up" (or down) to frequencies twice as high (or as low)? What weve done then is re-create the frequency/amplitude relationships starting at a higher frequency—changing the perceived pitch without changing the frequency.
The phase vocoder technique actually works just fine, though for radical
pitch/time deformations we get some problems (usually called "phasiness").
These techniques work better for slowly changing harmonic sounds and for
simpler pitch/time relationships (integer multiples). Still, the phase
vocoder works well enough, in general, for it to be a widely used technique
in both the commercial and the artistic sound worlds.
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