NOISE GENERATORS, FILTERS, AND SUBTRACTIVE SYNTHESIS
3.1 SOUND SOURCES FOR SUBTRACTIVE SYNTHESIS
PREREQUISITES FOR THE CHAPTER
In this chapter we will discuss filters, a fundamental subject in the field of sound design and electronic music, and subtractive synthesis, a widely-used technique that uses filters. A filter is a signal processing device that acts selectively on some of the frequencies contained in a signal, applying attenuation or boost to them. The goal of most digital filters is to alter the spectrum of a sound in some way. Subtractive synthesis was born from the idea that brand-new sounds can be created by modifying, through the use of filters, the amplitude of some of the spectral components of other sounds.
Any sound can be filtered, but watch out: you can’t attenuate or boost components that don’t exist in the original sound. For example, it doesn’t make sense to use a filter to boost frequencies around 50 Hz when you are filtering the voice of a soprano, since low frequencies are not present in the original sound.
In general, the source sounds used in subtractive synthesis have rich spectra so that there is something to subtract from the sound. We will concentrate on some of these typical source sounds in the first portion of this section, and we will then move on to a discussion of the technical aspects of filters.
Filters are used widely in studio work, and with many different types of sound:
> Sounds being produced by noise generators, by impulse generators, by oscillator banks, or by other kinds of signal generators or synthesis
> Audio files and sampled sounds
> Sounds being produced by live sources in real time (the sound of a musician playing an oboe, captured by a microphone, for example)
NOISE GENERATORS: WHITE NOISE AND PINK NOISE
One of the most commonly used source sounds for subtractive synthesis is white noise, a sound that contains all audible frequencies, whose spectrum is essentially flat (the amplitudes of individual frequencies being randomly distributed). This sound is called white noise in reference to optics, where the color white is a combination of all of the colors of the visible spectrum. White noise makes an excellent source sound because it can be meaningfully filtered by any type of filter at any frequency, since all audible frequencies are present. (A typical white noise spectrum is shown in Figure 3.1.)
Fig. 3.1 The spectrum of white noise
Another kind of noise that is used in similar ways for subtractive synthesis is pink noise. This kind of sound, in contrast to white noise, has a spectrum whose energy drops as frequency rises. More precisely, the attenuation in pink noise is 3 dB per octave; it is also called 1/f noise, to indicate that the spectral energy is proportional to the reciprocal of the frequency. (See Figure 3.2.) It is often used, in conjunction with a spectral analyzer, to test the frequency response of a musical venue, in order to correct the response based on some acoustic design.
Fig. 3.2 The spectrum of pink noise
In digital systems, white noise is generally produced using random number generators: the resulting waveform contains all of the reproducible frequencies for the digital system being used. In practice, random number generators use mathematical procedures that are not precisely random: they generate series that repeat after some number of events. For this reason, such generators are called pseudo-random generators.
By modifying some of their parameters, these generators can produce different kinds of noise. A white noise generator, for example, generates random samples at the sampling rate. (If the sampling rate is 48,000 Hz, for example, it will generate 48,000 samples per second.) It is possible, however, to modify the frequency at which numbers are generated – a generating frequency equal to 5,000 numbers a second, for example, we would no longer produce white noise, but rather a noise with strongly attenuated high frequencies.
When the frequency at which samples are generated is less than the sampling rate, “filling in the gaps” between one sample and the next becomes a problem, since a DSP system (defined in the glossary for Chapter 1T) must always be able to produce samples at the sampling rate. There are various ways of resolving this problem, including the following three solutions:
• Simple pseudo-random sample generators
Fig. 3.3 Generation of pseudo-random values
• Interpolated pseudo-random sample generators
Fig. 3.4 Generation of pseudo-random values with linear interpolation
Interpolation between one value and the next can be linear, as shown in the figure, or polynomial, implemented using polynomial functions to connect the values using curves rather than line segments. (Polynomial interpolation is shown in Figure 3.5, however, we will not attempt to explain the details here.) The kinds of polynomial interpolation most common to computer music are quadratic (which use polynomials of the second degree) and cubic (which use polynomials of the third degree). Programming languages for synthesis and signal processing usually have efficient algorithms for using these interpolations built in to their runtimes, ready for use.
Fig. 3.5 Generation of pseudo-random values with polynomial interpolation
• Filtered pseudo-random sample generators In this kind of approach, the signal produced is filtered using a lowpass filter. We will speak further of this kind of generator in the section dedicated to lowpass filters.
INTERACTIVE EXAMPLE 3A – NOISE GENERATORS – PRESETS 1-4
If the pure sounds composing a complex sound are not integer multiples of the lowest frequency component, we have a non-harmonic sound and the components are called non-harmonic components, or partials.
OSCILLATORS AND OTHER SIGNAL GENERATORS
In Section 1.2T, we examined the “classic” waveforms that are often found in synthesizers, such as the square wave, the sawtooth wave, and the triangle wave. Section 2.1T explained how these waveforms, when geometrically perfect (perfect squares, triangles, etc.), contain an infinite number of frequency components. The presence of infinitely large numbers of components, however, causes nasty problems when producing digital sound, since an audio interface cannot reproduce frequencies above half of its sampling rate. (We will discuss this topic in much greater detail in Chapter 5.) When you attempt to digitally reproduce a sound that contains component frequencies above the threshold for a given audio interface, undesired components will appear, which are almost always non-harmonic. To avoid this problem, band-limited oscillators are often used in digital music. Such oscillators, which produce the classic waveforms, are built so that their component frequencies never rise above half of the sampling rate. The sounds generated by this kind of oscillator therefore make a good point of departure for creating sonorities appropriate for filtering, and as a result, they are the primary source of sound in synthesizers that focus on subtractive synthesis. In Section 3.5 we will analyze the structure of a typical subtractive synthesizer.
It is, of course, also possible to perform subtractive synthesis using synthetic sounds, rich in partials, that have been realized using other techniques such as non-linear synthesis or physical modeling. We will cover these approaches in following chapters.
FILTERING SAMPLED SOUNDS
Beyond subtractive synthesis, one of the everyday uses of filters and equalizers is to modify sampled sounds. Unlike white noise, which contains all frequencies at a constant amplitude, a sampled sound contains a limited number of frequencies, and the amplitude relationships between components can vary from sound to sound. It is therefore advisable, before filtering, to be conscious of the frequency content of a sound to be processed.
Remember that you can only attenuate or boost frequencies that are already present. This is true for all sounds, sampled or otherwise, including those captured from live sources.
 Besides altering the amplitude of a sound, a filter modifies the relative phases of its components. top
 Another way to define the difference between white noise and pink noise is this: while the spectrum of white noise has the same energy at all frequencies, the spectrum of pink noise distributes the same energy across every octave. A rising octave, designated anywhere in the spectrum, will occupy a raw frequency band that is twice as wide as its predecessor’s; pink noise distributes equal amounts of energy across both of these frequency bands, resulting in the constant 3 dB attenuation that is its basic property. top It is for this reason that sampling rate of an audio interface is almost always more than twice the maximum audible frequency for humans. top
3.3 THE Q FACTOR
3.4 FILTER ORDER AND CONNECTION IN SERIES
Filters of the first order
Second-order resonant filters
Higher order filters: connecting filters in series
3.5 SUBTRACTIVE SYNTHESIS
Anatomy of a subtractive synthesizer
3.6 EQUATIONS FOR DIGITAL FILTERS
3.7 FILTERS CONNECTED IN PARALLEL, AND GRAPHIC EQUALIZATION
3.8 OTHER APPLICATIONS OF PARALLEL FILTERS: PARAMETRIC EQ AND SHELVING FILTERS
3.9 OTHER SOURCES FOR SUBTRACTIVE SYNTHESIS: IMPULSES AND RESONANT BODIES
from “Electronic Music and Sound Design” Vol. 1 by Alessandro Cipriani and Maurizio Giri