ADDITIVE AND VECTOR SYNTHESIS
2.1 FIXED SPECTRUM ADDITIVE SYNTHESIS
PREREQUISITES FOR THE CHAPTER
A sound produced by an acoustic instrument, any sound at all, is a set of complex oscillations, all produced simultaneously by the instrument in question. Each oscillation contributes a piece of the overall timbre of the sound, and their sum wholly determines the resulting waveform. However, this summed set of oscillations, this complex waveform, can also be described as a group of more elementary vibrations: sine waves. Sine waves are the basic building blocks with which it is possible to construct all other waveforms. When used in this way, we call the sine waves frequency components, and each frequency component in the composite wave has its own frequency, amplitude, and phase. The set of frequencies, amplitudes, and phases that completely define a given sound is called its sound spectrum. Any sound, natural or synthesized, can be decomposed into a group of frequency components. Synthesized waveforms such as we described in Section 1.2 are no exception; each has its own unique sound spectrum, and can be built up from a mixture of sine waves. (Sine waves themselves are self-describing – they contain only themselves as components!).
SPECTRUM AND WAVEFORM
Spectrum and waveform are two different ways to describe a single sound. Waveform is the graphical representation of amplitude as a function of time.)
Fig. 2.1 The waveform of a complex sound
In Figure 2.2, we see the same complex sound broken into frequency components. Four distinct sine waves, when their frequencies and amplitudes are summed, constitute the complex sound shown in the preceding figure.
Fig.2.2 Decomposition of a complex sound into sinusoidal components
A clearer way to show a “snapshot” of a collection of frequencies and amplitudes such as this might be to use a graph in which the amplitude of the components is shown as a function of frequency, an approach known as frequency domain representation. Using this approach, the x-axis represents frequency values, while the y-axis represents amplitude. Figure 2.2b shows our example in this format: a graph displaying peak amplitudes for each component present in the signal.
Fig. 2.2b A sound spectrum
Pre-recorded sound, which is then processed in realtime
In order to see the evolution of components over time, we can use a graph called a sonogram (which is also sometimes called a spectrogram), in which frequencies are shown on the y-axis and time is shown on the x-axis (as demonstrated in Figure 2.2c). The lines corresponding to frequency components become darker or lighter as their amplitude changes in intensity. In this particular example, there are only four lines, since it is a sound with a simple fixed spectrum.
Fig. 2.2c A sonogram (also called a spectrogram)
Now we will consider a process in which, instead of decomposing a complex sound into sine waves, we aim to do the opposite: to fashion a complex sound out of a set of sine waves.
This technique, which should in theory enable us to create any waveform at all by building up a sum of sine waves, is called additive synthesis, and is shown in diagrammatic form in Figure 2.3.
Fig. 2.3 A sum of signals output by sine wave oscillators
In Figure 2.4, two waves, A and B, and their sum, C, are shown in the time domain.
Fig.2.4 A graphical representation of a sum of sine waves
As you can easily verify by inspection, instantaneous amplitudes for wave C are obtained by summing the instantaneous amplitudes of the individual waves A and B. These amplitude values are summed point-by-point, taking their sign, positive or negative, into consideration. Whenever the amplitudes of A and B are both positive or both negative, the absolute value of the amplitude of C will be larger than that of either of the component, resulting in constructive interference, such as displayed by the following values:
A = -0.3
Whenever the amplitudes of A and B differ in their signs, one being positive and the other negative, the absolute value of their sum C will be less than either one or both components, resulting in destructive interference, as shown in the following example:
A = 0.3
“The largest part, indeed nearly the entirety, of sounds that we hear in the real world are not pure sounds, but rather, complex sounds; sounds that can be resolved into bigger or smaller quantities of pure sound, which are then said to be the components of the complex sound. To better understand this phenomenon, we can establish an analogy with optics. It is noted that some colors are pure, which is to say that they cannot be further decomposed into other colors (red, orange, yellow, and down the spectrum to violet). Corresponding to each of these pure colors is a certain wavelength of light. If only one of the pure colors is present, a prism, which decomposes white light into the seven colors of the spectrum, will show only the single color component. The same thing happens with sound. A certain perceived pitch corresponds to a certain wavelength  of sound. If no other frequency is present at the same moment, the sound will be pure. A pure sound, as we know, has a sine waveform.”
The components of a complex sound sometimes have frequencies that are integer multiples of the lowest component frequency in the sound. In this case the lowest component frequency is called the fundamental, and the other components are called harmonics. (A fundamental of 100 Hz, for example, might have harmonics at 200 Hz, 300 Hz, 400 Hz, etc.) The specific component that has a frequency that is twice that of its fundamental is called the second harmonic, the component that has a frequency that is three times that of the fundamental is called the third harmonic, and so on. When, as in the case we are illustrating, the components of a sound are integer multiples of the fundamental, the sound is called a harmonic sound. We note that in a harmonic sound the frequency of the fundamental represents the greatest common divisor of the frequencies of all of the components. It is, by definition, the maximum number that exactly divides all of the frequencies without leaving a remainder.
INTERACTIVE EXAMPLE 2A – HARMONIC COMPONENTS
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.
INTERACTIVE EXAMPLE 2B – NON-HARMONIC COMPONENTS
 In the case of periodic sounds, the waveform can be fully represented by a single cycle. top
 “The length of a cycle is called its wavelength and is measured in meters or in centimeters. This is the space that a cycle physically occupies in the air, and were sound actually visible, it would be easy to measure, for example, with a tape measure.” (Bianchini, R. 2003) top
Periodic versus aperiodic, and harmonic versus non-harmonic
2.3 CROSSFADING BETWEEN WAVETABLES: VECTOR SYNTHESIS
2.4 VARIABLE SPECTRUM ADDITIVE SYNTHESIS
from “Electronic Music and Sound Design” Vol. 1 by Alessandro Cipriani and Maurizio Giri