## Introduction

As previously discussed in Section 1, sine waves can be considered the building blocks of sound. In fact, it was shown in the 19th Century by the mathematician Joseph Fourier that any periodic function can be expressed as a series of sinusoids of varying frequencies and amplitudes. This concept of constructing a complex sound out of sinusoidal terms is the basis for additive synthesis, sometimes called Fourier synthesis for the aforementioned reason. In addition to this, the concepts of additive synthesis have also existed since the introduction of the organ, where different pipes of varying pitch are combined to create a sound or timbre.

A simple block diagram of the additive form may appear like in Fig. 6.1, which has a simplified mathematical form based on the Fourier series:

$f(t)=a_{0}+\sum _{n=1}^{\infty }a_{n}\sin(f_{n}t)$ Where $a_{0}$ is an offset value for the whole function (typically 0), $a_{n}$ are the amplitude weightings for each sine term, and $f_{n}$ is the frequency multiplier value. With hundreds of terms each with their own individual frequency and amplitude weightings, we can design and specify some incredibly complex sounds, especially if we can modulate the parameters over time. One of the key features of natural sounds is that they have a dynamic frequency response that does not remain fixed. However, a popular approach to the additive synthesis system is to use frequencies that are integer multiples of the fundamental frequency, which is known as harmonic additive synthesis. For example, if the first oscillator's frequency, $f_{1}$ represents the fundamental frequency of the sound at 100 Hz, then the second oscillator's frequency would be $f2=2f_{1}$ , and the third $f3=3f_{1}$ and so on. This series of sine waves produces an even "harmonic" sound that can be described as "musical". Oscillator frequency relationships that are not integer related, on the other hand, are called "inharmonic" and tend to be noisier and take on the characteristics of bells or other percussive sounds.

## Constructing common harmonic waveforms in additive synthesis Figure 6.2. The first four terms of a square wave constructed from sinusoidal components (partials).

If we know the amplitude weightings and frequency components of the first $x$ sinusoidal components or partials of a complex waveform, we can reconstruct that waveform using an additive system with $x$ oscillators. The popular waveforms square, sawtooth and triangle are harmonic waveforms because the constituent sinusoidal components all have frequencies that are integer multiples of the fundamental. The property that distinguishes them in this form is that they all have unique amplitude weightings for each sinusoid. Fig. 6.2 demonstrates the appearance of the time-domain waveform as a set of sines at unique amplitude weightings are added together; in this case the form begins to approximate a square wave, with the accuracy increasing with each added partial. Note that to construct a square wave we only include odd numbered harmonics- the amplitude weightings for $a_{2}$ , $a_{4}$ , $a_{6}$ etc. are 0. Below is a table that demonstrates the partial amplitude weightings of the common waveshapes:

Waveshape $a_{1}$ $a_{2}$ $a_{3}$ $a_{4}$ $a_{5}$ $a_{6}$ $a_{7}$ $a_{8}$ $a_{9}$ General rule
Sine 1 0 0 0 0 0 0 0 0
Square 1 0 1/3 0 1/5 0 1/7 0 1/9 $1/x$ for odd $x$ .
Triangle 1 0 -1/9 0 1/25 0 -1/49 0 1/81 $1/x^{2}$ for odd $x$ , alternating + and -.
Sawtooth 1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 $1/x$ A conclusion you may draw from Fig. 6.2 and the table is that it requires a large amount of frequency partials to create a waveform that closely approximates the idealised mathematical forms of the waveforms introduced in Section 5. For this reason, it should be apparent that additive synthesis techniques are perhaps not the best method for producing these forms. The strengths of additive synthesis lie in the fact that we can exert control over every partial component of our sound, which can produce some very intricate and wonderful results. With the constant modification of the frequency and amplitude values of each oscillator, the possibilities are endless. Some examples of ways to control the weightings and frequencies of each component oscillator are illustrated:

• Manual control. The user controls a bank of oscillators with an external control device (typically MIDI), tweaking the values in real time. More than one person can join in and change / alter the timbre to their whims.
• External data. Digital information from another source is taken and converted into appropriate frequency and amplitude values. The varying data source will then effectively be in 'control' of the timbral outcomes. Composers have been known to use data from natural sources or pieces derived from interesting geometric, aleatoric and mathematical models.
• Recursive data. Given a source set of values and a set of algorithmic rules, the control parameters reference the previous value entered into the system to determine the result of the next one. Users may wish to "interfere" with the system to set the process on a new path. See Markov chains.

There is, however, the major consideration of computational power: complex sounds may require many oscillators all operating at once which will put major demand on the system in question.