# 10 Additive Synthesis

## Table Of Contents

1 Basic Waveforms Revisited2 Helmholtz Piano3 Overtone Envelopes4 Nonharmonic Sounds## Explanation of the Concepts

Additive synthesis is the process of creating complex timbres by summing sinusoids of varying frequency, amplitude and phase. Theoretically, any periodic waveform can be created this way. The physical groundwork for the technique was laid by scientists in the 19th century. In 1822, the French mathematician Jean Baptiste Fourier discovered that the static temperature distribution on a cool metal plate with a single point-source of steady heat could be described thus as an infinite series of sinusoids that are harmonically related. Fourier and others then developed computational techniques for separating these harmonically related sinusoids from various complex waves (eventually including sound waves). The acoustic implications of this work were first proposed by Georg Simon Ohm in 1843. In an article entitled "On the definition of sound, along with its theory", He states that "...the elements of a sound ... must have the form a*sin2πmt or a*cos2πmt, ... so that a succession of impressions upon our ears, which is periodic according to the form given here, must also necessarily cause the sensation of a tone." This statement, known as "Ohm's Acoustic Law" implies that any complex, periodic timbre can be broken down into individual sinusoidal constituents, according to Fourier's equation, of which the complex timbre is the sum. In 1863, this law was empirically proven by Herman von Helmholtz in his treatise "Lehre von den Tonempfindungen". In a section entitled "Proof of Ohm's Law", Helmholtz describes experiments that employ the principle of sympathetic vibration in the analysis of the frequency content of complex timbres. He concludes that "the human ear perceives pendular [i.e. sinusoidal] vibrations alone as simple tones, and resolves all other periodic motions [i.e. complex timbres] of the ear into a series of pendular vibrations [sinusoids], hearing the series of simple tones which correspond with these simple vibrations." In other words, according to this view, the ear itself breaks down complex timbres into their sinusoidal constituents. This implies that the inverse is also true: that when the ear simultaneously receives a number of sinusoids, it will perceive a complex tone.

The first musical composition to use this technique was Karlheinz Stockhausen's*Studie I*, composed at the Westdeutcher Rundfunk during the summer of 1953.