## Fourier Series

According to the Fourier theorem, any periodic signal can be represented by an infinite sum of sinusoids with individual

• amplitude $a_i$
• frequency $f_i$
• phase $\varphi_i$

$\displaystyle y = \sum\limits_{i=1}^{\infty} a_i \ sin(2 \pi f_i \ t +\varphi_i )$

When applying this principle to musical sounds, a simplified model can be used to generate basic timbres. All sinusoidal components become integer multiples of a fundamental freuquency $f_0$, so called harmonics, with a maximum number of partials $N_{part}$. In a very reduced model, the phases of the partials can be ignored:

$\displaystyle y (t) = \sum\limits_{n=1}^{N_{part}} a_n(t) \ sin(2 \ \pi \ n \ f_0 (t) \ t)$

As following sections on spectral modeling show, a more advanved model is needed to synthesize musical sounds which are indistinguishable from the original. This includes the partials' phase, inharmonicities as deviations from exact integer multiples, noise components and transients. However, depending of the number of partials and the driving function for their parameters, this limited formula can generate convincing harmonic sounds.

## A Brief History

### Early Mechanical

Early use of the Fourier representation, respectively additive synthesis, for modeling musical sounds has been made by Hermann von Helmholtz. He built mechanical devices, based on tuning forks, resonant tubes and electromagnetic excitation for additive synthesis. Von Helmholtz used these devices for investigating various aspects of harmonic sounds, including spectral distribution and relative phases. Tuning forks with resonant tubes (von Helmholtz, 1870, p.183).

### Early Analog

The history of Elektronische Musik started with additive synthesis. In his composition Studie II, Karlheinz Stockhausen composed timbres by superimposing sinusoidal components. In that era this was realized through single sine wave oscillators, tuned to the desired frequency and recorded on tape.

Studie II is the attempt to fully compose music on a timbral level in a rigid score. Stockhausen therefor generated tables with frequencies and mixed tones for creating source material. [Fig.1] shows an excerpt from the timeline, which was used to arrange the material. The timbres are recognizable through their vertical position in the upper system, whereas the lower system represents articulation, respectively fades and amplitudes. [Fig.1] From the score of Studie II.

### Early Digital

As mentioned in Introduction II, Max Mathews used additive synthesis to generate the first digitally synthesized pieces of music in the 1950s. In the early 1960s, Mathews had advanced the method to synthesize dynamic timbres, as in Bycicle Built for Two:

• Hermann von Helmholtz. Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik, 3. umgearbeitete Ausgabe. Braunschweig: Vieweg, 1870.
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