Sine Wave: The Atomic Component¶

With Fourier in mind, the sine wave can be considered the atomic unit sound - more precisely of periodic signals and thus of musical sounds.
Additive synthesis and related approaches build musical sounds from scratch, using these integral components. When a sound is composed of several sinusoids, they are referred to as partials, regardless of their properties. Partials which are integer multiples of a fundamental frequency are called harmonics or overtones, when related to the first harmonic:

Partial: A single sinewave with arbitrary frequency in an additive model. It is a part of a sound.
Harmonic: A sine wave within a harmonic sound, including the fundamental frequency.
- The first harmonic has the frequency $f_{H_1} = f_0$
Overtone: Any sine wave within a harmonic sound, excluding the fundamental frequency.
- The first overtone has the frequency $f_{O_1} = 2 f_0$

Fourier Series¶

According to the Fourier theorem, any periodic signal $x_N(t)$ can be approximated by a finite sum of sinusoids. The general equation is as follows:

$$ x_N(t) = a_0 + \sum_{n=1}^N \left( a_n \cos \left(2 \pi \frac{n}{P} t \right) + b_n \sin \left( 2 \pi \frac{n}{P} t \right) \right) $$

Number of partials: $N$
Wavelength: ${\displaystyle {\tfrac {P}{n}}}$
Frequency: ${\displaystyle {\tfrac {n}{P}}}$

Additive Synthesis¶

With the time varying parameters

Amplitude: $a_i$
Frequency: $f_i$
Phase: $\varphi_i$

we can simplify this for an infinite numer of partials:

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

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

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

As following sections on spectral modeling show, a more advanced 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.