Overview

Additive synthesis creates complex sounds by adding together individual sinusoidal signals called partials. A partial's frequency and amplitude are each time-varying functions, so a partial is a more flexible version of the harmonic associated with a Fourier series decomposition of a periodic waveform.

In this module you will learn about partials, how to model the timbre of natural instruments, various sources of control information for partials, and how to make a sinusoidal oscillator with an instantaneous frequency that varies with time.

Partials

A partial is a generalization of the harmonic associated with a Fourier series representation of a periodic waveform. The screencast video of Figure 1 introduces important concepts associated with partials.

Figure 1: [video] Important concepts associated with partials

Modeling Timbre of Natural Instruments

Perception of an instrument's timbre relies heavily on the attack transient of a note. Since partials can effectively enter and leave the signal at any time, additive synthesis is a good way to model physical instruments. The screencast video of Figure 2 discusses three important design requirements for partials necessary to successfully model a physical instrument.

Figure 2: [video] Three design requirements for partials to model physical instruments

Sources of Control Information

The time-varying frequency of a partial is called its frequency trajectory, and is best visualized as a track or path on a spectrogram display. Similarly, the time-varying amplitude of a partial is called its amplitude trajectory. Control information for these trajectories can be derived from a number of sources. Perhaps the most obvious source is a spectral analysis of a physical instrument to be modeled. The screencast video of Figure 3 discusses this concept and demonstrates how a trumpet can be well-modeled by adding suitable partials.

The code for the LabVIEW VI demonstrated within the video is available here: trumpet.zip. This VI requires installation of the TripleDisplay front-panel indicator.

Figure 3: [video] Modeling a trumpet tone by adding together increasing numbers of partials

Control information can also be derived from other domains not necessarily related to music. The screencast video of Figure 4 provides some examples of non-music control information, and illustrates how an edge boundary from an image can be "auralized" by translating the edge into a partial's frequency trajectory.

Figure 4: [video] Skyline of Houston translated into a frequency trajectory

Instantaneous Frequency

The frequency trajectory of a partial is defined as a time-varying frequency f(t) f(t) . Since a constant-frequency and constant-amplitude sinusoid is mathematically described as y(t)=Asin⁡(2π f 0 t) y(t)=Asin⁡(2π f 0 t) , intuition perhaps suggests that the partial should be expressed as y(t)=a(t)sin⁡(2πf(t)t) y(t)=a(t)sin⁡(2πf(t)t) , where a(t) a(t) is the amplitude trajectory. However, as shown in the screencast video of Figure 5 this intuitive result is incorrect; the video derives the correct equation to describe a partial in terms of its trajectories f(t) f(t) and a(t) a(t) .

Figure 5: [video] Derivation of the equation of a partial given its frequency and amplitude trajectories