The Mathematics of The Mathematics of SoundSound

Professor Cross’s laser and video displays Professor Cross’s laser and video displays came out of a desire to visualize sound. came out of a desire to visualize sound. Sound, while itself a phenomenon Sound, while itself a phenomenon experienced through one’s sense of hearing, experienced through one’s sense of hearing, can be rendered visually on a medium such can be rendered visually on a medium such as an oscilloscope... or a laser light show. as an oscilloscope... or a laser light show. Knowing the qualities of sound waves, one Knowing the qualities of sound waves, one can create an endless number of unique can create an endless number of unique patterns, ranging from simple geometric patterns, ranging from simple geometric shapes to quasi–random “squiggles.” shapes to quasi–random “squiggles.”

Single-channel audio as a Single-channel audio as a function of time function of time

Sound can be represented Sound can be represented mathematically by the functionmathematically by the function::

When dealing with audio signals, it is When dealing with audio signals, it is common to refer to degrees of phase, so common to refer to degrees of phase, so 2·pi can be changed to 360°: 2·pi can be changed to 360°:

Where: Where:

tt = elapsed time; = elapsed time;AA = amplitude; = amplitude;nn = fundamental frequency or first = fundamental frequency or first harmonic;harmonic;fxfx = waveform. = waveform.

These qualities can all be represented These qualities can all be represented graphically by plotting the overall wave graphically by plotting the overall wave function against time: function against time:

An oscilloscope allows for the visualization of waveforms in real time.An oscilloscope allows for the visualization of waveforms in real time.

AMPLITUDE AMPLITUDE

Changes in amplitude result in changes in Changes in amplitude result in changes in the volume of the sound. When translated the volume of the sound. When translated into visualization, these are the extremes into visualization, these are the extremes reached by the crest and trough of the reached by the crest and trough of the waveform. The “height” increases and waveform. The “height” increases and decreases from changes in amplitude. decreases from changes in amplitude.

FREQUENCY FREQUENCY

Changes in the fundamental frequency Changes in the fundamental frequency (a.k.a. first harmonic) result in changes in (a.k.a. first harmonic) result in changes in the pitch of the sound. When translated into the pitch of the sound. When translated into visualization, the shape of the waveform will visualization, the shape of the waveform will appear to stretch or compress. appear to stretch or compress.

WAVEFORM WAVEFORM

When a piano, violin, or trumpet play When a piano, violin, or trumpet play the same pitch, their distinctive sound the same pitch, their distinctive sound is still apparent. is still apparent. TimbreTimbre is the is the distinctive quality that is unique for an distinctive quality that is unique for an instrument, singing voice, or generated instrument, singing voice, or generated tone. The timbre is a result of a specific tone. The timbre is a result of a specific waveform which is a summation of the waveform which is a summation of the fundamental frequency and higher fundamental frequency and higher harmonics (a.k.a. overtones). harmonics (a.k.a. overtones).

►The purest timbre is produced by a The purest timbre is produced by a sine wave tone generator; the tone sine wave tone generator; the tone consists only of the fundamental consists only of the fundamental frequency (no higher harmonics). As frequency (no higher harmonics). As expected, a sine wave tone appears as expected, a sine wave tone appears as a sine wave on an oscilloscope. a sine wave on an oscilloscope.

►Tone generators are also capapble of Tone generators are also capapble of producing square, triangle, and producing square, triangle, and sawtooth waveforms. All of these sawtooth waveforms. All of these waveforms visually translate to their waveforms visually translate to their respective patterns on an oscilloscope. respective patterns on an oscilloscope.

Two-channel audio in the Two-channel audio in the x–yx–y domaindomain

Stereo audio has two channels (left and Stereo audio has two channels (left and right), and the signals on the respective right), and the signals on the respective channels can be displayed simultaneously channels can be displayed simultaneously using an oscilloscope’s using an oscilloscope’s x–yx–y mode. In mode. In mathematics, this is analogous to a mathematics, this is analogous to a parametric plot. parametric plot.

If the same audio signal is sent over both If the same audio signal is sent over both channels (in essence being monaural), the channels (in essence being monaural), the resulting resulting x–yx–y plot is rather underwhelming: plot is rather underwhelming: a line rotated 45° between the two axes. a line rotated 45° between the two axes.

PHASE PHASE

Visualizations become more interesting as Visualizations become more interesting as deviations in phase are introduced. While deviations in phase are introduced. While two like channels in phase (or 180° out of two like channels in phase (or 180° out of phase) produce the 45° line, any other phase) produce the 45° line, any other phase difference will start to create true phase difference will start to create true two–dimensional images. two–dimensional images.

For a simple sine wave, a 90° phase difference For a simple sine wave, a 90° phase difference changes the pattern from a 45° to a circle.changes the pattern from a 45° to a circle.

While phase changes may not be audible, While phase changes may not be audible, they certainly are apparent (and they certainly are apparent (and important) in the visualization of sound. important) in the visualization of sound.

AMPLITUDE AMPLITUDE

Adjustments in amplitude have an Adjustments in amplitude have an effect on dimensions of the effect on dimensions of the visualization. In case of the circle visualization. In case of the circle plot, the circle will grow and shrink as plot, the circle will grow and shrink as the volume gets louder and softer. the volume gets louder and softer. The plot stays a circle as long as the The plot stays a circle as long as the amplitude in both channels is equal. amplitude in both channels is equal.

As the amplitudes between the left and As the amplitudes between the left and right channels vary, the plot becomes right channels vary, the plot becomes elliptical, with the ratio of its two elliptical, with the ratio of its two dimensions equal to the ratio of the dimensions equal to the ratio of the amplitudes in each amplitudes in each

FREQUENCY FREQUENCY

The input frequencies of the left and right The input frequencies of the left and right channel affect the speed of the trace in the channel affect the speed of the trace in the x–yx–y plot. At low frequencies, the trace of plot. At low frequencies, the trace of the circle plot will move slowly, often to the the circle plot will move slowly, often to the point where its movement is visible to the point where its movement is visible to the human eye. At sufficiently high human eye. At sufficiently high frequencies, however, the trace is moving frequencies, however, the trace is moving fast enough that it appears as a fast enough that it appears as a continuous line. continuous line.

In addition to trace speed, the ratio of In addition to trace speed, the ratio of frequencies between the left and right frequencies between the left and right channels has an effect on the shape of the channels has an effect on the shape of the plot, deviating from the circle or ellipse plot, deviating from the circle or ellipse pattern. The patterns created by different pattern. The patterns created by different frequency ratios was first investigated by frequency ratios was first investigated by Nathaniel Bowditch. Jules-Antoine Lissajous Nathaniel Bowditch. Jules-Antoine Lissajous studied the phenomenon in greater detail by studied the phenomenon in greater detail by reflecting light off mirrors affixed to tuning reflecting light off mirrors affixed to tuning forks with specific pitch ratios. These forks with specific pitch ratios. These patterns are known as patterns are known as Lissajous figuresLissajous figures. .

WAVEFORM WAVEFORM

►Patterns are also affected by the Patterns are also affected by the respective harmonics in the waveform respective harmonics in the waveform of each channel. The addition of of each channel. The addition of various harmonics in waveforms have a various harmonics in waveforms have a significant impact on the shape of an significant impact on the shape of an x–x–yy plot. Waveforms based on sine, plot. Waveforms based on sine, square, and other artificially–generated square, and other artificially–generated tones can be used to produce tones can be used to produce geometric patterns—however, they are geometric patterns—however, they are generally less pleasing to the ear than generally less pleasing to the ear than non–electronic sounds non–electronic sounds

On the other hand, music that may be On the other hand, music that may be enjoyable to listen to will usually result enjoyable to listen to will usually result in quasi–random “squiggles,” which are in quasi–random “squiggles,” which are a direct effect of the countless a direct effect of the countless harmonics produced by a group of harmonics produced by a group of instruments and/or voices. Often times instruments and/or voices. Often times when creating an audio visualization when creating an audio visualization experience, one may choose experience, one may choose electronically–generated sounds for the electronically–generated sounds for the x–yx–y plot, played in sync or mixed with a plot, played in sync or mixed with a non–electronic music soundtrack. non–electronic music soundtrack.

Knowledge of the mathematical quality Knowledge of the mathematical quality of sounds can allow an artist–musician of sounds can allow an artist–musician to create and visualize audio that can to create and visualize audio that can be appealing to the eyes (and the be appealing to the eyes (and the ears). ears).

Created by Ivelin Ivanon and Milen Valentinov from Bulgaria from school “SOU Jeleznik”

19.03.2010