SOME NOTES ON ACOUSTICS, ELECTRO-ACOUSTICS, AND SOUND SYNTHESIS
by Dr. Steve Paxton, Texas Tech University

I. Accoustics

    We hear sound because of rapid and relatively subtle changes in air pressure next to our ear drums.  These are pressure changes can be caused by doors slamming, strings vibrating, children screaming, bells ringing, jets taking off, columns of air being disturbed (flutes, trumpets, didjeridus, pop bottles, etc.) and by thousands of other natural and decidedly un-natural phenomena.
    These fluctuations of air pressure travel through air in waves, just as ripples travel through disturbed water.  These waves can be measured.  Such a measurement is often expressed graphically as a wave-form, a picture or graph of how the air pressure changes in time.
    All change is measurable and can be graphed.  We can measure how frequently or rapidly the changes occur,  how drastic the changes are, and how complex the changes are. In terms of the fluctuating air pressure that causes sound, we call the three measurements  frequency, amplitude, and  timbre, and they constitute the three parameters by which all sound can be analyzed. By observing  frequency  (pitch), amplitude (loudness), and timbre (tone quality), we can observe everything that is observable about sound.
 
Frequency
    Frequency is the measurement of how often a change occurs.  If a vibrating string causes the air pressure surrounding it to change very rapidly, then the string itself must be changing position (vibrating, that is) very rapidly.  Psychologically, we interpret different frequencies as different pitches.  Slowly vibrating strings, columns of air, drum heads, etc. create 'low-pitched' sounds, and rapidly vibrating strings, columns of air, drum heads, etc. create 'high-pitched' sounds.  Graphically, in the picture of a wave-form, frequency corresponds to wave-length.
 
Amplitude
    Amplitude is the measurement of how drastic a change is.  When you play on a swing at the park, you can make a small singing arc, traveling just a short distance before reversing your direction, or you can make a big arc, traveling a great distance before heading back to where you started (and beyond -> your momentum carries you even beyond where you started).  The small-arc swing has a low amplitude, while the big arc has a high amplitude.  Psychologically, we interpret different amplitudes of sound as having different levels of loudness, because of how drastically the air pressure is fluctuating.  Graphically, in the picture of a wave-form, amplitude corresponds to wave-depth.
 
Timbre
    Timbre is more difficult to visualize and understand than frequency and amplitude, partly because it is in many ways a resultant effect of frequency and amplitude.  (In fact, all three of these parameters of sound are inextricably linked together.  It is only in analyzing them that we even dare to think of them as being separate.)
    Change can be simple or complex, in the same way that a trip to California could be straight down I-40 or a complex combination of side trips over county and state roads.  To carry the analogy even further, a complex, convoluted trip to California could be analyzed as being merely the combination of many individual simple trips over these various county and state roads.  In the same way, sounds can be simple or complex, and the complex ones can be thought of as multi-layered combinations of the simple ones.
    The simplest sound is the result of a perfectly even and regular increase and decrease of air pressure.  A test tone on radio or TV produces such a sound.  The sound of a flute or of someone singing 'ooh' is almost as simple. If many of these frequencies and amplitudes occurring simultaneously are mathematically related to each other in fairly simple ways (whole-number ratios, for instance), the overall sound will meld together and we will perceive it as a unified, single tone of a specific fundamental pitch.  Devices such as oboes, trumpets, opera singers (usually), and sawtooth wave oscillators create such sounds.
    At other times, the relationships between the many different frequencies and amplitudes may be so mathematically complex that we cannot isolate any one fundamental pitch, but rather we hear a wide blur of pitches that we often describe as noise.  Bass drums, jet engines, rattlesnakes, and spoken consonants are more likely to create these relatively more complex and noisy types of sounds. In the picture form of a wave-form, timbre corresponds to wave-shape.
 
 

II. Electro-Accoustics

    Sounds can be produced electronically because acoustics and electricity behave in ways that are analogous to each other.  They operate in similar fashion.  Levels of electrical current, or voltage, can increase and decrease much in the same way the air pressure can increase and decrease.  If these fluctuations of voltage are introduced to a loudspeaker (a flexible membrane attached to an electromagnet), they can cause the loudspeaker to vibrate,  It of course vibrates with the same frequency, amplitude, and timbre (wave-form) as the fluctuating voltage driving it.  The vibrating loudspeaker causes the air pressure adjacent to its flexible membrane to change rapidly, with, again, the same frequency, amplitude, and timbre as the electrical current that made the whole system start vibrating in the first place.  And finally, if our ear drum is close enough to these changes in air pressure, we hear sound.
    A synthesizer is a device that can create fluctuating voltage, and that provides the user ways to alter the frequency, amplitude, and timbre of those voltage fluctuations.  Said another way, it generates sound waves, and allows the user  to alter the wave-forms.  (Similarly, a violin is a device that can create fluctuating air pressure, and that provides the user with ways to alter the frequency, amplitude, and timbre of those air pressure fluctuations.
 
Analog synthesizers
    Analog synthesizers contain electronic circuits that actually generate fluctuation voltage, and thereby generate sounds, as described above.  These are called oscillators.  Analog synthesizers also contain filters, devices that allow the user to alter the timbre of sounds generated by the oscillators.  Finally, they contain amplifiers, devices used to alter the amplitude of  the sounds  generated by the oscillators.
Digital Synthesizers
    Digital synthesizers do not normally contain actual oscillators, filters, and amplifiers.  Instead, they contain computers that calculate mathematical representations of sound waves.  In much the same way that a graphing calculator or a computer program can create a wave-form on its display.  These mathematically represented wave-forms can be altered as to their frequency, amplitude, and timbre to create more wave-forms.  In fact, whole tables of these "imaginary" sound waves, expressed only as numbers, can be stored in the memory of a digital synthesizer.  However, we can't really hear these so called waves.  We can just see the lists of numbers or the graphic diagrams that represent them.  And lists of numbers and pictures of wave-forms cannot drive loudspeakers.  Therefore, every digital synthesizer contains a rather sophisticated device that converts numbers into actual fluctuations of voltage.  This device is called a digital-to-analog converter (DAC), and is usually the last stage of a digital synthesizer.
 

III. Sound Synthesis

    Most musicians, whether they work with acoustic music, electro-acoustic music, or both, spend most of their time dealing with frequency and amplitude.  However, it is the parameter of timbre that more subtly and pervasively inhabits everything we do as musicians.  Perhaps it is because timbre is so often beyond our control that we tend to concentrate on it less.
    The most interesting things about timbre are the natural acoustical phenomena associated with it, specifically overtone  and the harmonic-series.
 
Overtones
    The simplest possible sounds discussed above (test-tones, some flute notes, etc.) exist singularly as very gradual and regular increases and decreases of energy, whether the energy be in the form of changing air pressure or changing voltage.  We hear these very pure sound waves as fundamental sounds, the frequency of which is very easy to focus on.  Because of this singularity, these pure tones can even be said to have no timbre, or no color, or to be wholly lacking in timbre and color.  A graphic representation of such a pure tone, its wave-form, would be in the shape of a sine-wave.
    A more complex sound, on the other hand, like a bass drum or a spoken consonant, consists of many simultaneously heard sound waves, each of which is in fact a sine-wave with different frequency.  These individual components of a complex sound are called overtones or partials, and when piled up on top of one another they form complex wave-forms.  The overtone sometimes obscure the frequency of the lowest, or fundamental sine-wave (the frequency which the overtones are "over"), and sometimes they reinforce it.
 
The Harmonic Series
    When a complex sound has overtones that strongly reinforce the fundamental sine-wave over which they are heard, they constitute a harmonic series  of overtones.  The reason they reinforce the fundamental are related to the very specific and mathematically precise relationships that exist between their frequencies and amplitudes:  their frequencies and amplitudes  relate to the frequency and amplitude of the fundamental in whole-number ratios.
    The frequency relationships are the most consistent  and easy to understand.  The first overtone (also called the second partial) is twice the frequency of the fundamental, the second overtone (third partial) is three times the frequency of the fundamental, the third overtone (fourth partial) is four times the frequency of the fundamental, and son on.  That means that the frequency ratios between the partials consist of ratios such as 2:1,  3:2,  4:2,  3:1,  etc. -> all whole number ratios.
    Unlike frequency, the amplitudes of the overtones in a harmonic series differ from wave-form to wave-form.  In some instances the second partial may be one-half of the amplitude of the fundamental, the third partial one-third the amplitude of the fundamental, the fourth partial one-fourth the amplitude of the fundamental, and son on.  For other wave-forms, the second partial may be one-fourth, the third partial one-ninth the amplitude of the fundamental, the fourth partial one-sixteenth the amplitude of the fundamental, the fifth partial one-twenty-fifth the amplitude of the fundamental, and so on.  Different wave-forms contain differing amounts of energy, or amplitude, in the upper partials, but these amplitudes are always related to each other in whole-number ratios, in inverse proportion to the amplitude of the fundamental:  1,2,  1:3,  1:4 in the first example;  1:4,  1:9,  1:16,  1:25  in the second.
    Certain types of physical objects produce this harmonic series of overtones when they are "excited" or disturbed, and air pressure around them is subsequently disturbed.  Such objects include strings and columns of air. Most of the sounds traditionally used in music are produced by instruments that use strings or columns of air, and these instruments therefore generate a harmonic series of overtones when played.
    Certain other types of physical objects do not produce a harmonic series of overtones when they are "excited."  This group includes struck membranes and struck wooden or metal plates.  Many of these sounds also find their way into traditional music, usually by way of instruments in the percussion family.  These instruments do produce overtones, but they are usually non-harmonic overtones. Electronic sound synthesis, of both the digital and analog variety, produces opportunities for musicians and sound designers to directly manipulate the frequencies and amplitudes of overtones.  This makes it possible to emulate the sounds of acoustic instruments by duplicating the frequency and amplitude ratios in nature, but also to create sounds that could not exist in the natural world of acoustics:  sounds whose frequency and amplitude relationships have their genesis in the laws of mathematics or in the imaginations if musicians or both.