A sustained discussion thread about the musical terms tremolo and vibrato developed in MMDigest in October 1999 [ ref. http://www.mmdigest.com/Archives/KWIC/T/tremolo.html ].   These terms allude to the quavering sound produced by a skilled violinist or singer, and also to the "effects" devices which can be applied to modify the sound of a pipe organ or modern electronic organ or synthesizer.  In the case when the underlying tone is a sinusoidal waveform, or sinewave, the result of applying tremolo or vibrato modulation is easily described with simple equations and demonstrated with common laboratory equipment such as the audio oscillator and oscilloscope.

Since the terms tremolo and vibrato are not rigorously defined by the musical world, engineers call the effects AM and FM, meaning amplitude modulation and frequency modulation.  The underlying unvarying tone is called the carrier frequency, and how fast the amplitude/frequency is undulated is the modulation frequency.  These precise terms (and the descriptive equations) come from telephone and radio technology.

A musical instrument that features amplitude modulation (AM) is the Deagan Vibraphone, a sort of marimba with motor-driven rotating shutters that vary the sound amplitude.  The steel guitar (or Hawaiian guitar) features frequency modulation (FM) as the player wiggles the heavy steel bar back and forth to vary the speaking length of the vibrating string.

To illustrate what different modulation forms sound like I have fabricated four synthetic auditory demonstration files. They all share these common parameters:

  duration:  1 second
  carrier frequency: A=440 Hz
  modulating frequency:  6 Hz   (the tremolo or vibrato rate)
  sampling for WAV file: 11 kHz (file size 22 kilobytes)

The basic signal, or carrier, is a pure sinusoid, so that the samples are devoid of any timbral character that would associate them with any real instrument. Instead, they are intended to point at the specific character evoked by the tremolo/vibrato or AM/FM in itself.  The figures below show the power spectrum (energy versus frequency) and the waveform envelope as would be seen on an oscilloscope (amplitude versus time).

Click on the underlined WAV-file name or the picture and the browser should launch a system program at your computer which will play the 1-second sound via the computers sound card and loudspeaker.  Then use the "Back" button to return to this text page.
 

SIN.wav (22kb) is the basic signal without modulation: no AM, no FM.  Listening to a single sinewave tone is very boring, but sonorous chords can be pleasing, especially when the tones are related by simple integral fractions (Just Temperament).
FM.wav (22kb) has a pure frequency modulation with an extent (modulation swing) of +/-60 cents, roughly 2/3 of a semitone, meaning that the frequency undulates between 424 and 456 Hz. Such values would be typical for classical Western style singing. The spectrum comes out fairly complex as a cluster of partial tones, centered at the nominal 440 Hz and spaced apart with the modulation rate. The amplitude is constant. In a practical situation this kind of signal will also acquire some amplitude modulation due to resonance effects in the instrument and the surrounding room.
AM.wav (22kb) has a pure 100% amplitude modulation. The spectrum has three partial tones, in radio terminology called the carrier and its two sidebands.  The three frequencies shown are 434, 440 and 446 Hz.
AMSC.wav (22kb) is a 6 Hz beat between two tones, 437 and 443 Hz. This alludes to the technique used in some organ stops (like Voix Celeste) where the tone is given by two slightly differently tuned pipes. In radio this corresponds to AM with suppressed carrier (AMSC).

Equations:  Amplitude of the carrier is normalized to +/-1.0 volt peak - with the AM variants the momentary amplitude varies up to twice this value.    t is seconds, pi =3.1416, f0=440 Hz, fm= 6 Hz.    SIN.wav:       A(t)=sin(2*pi*f0*t)    FM.wav:        A(t)=sin(2*pi*f*t) , where f=f0*(1+k*sin(2*pi*fm*t)) and k=0.036 (modulation index for +/-60 cents deviation)    AM.wav:       A(t)=(1+sin(2*pi*fm*t))*sin(2*pi*f0*t)    AMSC.wav:  A(t)=sin(2*pi*f1*t)+sin(2*pi*f2*t) , where f1=437Hz and f2=443 Hz