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 vocalist, 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, scientists call the effects FM and AM, meaning amplitude modulation and frequency modulation.  The underlying unvarying tone is called the carrier frequency, and the modifying frequency 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 three 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)

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 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.
 

FM.wav 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 has a pure 100% amplitude modulation. The spectrum has three partial tones, in radio terminology called the carrier and its two sidebands.


 
 

AMSC.wav is a beat between two tones, 6 Hz apart. 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), although with only 3 Hz modulation rate.


 

Equations:

  t is seconds, pi =3.1416, f₀=440 Hz, f_m= 6 Hz

  AM.wav:   A(t) = sin(2*pi*f_m*t)* sin(2*pi*f₀*t), where  f_m= 6 Hz

  FM.wav:   A(t) = sin(2*pi*f*t) , where f = (f₀+k*sin(2*pi*f_m*t)) and k=0.036 (modulation index)

  AMSC.wav: A(t) = sin(2*pi*f₁*t) + sin(2*pi*f₂*t) , where f₁=437Hz and f₂=443 Hz