t h o s e   t w i s t e d   b e a t s

f i r s t   p r i z e  -  g a l l e r y   o f   a c o u s t i c s
138th meeting of the Acoustical Society of America, Columbus, Ohio, November 1999


The Gallery of Acoustics is an annual exhibition/competition taking place at the meeting of the Acoustical Society of America. It is sponsored by the
Technical Committee on Signal Processing in Acoustics.
The Gallery consists of posters, videos, and audio clips generated by acoustic processes or resulting from signal processing of acoustic data.

A panel of referees judges the entries on the basis of aesthetic/artistic appeal, ability to convey and exchange information, and acoustical relevance and originality. A cash prize of $350 is awarded to the winning entry.  The top three entries, with the authors' permission and cooperation, are posted on the Gallery's web site.

The winning entry for 1999 (138th meeting of the Acoustical Society of America, Columbus, Ohio, November 1 - 5, 1999) was a poster depicting the propagation of sound-interference products by Pantelis Vassilakis (Musical Cognition and Acoustics Laboratory, Ethnomusicology, UCLA), titled "Those twisted beats".

For more information on the past award winners and submission instructions visit the Gallery of Acoustics website.

Those Twisted Beats

Two-dimensional signals, oscillation energy, and negative amplitudes

2D modulated signals are seen as deriving from the linear addition of 2D sines.  In the simplest case this would be a signal with fluctuating amplitude, resulting from the linear addition of two sines with slightly different frequencies.  For any such addition to be meaningful, all variables represented by each signal must share the same units. For 2D signals, these variables are:

  1. time (usually on the x axis),
  2. oscillating displacement, voltage, pressure, etc.; (usually on the y axis), and
  3. some representation of the oscillation energy, (i.e. the area outlined by the signal and the time axis).

In all cases, only two of the variables have consistent units within a single 2D coordinate system, with relative energy values being usually misrepresented.  Additionally, the mathematical expression describing a beating tone-pair indicates that the modulating amplitude will alternate periodically between positive and negative values, an alteration that the 2D signals do not represent.


Modulations as twists

The twisted-signal hypothesis is introduced as a reply to the above points and is illustrated below for the case of a two-component complex signal with f1 = 10Hz; f2 = 12Hz; A1 = A2 = 1; t = 1sec; , :

More Images In the figure (left):

a) fmodulation = 1Hz is represented by the number of twists around the x-axis,

b) the degree of the modulation is represented by the radius of the twist, and

c) fbeat = 2fmodulation = 2 is represented by the number of amplitude fluctuations formed when the 3-dimensional twisted signal is projected on two dimensions (projection angle represents phase relationship of the original sines).

This collapse from 3 to 2 dimensions represents the fact that, while 
mod represents the oscillation of Amod
beat represents the oscillation of (Amod)2.