The best known audio paradox is known as Shepard's paradox. It is the audio equivalent of the endless staircase illusion made famous by M. C. Escher. In this audio paradox a series of tones can be made to sound as if they are ascending or descending in pitch forever.
The following graph of Shepard's paradox shows frequency versus time. You can plainly see that the pitch appears to be increasing.
Spectrum of Shepard's ascending paradox
Note that if the same graph is looped you can see that the finishing pitch is the same as the starting pitch. It may be obvious to the eye, but the ear cannot perceive where the sample starts and finishes. If you listen to this sample the tones will appear to increase in pitch even if the sample loops back to the beginning and starts over.
The same spectrum looped three times
Two other audio paradoxes are included. One paradox demonstrates that it is possible to make the ear believe a series of tones are decreasing in pitch when they are actually increasing in pitch. The last paradox demonstrates that it is possible to make the ear believe that a tempo is getting quicker when it actually remains constant. The audio samples are provided in MP3 audio format.
This is a recording of Shepard's paradox synthesized by Jean-Claude Risset. Pairs of chords sound as if they are advancing up the scale, but in fact the starting pair of chords is the same as the finishing pair. If you loop this sample seamlessly then it should be impossible to tell where the sample begins and ends.
This is a recording of a paradox where bells sound as if they are falling through space. As they fall their pitch seems to be getting lower, but in fact the pitch gets higher. If you loop this sample you will clearly see the pitch jump back down when the sample repeats. This reveals that the start pitch is obviously much lower than the finishing pitch.
This recording is subtle. A drum beat sounds as if it is quickening in tempo, but the starting tempo is the same as this finishing tempo.
"Paradoxical Sounds" by Jean-Claude Risset; from "Current directions in computer music research"; edited by Max V. Mathews and John R. Pierce; MIT Press; 1989; ISBN-0-262-13241-9; Chapter 11; pages 149-158.
"Paradoxes of musical pitch" by Diana Deutsch; Scientific American; August 1992; pages 88-95.
Thierry Rochebois' paradoxical synthesizer: http://www.solorb.com/gfc/anhevn/paradox/paradox.html. This is Thierry Rochebois' Home Page. He has a link to a description of a fairly simple hardware implementation of a 12 key paradoxical synthesizer that he designed.
original edits (prewiki)
Sunday, December 3, 1995 Sunday, November 15, 1998