Shepard's tone and other auditory illusions

We are all familiar with optical illusions. They can be found in the visual arts, movies, video games, royalty free music , and even architecture. At the same time, there are illusions aimed at deceiving the other senses that are much less well known. In this article we would like to talk about examples of auditory illusions.

One of the most frequently cited examples of auditory illusions is Shepard's tone. It is named after the American psychologist Roger Shepard, who created it in 1964 and was engaged in research in the field of cognitive science.

This illusion is an infinitely ascending sequence of tones. It is created by superimposing two sinusoidal waves that are separated by an octave, that is, they differ from each other in frequency by a factor of two. To achieve the effect, it is necessary that the volume of the sounds of the upper octave gradually decreases as you move up, and the volume of the lower octave increases. The lower octave thus imperceptibly replaces the upper octave, after which the upper octave moves downward and continues its upward movement, slowly increasing in volume. With constant repetition, this sequence is perceived by the listener as an endless rise in pitch. A similar principle can be used to create the illusion of an infinitely descending tone.

The French composer Jean-Claude Risset, known for his work in acoustics and computer music, later created a version of the Shepard fret with a continuous, rather than stepped change in pitch as in the original, called the Shepard-Risset glissando.
https://youtu.be/iDSEB2HF11M

He is credited with another illusion, called Rhythm Risse, which gives the listener a feeling of endless acceleration or deceleration of the tempo. Just like Shepard's tone, this illusion uses the technique of substituting two sequences of sounds for each other in turn, but with twice the tempo.
https://www.youtube.com/watch?v=oQf_tS5WAP4
The practical application of Shepard's tone can be found in several films by Christopher Nolan, who can be considered a fan of this illusion. It was used in the music for The Prestige and Dunkirk and in the engine sound for Batman's motorcycle in the Dark Knight trilogy.
https://www.youtube.com/watch?v=LVWTQcZbLgY&feature=emb_imp_woyt
Other well-known examples of the use of the effect include the song Echoes by the progressive rock band Pink Floyd, and the first and last songs of the album A Day at the Races by Queen. In the now classic game Super Mario 64, Shepard's tone accompanies the ascent of the endless staircase blocking the way to the finale.

Many auditory illusions are by British-American psychologist Diane Deutch, who conducted research in the psychology of music. One of the illusions she discovered, based again on Shepard's tone, is called the Triton Paradox. It uses the alternation of two Shepard tones that produce notes that differ in pitch by the musical tritone interval. Although the pitch of the tones remains unchanged, listeners perceive their alternation as a downward or upward movement, and the same direction is maintained on subsequent listens.

Of great practical value is the illusion of a missing or phantom fundamental frequency, which we encounter regularly in everyday life. This illusion allows us to hear the low-frequency component of a musical sound even when it is not actually present, such as when listening to music through small speakers or in-ear headphones. It works by having the brain build up the missing fundamental frequency on the basis of its higher sounds. Sound processing effects are based on this illusion, and are designed to adapt the recording for playback on a wide range of devices.
https://www.youtube.com/watch?v=t-iWKvh6Fbw&feature=emb_imp_woyt