Strange Loops and Circular Tones

Abstract

Chapter 4 explores a class of musical illusions and paradoxes that involve the circular dimension of pitch. Pitch can be described in terms of two dimensions. The first is called pitch height, which can be experienced by sweeping one’s hand from left to right up a piano keyboard. The second is a circular dimension known as pitch class, which defines the position of a tone within the octave. Circularity effects in music are analogous to many of the visual works of M. C. Escher, and have been employed in music for hundreds of years. However, with the advent of computer music, striking pitch circularities became possible. The circular scales invented by Roger Shepard (based on Shepard tones) and circular glides invented by Jean-Claude Risset are explored. These remarkable illusions of ever-increasing (or ever-decreasing) pitch are presented as sound examples. They have powerful emotional effects, and their influence in musical compositions, such as the soundtracks and sound design of The Dark Knight and Dunkirk, is described. A new way of producing pitch circularity, which was invented by the author, is also discussed. This new algorithm can be used with natural instrument sounds, and so opens the door to new compositional opportunities.