Complexity

From Number Theories to Information: Theoretic Estimations of Complexity

During the Renaissance period and up until the 18th century, complexity in music was associated with number theories originating from the Pythagorean tradition (harmonic ratios that vary from simple to complex).

In the mid-20th century, information theory provided the formal means of defining complexity in music, wherein the amount of information carried by the discrete elements of the music is measured in bits. This measure originated from Claude Shannon's concept of entropy that determines the uncertainty of a signal. Another way to define complexity is to describe the absolute amount of information that is required to transmit the object, known as Kolmogorov complexity.

From the 1970s to the 1990s, one typically applied the information-theoretic calculation of complexity to melodies (pitches, intervals, or durations) and these predictions have been broadly similar to complexity ratings provided by listeners. These studies have also found support for the fractal-like qualities (1/f) of music, in which the frequency spectrum of discrete items is optimally complex for a particular region of the slope (between 1 and 2 in speech and music) describing the spectrum.

The problem with information-theoretic notions is that the probabilities of notes and durations are far from uniform because statistical regularities for each property exist in music. For this reason, after the 1980s in Western music, estimates of complexity took into account the probabilities of tones and intervals. For example, Dean Keith Simonton calculated the transition probabilities of a glossary of over 15,000 classical music themes were complied and this information was used to define the originality (close variant of complexity) of the individual themes. Interestingly, this measure of complexity was found to be associated with biographical and historical trends.

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