Figure C-1. Four key notes for the song “How Dry I Am” used by Bernstein to illustrate the infinite variety of music.
The tonality of Western music arises from its exclusive use of twelve pitch-classes. In spite of being constrained by this sparse set of basic musical elements, composer Leonard Bernstein argues that Western music is infinite in its variety (Bernstein, 1966). He observes that if one considers the twelve pitch-classes in a single range, and computes their possible melodic combinations, the result is 1,302,061,344. If one extends this approach to consider both melodic and harmonic combinations of these elements the result is 127 googols, where a googol is a digit followed by 100 zeroes. “The realm of music is an infinity into which the composer’s mind goes wandering” (Bernstein, 1966, p. 34).
Bernstein (1966) explores this theme with a particular example, the four note melody that starts the song “How Dry I Am”. These four notes are provided in Figure C-1. He notes the importance of this pattern, and the variations of musical effects that it can produce, by noting its presence in a huge range of compositions that begins with a French folk song and ends with the final movement of Shostakovich’s Fifth Symphony. Bernstein ends his discussion by proposing a variation of Figure C-1 that depicts “a motto of man’s infinite variety” (Bernstein, 1966, pp. 46-47).
Early in this book we saw another example of variety from small numbers of elements. The musical signal composed by John Williams for Steven Spielberg’s 1977 movie Close Encounters of the Third Kind (Figure O-1) was selected from a sample of 350 five-note compositions created by Williams. Had Williams composed all possible five-note melodies the movie’s signal would have been selected from among about 134,000 possibilities. The calculation of possibilities is conservative because it fails to take into account rhythmic variations; not all of the notes in Williams’ signal have the same duration.
The infinite possibilities of Western tonal music are reflected in music’s constant evolution. American composer Aaron Copland wrote Our New Music (Copland, 1941) to explain the circumstances that had led to modern classical music. His goal was to alleviate his readers’ bewilderment with modern music. “Being unaware of the separate steps that brought about these revolutionary changes, they are naturally at a loss to understand the end result” (Copland, 1941, p. v). He traced modern music’s development as a move away from a century of Germanic musical influences. This move begins with explorations of folk music in the late 19th century, proceeds through explorations of new views of harmony, rhythm, and tonality. Copland argues that it ends by coming full circle, in Stravinsky’s compositions of the late 1920s and early 1930s, and returning to melodic forms from the 18th century.
The infinite possibilities of Western tonal music make it nearly impossible to predict its future too. In the early 1940s one could analyze existing modern music and describe a neoclassicism that had roots in the 18th century (Copland, 1941). However, Copland’s analysis of modern American music does not even hint of the radical developments that would flourish there beginning in the 1960s with, for example, the invention of mimimalism (Glass, 1987; Griffiths, 1994, 1995; Hartog, 1957; Nyman, 1999; Pleasants, 1955; Potter, 2000; Reich, 1974, 2002).
Western tonal music has infinite variety and unpredictability. However, it is neither accidental nor unsystematic. When a composer’s mind goes wandering into the infinite musical realm, it does not randomly move from one musical entity to another. Its search through this realm is guided by new ideas concerning musical structure – new notions of melody, harmony, rhythm and the like – in short, new music theory. Rather than being “dusty abstract rules of form and harmonic structure” (Bernstein, 1966, p. 24), music theory itself seems both vast and dynamic. When violent upheaval is heard in classical music, its root cause must be changing conceptions of music’s structure.
Does musical theory itself exhibit infinite variety? I have no idea. However, historical examinations reveal enormous changes in basic ideas, such as whether different inversions of a chord are the same chord, or what is the root note of a major or minor triad (Damschroder, 2008; Rehding, 2003; Riemann, 1895). We saw in Chapter 1 that the psychophysical study of music that began in the late 19th century faced the tension between the physics of sound and individual differences in aesthetics that permitted just intonation to be replaced by equal temperament (Hui, 2013).
As well, evolving notions of consonance have permitted new musical intervals to become accepted in music. The dissonance of the flattened seventh note led Helmholtz to reject its use in his advice to composers (Helmholtz & Ellis, 1863/1954); now it is definitive to the blues and plays a central role in Gershwin’s classic Rhapsody In Blue (Adams, 2008). Later, seasoned jazz musicians who were completely comfortable with the flattened seventh were jarred and puzzled by the flattened fifth interval when it was introduced to jazz via bebop (Kelley, 2009).
Clearly there is no single, unified theory of music. A multitude of music theories have existed; many different theories can exist at the same time; new theories can be invented or discovered. One approach to composing innovative music involves taking a new musical theory an examining the compositions that it can pick out of the infinite realm of music. Where might one find a new musical theory to exploit in this fashion?
There are many, many possible answers to this question. One reading of the current book suggests one: train an artificial neural network to map some musical inputs to some other musical outputs. The kind of training that we have seen in preceding chapters informs networks about their progress, but does not inform them how to construct the mapping. As a result, these networks can discover new musical regularities or ideas for performing the mapping. We have seen many instances of this in the current book, even when networks are trained on basic, traditional musical tasks.
Crucially, for a network to deliver a new musical theory its internal structure must be explored. Artificial neural networks can only inform the study of music if we first reject the romanticism that characterizes much of connectionist cognitive science.