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.