Mind, Body, World- Foundations of Cognitive Science, 2013a

of staff paper. Since the time of Pythagoras, scholars have understood that music
reflects regularities that are intrinsically mathematical (Ferguson, 2008). There is
an extensive literature on the mathematical nature of music (Assayag et al., 2002;
Benson, 2007; Harkleroad, 2006). For instance, different approaches to tuning
instruments reflect the extent to which tunings are deemed mathematically sensible
(Isacoff, 2001).
To elaborate, some pairs of tones played simultaneously are pleasing to the ear,
such as a pair of notes that are a perfect fifth apart (see Figure 4-10)—they are consonant—while
other combinations are not (Krumhansl, 1990). The consonance of
notes can be explained by the physics of sound waves (Helmholtz & Ellis, 1954). Such
physical relationships are ultimately mathematical, because they concern ratios of
frequencies of sine waves. Consonant tone pairs have frequency ratios of 2:1 (octave),
3:2 (perfect fifth), and 4:3 (perfect fourth). The most dissonant pair of tones, the tritone
(an augmented fourth) is defined by a ratio that includes an irrational number
(√2:1), a fact that was probably known to the Pythagoreans.
The formal nature of music extends far beyond the physics of sound. There
are formal descriptions of musical elements, and of entire musical compositions,
that are analogous to the syntax of linguistics (Chomsky, 1965). Some researchers
have employed generative grammars to express these regularities (Lerdahl
& Jackendoff, 1983; Steedman, 1984).
For instance, Lerdahl and Jackendoff (1983) argued that listeners impose a
hierarchical structure on music, organizing “the sound signals into units such as
motives, themes, phrases, periods, theme-groups, sections and the piece itself ”
(p. 12). They defined a set of well-formedness rules, which are directly analogous to
generative rules in linguistics, to define how this musical organization proceeds and
to rule out impossible organizations.
That classical music is expected to have a hierarchically organized, well-formed
structure is a long-established view amongst scholars who do not use generative
grammars to capture such regularities. Composer Aaron Copland (1939, p. 113)
argued that a composition’s structure is “one of the principal things to listen for”
because it is “the planned design that binds an entire composition together.”
One important musical structure is the sonata-allegro form (Copland, 1939),
which is a hierarchical organization of musical themes or ideas. At the top level of this
hierarchy are three different components that are presented in sequence: an initial
exposition of melodic structures called musical themes, followed by the free development
of these themes, and finishing with their recapitulation. Each of these segments
is itself composed of three sub-segments, which are again presented in sequence.
This structure is formal in the sense that the relationship between different themes
presented in different sub-segments is defined in terms of their key signatures.
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