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

theorize about mental processing before understanding the organization to which
the processing leads. (Lerdahl & Jackendoff, 1983, pp. 3–4)
One consequence of ignoring mental processing is that the generative theory of
tonal music is generally not applied to psychologically plausible representations.
For instance, in spite of being a theory about an experienced listener, the various
incarnations of the theory are not applied to auditory stimuli, but are instead
applied to musical scores (Hamanaka, Hirata, & Tojo, 2006; Lerdahl, 2001; Lerdahl
& Jackendoff, 1983).
Of course, this is not a principled limitation of the generative theory of tonal
music. This theory has inspired researchers to develop models that have a more
algorithmic emphasis and operate on representations that take steps towards psychological
plausibility (Temperley, 2001).
Temperley’s (2001) theory can be described as a variant of the original generative
theory of tonal music (Lerdahl & Jackendoff, 1983). One key difference between
the two is the input representation. Temperley employs a piano-roll representation,
which can be described as being a two-dimensional graph of musical input. The
vertical axis, or pitch axis, is a discrete representation of different musical notes.
That is, each row in the vertical axis can be associated with its own piano key. The
horizontal axis is a continuous representation of time. When a note is played, a horizontal
line is drawn on the piano-roll representation; the height of the line indicates
which note is being played. The beginning of the line represents the note’s onset,
the length of the line represents the note’s duration, and the end of the line represents
the note’s offset. Temperley assumes the psychological reality of the piano-roll
representation, although he admits that the evidence for this strong assumption is
inconclusive.
Temperley’s (2001) model applies a variety of preference rules to accomplish
the hierarchical organization of different aspects of a musical piece presented as a
piano-roll representation. He provides different preference rule systems for assigning
metrical structure, melodic phrase structure, contrapuntal structure, pitch class
representation, harmonic structure, and key structure. In many respects, these
preference rule systems represent an evolution of the well-formedness and preference
rules in Lerdahl and Jackendoff ’s (1983) theory.
For example, one of Temperley’s (2001) preference rule systems assigns metrical
structure (i.e., hierarchically organized sets of beats) to a musical piece. Lerdahl
and Jackendoff (1983) accomplished this by applying four different well-formedness
rules and ten different preference rules. Temperley accepts two of Lerdahl and
Jackendoff’s well-formedness rules for metre (albeit in revised form, as preference
rules) and rejects two others because they do not apply to the more realistic representation
that Temperley adopts. Temperley adds three other preference rules.
This system of five preference rules derives metric structure to a high degree of
278 Chapter 6