Figure I-5. Two very different theories
can be used to define a major scale.
One, on the left, involves identifying patterns of musical intervals
between adjacent scale pitches. Another,
on the right, involves measuring tritone balance – a property that major scales
have little of.
When the cognitive revolution occurred,
researchers used the digital computer as a metaphor to provide insight into the
workings of the mind. Their working
hypothesis was that thinking involved the same kind of operations involved when
computers performed computations. As a
result, one could attempt to explain cognition in exactly the same way that one
explains how a computer works.
How does one explain a computer? From the perspective of a science grounded in
physical, causal laws one might expect to explain computing by describing the
workings of the various physical or electronic components from which a computer
is constructed.
However, computer explanations are not
merely physical, but are also more abstract and functional (Cummins, 1983). That is, explaining a
computer does not focus exclusively on the stuff it is made of. Instead, it focuses on what this stuff does.
For instance, one might detail the algorithm
or program that is being carried out by a computer. This involves describing the function of
various processing operations (first the program reads in some data, then it
transforms the data according to this formula, and finally it prints the
results). This sort of account rarely
involves explaining how the various operations of a computer are brought to
life by the intricacies of its hardware.
Similarly, one might provide a very general
account of the information processing problem that a computer solves when it
runs a particular program. For instance,
perhaps the program’s purpose is determining the minimum value of some
equation. Again, this kind of account
does not appeal to hardware.
David Marr is best known for his convincing
arguments that a complete account of an information processing system like a computer
requires three different levels of analysis, each of which answers different
kinds of questions using distinct vocabularies and methods (Marr, 1982). At the computational level,
mathematical proofs are used to answer the question “What information processing
problem is the system solving?” At the
algorithmic level, experiments are conducted to answer the question “What information
processing steps are being used to solve the information processing
problem?” At the implementational level,
physical properties are examined to answer the question “What physical
properties are responsible for bringing particular information processing steps
to life in a specific information processing device?”
Central to Marr’s (1982) theory is that a
complete explanation of an information processor requires examining at the
computational, algorithmic, and the implementational levels of analysis. Furthermore, systematic links between each
level of analysis must also be established.
Establishing the links between levels is
what makes explaining information processing both challenging and
exciting. Imagine that it has been
established at the computational level that a particular system is solving
Problem X. It turns out that there are
many different algorithms that can be used to solve this problem. In other words, there is a many-to-one
relationship from the algorithmic level to the computational level. Similarly, any one of these algorithms can be
brought to life on computing machines based on very different physical
principles (Hillis, 1998). There is a many-to-one relationship
from the implementational level to the algorithmic level.
The phrase ‘multiple realizations’ is often
used when many-to-one relationships are at play. That is, one computation can be realized by
multiple algorithms, while one algorithm can be realized by multiple physical
systems.
One exciting aspect of interpreting the internal
structure of a musical network is that one might be confronted with multiple
realizations. That is, while one might
expect that identifying some musical property involves one procedure,
interpreting a network can reveal a completely different approach.
The scale mode network provides an example
of this. Its task is to turn its output
unit on when it is presented a major scale.
Traditional music theory dictates that a major scale is defined by a
particular pattern of musical intervals between adjacent pitches in a scale, as
illustrated on the right side of Figure I-5.
However, the interpretation of the multilayer perceptron for this task revealed
that it uses a completely different musical property, tritone balance, which major
scales (unlike harmonic minor scales) tend not to exhibit. This alternative theory is depicted on the
left side of Figure I-5.
Multiple realizations in music should not
be surprising. The history of music, and
in particular the history of musical analysis, reveals that musical theory is
constantly evolving. As a result, one
can find many different theoretical accounts of the same musical phenomenon.
For instance, in modern music theory it is
typical to view that different inversions of a chord are all instances of the
same chord. However, prior to 1771 music
theory held that different inversions of a chord were all different chords (Damschroder, 2008). Similarly, in modern music
theory it is accepted that the A major triad (built from the pitch-classes A,
C#, E) and the A minor triad (built from the pitch-classes A, C, E) both have
the same tonic – the pitch-class A. However,
according to the music theory of Hugo Riemann, the root of the A major triad is
A, but the root of the A minor triad is E (Rehding, 2003; Riemann, 1895).
There are two key points to highlight
here. The first is that current theories
of Western tonal music are not the only ones possible; alternative theories
exist, and many have been proposed at various times in history. The second is that artificial neural networks
are not constrained by current theories of music, and therefore may be quite
capable of discovering viable and interesting alternatives. In short, musical multiple realizations
exist, and artificial neural networks may be able to reveal them.
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