Sunday, April 05, 2015

Multiple Realizations and Music

As described in this previous post, the  text below is a draft of one of several "interludes" to be included in a book that I am working on concerned with music and artificial neural networks.

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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