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The troubles encountered in this paper represent those which potentially affect any user of neural networks: underfitting, data starvation, and patternless data. Once we progressed to our MLP of choice, though, most interesting is the comparison it, the theorists' analysis process in varying degrees of depth, and the complexity of the music itself.

For example, MLPFreqFinder gets no answer wrong that our 1st-order human simulator does not also. Despite using the relatively abstract concept of scales, ScaleFinder cannot make fine distinctions between keys that end up with equal point values. In this event of a tie -- certainly common in only the first few measures of a piece -- ScaleFinder always chooses the first scale to fit the notes, which is rarely the best. Meanwhile the MLP, while not always correct, at least chooses a good approximation*; when the key center is "obvious" enough for even lowly ScaleFinder to succeed, it never falls behind.

The history of the sonata form up to Beethoven's time was already introduced, the core observation being the increasing deviations from the norm he made during his career. We see this most clearly in the final example, where TheoryFinder at last makes an incorrect (although close) prediction. Fascinatingly, the neural network makes the same mistake as the well-trained "human", bypassing the "obvious" answer that Scale finds, although it knows nothing about the concept of intervals a priori.

Certainly we could have constructed a feature set that preprocessed the data enough that the MLP would surpass any of the algorithms so far (which remain quite simple, all things considered), but the aim of the project was to determine how much of the legwork the AI portion of the code could accomplish.** While I won't continue to laud an incorrect answer, I believe that in the correlation of errors between the final MLP and the TheoryFinder a healthy middle ground has been struck. The feature selection step added between Circle and Freq relied on no abstract concepts of music theory, yet provided a function that, while far from perfect, held unexpected insight into the problem space.


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*There is one anomaly when measuring "closeness" in this way: in the first example of the second "test" set (the one labeled "training"), FreqFinder is way off. The most common transformed values are {0, 11, 10}, but the MLP has no way of recognizing the mod-12 arithmetic that makes these chords actual neighbors.

**One extension suggested by a colleague was to add the ability to import MIDI files, which are score-like data structures found throughout the Net's classical music fan pages. They vary drastically in content and quality, making patterns much more difficult to see (even for a human) than my carefully inputted Beethoven, but the ability to suddenly feed thousands of data points into the algorithms might produce interesting results.