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beyond this the mathematicians' contributions to art will have little value. Its direct benefits will be found to be negative rather than positive. While it may prevent the perpetration of jarring discords, it will hardly make possible the creation of masterpieces; for here again the personal equation comes into play. Lacking the note of personality, no real art is possible. A musician of my acquaintance, having discovered that when the law of mathematics was applied to a sonata by Beethoven, the theme worked out faultlessly to a seemingly inevitable conclusion, decided that the process could be reversed, and that a given theme, if correctly figured out, would undoubtedly produce a musical number of faultless beauty. He put his theory into practice and made a sonata according to this system. His production was
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