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I like the application to the study of equal tempered musical scales, which, for example, leads to an the explanation of why (and in what sense) the standard scale of 12 semitones is optimal. A web search for "music, continued fractions" reveals lots of references. This one has some ideas that were new to me: http://doctroid.wordpress.com/2009/04/13/generalized-continued-fractions-and-equal-temperament