Beauty and structure? See, e.g., Douglas Hofstadter's discussion of crab canons and the Ricercar fugue of Das Musikalische Opfer in his book Gödel, Escher, Bach: An Eternal Golden Braid. Or, in line with my interest in continued fractions, one could consider overtones of a vibrating string or open pipe. These are integer multiples of some fundamental frequency. There is a natural equivalence relation on frequencies which identifies octaves as the same "note". This means that two frequencies give the same note when the base 2 logarithm of their ratio is an integer. In this way, the first new note we get from considering overtones corresponds to three times the fundamental frequency, and there is exactly one frequency (a "perfect fifth") within an octave above the fundamental frequency which gives that new note. The distance of a perfect fifth is therefore log2(3/2), but this is an irrational number, so stacking perfect fifths will never lead to an equal division of the octave. The "best" rational approximations of an irrational number, however, come from convergents of its continued fraction expansion:
Take note of the denominators of the rational approximations; the 5 + 7 = 12 corresponds to the well-known chromatic twelve-tone equal temperament scale, as seen in an octave of a standard keyboard.
Grammy winning composer Wendy Carlos discovered an equal temperament tuning consisting of twenty steps. Instead of using the octave as her base interval, she used the perfect fifth and divided it according to divisions which closely matched a stacking of a major thirds (i.e., a frequency ratios of 5/4).
Take note of the denominators of the rational approximations; in analogy with the 12 tone scale above, the 9 + 11 = 20 corresponds to this tuning discovered by Carlos, as seen in a perfect fifth of a "Carlos keyboard".
For more info, please see the slides and associated files for my talk The Role of Continued Fractions in Rediscovering a Xenharmonic Tuning.