The Myth:
Mozart used the golden ratio to determine the relative lengths of the two thematic sections that make up a piano sonata.
The Reality:
While it is true that Mozart was apparently a fan of mathematics as recreation and may have been familiar with the golden ratio, there is no historical evidence that he used it as a compositional tool. The two components of a movement in a sonata measured are the exposition and the development and recapitulation. The former introduces a theme, and the later develops and revisits the theme and is the longer of the two. Linear regression on the pairs of lengths associated with each of these two elements (pairs of the form (long,short)) yields the line y=1.360+0.6260x, with impressively high r2 value of 0.938. The slope is within 2% of the golden ratio, and the vertical shift is only a single measure. At first blush this is pretty convincing evidence until one considers that the movement of a piano sonata imposes some natural limitations on the form. In particular, the exposition needs to be long enough to fully introduce a theme, and the development and recapitulation needs to be longer than the exposition. It isn't too unreasonable then to expect our values for the length of the exposition to lie somewhere between a fourth to one half of the length of the two measures together. As seen in the discussion of the golden ratio our expected value would be about 0.6219, which is within about 0.6% of the golden ratio. A careful consideration of a histogram of the ratio m/M (length of exposition divided by length of development and recapitulation) shows significant variation from the reciprocal of the golden ratio in choice of the division and lends credence to the idea that this is likely as not an artifact of the structure of the sonata as opposed to some sort of conscious choice on the part of Mozart, and certainly not a consistent choice.
References:
[Put95] John F. Putz, The golden section and the piano sonatas of mozart, Mathematics Magazine 68 (1995), no. 4, 275–282.
Index Submit Suggestions Math Myths Home Sunday 05/21/2006 08:03:53 PM