In 1995, Alma College mathematician John F. Putz counted the measures in Mozart’s piano sonatas, comparing the length of the exposition (a) to that of the development and recapitulation (b):

He found that the ratio of b to a + b tends to match the golden ratio. For example, the first movement of the first sonata is 100 measures long, and of this the development and recapitulation make up 62. “This is a perfect division according to the golden section in the following sense: A 100-measure movement could not be divided any closer (in natural numbers) to the golden section than 38 and 62.”

Ideally there are two ratios that we could hope would hew to the golden section: The first relates the number of measures in the development and recapitulation section to the total number of measures in each movement, and the second relates the length of the exposition to that of the recapitulation and development. The first of these gives a correlation coefficient of 0.99, the second of only 0.938.

So it’s not as impressive as it might be, but it’s still striking. “Perhaps the golden section does, indeed, represent the most pleasing proportion, and perhaps Mozart, through his consummate sense of form, gravitated to it as the perfect balance between extremes,” Putz writes. “It is a romantic thought.”

(John F. Putz, “The Golden Section and the Piano Sonatas of Mozart,” Mathematics Magazine 68:4 [October 1995], 275-282.)

In 2013, University of Minnesota geography student Daniel Crawford composed “A Song of Our Warming Planet,” a solo cello piece built on climate data. The pitch of each note corresponds to the average annual surface temperature of a year in the range 1880-2012 in data from NASA’s Goddard Institute of Space Studies; each ascending halftone represents roughly 0.03°C in planetary warming.

“Climate scientists have a standard toolbox to communicate their data,” Crawford said. “We’re trying to add another tool to that toolbox, another way to communicate these ideas to people who might get more out of music than maps, graphs, and numbers.”

Below: He later applied the same method to create a string quartet using data from 1880 to 2014. The four parts reflect the average annual temperatures in four regions: the cello the equatorial zone, the viola the mid-latitudes, and the violins the high latitudes and the arctic.