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):

Köchel and movement a b a + b
279, I 38 62 100
279, II 28 46 74
279, III 56 102 158
280, I 56 88 144
280, II 56 88 144
280, II 24 36 60
280, III 77 113 190
281, I 40 69 109
281, II 46 60 106
282, I 15 18 33
282, III 39 63 102
283, I 53 67 120
283, II 14 23 37
283, III 102 171 273
284, I 51 76 127
309, I 58 97 155
311, I 39 73 112
310, I 49 84 133
330, I 58 92 150
330, III 68 103 171
332, I 93 136 229
332, III 90 155 245
333, I 63 102 165
333, II 31 50 81
457, I 74 93 167
533, I 102 137 239
533, II 46 76 122
545, I 28 45 73
547, I 78 118 196
570, I 79 130 209

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.)

Climate Music

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.


Inspired by his wife’s art studies, physicist David C. Roy turned his training to sculpture and began fashioning moving mechanisms of birch, not clocks themselves but clocklike in that they’re wound by hand and then run unpowered, sustaining their motion through escapements, suspended weights, and constant force springs.

“I saw it as another type of creative problem solving, not all that different from my advanced physics courses, but with a completely different goal,” he writes. “To this day, I find art and science to be closely linked.”

More on his website and YouTube channel.



French painter Joseph Ducreux (1735–1802) was fascinated with physiognomy, the notion that a person’s character is reflected in their outward appearance — and this led to some decidedly unconventional self-portraits.

At the same time, Franz Xaver Messerschmidt (1736-1783) was doing similar work in three dimensions.

Podcast Episode 314: The Taliesin Murders


By 1914 Frank Lloyd Wright had become one of America’s most influential architects. But that August a violent tragedy unfolded at his Midwestern residence and studio. In this week’s episode of the Futility Closet podcast we’ll describe the shocking attack of Julian Carlton, which has been called “the most horrific single act of mass murder in Wisconsin history.”

We’ll also admire some helpful dogs and puzzle over some freezing heat.

See full show notes …

A New Lease

Brazilian pianist João Carlos Martins won worldwide acclaim but had to retire in March 2019 after 24 surgeries could not relieve the pain caused by a degenerative disease and a series of accidents.

But designer Ubiratã Bizarro Costa proposed making some neoprene-covered bionic gloves that lift Martins’ fingers after they depress the keys, and by December they had perfected them.

“I might not recover the speed of the past,” Martins told the Associated Press. “I don’t know what result I will get. I’m starting over as though I were an 8-year-old learning.”

But his goal now is to play an entire Bach concert perfectly. “It could take one, two years. I will keep pushing until that happens. I won’t give up.”

Living Sculpture

Crop artist Stan Herd plants, mows, and plows land to create large-scale images visible from the air.

“All over the world farmers draw with the plough, harrow, and harvesting combine, and paint with the colors of their crops,” he says. He applies the same techniques to create portraits, still lifes, and (somewhat recursively) landscapes.