Antonin Dvorak, a 19th-century Czech composer, said that "Mozart is sunshine." Although most people agree that Mozart's music sparkles brilliantly, no one knows for sure how Mozart created those shimmering sounds. Perhaps he relied on musical genius or inspiration from daily events. On the other hand, he might have composed measures of music with mathematical equations.
Considerable evidence suggests that Mozart dabbled in mathematics. According to his sister, Wolfgang "talked of nothing, thought of nothing but figures" during his school days. Moreover, he jotted mathematical equations in the margins of some of his compositions, including Fantasia and Fugue in C Major, where he calculated his odds of winning a lottery. Although these equations did not relate to his music, they do suggest an attraction to mathematics.
The structure of Mozart's music attracted the attention of John F. Putz, a mathematician at Alma College. "My son--who is a composer and pianist--told me that Mozart's piano sonatas are divided into two distinct sections," Putz recalls. "I knew that Mozart's music is highly regarded for its elegant proportions, among other things, so I thought it would be interesting to check whether the divisions Mozart used were very close to golden-section divisions."
The golden section--a precise way of dividing a line, music or anything else--showed up early in mathematics. It goes back at least as far as 300 b.c., when Euclid described it in his major work, the Elements. Moreover, the Pythagoreans apparently knew about the golden section around 500 b.c. The oldest examples of this principle, however, appear in nature's proportions, including the morphology of pine cones and starfish. Moreover, Putz said, "The golden section is thought by some people to offer the most aesthetically pleasing proportion."
To describe the golden section, imagine a line that is one unit long. Then divide the line in two unequal segments, such that the shorter one equals x, the longer one equals (1 - x) and the ratio of the shorter segment to the longer one equals the ratio of the longer segment to the overall line; that is, x/(1 - x) = (1 - x)/1. That equality leads to a quadratic equation that can be used to solve for x, and substituting that value back into the equality yields a common ratio of approximately 0.618. That value has been given many names, including the golden ratio, the golden number and even the divine proportion.
In the October 1995 issue of Mathematics Magazine (68(4):275-282), Putz described his investigation of whether the golden ratio appears in Mozart's piano sonatas. According to Putz: "In Mozart's time, the sonata-form movement was conceived in two parts: the Exposition in which the musical theme is introduced, and the Development and Recapitulation in which the theme is developed and revisited.... It is this separation into two distinct sections ... [that] gives cause to wonder how Mozart apportioned these works." That is, did Mozart divide his sonatas according to the golden ratio, with the exposition as the shorter segment and the development and recapitulation as the longer one?
Putz represented the two sections--the exposition and the recapitulation and development--by the number of measures in each. In the first movement of the Sonata No. 1 in C Major, for instance, the exposition and the recapitulation and development consist of 38 and 62 measures, respectively. "This is a perfect division," Putz writes, "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." An equally good approximation to the golden section exists in the second movement of that sonata. The third movement, however, deviates from the golden section.
A clear answer to Putz's question required looking at more than one sonata. So Putz examined 29 movements from Mozart's piano sonatas-the ones that consist of two distinct sections. Then he plotted the number of measures in the development and recapitulation versus the total number of measures in each movement, which is the right side of the golden--section equality as given earlier. The results reveal a stunningly straight line-so straight that its correlation coefficient equals 0.99, or nearly the 1.00 of a perfectly straight line. Moreover, the distribution of the ratios of the number of measures in the development and recapitulation to the total number of measures in each movement lies tightly packed and virtually on top of the golden ratio.
Although those results might seem like solid evidence that Mozart did use the golden ratio when he divided the sections of his piano sonatas, Putz knew that another comparison must be made. If Mozart used the golden section, then the other ratio from the golden--section equality--in this case, the ratio of the number of measures in an exposition to those in the recapitulation and development--should also equal the golden ratio. A plot of those measurements also produces a very straight line, but one with a lower correlation coefficient of 0.938, which Putz interpreted as "somewhat less goodness of fit." In addition, the distribution of the ratios of the number of measures in the expositions to those in the recapitulation and development peaks near the golden ratio of 0.618, but it also covers a considerable spread, ranging from 0.534 to 0.833.
The results from the two analyses seemingly conflict. The first analysis suggests that Mozart probably did use the golden section, but the variability in the ratios from the second analysis suggests that he did not use the golden section. That disagreement, however, did not surprise Putz, who wrote that the mathematics behind the golden section predict that "what we have observed in these data is true for all data...." That is, the ratio of the longer segment to the overall length is always closer to the golden ratio than is the ratio of the shorter segment to the longer one. As such, Putz concentrated on the distribution of the latter ratio as constrained by sonata form, and the spread in the distribution of ratios from that analysis suggests that Mozart did not apply the golden section to his piano sonatas.
In the end, we may never know if Mozart composed his sonatas, even in part, from equations. "We must remember," Putz writes, "that these sonatas are the work of a genius, and one who loved to play with numbers. Mozart may have known of the golden section and used it." Nevertheless, Putz thinks that the considerable variation in the data "suggests otherwise." In any case, Mozart did create divine divisions in his piano sonatas-making the interplay of sections shine like sunlight. Yet he apparently timed those divisions with his mind--not with math, or at least not with the golden section.