No. 107: Sep-Oct 1996
Does the brilliance of Mozart's compositions derive entirely from his musical genius, or did he rely in part upon a mathematical construct: the famous Golden Section or Golden Ratio? The Golden Section is a mathematical formula for dividing into two parts: a geometrical line, a musical composition, or anything else possessing the property of length. The ratio of the two divided parts is the Golden Section, which equals 0.618.* For some artists, musicians, architects, the Golden Section is the most esthetic way of dividing the length of anything.
For humans, the history of the Golden Section goes back at least as far as Euclid in 300 BC. For nature, it began eons ago: The shapes of pine cones, starfish geometry, and other dimensions of living things incorporate the Golden Section. The questions we address here are: (1) Did Mozart consciously make use of this ratio, 0.618, in his music? (2) Why is the Golden Section esthetically pleasing?
It is not well known that Mozart was fascinated by mathematics as well as music. He even jotted down equations in the margins of some of his compositions. Chances are excellent that he knew of the Golden Section and its reputation for conferring elegance on structures -- even musical compositions.
J.F. Putz, a mathematician, has measured some of Mozart's works. Mozart's piano sonatas were convenient targets, because in Mozart's time they were customarily divided into two parts: (1) Exposition; and (2) Development and Recapitulation. Sure enough, the first movement of Mozart's Sonata No. 1 in C Major consists of 100 measures that are divided into the customary two parts; 38 in the first, 62 in the second. This ratio 38/62 (0.613) is as close as one can get to 0.618 in a composition of 100 measures. The second movement of this sonata is also divided according to the Golden Section, but the third movement is not. Many other Mozart piano sonatas seem to employ the Golden section, but some deviate considerably. So Putz could not really claim that Mozart consciously used the Golden Section to "improve" his music (Question #1 above), but there are certainly a lot of "coincidences."
(May, Mike; "Did Mozart Use the Golden Section?" American Scientist, 84:118, 1996)
Question #2 above. Why is a particular ratio exceptionally pleasing to humans -- and to nature in general? The ratio 0.5 seems neater! If Mozart used the Golden Section unconsciously and frequently, the Golden Section may somehow be encoded in the human brain as it is in the biological machinery that controls the developing pine cone and starfish. In humans the Golden Section manifests itself in artistic creations rather than boldily morphology! Obviously, we cannot answer Question #2.
*To calculate the Golden Section, the length of a line (or musical composition) is made equal to 1, and then divided into a short section, x, and a longer section, (1 -x). The ratio of the short section to the long section is then made equal to the ratio of the long section and the length of the whole line:
x/(1 - x) = (1 - x)/1
This can be solved for x, and the Golden Section:
x/(1 - x) = 0.618
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