MOZART S PIANO SONATAS AND THE THE GOLDEN RATIO. The Relationship Between Mozart s Piano Sonatas and the Golden Ratio. Angela Zhao

Transcription

1 The Relationship Between Mozart s Piano Sonatas and the Golden Ratio Angela Zhao 1

2 Pervasive in the world of art, architecture, and nature ecause it is said to e the most aesthetically pleasing proportion, the golden ratio, also known as the golden mean, golden section, or divine proportion, is a numer encountered when the ratio of two quantities is equal to the ratio of their sum to the larger quantity. The golden ratio has een studied extensively, especially its psychological effects, and its unique properties have given it a uiquitous presence in the world, ranging from Leonardo DaVinci s Mona Lisa to the Greek Parthenon. The golden ratio is denoted with the Greek letter phi ( or φ ) (Livio, 2002). As seen in Figure 1 showing the golden rectangle and Figure 2 showing two quantities, a and, in which is the larger quantity, are said to e in the golden ratio if = = (May, 2014). 1 a + a To find the value for, one can egin with the fraction. Because is equivalent to 1 a + a and can e sustituted as, = 1 + = 1 +. Therefore, 1 + =. Multiplying the a + a 1 1 equation y will result in a new equation: φ + 1 = φ 2 which can then e rearranged into a quadratic equation: 2 1 = 0. Using the quadratic formula, = 2 (1) = a 2 1 ± 1 4 (1) ( 1) 1 ± 5 Therefore, = = Its conjugate, denoted with a lowercase phi ( φ ) is also known as the golden ratio. φ = = (Weisstein, 1995). I have een playing piano since I was five years old, and since then, I have learned a total of eleven of the eighteen Sonatas written y Wolfgang Amadeus Mozart, a renowned and 2

3 influential composer of the Classical era in the late 18th century. Before I learn a classical piece, I always study the structure, form, and melody of the music, as studying the music helps me to have a etter understanding of the composer s intentions and the overall expression of the piece. Because I am well acquainted with Mozart s music due to my previous studies, I am conscious that his sonatas, which are mostly three movements long, follow a specific form known as the sonata form. The sonata form consists of two main parts: the Exposition and the Development and Recapitulation (see Figure 3 elow). The Exposition is the eginning half of the piece during which the main theme is introduced. The Development and Recapitulation consist of the second half of the piece; the theme is developed, expanded, and reintroduced in the end (Cummings, 2012). His sonata form comined with the exquisite alance of his pieces that are undoutedly pleasing to the ear triggered my curiosity in the golden ratio. After all, Mozart s music is highly regarded for its elegant proportions, and many critics have noted that in many of his compositions, he wrote down mathematical equations in the margins, showing that he had a penchant for mathematics (Barua, 2011). Therefore, it would e interesting to assess whether Mozart used the golden ratio when he composed his eighteen Piano Sonatas. Tale 1 provides a collection of data for all of Mozart s sonatas that follow the sonata form. Only the ones that utilize the sonata form were taken into account ecause the sonata form allows the piece to e roken down into two sections, a and. The data in Tale 1 represent the numer of measures in each section: the numer of measures in the Exposition is represented y a and the numer of measures in the Development and Recapitulation is represented y (Mozart, 1789). The first column represents the sonata and its movement using the Köchel cataloging system, a unique Austrian numering system used solely for Mozart s compositions. 3

4 Looking into Mozart s first sonata (K. 279), the first movement has a total of 100 measures and the Development and Recapitulation section contains 62 measures. If 100 φ is rounded to the nearest natural numer, it corresponds quite closely to 62. Therefore, the first 4

5 movement is a perfect proportion according to the golden ratio, for a 100 measure piece cannot e divided more perfectly into the golden ratio than 38 and 62. This is also apparent in the second movement of the sonata; a 74 measure piece cannot e divided any closer to the golden ratio than 28 and 46. However, in order to assess the consistency of the ratio in all of Mozart s piano sonatas that follow the sonata form, a linear regression was performed (see Figure 4 elow). On the x axis was the numer of measures in the whole piece and on the y axis was the numer of measures in the Development and Recapitulation section. If Mozart really did use the golden ratio, then the equation of the line should approximate y = φ x. a + According to the linear model aove, the linear regression equation for is y = x This was done y creating a scatter plot and finding the line of est fit. The slope of the line is extremely close to the value of φ and the correlation coefficient, r, is equal to 5

6 , which is extremely close to 1. This shows that there is a strong linear relationship. Furthermore, there are no outliers and there is definitive evidence to suggest that Mozart did, in fact, use the golden ratio. To further sustantiate the evidence, a frequency distriution of the ratio a + (see Figure 5) was performed to indicate the centrality of φ. According to the frequency distriution, the mode is around 0.62 and the ratios are skewed towards the golden ratio as well. Using a TI 84 Plus calculator, the 1 Vars Stats command was also used to determine the centrality of the data; the mean was 0.608, which is extremely close to φ. Furthermore, the standard deviation was calculated as ± 0.026, showing that the data did not fluctuate greatly. This, again, corroorates the theory that Mozart used the golden ratio when composing his sonatas. If a + is close to φ, then a should e close to φ as well. To test this, linear regression was performed again (see Figure 6); this time, however, was put on the x axis and a was put on 6

7 the y axis. The data still looks linear, and the equation of the linear regression is y = x The slope of the line is less close to φ and the correlation coefficient, r, is , showing that the linear relationship is less strong. To further analyze the evidence, a frequency distriution was performed for this set of data. According to the frequency distriution, the ratio of a to is widely dispersed. Although the mode is still around 0.62, the data display a large variance that was not seen in the frequency distriution chart of a +. The ratio ranges from 0.50 to Using a TI 84 Plus calculator, the 1 Vars Stats command was used again to determine the centrality of the data; the mean was 0.646, which is not as close to φ. The standard deviation was ±0.074, showing a large range in data, which is also supported y the frequency distriution. 7

8 It can e concluded that Wolfgang Amadeus Mozart did not consciously use the golden ratio when he composed his eighteen piano sonatas. The results from the two analyses seem conflicting ecause the linear regression model and the frequency distriution chart in the first analysis point to evidence that Mozart did, indeed, use the golden ratio. However, the second analysis contradicts the theory. The elegant alance in Mozart s sonatas may solely e a result of his prodigious talent for composition, which is why he is dued as one of the world s most recognized musical geniuses. A limitation in this exploration is the sample size of the data I collected. The sample size is relatively small; only twenty nine data points were used. As an extension in the future, I could analyze not only Mozart s piano sonatas, ut his other compositions that are also divided into two sections. Moreover, I could analyze his violin sonatas, orchestral works, and his operas. Moreover, I could analyze other renowned classical composers such as Ludwig van Beethoven and Frédéric Chopin. From this exploration, I learned to e more aware of the mathematics ehind nature and art. The golden ratio is prevalent in nature, from the spiral arrangement in leaves to the proportions of the Notre Dame, and it is an understatement to say that art and music imitate nature. Therefore, I have come to a greater appreciation of the structural forms of classical compositions. Although Mozart may not have used the golden ratio, other composers during his 8

9 time may have used it, as music and mathematics are more interconnected than one may think. In the future when I am studying a piece, I will take the golden ratio into account and see if there is an underlying pattern in the piece that is interconnected with φ. References 9

10 Barua, R. (2011). Music, mathematics and Mozart. Retrieved Decemer 9, 2015, from mathematics and mozart/ Cummings, R. (2012). Wolfgang Amadeus Mozart. Retrieved from Livio, M. (2002). The golden ratio and aesthetics. Retrieved Decemer 9, 2015, from ratio and aesthetics May, M. (2014). The golden ratio. Retrieved Decemer 13, 2015, from Mozart, W. A. (1956). Mozart Sonatas and Fantasies for the piano. King of Prussia, PA: Theodore Presser Company. (Original work pulished 1789) Weisstein, E. W. (1995). Golden ratio. Retrieved Decemer 1, 2015, from 10