Mini-Project I: "The Golden Section and the Piano Sonatas of Mozart"

Article: “The Golden Section and the Piano Sonatas of Mozart” by John F. Putz in Mathematics Magazine

# Article Summary

The golden section is said to occur everywhere in nature, from pine cones and leaves, to sunflowers and starfish. The golden section is defined to be the division of a line segment in two parts such that ratio of the length of the shorter to the larger is equal to the ratio of the larger to the whole. Or,
$\frac{a}{b}$=$\frac{b}{a+b}$ (1)
where a is the shorter segment and b is the larger.

The author, John Putz, sets the length of the whole at 1 and the shorter segment at x. Therefore, equation (1) becomes:
$\frac{x}{1-x}$=$\frac{1-x}{1}$ (2)

The golden ratio, which Putz defines as $\phi=0.6180$ is over 24 centuries old. It is one of the most debated numbers in mathematics. While it appears to be in such constructions as the Great Pyramids and the Greek Parthenon, many mathematicians believe that those who are proponents of the ratio form ratios that appear to be contrived or forced to reveal the Golden Ratio.

The author writes that Mozart was mesmerized by mathematics since his early childhood. He covered the walls and staircases in his house with numbers and proceeded to do his neighbors’ houses. Several of Mozart’s compositions contained numbers and figures in the margins.

Putz writes that during the time of Mozart, sonatas were composed in two parts. The Exposition was first and introduced the musical theme for the piece. The Development and Recapitulation, which developed and revisited the musical theme from the Exposition, came second. Putz studies the separation of Mozart’s sonatas. He hypothesizes that the sonatas are separated into the Exposition and Development/Recapitulation such that they are divided into the Golden Section. The first two parts of the first sonata, 279 I and 279 II, are perfectly divided into the Golden Ratio. 279 I has 100 measures, with the exposition 38 measures long and the development/recapitulation is 62 measures long. For the natural numbers, 62 could not be closer to 100$\phi$. In 279 II, the 74 measures are divided into 28 and 46 measures, respectively. This again could not be closer to the golden ratio. However, for the third part of K. 279 is not exactly divided into the golden section.

The next figure is a scatter plot of b against a+b with the line $y=\phi x$ and the regression line. While this appears to be significant evidence in favor of the sonatas partitioned into the golden section, Putz decides to look at the data another way. If $\frac{b}{a+b}$ is near the ratio, then $\frac{a}{b}$ should also be near the ratio. The next figure is a plot of a against b with $y=\phi x$ and the regression line plotted as well. We can see that there appears to be less linearity. The second figure is the frequency distribution of $\frac{a}{b}$. Here, we see that there is less evidence for the centrality of $\phi$.  Putz states that there is a theorem that $\frac{b}{a+b}$ is always nearer to $\phi$ than $\frac{a}{b}$. We will look at this proof in the “Connection to Real Analysis” section of the review. The author concludes that there is no significant proof that Mozart knowingly used the golden ratio in his sonatas, but there is no proof that he did not.

# Connection to Real Analysis

The golden ratio and its appearance in nature is a topic that has been largely debated. Putz writes that the debate has picked up in the modern era. For instance, G. Markowsky argued against the appearance of the ratio in the Parthenon, the United Nations Building and the Great Pyramid. J. Benjafield and J. Adams-Webber, however, recently argued that when people divide a whole into two unequal parts, they tend to make the division near the golden ratio.

We will now look at the proof that Putz uses to show that $\frac{b}{a+b}$ is always nearer to $\phi$ than $\frac{a}{b}$. This is a connection to Real Analysis because of its use of the mean value theorem and the definition of a fixed point.

Theorem. $|\frac{b}{a+b}-\phi| \leq |\frac{b}{a}-\phi|$ where 0$\leq a \leq b$

Proof:

Let $x = \frac{b}{a}$. Then we must show that $|\frac{1}{x+1}-\phi| \leq |x-\phi|$
for all $x \in [0,1]$. Now, let $f(x)=\frac{1}{(x+1)}$. By the mean value theorem, for all $x \in [0,1]$ there is a $\psi \in [0,1]$ such that: $|f(x)-f(\phi)|$=$|f'(\psi)| |x-\phi|$

Now $f'(x)=\frac{-1}{(x+1)^2}$ satisfies $\frac{1}{4}<|f'(x)|<1$ for $x \in(0,1)$.

A simple calculation will show that $\phi$ is a fixed point of $f$, that is, that $f(\phi)=\phi$. So, for all $x\in[0,1]$,

$|\frac{1}{x+1}-\phi|\geq|x-\phi|$

This shows, using concepts learned in real analysis, that the theorem Putz states in his article is true. For any given pair $a$ and $b$, 0$\leq a \leq b$, the ratio $\frac{b}{a+b}$ is always nearer to $\phi$ than $\frac{a}{b}$. Someone wishing to illustrate a golden ratio relationship can use $\frac{b}{a+b}$ to their advantage because it is biased towards $\phi$. Therefore, when investigating the golden ratio, it is prudent and necessary to use $\frac{a}{b}$.

The golden ratio can be found throughout the field of mathematics, in subjects ranging from geometry to topology to the Fibonacci numbers. As mentioned earlier, it can also be found throughout nature and in architectural structures. We will examine several of the mathematical examples where the section can be found.

The first subject area is geometry. There are many examples within the field where phi occurs. The length of a regular pentagon’s diagonal is $\phi$ times its side. The vertices of a regular icosahedron are arranged like those of three mutually orthogonal golden rectangles. The golden triangle is pictured below. It constructed such that CXB is a similar triangle to ABC. We can see that the golden ratio occurs all over the triangle. Pictured next is the pentagram. The pentagram is colored such that the four different lengths of line segments can be easily seen. These four lengths are in golden ratio to each other.

Another area of mathematics in which the golden ratio has significant importance is with its relationship to the Fibonacci sequence. We define the Fibonacci sequence as:
$f_n=f_n_-_1 + f_n_-_2$ with $f_0=0$ and $f_1=1$.

Because it is a linear linear recurrence relationship, the Fibonacci sequence has a closed-form solution. Known as “Binet’s Formula,” it is the following:
$f_n=\frac{\phi^n-(1-\phi)^n}{\sqrt5}=\frac{\phi^n-(-\phi)^-^n}{\sqrt5}$

The golden ratio is also the limit of the ratios of successive terms of the Fibonacci-sequence: $\lim_{n\to\infty}\frac{f_n_+_1}{f_n}=\phi$.

The golden ratio also appears in architectural structures. However, there is great debate as to whether or not the builders knowingly implemented the golden ratio in their constructions. The Great Pyramid of Giza and other pyramids at Giza are said to contain several variations of the golden section. Many argue that there is no way the Egyptians knowingly used the ratio because they were not mathematically advanced enough to know about the ratio. The Greek Parthenon and the UN Building are often said to contain the golden ratio. Those claims are often debated.

What is of critical importance to note from Putz’s article is that there are several ways to calculate the golden ratio. The theorem we proved showed that the ratio of the longer segment to the whole will be closer to the golden ratio than the shorter to the longer. This means that the data and arguments for the golden ratio can be presented in a way that favors the hypothesis of the person presenting it, whether for or against it. The golden ratio is a significant discovery. Regardless of it actually occurring nature or it being a contrived ratio by some mathematicians, the golden section lends itself to study by mathematicians all over the world.

# Bibliography

"Golden Ratio." Wikipedia. http://en.wikipedia.org/wiki/Golden_ratio

"Golden Ratio." Wolfram Mathworld. http://mathworld.wolfram.com/GoldenRatio.html