Fibonacci sequences provide many examples that link math with music. The Fibonacci sequence of numbers looks like this:

 0 + 1 1 + 1 1 + 2 2 + 3 3 + 5 5 + 8 8 + 13 13 + 21 ... 1 1 2 3 5 8 13 21 34 ...

For example, A very special application of Fibonacci numbers is called golden proportion. As we divide any Fibonacci number by one adjacent to it in the sequence, it becomes a ratio like this:

 1/1 = 1.000000 1/2 = 0.500000 2/3 = 0.666666 3/5 = 0.600000 5/8 = 0.625000 8/13 = 0.615384 13/21 = 0.619047 21/34 = 0.617647 34/55 = 0.618181

These ratios become equal to 0.618. The ratio of this number and one is called the golden proportion.

Find the positioning of the famous 'motto' in Beethoven's Fifth Symphony.

 motto 5 measures 372 measures motto 5 measures 228 measures motto 5 measures X Y

The first movement is divided into a golden proportion by the motto.

X = 372 + motto = 372 + 5 = 377

Y = 228 + motto = 228 + 5 = 233

Y / X = 233 / 377 = 0.618  (the golden proportion)

A musical composition consists of 144 measures as shown below. Use the golden proportion to find the number of measures A and B.

 34 measures A B 34 measures 0.618 1 0.618 1 Theme Slow, soft Fast, loud Repeat of theme