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 |
Estimate the answer:
A = 34 and B = 34
A = 55 and B = 21
A = 68 and B = 34
|
