Ideal Proportions

The math behind music, art, and games

Golden Rules

A golden rectangle is a rectangle whose sides are proportional in what is known as the golden ratio. The pleasing, precise proportions of the golden rectangle have been present in the works of architects, artists, and sculptors throughout the ages. Even photographs and index cards often come in these same golden proportions. As shown at right, in a golden rectangle, when you lop off a square (the green shaded area), the remaining piece is another, smaller golden rectangle.

You can derive the golden ratio from the golden rectangle above, which comprises a green square of side p and a smaller yellow rectangle that has a short side one unit in length and a long side p. By definition, since both the overall rectangle and the small yellow rectangle are golden rectangles, the ratio of the longer side to the shorter side of both rectangles must be equal. Therefore

*p/*1 = (

*p*+ 1)

*/p*, which equals

*p*=

^{2}*p*+ 1. With a bit of algebra, we can see that

*p*= (1 + √5)/2, or 1.61803 . . . This is the golden ratio.

Here are some other interesting rectangles. In each case the green rectangles are squares, and the yellow rectangles have the same proportions as the whole rectangle. Can you figure out the numerical value of each rectangle's proportions? You might need to pull out your old algebra book for problems 4 and 5.

Tuning Up

Pythagoras was one of the first to discover that harmonious musical intervals correspond to whole-number ratios. Try it: Pluck a guitar string. Press the string to the fret board at the halfway point and pluck again. This note is an octave higher than the first and corresponds to the ratio 2:1—the first two notes of "Over the Rainbow." The interval between one string and another two-thirds as long, 3:2, is a perfect fifth—think of the

*Star Wars*theme. A major third—the third and fourth notes in Beethoven's Fifth Symphony—corresponds to the ratio 5:4.

**[Easy] 1.**You can determine the interval between two strings by dividing the longer length by the shorter length. Intervals are equivalent if the fractions are the same, or if one fraction can be made to equal another by repeatedly multiplying it by 2 (which changes the pitch by an octave), by inverting it (which reverses the order of pitches), or by doing both operations. The intervals 3/2 and 4/3 are equivalent: A string 3 units in length and another 2 units in length, played in reverse order (2/3), are equivalent. Double the shorter string, and the resulting interval (4/3) is the same, with one note an octave lower. How many distinct intervals do these six strings comprise? For example, the interval between the two left strings is 243/216, or 9/8.

**[Medium] 2.**Scales in what is called just intonation are tuned so that every note is an exact whole-number ratio when compared with the starting note. Below are the strings of one 12-note scale in just intonation. Pitches are expressed as fractions; the string labeled 9/5 is five-ninths the length of the longest one. Differences in length between adjacent strings, called half steps, vary. How many distinct half steps are in this scale?

**[Hard] 3.**The equal-temperament tuning system adjusts the intervals in the 12-tone scale so that all half steps are equal. A half step is now the 12th root of 2—the number that equals 2 when multiplied by itself 12 times (written as 2

^{1/12}= 1.05946 . . . ). The perfect fifth is almost exactly 3/2 (2

^{7/12}= 1.49831 . . . ). Unfortunately, the major third (2

^{4/12}= 1.25992 . . . ) is not as close to 5/4 as one would like.

What if an octave comprised more than 12 notes? What is the smallest number of notes in an equal-tempered octave that would include a pitch that (a) better approximates the ratio of a major third in just intonation (5/4), (b) better approximates the ratio of a perfect fifth in just intonation (3/2), and (c) approximates both the major third and the perfect fifth better than the 12-tone scale does?

Working the Angles

Every game has its own shape—baseball is played on a diamond, soccer on a rectangle, and marbles in a circle. Pool is played on a table made up of two squares, and to play the game well requires a knowledge of angles. Place a ball at the center of one of the two squares. Where should you shoot the ball so that it bounces off each of the four sides of the table exactly once before landing in pocket A? How about pocket B or C? Which of these three problems has two different solutions? Of the two alternate solutions, which is shorter? Assume that the ball bounces off a cushion at the same angle it approached it, like light bouncing off a mirror. A ball can land in a pocket only if it is aimed exactly at the center of the pocket.

**Solution**

Want to see the solution to this puzzle?

Got new solutions for the puzzle? Want to see other people's solutions? Talk to the puzzle master in his discussion forum at www.scottkim.com.

These puzzles were inspired by

*(Disney, 1959), a half-hour mathematics documentary available at educational stores and Web sites. Also see the Just Intonation Network Web page (www.dnai.com/~jinetwk) and a page compiled by Ron Knott of Surrey University: www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci.*

**Donald Duck in Mathmagic Land**© Copyright 2002 The Walt Disney