Falling Rhythm

1. Taking one of the strings (String 1), measure, mark, then attach weights at the following distances from one end (which we'll call "the bottom")—50 cm, 100 cm, 150 cm, 200 cm, and 250 cm—finishing at "the top."
2. Taking the remaining string (String 2), again measuring up from the “bottom,” mark and then attach weights at: 10 cm, 40 cm, 90 cm, 160 cm, and 250 cm (finishing again at the “top”).
3. Arrange the cookie sheet or piece of metal or wood so that it sits securely at a 45-degree angle and you can drop the weights onto it from above. Besides increasing the sound as the weights hit the cookie sheet or board, the angled surface allows the weights to slide off, which helps to smooth the descent of each weighted string.

Hold String 1 at the 250-cm point (closest to the “top” of the string, where you marked it), so the bottom of the string barely touches the cookie sheet.

Now drop the weighted string onto the angled piece of metal or wood and listen to the sound made by the falling weights. Notice that the rhythm gets faster and faster as the weights fall.

Now repeat with String 2. Hold it by the 250-cm point (closest to the “top” of the string, where you marked it), so the bottom of the string barely brushes the cookie sheet. Drop the weighted string onto the angled piece of metal or wood, as you did before, and listen for the rhythm. Notice that it has a constant beat: Even though the weights are not spaced evenly, they make a regular rhythm as they fall.

The weights fall under gravity, accelerating downward—that is, both strings go faster and faster as they fall. Specifically, each weight falls a distance proportional to the square of the time that it falls. As a result, String 1, with its equally spaced weights, hits the surface with shorter and shorter time intervals between the weights.

In order to hit at equal time intervals, the weights must be spaced so that their distance from one another increases proportional to a square. If you look at where the weights were placed on String 2, you’ll notice that the distances between the weights are proportional to the squares of the number of each weight in the sequence: 1, 4, 9, 16, and so on.

The weights on String 1 are spaced at equal distances. Let them fall, and you can hear the way objects accelerate under gravity. The weights on String 2 are spaced to equalize the time it takes for each weight to hit the floor, so the rhythm of the weights hitting in free fall is nice and even.

Notice that the distances 1, 4, 9, and 16 (which are proportional to the distances in centimeters between the weights on String 2) have the following interesting properties: they are all perfect squares, and the spacings between them are 3, 5, 7. Simply by starting at 1 and adding the odd integers one at a time, you produce the perfect squares.