Number Sense: Insights from Ratios

HS #46 2019.5.9

Number Sense: Insights from Ratios

A few years ago while singing with the College Chorus at a Hope College Vespers, I got wondering how many Dimnent Chapel’s filled with sand are needed to fill St. Peters Basilica. So I counted the blocks to the ceiling, paced off the length and width, and returning home looked up the volume of St. Peters. You might be surprised - the answer is 147.

And sitting in the Hope Church choir loft during a children’s sermon, I wondered how I might illustrate to children the 250 million year age of my colorful petrified rock (which I call “God’s artwork”).  It turns out that’s how long it would take to fill the sanctuary if a person brought a quarter teaspoon of sand with them each Sunday. 

Number sense, in particular understanding the relative ratios of things, helps give an appreciation for the world around us. For example, Lake Michigan would fill the Grand Canyon with a little left over. And bottled water costs about 2000 times as much as tap water – without the prophylactic benefits of fluoride. 

Relative rates are interesting: How long does it take a top athlete to go a mile? Swimming: 10 minutes. Walking and kayaking: 6 minutes. Running, cross-country skiing, and roller blading: 4 minutes. Bicycling and ice-skating: 2 minutes. A bit slower, the rate the continents are drifting apart is about the same rate your fingernails grow. They take about 40,000 years to go a mile. 

Relative distances are also interesting: One late night walking home from the office I noticed the blinking lights of a jetliner, the moon, a comet by the sun, and some stars. Extending my arm, I estimated the jet is 25,000 times farther from my eye than my hand, and the moon is also 25,000 times farther than the plane. In contrast, the sun is only 400 times farther than the moon. But then the next nearest star, Alpha Centari, is 15 million times farther away than the sun, and the farthest thing we can see in the night sky, the Andromeda Galaxy, is almost a million times farther than that. 

Owen Gingerich, Harvard University and Smithsonian astronomer, once told me that if your hand were a star, the next nearest star would be in Washington, DC. But if your hand were a galaxy, the next nearest galaxy would be the distance of your other outstretched hand. 

Music is all about ratios. From one C to the next (an octave) the higher note vibrates exactly twice as fast as the lower: a 2:1 ratio. Since there are twelve notes between them, each of those notes is the same ratio higher than its neighbor.  

So that ratio must be the number that when multiplied by itself 12 times gives 2. That number is about 1.06. Multiplying this number by itself, you can see that the frequency of G (fifth of the chord) is almost exactly in a 3:2 ratio with the root (C). The F (the fourth) is almost exactly in a 4:3 ratio with the root, and E (third of the chord) is almost exactly in a 5:4 ratio with the root. These nice ratios mean that the wave vibrations match up often. (If one visitor comes every 4 days and another comes every 3 days, they will meet every 12 days.) That is why C-E, C-F, and C-E-G sound so nice together. It is entirely serendipitous that twelve steps from C to C give such nice results, and is undoubtedly why we use the twelve-tone system. How cool is that. 

But most enlightening is what ratios reveal about the place of humans in the universe.  Imagine a number line with equally spaced tics. The leftmost tic is the smallest possible (quantum) distance and each subsequent tic represents the number 10 times larger than the previous until the rightmost tic is the distance across the universe. It turns out the size of the human body falls exactly halfway along the line. That is, by ratios, our body size is halfway between the smallest thing and the largest.  

However when the same is done with the shortest (quantum) time and the longest time (the age of the universe), the human lifespan is a full three-quarters of the way towards the longest. That is, in terms of ratios, each person reading this has lived or can expect to live three quarters of the age of the universe.  That’s enough time to make a difference in the world.