Pythagorean Thinkings 1: Just Who Was He?

Hello all, and welcome back to The Wandering Mathematician.  Today I’m starting a three-part series on Pythagoras, the ancient Greek philosopher and mathematician.  Next week, we’ll look at all the different ways you can prove his famous theorem, and the week after that, we’ll see some ways we can take it further.  Today, though, I thought we’d ask what other mathematics he created over his lifetime.

Sadly for us, the answer is: not loads.  He was mainly a philosopher, who was credited with putting forward a way of life based around friendship and communal spirit.  Aside from that, the Pythagorean Theorem, and marrying his student Theano, he is mainly known for inventing the idea of harmonies.

What, I hear you ask, are these?  Everyone has noticed, at some point, that two tones seem to sound good together and two others don’t.  Generally, musicians tend to prefer to use the tones that sound best together (“concordant”), although sometimes they use what are known as “discordant” (not good-sounding) notes for effect.  The question that Pythagoras sought to answer was whether there was a mathematical way of predicting which notes would be concordant and which would be discordant.

There is a legend, which is certainly false and probably originated in the Middle East, that Pythagoras first discovered his system of harmonies when he heard the hammers inside a blacksmith shop and noticed that some sounded better together than others.  We don’t actually know how he discovered this, but we do know that he found that notes sound best together when their frequencies (the speed of the sound wave through the air, which regulates pitch) are in the ratio .  In other words, the pitches are the same.  The next most pleasing ratio is , where the pitch of the higher note is double that of the lower note, followed by (or ), and so on.  In general, Pythagoras found, the human ear likes small numbers and simple ratios, which makes intuitive sense: fractions like look ugly, and generally indicate that you’ve gone wrong somewhere, while fractions like do not.

Musicians have names for all of these ratios.  If two notes have pitches in the ratio , that’s a unison.  The ratio gives an octave, while the ratio gives a perfect fifth.  If you haven’t heard those terms before, don’t worry.  They aren’t mathematical; they’re musical.

This seems fantastic, almost otherworldly, but it isn’t really.  The mechanism is actually quite simple.  Sound waves oscillate, that is, they go up and down relative to the direction of motion.  Musical notes contain “harmonics”, quieter waves that oscillate twice, or three times, or four times, or more as fast as the main wave. Notes sound good together when they have the same harmonics, and the simpler the ratio between the waves, the more likely it will be that any given harmonic is shared by both of them. For example, let’s say we’ve got a wave at hz, and another at hz: an octave ratio. The first has harmonics at hz, of which exactly half (hz) are shared by the second. If we make the ratio less simple and make the second wave hz, only one third (hz) of the harmonics are shared.

It is often tempting to see a mathematical result and say that it must be magic.  It never is.  Mathematicians are usually very skeptical, preferring to consign such things as magic to the realms of unprovable conjecture and concentrate on what they can show to be true or false, such as why simpler ratios between notes sound good.  Pythagoras didn’t just mathematize the study of music.  In doing so, he legitimized it for most mathematicians.