Pythagoras and Me

As someone who is interested in both music and science, I find myself fascinated with the concept of scales. Why are they the way they are? If music is so mathematical, why does it seem so arbitrary. There are 12 notes in an octave, but what’s so special about this number 12? Then the major scale is formed from seven of those notes, which is strange. If the major scale is so nice and melodious and all that, why aren’t the notes evenly spaced? Well, I set out to find some answers. I spent hours scouring the internet, visiting the library, reading books, scratching figures in a notebook. You don’t have to do that, because here’s what I found out, all nicely summarized.

The random part

Turns out my instincts were correct, to a certain extent: our system of music is indeed arbitrary, but there is also some hidden magic. If you traveled back in time and listened to an orchestra from 400 years ago, it would sound flat, but the tunes would still be familiar. That’s because it’s not the notes themselves that are special, but the relationship between them.

Not too long ago (relatively), the International Organization for Standardization decided that an audio tone at a frequency of 440Hz would be called A. That’s the A above the C in the center of a piano keyboard, and from this frequency we get all the rest. The choice of 440 had no basis in cosmic laws or the circumference of the Earth or anything like that. In fact, in previous times different countries used different frequencies, which is why a standard had to be set. And 440 was as good as any other frequency. So there’s the start.

The magic part

Now where do we get the rest of the notes? That’s where Pythagoras comes in.

Pythagoras was a fascinating character. He’s probably most known for his theorem about the proportions of the sides of a right triangle, but in his day he was prominent in the philosophical, religious, and musical areas as well. He founded a group, first in Greece, then in the south of Italy, a group that was somewhere between a religious cult, university study group, and a garage band. He and his students, and after he died his followers, came up with many significant ideas in mathematics and astronomy, as well as some kooky religious ideas. But what we are concerned with here are their musical contributions.

First the word frequency, what does it mean, anyway? It’s just how often something happens, right? As in, “I drink with a frequency of five beers per day.” When we’re talking about music, it refers to how fast a string or a column of air or a drum head vibrates. A guitar string playing that middle A moves up and down 440 times per second. 440 Hertz (named after a guy named Mr. Hertz), written as 440Hz. Sounds pretty fast, and it is, to look at, but the human ear can hear frequencies up to around 20,000Hz. Try that with beer.

Ok, next thing is the idea of harmonic frequencies. That’s where the real magic comes in. Turns out, when you pluck a string that’s tuned to 440Hz, that’s not the only frequency of sound that comes out. You get the 440Hz, but you also get a slightly quieter 880Hz tone (which is 440 times two), as well as another one, even quieter, at 1320Hz (440 times three), and so on. These “extra” notes are called harmonics, and they occur everywhere in nature, when you play an instrument, when you bang two sticks together, when birds sing. They are natural, they are cosmic, and we hear them so much we figure they must sound nice. (Pythagoras loved them so much he had a theory that the planets “sang” in harmonic frequencies as they orbited around the Earth, but we’ll ignore that kookiness.)

Here are the harmonic frequencies produced when plucking an A. (Don’t worry, you don’t have to memorize them. I’ll tell you why they’re important in a second.)

1x

2x

3x

4x

5x

6x

7x

8x

440

880

1320

1760

2200

2640

3080

3520

The next thing Pythagoras discovered is that if you cut the string in half the frequency doubles, from 440 to 880. This is the same thing you’re doing when you hold down a guitar string at the 12 fret. The new harmonic frequencies are:

1x

2x

3x

4x

5x

6x

7x

8x

880

1760

2640

3520

4400

5280

6160

7040

 Now look back at the shaded columns in the first chart. Notice something? The numbers in the second chart are the same as every-other column in the first chart! That really is magic. There must be something special about the relationship between these two notes. In fact, they sound so similar we give them the same name: A. That’s right, they’re both As. The second one is an octave higher than the first. So now we know why the scale is as long as it is: it’s just a doubling of the beginning frequency. Now we can work on filling in the middle notes, the ones between A at 440Hz and A at 880Hz.

Try and stay awake for this part

What do we know so far? Harmonic frequencies sound good together, and doubling a frequency gives a similar sounding note. Ok, well cutting something in half is the same as doubling, but backwards, so let’s try this: take the number from column three of table 1, the 1320Hz, which we know sounds good with the A, and cut it in half, so it fits between 440 and 880. We get 660Hz, another nice looking number, if I do say so myself. So we tune up our little test strings, one to 440Hz, another to 660Hz, and we pluck them together. What do you know, they sound sweet! Let’s give it a name. I’m gonna go with E, just because. Now we have three notes in our scale: A, E, and another A.

Well, if we can do that with the third harmonic, we can do it with the fifth, which is 2200Hz. Dividing it in two gives us 1100Hz, which is still too high, so divide it in two again, giving us 550Hz. I like that number, too. Let’s call it D. (Don’t worry about where these names come from, for now. They’re basically arbitrary.) Now we’ve got A, D, E, and another A. We’re on our way to scale-hood!

Do you see what we’re doing here? We’re taking the harmonics of the starting frequency and dividing by two until they fit into the octave. We can do the same thing starting on D, or E, or any of the new frequencies we come up with. I’m doing this with the help of a calculator and modern technology which allows us to measure how many times a piece of string vibrates in a second. (Let’s pause for a second to consider just how cool that is… Ok, back on topic.) Pythagoras and crew did the same thing by cutting and measuring strings. I think he wins.

I could keep going with the math, but that would be boring and tedious. Just trust me that if we were to keep going we’d end up with something pretty close to the notes of the major scale we still use today, and if you start in different places, you get pretty close to all twelve notes we use today. “Hold on a second, Eddy!” I hear you saying. “What do you mean, ‘pretty close?’ If these ratios are magical, why aren’t they exactly what we use?” Ok, you caught me. That’s where even tempered tuning comes in.

Even tempered tuning

Turns out the numbers don’t work out exactly as perfectly as the Pythagoreans thought. The problem is, we end up with slightly different numbers when we start on, for example, an E, instead of an A. That wouldn’t be a big deal if you and all the members of your band could agree to always play every song in the same key. But then if one day you got sick of A and wanted to try a song in the key of E, you’d have to retune all your instruments slightly. That’s no fun, so somebody came up with “even tempered tuning.”

When Pythagoras followed the above exercise, he came up with twelve notes all together. He could have stopped earlier or kept going to get a different number of notes in the scale, like they have in India or Africa, but he didn’t. Twelve is what he came up with, and that’s what we still use here in what we for some reason call the “western world.” Anyway, as I said the notes were not quite evenly spaced. Maybe that’s why the planets don’t actually sing. We can make them evenly spaced, but that messes with Pythagoras’s beautiful ratios, and the planets, and yada-yada-yada, so it took thousands of years for someone to get over the fear of being struck by lightening and actually space them evenly. They came up with a ratio of around 1.059 between each note, which when multiplied by itself 12 times gives you 2, which is the octave we’re looking for. Some dude named Bach wrote a piece of music called The Well-tempered Clavichord using this new-fangled tuning technique, and the rest, as they say, is rock-and-roll history. It caught on, and we’ve been using even tempered tuning ever since.

Rock on

So next time you find yourself thinking, “Hmm, I wonder why the scale has seven notes,” you’ll know you have a Greek named Pythagoras and his cult to thank. Maybe that won’t turn your guitar solos into blistering works of pure genius, but it’s a step towards understanding more about why music is the way it is. I hope you find the topic as interesting as I do, and my explanations made sense. There is a lot more detail to be discovered, so if you’re still interested check out one of the references in the list that follows.