The Scales of Music

To understand music, we must understand scales.

Our everyday lives are generally full of units of measurement, and many of those units are essentially arbitrary. If you’re driving down a quiet street and a cop pulls you over for doing 90 miles per hour, the thing he’s measuring (your speed) may be real and meaningful but the units he’s using (miles, hours, miles per hour) are essentially arbitrary, the products of a quirky set of historical accidents. There is nothing fundamental about the length of an hour that makes it an especially meaningful unit of time: if our Babylonian forerunners had done things differently we could easily be using a basic unit that was some other length. Equally, there is nothing fundamental about the length of a mile that makes it an especially meaningful unit of distance; for that reason, a lot of countries have in fact chosen a different (equally arbitrary) unit called “a kilometre” to do the same work.

By contrast, some of our common units have a deeper logic to them. The length of a day was not decided for us by some ancient civilisation; “a day” is the non-arbitrary length of time that it takes the earth to spin once around its axis. This gives days some very useful and meaningful properties.

How about music? In the same way that time (on earth) is organised around these things called “days,” music is organised around things called scales: sets of musical notes lined up in order. But is the length of a scale arbitrary (like the length of a metre) or fundamental (like the length of a day)? Let’s find out.

You might (or might not) know that there are seven notes in a Western scale — this, for example, is why the pattern of white-keys on a piano repeats in sets of seven. Following the famed musicologist Fraulein Maria (a.k.a. Julie Andrews in The Sound of Music, and more seriously following a Medieval Italian monk named Guido), we will name these seven notes as follows:

Do - Re - Mi - Fa - So - La - Ti

You might recall the scene from the famous movie/musical where Fraulein Maria, a suspiciously decked-out nun, asks the Von Trapp children how they started learning to read. “A, B, C!,” replies the adorable pig-tailed Marta, and Fraulein Maria promptly launches into a song about the ABC’s of music. Do, a deer, a female deer, Re, a drop of golden suuuuun.... Mi, a name I call myself, Fa, a long long way to ruuuuun....” Maria teaches us that there are seven such syllables in the “alphabet” of music: Do, Re, Mi, Fa, So, La, Ti.... which brings us back to Do. The cycle repeats: after Ti we go back to Do, and then up through Re-Mi-Fa-So-La-Ti again, and then back to Do, onwards and onwards.

The jump from one Do to the next is called a musical octave, since oct— is a Latin root for eight and there are are eight notes of a scale between them (counting both Do’s). An octave is one example of something called a perfect interval, and we’re not just talking ‘perfect’ the way your grandmother thinks you’re perfect — there really is something deep and fundamental about the perfectness of an octave, mathematically speaking. We’ll get to that properly soon.

The second Do — the “higher” Do — is exactly double the frequency of the lower Do. They are, in a fundamental sense, the “same note.” In a similar way, “a day” could be called a “perfect interval” in time: if you start at a particular time and add a day to it, you are fundamentally back to “the same time” again. If I tell you it’s 11:11 right now, and ask you what time it will be exactly one day later, you’ll say 11:11 without missing a beat. (I hope your wish was about learning music theory). Similarly, if I tell you that we’re starting at the note Do, and ask you what note will occur seven notes above it, the answer is that it will be Do again. The new Do will just be one octave higher.

To understand what it means in physical terms to have two notes be “the same note,” let’s have a quick look at how sound works. Very basically, the physical reason you can hear music is because sound-waves come crashing out of instruments and into your ear.

The thing that determines which sound you hear is the “shape” of the wave —  essentially, how far apart its peaks and troughs are. For example, the red curve below represents a wave that might produce the sound Do in your ear — the first Do Fraulein Maria sings in her song.

If the red wave is the “lower Do,” the wave that creates the “higher Do” would look like this:

If we look at them together, we see that the distance between the peaks and troughs of lower Do is exactly twice the distance between the peaks and troughs of higher Do.

There’s no reason to stop there; we might find a blue wave that looks like this:

Of course, we can superimpose that wave on the green and red ones — the blue wave is still the same note as the green and red, it’s just an “even higher” Do than the green one. The distance between the peaks and troughs of the blue wave are twice the distance between the peaks and troughs of the green wave, and four times the distance between the peaks and troughs of the red wave (that’s 2x2, or 22).

Why exactly this “double frequency” trick works — why exactly those waves create a feeling in our heads that makes these notes meaningfully “the same” — is an awesome neuroscience question that there’s no way we’re going to even think about answering in a friendly little book like this one.  Of course it doesn’t seem crazy, though, that doubling and halving something (however many times) is a “non-arbitrary” thing to do.

Your ear is very used to hearing (and even singing!) the octave — well, your ear isn’t doing the singing, but you know what we mean. Some very famous examples: in the opening of “Somewhere over the Rainbow” (picture a pig-tailed Judy Garland), the octave is the distance between the two notes that Judy sings for “Some-” and “-where.” It’s a big leap from “some” to “where” — Judy jumps an entire octave in a single breath, but she’s singing the “same” note both time. If we could see the waves she is singing they would look like this:

Perhaps the octave-jump that we are most accustomed to singing (and messing up) is the distance between “Happy” and “birth—” in the third line of “Happy birthday to you.” This devious song gives us two chances to feel confident in our singing abilities in the first two “Happy birthday to you’s” — the first two jumps from “Happy” to “birth” are just jumps from So to La on our seven-note scale, which is easy enough to do:

But then, on the third line, the song suddenly throws us an octave-jump. On the third line, “Happy” and “birth” are now actually the same note, but “birth” is a whole octave higher.

It is at this point that most people become incredibly quiet and let someone else carry the tune, or let out a little squeal, or try to cheat the octave and land on a more comfortable note putting everyone in a different key altogether (this is why your musically-inclined friends tend to wince during most singings of Happy Birthday). But that brings us to the topic of keys — and with that, on to the next chapter.