Music Theory for Conworlders: Introducing Overtones and the Octave

As an introduction, we'll look at the basic concepts of western music theory. Most people know there's seven distinct tones in the major or minor keys, and there's twelve distinct tones in total per octave. People might also know of chords.

But what do these things even mean, and what kind of structure is there to them? Why twelve? What is an octave? What are tones, even? [Assuming most readers come here from a facebook discussion, this will basically have been covered already. If people come here from other places, I might add a more detailed definition somewhere later.

Now, I will probably return to the physics of tones a bit later, but for now, we'll observe that a tone has a frequency. What we mean to say when we say that in the ISO16 tuning standard A₄ is 440hz we mean to say that the sound wave is characterized by something that repeats 440 times per second.

We could make a wheel with equally spaced spikes that beat against a flexible piece of plastic, and have 440 of those spikes. If it rotates a whole turn in one second, the sound generated by those impacts would sound like a tone of 440hz to us.

There is nothing intrinsically special about 440hz, it is an arbitrarily picked standard. Baroque organs in Germany have A varying from roughly under 400 to about 500 hz. It's only fairly recently A440 has been standardized (basically sometime during the 20th century), and we find music that deviates from it as well.

Almost all instruments have a secondary important frequency-related fact going on - overtones. Most things will not just have one frequency, they will emit sounds of several frequencies simultaneously. For instruments, it's quite common for these overtones to be integer multiples of the lowest frequency, the 'fundamental'. (And these will later on be important when we look at concepts like consonance and dissonance, and therefore also when we look at chords and scales.)

So, when you play A440 on a guitar, you also cause generate the frequencies 880hz, 1320hz, 1760hz, 2200hz, ... . These contribute to the sound of the instrument - in fact, it's one of the things that distinguishes the sound of an organ from that of a guitar or that of a saxophone from that of a trumpet. We will start out by looking at instruments with this integer-multiples structure to their overtones. Other possibilities exist - pianos have near-integer overtones that tend to be slightly too large for integers (1, 2.01, 3.009, ...), and bells and gamelans have weird non-integers where multiple numbers close together may appear combined with large gaps as well.

To hear a synthesized version of this phenomenon, listen to the sample here below:
Play Music - Audio Hosting - Overtones

The very first overtone, the doubling of the frequency, has a rather special role in music. It corresponds to the octave, and it is nearly an universal interval in human music. Let us consider what happens with the overtones when you play two tones, an octave apart.
We have two tones, frequencies X and 2X. Each of these further is multiplied by a series of integers:
X, 2X, 3X, 4X, 5X, 6X, 7X, 8X, ...
2X, 4X, 6X, 8X, 10X, ...

We find that each overtone of 2X is also present in the overtone series of X. Thus. they will resonate together. For this reason, Xhz and 2Xhz will interact in similar ways with some other tone at Yhz as well. This, in part, is why we perceive tones where one is double (or quadruple, or multiplied by 2ⁿ for any integer n) the other. This, we call octave equivalence, and it'll be somewhat important later on.