Music Theory, part 1

I think theory is a really important thing to learn, and while the study of theory comes with a ton of caveats and exceptions, I do believe that it helps you expand your knowledge of what is possible to achieve through music, and more importantly, shows you how to achieve it. There is more to say regarding the value of music theory, but I will make references it as time goes on rather than try to put it all in one big paragraph.

So let's get started.

Lesson 1 - fundamentals
1.1 The Harmonic Series
I am sure you all have heard the terms Perfect 5th, Major 7th, etc a lot of times before. It's going to be necessary to understand these intervals backwards and forwards in order to grasp some of the more advanced concepts we will come across. But instead of just teaching them by rote memorization, I would like to discuss WHERE the intervals come from and why they form the basis of Western harmony.
All music comes about via sound, which is a vibrating air column, which can be described mathematically as a wave. This is most obvious on a stringed instrument like a guitar, where you can physically see the shape of a wave form when you pluck it. Note that the two ends of the wave need to be clamped down, or zeroed, in order for it to vibrate; if one end was just hanging you wouldn't get sound. Those points where the amplitude is zero and the string is not vibrating are called nodes.
Now, what is not immediately apparent is that on any vibrating object, there are actually multiple waves superimposed upon each other. Essentially, as long as the vibrating string has nodes at both ends, any waveform can happen in between the two. In the below picture you will see many different possibilities of waves that can (and do) exist simultaneously on a vibrating guitar string.

Each one of these different vibration possibilities after the is called a harmonic, or partial. While the fundamental or first harmonic (top wave) is the loudest, the relative loudness of the higher harmonics is what gives every instrument its distinctive tone color. So while a guitar and a trumpet might be playing the same note, the higher-order harmonics are different and that's why they sound like different instruments.

Let's listen to what those individual harmonics actually sound like. To do that, we will artificially CREATE a node (say, by lightly touching the string with our finger) that silences the harmonics below (in the picture, those above) the one we want to hear. So if we touching the vibrating string at exactly one half the length, it will silence the fundamental but not the second harmonic, since at that point the fundamental has a nonzero amplitude, but the second harmonic is already at zero so it doesn't matter. What note do we hear? An octave above the fundamental. K, let's try isolating the third harmonic, at 1/3 the length of the string. Silences both the fundamental and the second harmonic, and we hear a fifth above the note that was the second harmonic. Now touch at 1/4 the length... we hear a note that's a perfect fourth above the third harmonic. It goes up and up until they are barely audible. This series of notes you hear is called the harmonic series.

1.2 Generating Intervals

But let's not yet assign the intervals between successive notes labels like "fourth" and "fifth." Instead I will refer to them as such: the interval between two notes is the ratio of their multiplicative factors relative to the fundamental. That is, the interval between harmonics 2 and 3 is, (2x as fast) to (3x as fast), or 2/3. [I am allowed to say that the second harmonic is twice the speed as the fundamental because the speed of vibration, or frequency, is inversely proportional to its wavelength, as long as the speed of sound remains constant (which it does).] Essentially, what we are saying is that the second harmonic is vibrating at 2/3 the speed as the third harmonic. Or: the third is vibrating at 3/2 the speed of the second, which is the convention I will be using from here on out.

K, so we will call the interval between the fundamental and second harmonic = 2/1. Between the second and third harmonics = 3/2. Third and fourth = 4/3. Do you see a pattern? We are adding a whole number to both the numerator and denominator for each successive interval. Now that we have a set of intervals that we can describe mathematically, let's see what we can do with them. How about multiplying them together? If I know that multiplying one frequency by say, 4/3, gets me a note a certain interval above the original, can I multiply AGAIN and get the same thing but above my new note? (Yes.) So finding the frequency of a note TWO "4/3 intervals" above the original would be 4/3 * 4/3 = 16/9 times the frequency of the original.

As you no doubt can predict, the more we multiply intervals (which are improper fractions) together, the larger and larger the ratios become. Is there any way to simplify them? Here is where I admit our first non-derived assumption: two frequencies that vibrate a power of 2 apart from each other (ie 2/1, 4/1, 8/1) are the same pitch. Essentially what I am saying here is that notes an octave (or multiple octaves) apart are still considered the same note. So, if the value of a ratio exceeds 2 (numerator is more than twice the denominator) then I know that interval is larger than an octave. And so if I divide it by 2 (or multiply by 1/2) until the numerator is no longer more than twice the denominator, I know that that interval is smaller than an octave. If the ratio between two pitches equals a power of 2, then they are the same note.

So armed with all these tools, I am going to try using any of these ratios as a Generating Interval to see how many distinct pitches I can get until I hit a power of 2, which I have assumed means that I have reached the 'same note' as the starting pitch. Let's play.... first up: 5/4. I don't know what interval this is, but how many times can I repeat it until I get back to the octave? 5/4 x 5/4 x 5/4 = 125/64, which is roughly 128/64 = 2 (a power of 2). So it looks like we could only use 3 of those intervals until we hit the first octave. OK let's try the next one, 6/5. I multiply this four times, and hit 1296/625 which again roughly equals 2. Damn, can't I break past 2? Maybe I can try using a simpler ratio, like 4/3.

K, I'm multiply (4/3) by itself again and again and still not hitting anything that looks like a power of 2. By the 12th time, my ratio is 16,777,216 / 531,441. Let's divide ... 31.57 .... Oh yeah! That's so close to 32, or 2^5! It seems that after ascending the interval of 4/3 12 times, I have finally landed on a power of 2, that is, landing on the same note as I started.

Fine, so if changing our generating interval from 5/4 to 4/3 helped us get 12 (as opposed to 3) distinct pitches before coming full circle, maybe I can use an even simpler ratio and get even more pitches! Let's try 3/2. (3/2)^12 = 129.76 = 2^7. Still only 12 pitches came out of that effort. So 12 seems to be the maximum number of distinct pitches we can derive from the intervals found in the harmonic series, and since we ascended the same interval each time as we calculated those 12, it follows that each one of those 12 pitches are equally spaced.

And that is why Western music has 12 pitches. Yes Western. Other cultures have different scale-construction systems which I'm not going to get into here because frankly, I don't understand them. Tomorrow we will discuss the intervals as you know and love them, by their actual names! Yaaaaay!

Note: You saw that when I landed close to, but not on, a power of two, I deemed it 'good enough' to call it the same note. If we were to be sticklers and say that we only hit the same note once we straight-up get a real power of 2, then we would be waiting forever. As long as we're using a generating interval derived from the harmonic series (that is, a whole number ratio), there is no way to truly bound the number of notes produced within the octave. This is because the harmonic series diverges! It explains why notes can go infinitely high and why it's possible to create a smooth 'slide' between two pitches. There are ways to use non-harmonic generating intervals to create 21-, 23-, 27-note systems, or really whatever you want. But if we want stop the madness, base our note system in the harmonic series and keep the notes more-or-less equally spaced, we have to accept that 31.57 = 32. Attempting to reconcile the inherent mathematical imperfections with the idea that all notes should be equally spaced is what led to the development of 12-Tone Equal Temperament, which Bach demonstrated with the Well-Tempered Clavier.


Edit: I received a couple questions asking to clarify what makes an octave so special and to re-explain the ratios between different harmonics. As follows...

The fundamental frequency can be any arbitrary frequency. I intentionally wasn't using any specific one to define our fundamental, to drive home the point that these properties are universally true. However, the upper harmonics that can be generated from that wave are dependent on our choice of fundamental. That is because the frequencies of the harmonics are multiples of the fundamental frequency.

When we isolate the (naturally occuring) second harmonic, we are in essence muting the first harmonic and hearing all the harmonics that are left over. (Those harmonics were actually there the whole time, but the first harmonic is much louder.) The second harmonic has a wavelength half as long as the first harmonic, as you can see in the diagram. Since v=f*λ (and v=the speed of sound which is constant), that means that the second harmonic has a frequency double that of the first harmonic. Through some mix of pitch perception and cultural context, we have come to identify the interval formed between a note of frequency f and another with frequency 2f as "the same note" despite the fact that they are obviously different frequencies. That is why we have the concept of octaves - it's a way to express the difference between two notes that have the same same pitch class (i.e. C, Eb, F#, etc) but different frequency. Mathematically, we say that doubling a frequency has the effect of adding an octave.

You can demonstrate this by playing an open string and then touching it lightly at the 12th fret, which is exactly half the length of the string. You'll hear the octave come through - that is the second harmonic you're hearing. Note that you can do this on any of the strings, which shows that it doesn't matter what our fundamental frequency (can be E, A, D, G, B whatever) is, the second harmonic is still always an octave above the fundamental.

"so, does "original" mean the first of those two notes, and the note you get by multiplying by 4/3 is the second note, and so you have an interval between those two notes"

this is completely correct. But I will go over it again to make sure you understand.

Let's say our fundamental frequency (aka first harmonic) has the frequency f. We just determined that the second harmonic has frequency 2f.

If you look at the diagram, you will see that the third harmonic has a wavelength 1/3 the size of the fundamental. Therefore its frequency is 3f. Fourth = 4f, Fifth = 5f, etc.

So if multiplying the fundamental frequency by a whole number gets us the pitch of whichever upper harmonic corresponds to that number, then we can choose to define the interval between the two notes (in this case, the fundamental and the upper harmonic) by the factor by which we multiplied. That is, the interval between the 2nd harmonic and the fundamental can be called "2:1." Or the interval between the 3rd harmonic and the fundamental can be called "3:1." OK great - but what if we want to find the interval between harmonics 3 and 2? That would be the ratio of the frequencies 3f and 2f, otherwise known as 3:2. That's the interval we call a fifth. (On the guitar, it's the interval between the 12th fret harmonic and the 7th fret harmonic)

Therefore if you multiply any frequency by a factor of 3/2 you will get a second note that forms that interval (a fifth) with your starting note. Again, your starting note can be completely arbitrary. Let's think of it this way - I want to find the frequency of a note a fifth above frequency m. Via the above means, it would be 3m/2. But we could also say, if we call m our fundamental, the 3rd harmonic will sound like a fifth, just an octave too high. So we'll take that frequency, 3m, and then subtract an octave by dividing the frequency by two. Voila, 3m/2.

3/2 is the multiplicative factor for a fifth. The other intervals are represented by other multiplicative factors... the so-called "Perfect intervals" are whole number ratios like 5/4 and 7/6 etc. However if we use the interval of a fifth to GENERATE those other notes (ie- start on C, go up by 2 fifths and you'll get the note D, so you have created a major second by using a fifth to get there) the ratios will be slightly different, this is what I allude to in blue at the bottom of part 1.