On my plane ride back from Seattle last night, I started a post about one of my favorite albums. I unfortunately quickly ran into a problem. It’s impossible for me to write about music without dipping into theory. Rather than hold you hostage ranting about chords without context, I thought it would be helpful to explain some theory before getting into serious music writing.
We’ll start with understanding some of the basic physics, move to explaining a common topic in music theory, and end with applying our knowledge to a specific chord progression.
(I’ll probably end up linking this article in every music post that I write.)
Sound is a wave. And a wave at its purest is modeled with a sine function.
A note’s pitch is determined by the frequency of the waveform. Notes with a longer wavelength have a lower pitch, and notes with a shorter wavelength have a higher pitch. The amplitude of a wave determines its volume — the larger the amplitude, the louder the note.
This is an example of what a sine wave sounds like. Please listen to the whole video.
If you listened to the whole video, you are too trusting. Don’t listen to anything I ever say.
But two things I hope you get out of listening to this:
Sine waves sound very “pure” (we’ll get to what this really means)
This doesn’t sound like anything that naturally occurs in real life.
So how do we get from this synthetic sound to the sounds we hear around us every day? An explanation lies in the concept of overtones.
Overtones and Timbre
Overtones are essentially higher frequencies that make up the sound that you’re listening to. If you sing the note “A”, it isn’t a pure wave that looks like the sine curve above. There are actually many higher notes that your ears don’t recognize as a separate tone. This isn’t the most intuitive concept in the world, so let me walk you through a demonstration.
There are a few other basic waveforms that producers use to create synthetic sounds.
Rather than model a complex sine wave with capacitors, the first synthesizers could utilize easier circuitry to produce square, triangle, and sawtooth waves. These may sound a bit more familiar to you — think of your typical 80’s era synth pop, or a chiptune retro video game soundtrack.
Below is a demonstration of a homemade synthesizer generating these four waves. Look out for the waveforms for sine, square, triangle, and sawtooth, and try to think about how each of them sound when they look the purest in the oscilloscope. Actually try to watch the whole video, for real this time. (Headphones warning — there are some high frequency noises.)
So what do these different waves have to do with overtones? Turns out, there are ways to approximately model these new waveforms by adding together different sine waves with different amplitudes and wavelengths. This concept is called additive synthesis.
People much smarter than me have calculated how to model all of these waves with infinite sine functions. I don’t want to get into the math of it all, as I am just so overwhelmingly unequipped to do so. (If you are an electrical engineer, you may notice a parallel between this concept and the Fourier Transform. Unfortunately, I am not an electrical engineer). The main takeaway, however, is that to generate these non-sine waves, you can combine a specific combination of higher and higher frequency sine waves to approximate the waveform. These sine waves are the overtones.
I think a really great way to illustrate this is by actually seeing & hearing it in action. Below are videos where a pure sine wave is modulated into each unique wave by adding more and more sine waves. You can hear each sine wave being added — and as more and more stack, you lose track of individual notes and start to hear the tone as one with its overtones.
This principle applies to real world instruments as well. Below is a graph of the sound of a violin. Notice that there is a general flow up and down. The frequency of the general curve (or to be fancy, the principal frequency) would be what we would identify the note as. But the imperfections in the wave from the overtones are what makes the violin sound like a violin.
These imperfections come from a variety of factors. Where the bow strikes the string, the shape of the instrument, even the room that its being played in can change the shape of the wave.
This is how we can differentiate different instruments from each other, even when they play the same note. Different overtones affect the frequency of the note being played for different instruments — simple as that! The term for this quality of sound is timbre (pronounced TAM-brr).
Quick tangent: Timbre can even help us understand spoken language. When we speak, our mouths and tongues act as filters to let specific overtones resonate more, while dampening other overtones. The result of this is different vowels. “E” emphasizes higher frequencies, while “O” keeps the tone relatively pure. You can even try to listen for it — try singing the letter “E” (not the note), then slowly transition to singing “O” while keeping the pitch consistent. Other than sounding absolutely crazy, you can hear the overtones gradually shift down as your mouth widens. We instinctively understand this as we develop our language processing skills, but this is rarely ever concretely acknowledged unless if you study linguistics.
This all is well and good, but it’s hard to connect these physics concepts to what you may have encountered about music theory. Here is my attempt to connect all of the dots.
If you’ve studied music at all, you’ve probably heard of the Circle of Fifths. It’s the cornerstone of modern music theory, and something that you probably slept through and don’t quite get.
I get frustrated because this is ordinarily taught as a concept that is inherent to music, where in reality it’s really the byproduct of some cool physics combined with a bit of number-smudging. The surface-level explanation is extraordinarily unintuitive: the twelve note scale can be re-ordered into “fifths”.
If you’re not sure what a fifth is — don’t worry! This isn’t necessary to understand, but if you’re curious, each note next to each other is 5 notes away in the Western major scale. For example, the Western C major scale goes C → D → E → F → G → A → B → C. If C is 1, D is 2, etc. then we arrive at G being 5. You can read the chart above as G being the fifth to C if C is 1. Then if G is 1, then D is the fifth. So on and so forth — we can arrange the 12 notes of the piano in this way. If this made no sense, please do not worry.
But why is this “fifth” so important?
Let’s take a listen to what a fifth actually sounds like.
Unscientifically, this just sounds good. Correct. Maybe even perfect. But why? Some might chalk it up to just how culture works — we’ve heard this interval in so much music that always sounds good. Others might think this is some weird thing where humans are wired to like this. An alien might come down and be repulsed by its dissonance.
Maybe, just maybe, all notes just sound good together! Let’s listen to some other intervals and test this theory.
Oh no.
Oh god.
Maybe there is something special about the fifth after all. “Leon,” you’re probably thinking, “you already wrote a transition from overtones into this section. It’s obviously about overtones!” And here is a gold star for you, you observant lil’ freak.
For a wave that is created with additive synthesis, it’s ideal for its subcomponent waves to not interfere with the principal frequency (the main wave). The best way to do this is to have the frequency of the subcomponent waves fit cleanly within the parent wave. That way, when the parent wave hits “0”, the subcomponent waves can also be 0. The way to do this is to have the wavelength of the subcomponent waves be 1/n of the parent wave. It’s way easier to visualize this:
Notice how in each of the waves, each of the subcomponent waves is 0 when the parent wave is 0, on each of the ends. Each wave’s frequency fits evenly & cleanly inside the parent wave. Another way of putting this - by the time that the main wave finishes one oscillation, the children wave is perfectly timed to have finished an oscillation.
This means that there are naturally occuring overtones that are common because of physics. These are special overtones, called harmonics. (Note: you may remember from high school math that the Harmonic Series is 1, 1/2, 1/3, 1/4, etc. Love love love these connections between math and music!)
Remember from our first section that the frequency of a wave determines its pitch. The frequency of a wave is simply the inverse of its wavelength, so if the wavelengths of the harmonics are 1/2, 1/3, 1/4, 1/5… of the wavelength of the initial wave, the frequencies of the harmonics are 2, 3, 4, 5… times the frequency of the initial pitch! So let’s pick a note and figure out what its naturally occuring harmonics are.
For some reason, 440 Hz is considered to be A. Don’t ask why, no one knows, it’s just something that we all got together and agreed upon. (Fun fact - the frequency of “A” has actually naturally risen over the years, and it will probably continue to rise.) So, if we multiply 440 by 2, 3, 4, 5, etc. we’ll get to hear the harmonics of A.
Here’s 440 Hz. If Substack lets you, I might leave this video on as we hear other frequencies in videos below, just to hear how it sounds. Feel free to turn down your volume so you don’t go insane.
Here is 880 Hz:
Notice how it sounds like the same note, just higher? This is an octave. Octaves work by multiplying waves’ frequencies by 2. That means not only 440 x 2, but also 440 x 4 and 440 x 8 are just the same note in different octaves. Let’s keep track of our harmonics with a crappy chart made in MS Paint.
Next, let’s listen to 440 x 3 - 1320 Hz.
As it turns out, this frequency is E, which is the fifth of A! If we play the videos at the same time we hear that it’s the same interval as the harmonic fifth. This explains the physics behind why the fifth is prevalent in music — it’s the first naturally occuring overtone. For all intents and purposes, you can think of the fifth as part of the initial note.
Let’s fill in our hastily-created chart. We know that 3 x 440 is E, and we can extrapolate our octave principle from before to fill in 6 x 440 as E as well.
If we break down any naturally occuring A, six out of the first eight overtones are composed of the initial note, A, and the fifth, E. This is why the fifth is important. It’s an inherent component of the original note.
What about 5x?
This note is C# — and if you know anything about music theory you’ll know that A, C#, and E form a major triad. Major chords aren’t a human-created construct; it’s inherent in the physics of waves. The relationship between the frequencies are naturally resonant! The universality of this is something that personally strikes me with awe — it’s truly beautiful that physics allows us to intuitively understand and appreciate major chords.
Let’s add this to the world’s worst note chart and move on to 7x.
7 x 440 is 3080 Hz, and I decided that that’s too high. I divided this by 2 so that it’s the same note, just at a lower octave — here’s 1540 Hz.
So far, you’ve been trusting me to tell you what note each frequency corresponds to. Every note so far has lined up quite well with a note on the keyboard. Or I could honestly be making this all up, since you’re probably not checking my work. (I’m not lying, but it would be funny.) Unfortunately, however, this lucky streak of fitting notes into the western scale unfortunately comes to an end here.
This note is between G and G#. It’s closer to G, but if we understand it as between these two, we can understand the natural occurence of two prevalent seventh chords: the A7 (A - C# - E - G) and AM7 (A - C# - E - G#), which are each major triads with one of the notes this tone is between. These two chords are both widely used in Western music. Although I’m not super versed in world music, I know that some non-Western scales use the pure harmonic instead of the Western approximation.
Here’s our filled out artisan hand-crafted draft IPA chart:
To summarize, sounds in the real world are made up of many different frequencies, known as overtones. Each distinct sound-producing object has a consistent set of overtones, known as its timbre, that allows us to unconsciously distinguish what it sounds like. Overtones that are resonant are typically of frequencies that fit within completely within its parent frequency, and the set of overtones that achieves this are called harmonics. The natural resonance of harmonics leads to inherent patterns between notes like the octave, fifth, major triads, and seventh chords.
Here’s a fun website to play with harmonics if you want to mess around more. And if things still aren’t clear, here’s legendary conductor Leonard Bernstein to explain it 20x better than I did:
Whew. Ready to move on?
Understanding Chord Progressions
Now that we have concepts of harmony in order, we can start talking about chord progression. Interpreting chord progressions can be relatively subjective, but one I’d like to try my hand at is the two-five-one progression. This is a progression that emerged from jazz, and because we now know about how harmonics relate to the circle of fifths, we can even discuss how this progression generates an emotional reaction. There’s a very specific feeling of setup, tension, and release that’s associated with the 2-5-1.
First, some examples of what we’re listening out for. Surprisingly, our first example comes from the land of video games — the Game Over screen from Super Mario World has a strangely beautiful and melancholy rendition of this exact chord progression. (It’s modulated to be a bit more complicated — the 2 and the 1 are both 9ths instead of 7ths, and the 5 is a minor 9th with a sixth added to keep the melody, but I think this is a perfect example to just get you listening for the progression). If you’re having trouble listening to when the chords switch, try listening to the lowest notes — when they switch, it’s the next chord.
Here’s another example of three 2-5-1 progressions in a row. This is “Tune-Up”, a song off of Miles Davis’s 1956 record, Blue Haze.
Another example - the main theme of Hayao Miyazaki’s Howl’s Moving Castle, composed by Joe Hisaishi, starts out by basically just stacking 2-5-1s on top of each other until the melody climaxes. (they start at 0:15)
Last example - THANK YOU by Tyler, The Creator off of IGOR basically repeats a 2-5-1 with a 3 chord afterwards to bring us back to the 2.
These are just four examples off the top of my head. There are so many in popular culture — once you start listening for it, you can’t unhear it.
To understand what the numbers of 2-5-1 actually mean, we need to return to the Western scale. Again, C Major is C→D→E→F→G→A→B→C. If C is 1, that means that 2 is D, and 5 is G. To explain it with our physics intuition, this means that the chords during each numbered section are going to be rooted in the overtones of the base note:
during the 2, we are going to use notes in the overtones of D
during the 5, we are going to use notes in the overtones of G
and in the 1, we are going to use notes in the overtones of C
How is this related to music theory? How did we chose 2, 5, and 1? Let’s take a look at where D, G, and C are in the Circle of Fifths:
We’re walking backwards in the circle of fifths! But why does this matter, and what does this mean? I mentioned before that there’s a feeling of setup, tension, and release associated with the 2, the 5, and the 1 respectively. By using our knowledge of harmonics, we can uncover how this exactly this is emotional response is generated.
To understand this we have to first discuss a principle associated with the harmonic discoveries we made earlier. Seeing the circle of fifths above, we may be tempted to think of it as a rotating wheel. C gets along with G and F, E gets along with B and A, and so on.
However, this isn’t exactly right. We discussed A’s notes in depth — particularly how it highlights the note E as the fifth. Nowhere in A’s related harmonic notes can we find D. In fact even if we keep going beyond 8 harmonics — all the way to 20 — we don’t even find D! Therefore, there’s an implied directionality to the circle of fifths. C gets along great with G, but hates F. D gets along great with A, but can’t stand G, and so on.
So we’ve established that the 2-5-1 is going against the grain of the natural direction of fifths. It’d be easy to move the opposite direction; C bleeds easily into G, and we could keep going that direction without much interference. But there’s no emotional payoff, no subversion, and no conflict. If we start at the 2 and go against the grain, we can hide hints throughout the progression to where we’re landing without overtly telling where we’re going to land.
This is why the 2-5-1 has emotional payoff. You can think of it as uncovering slowly where the harmonic root of the song is. The two is a hint — hmm, something’s building up, the five is building towards it but it’s not quite thereeeeuhhhhh, and the one is the whew! I see where that was going; the payoff of all of the setup. It tells a story. Because we’re building towards the 1, none of the chords beforehand have true harmonic resolution — they borrow elements from the 1 to create tension and dissonance within their own harmonies.
As a concrete example, instead of using the major third of D (F#), we use its minor third, F, in order to keep tension in the chord. We can’t use everything in D’s harmonies, because there’s no tension inherent to the chord. It’s too harmonious if we do this. In a book, if a setting is harmonious, there’s no point to the story. Same thing with music — if we give away all the harmonies too fast, there’s no payoff to the progression.
This is all kinda hand-wavey and woo-woo by my standards, but I think that this is the best way to illustrate technical theory without getting too into the weeds. We covered a lot in this post — all the way from physics to the emotional subversion of a specific chord progression. Hopefully you can hear what I’m talking about in the 2-5-1. I haven’t explained minor chords yet and I don’t particularly care to as this post is getting long enough. Let me know if this kind of content was interesting and I can do more music theory deep dives.
I’ll leave this as a puzzle for everyone left. There are 88 keys in a keyboard. Riddle me this — why is “perfectly tuning a piano” impossible?
Hint 1: Why did I say that number-smudging was involved with the circle of fifths?
Hint 2: How do I get from C to C via the circle of fifths? How do I get from the same C to C via octaves?
I’ll put the solution in my next post :)
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