#harmonic series

[galaxianmothman]

Introducing the cutest song I've ever written.

[victusinveritas]

The top image is a graph of the first nine harmonics of a harmonic series. The bottom image is an x-ray of the inner structure of a conch shell.

[orchidsslideontongues]

[oneefin]

i usually find youtube comments sections pretty low-brain, but one of my favorite bits of cleverness i've ever seen came from a youtube comment under a math video of someone explaining how they did it using music theory. and this is right up my alley so i'm going to break it down.

the problem in the video was "approximate log_2(3) without a calculator", and they did it using purely mathematical methods and a tiny bit of memorization. but you can actually do this with a basic understanding of tuning theory:

basically, there's this thing in music called the harmonic series. its a sequence of notes starting from some root pitch and ascending in whole number multiples of that pitch. for example, if you start from C:

each note in this sequence has a frequency n times the frequency of C5. get it?

now, even though this would be the "perfect" way to tune things, we don't actually tune notes this way - because it doesn't work if you change your root note. instead, western music is tuned to what's called 12-tone equal temperment (12tet), in which the ratio between two adjacent notes is exactly the same:

the ratio x is picked such that an octave is still x2. so x turns out to be the 12th root of 2.

anyway, how does this help with the original problem? well, these concepts give us two ways to think about a similar musical interval. consider the difference between C and G:

in the harmonic series, this distance is a x3 multiplication. however, in 12tet, this is 19 steps, each multiplying by the 12th root of 2. written another way, that gives 19/12 = log_2(3).

19/12 is 1.58333..., while log_2(3) is about 1.585. that's accurate to two decimal places.

in fact, the approximation isn't just really close - it can be considered optimally good among other fractions of a similar length. that's because mathematicians have a way to derive best fractional approximations for any irrational number, and on that list for log_2(3), 19/12 is there:

that's a cool solution to an arbitrary math problem, but also it's quite cool from a music theory perspective! it gives a reason for why it's not only appropriate for the western scale to has 12 notes in it, but that it'd be surprising for it not to have 12 notes in it, given that 19/12 approximates the 3rd note in the harmonic series so well.

[kelemengabi]

Coherence octave entanglment

[kelemengabi]

golden ratio

[matriarca-inodora]

Ok this will probably be really random, but here it goes: would anyone be interested in a post about how Apollo is in some respects a god of abstraction and mathematics?

(also, I´m currently working on some posts on epithets so it would be nice to have a bit a of a crazy theory post to balance my brain)

[matpisound]

ok so my mind was just blown

Music always finds new ways to be the most amazing thing of all time, and today I learned about what could be the most mind blowing thing in music theory that exists to this date, and it's been around since the birth of the universe itself. Yes, I am talking about the harmonic series, the almighty combination of math, physics, and music that gives us the foundation of every pitch we hear.

But first, what is the harmonic series anyway? Before we get into that, let's discuss how we even hear pitch in the first place. The actual mechanics of it are simple; a thing vibrates, which causes the air to vibrate, and if that vibrating air reaches our ears, our eardrums vibrate, which our brain then interprets as pitch. The frequency of the vibrating air determines how high or low the note is.

Now that we know how pitch works, let's take a listen to a piano, playing A4, which is at a frequency of 440 hertz:

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Simple enough right? Not much going on here, or so you thought. Every single pitch we hear creates overtones, which are additional frequencies dictated by the harmonic series. Before we go any further though, there is one notable exception to this rule: sine waves. Sine waves are the purest form of pitch that exist, and they don't produce any overtones. Here's an example of a sine wave at 440 Hz, the same pitch as the piano above:

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The overtones (or harmonics) are sine waves, which means the overtones don't have overtones of their own.

These harmonics naturally occur everywhere, and there is a real simple formula for how they're created. Let's take a look at that pitch from earlier, 440 Hz. We'll call this the first harmonic, or the fundamental harmonic. If we multiply this by 2 to get 880 Hz, the pitch goes up an octave to A5, the first overtone or the second harmonic. Multiplying the fundamental by 3 gives us our third harmonic or second overtone, which is about E6, a fifth higher than the last harmonic. Each overtone has a frequency of an integer multiple of the fundamental frequency. If we visualized this as a vibrating string, this is what we get:

Okay, cool. What does this have to do with anything? Well I'm glad you asked! The volume of these overtones is what gives each sound its timbre, or musical texture for lack of a better term. For example, here's an audio of the piano sound from earlier followed by the same note on an electric piano:

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They sound completely different right? The majority of the differences in sound are because of the presence of these overtones and how present they are or aren't. It applies to sounds on a single instrument as well. Stringed instruments employ this a LOT. On a violin, viola, cello, or bass (the orchestral kind), where you drag the bow is extremely important to getting the right sound because of the overtones. Electric guitars and electric basses exploit the sound changes by allowing musicians to select where the vibrations of the string are being picked up. The pickups (see below) are placed to allow some variation of tone when the guitar is plugged in. By switching the pickup selector to only use the pickup closer to the middle, it creates a sound where the overtones are less present, and using only the pickup closer to the end creates a sound where the overtones are more present.

But wait! There's more! I hinted at this earlier, but the harmonic series gives us musical intervals that help us make nice sounding harmonies. The interval between the first and second harmonics is an octave; the second and third, a perfect fifth; the third and fourth, a perfect fourth; the fourth and fifth, a major third -- the list goes on. Essentially, these are natural intervals that sound super pleasing to the ear because nature was just built like that.

The ratio between the frequencies of each harmonic plays a role as well. This is where the math comes in. The third and second harmonics have their frequencies in a 3:2 ratio, and that ratio always designates a perfect fifth. The sixth and fourth harmonics have a 6:4 ratio, which can be reduced to 3:2, so they also make a perfect fifth. The same principle applies to any pairs of harmonics that share a common frequency ratio. Hoo boy that was a mouthful. Here's an interactive program that demonstrates the harmonic series with sounds and a lot more other details that would make this post way longer than it already is.

One thing that I feel definitely should be here, however, is that modern music doesn't use the nice sounding intervals of the harmonic series; at least not all of them. You see, the harmonic series is the foundation for what we call "just intonation." The intervals between notes are taken from the harmonic series. This creates a problem when writing music, because these intervals are all relative to a single fundamental frequency, meaning simple harmonies within the key of the fundamental are the only ones that actually sound good. In order to write more interesting music that still sounds good, musicians decided to adopt a system known as "12-tone equal temperament," or just "equal temperament." What this does is it takes the ratio of the octave, 2:1, and uses that as a base. Then, they space 12 frequencies equally between the octave, which gives us something with a lot more consistency between different keys and funky harmonies. As music evolved during the periods following the Renaissance in Europe, the 12-tone equal temperament system came about as composers began writing more complex music. That doesn't mean that this new system is dissonant; there's a lot of beautiful music that uses this system, and by "a lot" I mean pretty much everything you hear. It also doesn't mean that we shouldn't use just intonation either. It can be useful for chord voicing when in an ensemble setting. When my band director tunes major chords, he has the people playing the third play slightly flatter than equal temperament and the people on the fifth play slightly sharper, as is in the just intonation system.

Anyway, the harmonic series is something that really fascinates me because it's so interesting and so fundamental. There's so much about it that I couldn't get into here and so much more I don't even know yet. But yeah, thanks for reading all the way through and I'd love to talk with y'all about this more if you have anything to add or anything you want to learn more about. I have a lot more ideas for posts so get ready for those in the near future, and with that, I'll catch you at the double barline!