Kpopalypse’s music theory class for dumbass k-pop fans: part 8 – harmonics, equal tempered tuning and the circle of fifths

The Kpopalypse music theory post is back!  This post takes a look at harmonics, tuning systems and the circle of fifths!  What do harmonics have to do with tuning and the circle of fifths, how is the circle useful, and how would readers rank the Loona members if they had to allocate each one a pitch?  At least two of these questions are probably answered in this post!

This post combines some knowledge already shared in the posts about determining keys, harmony basics and timbre, and expands on these areas, so if you’re a music theory noob you may wish to read over those posts again so this one might then make more sense.  Of course if you’re just here for the k-pop references and stupid humour this may not bother you, and if you’re super-clever and already know this stuff you may not give the ass of a rodent – the choice is yours!

In the post on timbre I briefly discussed harmonics, and their relationship to how we hear different types of sounds differently.  We learned that when touching a vibrating string at various mathematically equally divisible points, instead of the vibration stopping completely, it divides.  This physical interference with the string removes the original “fundamental” vibration, but leaves a higher vibration intact.

Fig. 1 – Dividing a string’s vibration to produce harmonics. The “fundamental” is the original vibration, at the top, before any interference.

The above diagram shows where you would place your finger on a string, and the below one shows the relative relationship of notes.

Fig. 2 – the difference in notes between each vibration. The fundamental is at the bottom of this chart. The “first overtone” in this chart correlates to “2x harmonic” in the diagram above this one, and so on.  Note that it’s not the absolute pitch but the relative distances between the pitches that is important here – the note C is chosen as the starting point, but the starting point could be any note.

The note produced by this higher vibration is called a harmonic, and there are a theoretically infinite amount of these, however in practical terms once you get above about a dozen or so they become impossible to isolate or hear clearly.  They also begin to have less and less relationship to the equal-tempered musical scale (I’ll explain what that is shortly).  Below is the fundamental pitch plus the first 19 harmonics.  Above each note is the “error margin” in cents away from equal temperament.  Cents = an interval measure which divides the semitone into 100 steps, so the +2 about note 3 (the second harmonic) means that this note is 2% sharp.

Fig. 3 – the harmonic series fromt he fundamental up to the 19th harmonic.

When Pythagoras was dicking around in his religious cult figuring out all this vibration stuff in 234686 BC, he noticed that the first overtone was an exact doubling in frequency of the fundamental pitch, but the second overtone (the 2% sharp one) was not exactly a doubling of the first harmonic, but significantly less vibration.  It wasn’t exactly halfway between the first and third overtone either, but instead was a ratio of 3:2.  His next step was to get that second overtone, treat that note as if it were the first overtone, and produce another second overtone out of this note to make a series of notes.  The result was this:

Fig. 4 – each note is the “second overtone” of the “first overtone” note before it.  The final note is a B#, which is the same note as C, meaning that the series has wrapped around to the start.

This produced twelve unique steps of notes before wrapping around to the start, and this is how the twelve notes in modern music were created.  However, there was a problem, which is that +2 error margin described earlier.  As each note was built on the overtone of the note before it, those +2 errors started to stack up, so by the time you got to note 11 it actually started sounding a bit less harmonious – the pleasing synergy when the notes were played together was lost the further one strayed from the original fundamental.  Worst of all, but the time you got to the very end, the final note didn’t really sound like the starting note, but slightly off – off enough to definitely sound “bad” or “out of tune” even if every note did have a sound mathematical relationship to the note before it.  The super-disgusting final jump from the very wrong note 11 back to the start in some tuning systems is actually called a “wolf interval“, that’s how drastic the effect is, that composers back then felt justified in comparing it to one of Exo’s worst songs.  The original solution to this dilemma was to start a mirror series in the other direction – rather than making the second harmonic the first harmonic and then creating a new second harmonic from that starting point, what would happen if we instead made the fundamental note the second harmonic, and created another first harmonic underneath it?

Fig. 5 – the same series, but stepping in the other direction. The bb is a “double-flat”, or two semitones below the note specified. For readability reasons, double-sharps and double-flats don’t tend to pop up in sheet music very much, but they’re useful for theoretical discussions like these just to demonstrate the note relationships and that each note is technically still a “fifth below” the previous note.

So now we have two different series, one going in each direction.  These series obviously meet in the middle, halfway through at the F#/Gb.  The result is this:

Fig. 6 – the circle of fifths. Regardless of which note you use as the starting position, each note is the second harmonic of the note immediately anti-clockwise in the circle.

Using the system of pure 3:2 ratio intervals, we have a bunch of +2 cent errors accumulating on one side of the circle, and a bunch of -2 errors accumulating on the other side.  This is because C to G is 3:2, but G to D using the same system means that C to D is 9:8, C to A ends up being 27:16, and so on, eventually you get to stupid numbers like 1024:729 for C to Gb, which meant that if you tuned using C as your base, playing a song in C Major sounded “nice” but playing a song in Gb major sounded “odd” because you were touching on more of the “high ratio notes” that tended to sound uglier.  Anything beyond seven notes was considered to sound even worse than Little PSY’s “Show Time“, and that’s why we have seven-note or “diatonic” scales in western music.  Pythagoras had a system for trying to fix the suckitude by fiddling with the ratio calcuations, and so did others, but no matter which ratio was used to base it all on, it can never be “perfect” because the fractions don’t add up neatly.  I won’t go into it here, but click the links if you want to get into the maths of it, it’s one of those weird mathematical things that doesn’t have a straight answer, like “recurring” decimals or the exact value of pi (π).

By the time the 19th century rolled around, musicians finally said “fuck it” and devised the “equal tempered” scale by just dividing the whole octave into twelve equal parts, so each of the twelve notes is exactly the same mathematical difference apart.  This makes the fifth sound “less pure” because it doesn’t exactly correlate to the sound produced by the fifth harmonic (it’s 2 cents out) and each note is in fact slightly “wrong” in terms of the way the harmonic vibrations relate to each other.  It’s a compromise system which is what all modern music is now based on, but it has the benefit that now each key sounds equally slightly dogshit instead of C Major sounding great and each subsequent step around the circle sucking more and more heavily.  You don’t need this knowledge to be able to play music or hear it, but it’s helpful to know that the circle of fifths was originally constructed in this way, and it’s not a completely arbitrary random construction but one that was originally based on the harmonic content of vibrations.

While Pythagoras was fucking around with this shit, so were the Chinese, and nobody really knows who discovered all of this vibration stuff first.  However the Chinese also noticed the accumulation of “sonic errors” and had a lower tolerance for dogshit-sounding notes than the Greeks, so they didn’t bother with “diatonic” scales, and stopped their series at five notes instead of seven – anything beyond the fifth note was considered to be undesirable to the ears.  Hence traditional Chinese music was all built around five note “pentatonic” scales.  If you want to hear the sound of a song written with pentatonic scales, JYP likes to write melodies using them a fair bit.

So now that we have our circle of fifths, what can we do with it?  Well, you can make a funky diagram of key signatures like this:

Fig. 7 – The circle of fifths, with key signatures. The letters around the outside are the major keys, the letters on the inside are the minor keys.

By starting on the note shown and by either applying the T T S T T T S pattern, or the T S T T S T T pattern (from the keys post), you will get major and (natural) minor scales respectively, and the above key signatures.  Having a chart like this takes a lot of the brain-work out of figuring out key signatures and relative major/minor keys, and that’s why music teachers generally just get you to memorise a chart like this, plus the circle of fifths, and just kind of ram it down your throat mindlessly without bothering to tell you why, it’s because they either don’t understand or just don’t care about everything else I went through in this post.

It’s worth noting that while all keys do sound the same these days thanks to equal temperament, keys do sound a little different when tuning with other systems.  That means that since all keys are potentially different, and all Loona girls are different, and there are twelve of both, we should be able to accurately allocate a key to each Loona girl.  So now is the time for you the readers to make an important selection!  Which Loona girl best fits which musical key?  Fill out the form below, or if the embedded form doesn’t work, click the picture of the twelve Loona girls below to bring up the form as a separate webpage!

Fig. 8 – The Loona girls. First row: Heejin, Hyunjin, Haseul, Yeojin. Second row: ViVi, Kim Lip, Kinsoul, Choerry. Third row: Yves, Chuu, Go Won, Olivia Hye.

That’s all for this post!  The music theory series will return with more episodes in 2020!

4 thoughts on “Kpopalypse’s music theory class for dumbass k-pop fans: part 8 – harmonics, equal tempered tuning and the circle of fifths

  1. I remember that fuzzy cats go down alleyways eating birds.

    I think we should return to 18th century well tempering. Having everything be out of tune so that some chords are less out of tune doesn’t seem to me to be a good compromise. Also, well tempering gives each key its own unique color, which I like. It makes stuff like Bach’s Well-Tempered Clavier more interesting.

    • “Fuzzy cats…” never heard this before. If my high school music class had used this I reckon our class may have had a hope in hell of actually learning something.

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