It was King Solomon who lamented that “there is nothing new under the sun”.  This is borne out again and again: story plots, terrorism, facial hair, blog posts, popular music.  Let me demonstrate the latter to you, or rather, let Axis of Awesome demonstrate it:

Next time you listen to the radio, you’ll hear those chords.  In fact, you’ll notice it a lot now you’re aware of it.  This is called the Baader-Meinhof phenomenon, like when you settle on which new car to buy, and suddenly it’s everywhere.

Musically, that chord sequence (I-V-vi-IV) just seems to work, and is seemingly infinitely reproducible.  Mathematically, harmony makes perfect sense, but why?  How do notes (and, by extension, chords) actually work?

You probably know that sound travels in waves which are created by vibrations.  We describe and analyse waves using trigonometry – sine and cosine – which you first meet in school in the context of right-angled triangles (triangles and music: who knew?)  A sine wave looks like this, I’ve labelled some useful terms:

Louder notes come from waves with bigger amplitude, higher notes from those with shorter wavelength, which means they have a higher frequency (more cycles per second).  You can see these ideas demonstrated beautifully in the following video, which you can reproduce yourself by putting your phone inside a guitar and facing it towards the sun.  Notice how the vibrations producing the lower notes make wider waves (bigger wavelength, shorter frequency) and how the waves are taller when the strings are first plucked, but as the sound quietens the amplitude decreases:

Now the frequency of a wave is the important thing when considering how notes work together.  Notes one octave apart (like “some” and “where” in Somewhere Over the Rainbow) work together so beautifully because the higher one has double the frequency of the lower one.  You can test this on a violin or guitar – play an open string, then put your finger in the place that divides the string exactly in half and you will now hear the note exactly one octave higher *.  What this means is that the two notes complete a cycle together, and hence reach your ear at exactly the same time – they work in perfect harmony.

The notes in a scale of C major are C, D, E, F, G, A, B, C.  Our roman numerals above correspond to notes in the scale, and chords built upon these notes (I-C, II-D, etc).  A major chord, such as C major, is made up of the first (or root) note (C), the third note (E), and the fifth note (G).  Let’s look at the waves of these notes together:

You can see that the cycles coincide almost completely when the C wave has done four cycles, the E wave has done five cycles and the G wave has done 6.  This means the waves coincide fairly regularly, the notes reach our ears together fairly regularly, and they sound good together.  These regular coincidences mean the wavelengths of each note are in ratios that are close to “quite nice” fractions.  The wavelength of C is roughly $\frac{5}{4}$ times bigger than that of E and $\frac{6}{4}$ or $\frac{3}{2}$ times bigger than that of G.

In contrast, look at the waves of C and D, they first come close to coinciding when the C wave has done 8 cycles and the D has done 9.  The irregularity of this coincidence means those notes aren’t half as pleasing together.

So, when wave cycles coincide more frequently, we hear more harmony, which explains how the most common chords are formed, but why is the I-V-vi-IV progression used so often?  I don’t have such a mathematically precise answer here, but let’s think about it for a moment.  In the scale of C major you start by playing the root chord of C major (C-E-G, remember).  It feels natural to use that fifth note of G again, due to the harmony in the first and fifth (three cycles of G coincide with two cycles of C as we saw above), so we play G major (G-B-D).

There’s some lovely symmetry in the IV chord (F major): including C you play 5 notes ascending to reach G and 5 notes descending to reach F, so that interval of a fifth, which is so pleasing to the ear, appears again in the other direction.

The third chord is A minor (the lower case roman numerals are used for minor chords, the “sad” chords) and this minor chord starting on the sixth note is significant in music – it is the relative minor, the minor key that contains exactly the same notes as the major key (A minor proceeds A-B-C-D-E-F-G-A, hence it is the relative minor to C major).  The introduction of this minor chord works as it is so intrinsically related to our first chord and creates a short change of mood.  After all, variety is the spice of life, but we don’t want too much of it, we are creatures of habit of course!

So it would seem that the progression I-V-vi-IV is the simplest way to use the most pleasing harmonies according to the mathematical principles underlying acoustics.  Either that, or someone somewhere came up with a catchy sequence which everyone else copied.  Just like King Solomon said they would.

* Now divide this half length of string in half again, you will have a note one octave up.  Can you see why the frets on a guitar get smaller as you go up the neck?