foo for thought

Science explains quite a lot in the modern world. I’ve been pretty fascinated with music throughout my life so I thought, by extension, this must hold true for music too? I know that pitch is simply just a vibrating wave, but in society, music holds lots of emotional value and depth. Music conveys a myriad of emotions, and is one of the only universal languages understood by everyone on the planet. How can certain songs sound better than others? Today I’ll provide some insight on why that is by explaining it through science and mathematics.

Sound waves are sinusoidal, which means they follow an oscillation similar to the sine graphs you learned in high school. The higher the frequency, the shorter the wavelength and the higher the pitch. In fact, we can start with recognizing that doubling the frequency of a sound wave raises the pitch an octave. Multiply any Hz by two, and you’ll hear that note but an octave higher. Similarly, halving the frequency lowers the pitch an octave.

This is what two pitches an octave apart look like when plotted. Notice how they intersect at 2pi, 4pi, and repeat the pattern. Because these two waves share this commonality, this is what I believe is what causes the sound of resonance. An octave sounds extremely pleasing to our ear, because it is the lowest whole number multiple of the base frequency, that number being 2.

Musically, this is what this interval looks like, using C as an example.

What happens if we increase this multiple to 3? What happens if we have a wave that oscillates 3 times as fast as the base wave? This is the perfect 5th interval. This is why a perfect 5th sounds so good to us, is because it’s literally just the second lowest whole number multiplied by our base frequency.

Even when plotting all three frequencies (the base, 2 times the base, and 3 times the base), we can see how they all intersect still at multiples of 2pi! Ah, resonance.

Similarly musically, we can see from moving to the 2nd harmonic (C up an octave) to the 3rd harmonic (G) gives us a perfect fifth chord. Already we can see a chord forming – looking pretty similar to how scores are orchestrated nowadays with those separated octaves in the bass.

Multiplying the base frequency by 4 is just multiplying it by 2 twice, so it’s 2 octaves above the base. More interestingly is if we multiply the base frequency by 5, a prime number! This results in a wave that moves five times as fast as our base frequency, what pitch would that be? The 3rd most satisfying interval to listen to – the major third! This is why major chords sound happy, because they make our ears happy! They follow the rules of physics and are literally the lowest multiples of a frequency you can get to in whole numbers.

Moving from the 4th to 5th harmonic, we can see how this is literally just a major triad forming! Much higher up the harmonic series is where we get to be a bit more dissonant – the 6th harmonic is again our perfect fifth, but the 7th harmonic (base frequency times 7) is where we discover our deviant – the minor 7th interval. Interestingly enough, it’s used a ton as a leading tone in music theory, but this is where the “resonance” seems to end in my mind. Major chords sound happy, dominant chords do not sound happy.

This is the final graph of the harmonic series I’ll subject you to – but just look how neatly up to the 5th harmonic even looks at the 2pi interval!

The really really cool part about this series is it actually has a name – the overtone series! Anyone who plays a brass instrument should at least be a bit familiar – with the same fingering, you can produce many different notes just by changing your embouchure, bringing you up the overtone series! The cool thing about the overtone series is that every sound that we hear is actually a combination of pitches from the overtone series. The severity of each of the overtones is what gives certain musical instruments their own specific timbre! This is a really cool demo that demonstrates this effect – you’ll see as you play around with the overtones, even if you don’t select the base frequency you’ll actually still hear it being played – our brains are pretty cool, huh?

So onto the interesting part of my discovery – I wanted to see why certain chord progressions sound good. I thought if we could figure out what makes certain chord progressions sound good, we can begin to understand what makes music sound good.

We already saw how easily we can reproduce a major chord just by using the overtone series – the 2nd, 3rd, and 5th harmonics already produce our major triad. With respect to our fundamental frequency, I was wondering, how does a V chord fit into the overtone series? I wasn’t really expecting to find anything, but I found something:

In blue we have highlighted all of the notes in the overtone series that cover our major I chord, the C major triad, and in red we have all of the notes in our V chord (G major). I noticed something unsurprisingly surprising – you can actually map the system of harmonic numbers to the V chord by multiplying all of them by 3. Or in more nerdy number theory terms, the set of harmonics that comprise a V chord = 3 times the set are harmonics that comprise a I chord – ie the sets are isomorphic.

Unsurprisingly, you can also do this with every number if you multiply it by the original set:

1	1-C	2-C	3-G	4-C	5-E	6-G	8-C	10-E	12-G
2
3	3-G	6-G	9-D	12-G	15-B	18-D
4
5	5-E	10-E	15-B	20-E	25-G#
6
7	7-Bb	14-Bb	21-F	28-Bb
8
9	9-D	18-D
10

You kinda start to see how we walk around the circle of fifths – disappointingly not in order, but sorta. The point that should be taken from this is not how it walks down some arbitrary circle we’ve constructed in western music, but recognize the relativity between pitches and their origin.

We can start to see this with the most similar resonant series to our root – the resonant series received starting on the fifth, which gives us our major 5 chord. It almost seems like in music, multiplying a frequency by 3 gives you the next pitch in the circle of fifths, making 3 a surprisingly important number. Dividing a set of frequencies by three gives you that chord a fourth a way, creating that nice resolution we all are aware of in the V-I cadence. Similarly, taking the 9th resonant series on our chart gives us a D major chord, which is the V/V chord – resolving downwards to G major, our 3rd resonant series.

It’s important to note that we can actually create “leading tones” by adding additional notes in the harmonic sequence. And we know they’ll sound good because science says so! This is why the V7/I resolution sounds just as satisfying as the V/I – why does a random accidental sound good? Because it’s part of the harmonic series!

Anyways, this was my exploration on why music sounds good. Whole numbers multiplied by frequencies create more frequencies that sound good when played together to our ear. This explains fundamental music theory on a mathematical level – if you managed to make it this far, hopefully you’re alright, and I’m glad someone else can nerd about math and music in the same capacity that I am.