Music Math (pt 1, major chord harmonics)

By now it’s out in the open, I am a music theory nerd and I spend far too much time over analyzing things.  With my engineering degree comes a large amount of math and mechanics, I’m not really that good at it, but I learned a lot about it. While my psychoacoustics post focused on the brains understanding of sound, this post will do the opposite, and I will talk about the physical properties of sound and how harmonics and chords can all be explained with math and physics.

A musical chord is two or more notes happening at the same time. Easy enough to understand, but some fiddling around with a piano will make you quickly realize that not any two notes can be played together and sound “good” to our ear. For example, playing two notes that are chromatically adjacent (one half step apart, the smallest unit of distance in harmonics) will produce a sound that, without context, sounds quite terrible. It doesn’t mean it isn’t a chord, it just means it doesn’t make sense mechanically. << I used this table for all of my information.

For example, lets say you play middle C, (C4, 4 referring to the octave it falls in across the entire range of musical notes) and at the same time play D flat or C sharp (which are sonically identical). These notes are chromatically adjacent, so it will not sound like a very good chord. C4 has a frequency of 261.63 Hz, D flat4 has a frequency of 277.18. Imagine the sound waves traveling through the air. As they oscillate, they are going to bump into each other a lot and interfere with each other path; this is what our brain perceives as dissonance. Our ear drum is attempting to resonate at two frequencies that don’t have many nodes or peaks in common; this is why it sounds “bad.”

Now let’s say you play C4 and G4 at the same time. C4 has a frequency of 261.63Hz, G4 has a frequency of 392Hz. At first this seems to be the same case as before, the waves will interfere with each other right? Actually, this is what we call a perfect 5th, the most magical and favorable chord to the human ear. Now it’s time for some math to explain why.

An octave is a doubling of frequency. C0 is 16.35 Hz and C1 is 32.7 Hz etc. When you play these two notes in unison, they sound different, but the same.  This is because these two notes have nodes and peaks that fall at exactly the same time. Two wavelengths will fit perfectly into one wavelength of the lower octave, so our ear is actually resonating exactly the same way, but faster. It works no matter how many octaves apart the notes are. So back to the perfect 5th, these frequencies don’t appear to have anything in common like the octave does. I digress, I have been deceptive. It becomes more obvious when you look at the lowest octave, and therefore the lowest frequencies each of the 12 distinct tones can occur at. C0 is 16.35, G0 is 24.5. These numbers do have something in common; they are both a multiple of 8.175. The next multiple of this number is 32.7, which is C1, the next octave up. To get to the next perfect fifth above C1, you have to double the distance between the perfect fifth of the previous octave, not only do C1 and G1 have a multiple of 8 in common, they also have the multiple of 16 in common. This doubling continues up as far as the human ear can perceive sound, with the amount of multiples in common increasing each time also.

This is why a perfect 5th sounds the way it does, as the waves are traveling through time and space, instead of the peaks and nodes interfering with each other, they fall within each other at certain points (depending on the interval). Since two notes that are an octave apart from each other fall perfectly in line with their peaks and nodes, the higher octave simply having twice as many, a perfect fifth has 1.5 wavelengths within one wavelength of the root or bottom note. So after three wavelengths of the root, the 5th will hit a node/peak at exactly the same time. This is why it sounds “good” to our ears. This is the most frequently re-occurring node/peak match that can happen in harmonics, hence perfect fifth.

So what about three notes at the same time? Well I’m sure you have already thought that far ahead. Let’s take a typical major chord, on the root of the musical key of C for consistency. This is C0 and G0 played at the same time, with the addition of E0 being the 3rd of the chord. E0 has a frequency of 20.6, which does not share the multiple of 8.125 with C0 and G0. This is actually just about halfway in-between the difference of the two frequencies, which again is always true no matter what octave you are in. We have 16.35Hz, 20.6Hz, and 24.5Hz.

This is where it gets a little weird. You may have gotten your calculator at this point to check my math, and you will notice that the 3rd is NOT perfectly in-between the root and the 5th. It’s really close (.3Hz off), but not exact. If you have studied math and mechanics like I have, this should bother you. The reason for this, is that there are 12 distinct pitches in music per octave, but there are only 7 distinct pitches that fall into each key (root, 2,3,4,5,6,7 then you are back to the same pitch as the root but an octave higher.)

7 is an odd number.

7 is also a prime number.

You may be wondering, who the hell decided that there should be 7 notes in a key when harmonics is based on notes sharing the same multiples of frequencies?

More on that next time.

Since there is an odd number of notes between a range of frequencies that doubles in value, the notes are not evenly spaced within an octave, except for the perfect 5th, hence perfect 5th.

Back to our typical major chord of C0, E0, and G0, they share a common multiple of ABOUT 4, which is ABOUT half of the multiple that C0 and G0. I won’t bore you with the math this time; I think you can see how those sound waves will line up pretty nicely. With every 3rd root wavelength sharing a node with the wavelength of the 5th, and every 6th wavelength of the root will share a node with all three notes. As long as every note in the chord shares at least one multiple, you can add as many notes as you want and it will still have the same satisfying resonating sound in your ear drum. However, the less multiples these notes have in common the more dissonant the chord will sound.

This brings us back to the two chromatically adjacent notes from the first example. Once you get to frequencies that high, every note is bound to share at least one multiple with the one next to it. Up until now, every interval has shared the multiple of 2 (4, 8, 16 etc.). 2 is a prime number, which means the only multiples it has are itself and 1 (2×1 = 2, there is no other way to combine two numbers and get 2. Sorry if you already know what a prime number is). What if the only shared multiple is 3, 5, 7 or any other prime number?

Find out next time.


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