Harmony of Numbers

You may not be the next Beethoven, or even the next Bieber, but chances are you have a fairly strong preference between the following two-note chords:

Why is it that we instinctively prefer one of these chords to the other?  This is a question dating back to antiquity; in fact, Pytahgoras is frequently credited with discovering the answer.  Nowadays, the reason is fairly well-known to folks who have studied music;  even Donald Duck has heard an explanation.

It all comes down to ratios, my friend.  All else being equal, if you pluck two strings on a stringed instrument, any difference in the sound they produce is due to differences in the lengths of the strings.  Longer strings will vibrate more slowly, producing deeper sounds, while shorter strings will vibrate more quickly, producing higher sounds.  So, when a pair of notes sound pleasing to our ear, what this really means is that the strings producing the notes have lengths that are in a pleasing ratio.

But what's a pleasing ratio?  Well, you could make an argument that simple ratios are pleasing.  Certainly I would rather deal with small numbers than big numbers.  By this logic, the simplest ratio I can think of is 1:1.  And indeed, if you pluck two strings of the same length, they will produce the same note; what's more harmonious than that?

If you're looking for something a little more interesting, you could consider the ratio of 2:1.  As Donald learned in Mathmagic Land, strings whose lengths have this ratio will produce an octave when played together: each will produce the same note, but at different pitches.  If the long string produces an A, the short string will produce a higher A; if the short string produces a D, the long string will produce a lower D.

Let's do one more.  What about a 3:2 ratio?  This is still fairly simple, because the numbers are relatively small.  And indeed, if you play notes whose string lengths are in this ratio, it sounds pretty good (here are a C and a G):

So what about the notes I started with?  The first pair is an octave, corresponding to a ratio of 2:1 (both notes are C).  The second corresponds to a ratio of 16:15 - not such a simple ratio, and not such a pleasing sound (the notes this time are B and C).  Maybe Pythagoras was on to something - if the ratio involves small whole numbers, then what we hear tends to sound good.

A few caveats are in order, however.  First, string length isn't the only thing that determines sound quality.  String tension and thickness matter, too:

It's good that string length isn't the only thing we can vary in order to affect sound quality.  If it were, then on instruments like the piano, the string for the lowest note would need to be more than 128 times longer than the string for the highest note!

The other thing to note (pun intended) is that even though simple ratios sound nice, that doesn't necessarily mean more complicated ratios will always sound bad.  For example, here's the sound of a 40:27 ratio (the notes are D and A):

Play this a few times - doesn't sound so bad, right?  Notice, though, that if you play the 3:2 ratio above, and then 40:27 ratio immediately afterwards, the latter my sound a little sour.  We can explain this mathematically by converting the ratios to decimals and comparing.  A 3:2 ratio corresponds to a fraction of 3/2, or a decimal of 1.5, while a 40:27 ratio corresponds to a fraction of 40/27, or a decimal of around 1.48.  So even though 40:27 isn't a "simple" ratio, it's fairly close to one, which is why the D-A note pair sounds okay in spite of the larger numbers involved in the ratio.

So, while there may be truth in the maxim that small number ratios sound nice, large number ratios may not sound too bad either.  But if you hear music that's particularly jarring, chances are good that there are some big numbers lurking in the ratios.

Teachers: want to explore this topic with your students?  Then check out our new lesson, Harmony of Numbers!


One thought on “Harmony of Numbers”

  1. This is really interesting! When I took music theory, I was taught to compose chords in a 1-3-5-7 pattern and vary their order and what octave they are. Now that I am studying mathematics, it is fascinating to see that chords can by built using fractions instead, trying to get to "simple" numbers. It would be cool if teachers would show students this in class; connecting math to music might pique their interest in one subject or another.

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