The understanding of music begins with the physics of the vibrating string. At the left are shown the first three modes of vibration of a string that has its ends pinned in place. Looked upon as a math problem (known as a boundary value problem), the key condition is that those two ends of the string are not allowed to move. The solution of the motion equation must produce zero motion at those locations at all times.

The top image shows the lowest frequency mode of vibration. The precise frequency value associated with this motion depends on how thick and heavy the string is, and also how tightly it’s stretched between the pins, but on nothing else. That frequency will sound like something to you – let’s just say that it happens to be the musical note C.

In theory, if you gave every little bit of the string precisely the correct starting displacement, you could get a solution that consisted of only that one frequency. But in practice that’s not going to happen. To accommodate any arbitrary starting condition, you have to recognize that the string can also shake in halves, in thirds, in fourths, and so on, as shown by the other images. The frequency associated with vibration of the halves is exactly double the one we just talked about, so it’s also a C note, one octave higher. That’s not terribly interesting, though it does define the span of a single octave of musical notes. Remember, you hear both of these C notes when you pluck the string. They’re blended together, but your brain is perceiving both of them.

It’s also perceiving the frequency associated with the string shaking in thirds, as shown in the bottom part of the image. This action has a frequency of three times the original frequency. So it’s up above the second octave we just discussed, but it also has a corresponding note between the first two C notes, which we can find by dividing its frequency by two. So that frequency is exactly one and a half times the frequency of the low C we discussed. That corresponds to the note of G.

If you play around with a piano or a guitar, you’ll find that C and G sound good together – they’re consonant. The reason our brains like them together is because we hear tones with that frequency ratio every time we hear a string vibrate. We’re used to them going together – we’ve learned to like it. It just “sounds right.”

This whole process of shaking in fractional pieces goes on. Shaking in quarters produces yet another C, another octave up. But shaking in fifths produces a frequency that can be divided by four to bring it down to a matching (octave-related) note within our first octave range. That’s 1.25 times the original C note frequency, and that represents the note we call E. C, E, and G also sound good together, and in fact they form the C major chord. You can go on, using smaller and smaller pieces of the string, until you obtain the entire 12-note scale common in Western music. You could keep going further, but the more pieces you split the string into the weaker the associated sound becomes, and eventually you get to notes that aren’t “there enough” to affect what you hear.

This is a wonderful story. It’s a shame it’s not precisely true. It is, actually – it’s how the whole story starts. One can derive a 12-note scale using this method which is called the just intonation scale, and it will sound nice. Unfortunately, the 12 notes of a D major just intonation scale are completely different frequencies from the 12 notes of a C major just intonation scale, and so on. A piano tuned to just intonation could only play music in a single key, and would have to be completely re-tuned in order to work in a different key. Not a nice situation if you’re interested in key transpositions.

Fortunately, someone realized that those 12 notes could be adjusted ever so slightly – slightly enough that our ears “don’t mind” – in a way that allows fully flexible key transpositions. No note frequency moves by even so much as 1%. The result is known as the equal temperament scale, and is the basis of modern Western music. The notes no longer have that precise relation to the physics of the vibrating string, but they are “close enough that things work.”

So this is where the story of music starts. My main interest is in guitar and piano, though I also dink around with the harmonica a little bit. I love the theory of it all, and I’m not terribly good at any of those three instruments (yet, I hope). I’ll keep you filled in on how I’m doing.