Music, Math and Models
I've said a few times that music is math. I've also said a few times that I'm a Chinese jet pilot and that Halle Berry wants to bear my children so it's probably hard to know when you should take me seriously and when you shouldn't.
But let's go back to this "music is math" stuff. On that part I'm not kidding. At the end of the day, a lot of things can be math, including music and supermodels. A few months ago I did an article on how Phi, the famously named Golden Ratio, could even tell us who would be a supermodel. It contained excellent science, humor, Elle MacPherson, Carol Alt and Heidi Klum , yet still left some math questions unanswered.
In that article I mentioned the golden ratio in music, namely that the octave, fifth, and major and minor sixths are ratios of consecutive numbers of the Fibonacci* sequence, making them the closest low integer ratios to the golden ratio. But that doesn't make a ton of sense until we get a little more basic, mostly because math doesn't exist in the real world, it isn't a hard science like physics.
So we'll figure out how math is music but first we'll show how music is physics. And, in proper attention whore fashion, I can show off some of my guitars.
As a stringed instrument the guitar shares common harmonics with all of the other stringed instruments. What makes the guitar different in sound from the others are shape, materials, the nature of the strings - oh, and being in the hands of Eric Johnson.
Because a guitar string is elastic and fixed at both ends, it creates waves. It doesn't matter so much what waves are but if you want to know, a wave is the frequency times the length of the wave.
v = f x λ
Let's talk about the speed of that wave, v. That v is determined by string tension and linear mass density (mass/length) µ, measured in kg/m. Here's the concept you need to know to figure out how a string will make specific sounds:
The wavelength of a standing wave on a guitar string is twice the distance between the bridge and the fret.
Guitar strings are basically the same so you have to vary the tension and the size of the strings if you want cool sounds. You have to vary both because if you only varied the tension, for example, the high strings would be tight and the low strings would be loose. It's just better for playing to have all of the strings with the same tension (T ).
Some mathematical magic: the strings are a perfect fourth apart in pitch ( except for the G and B, 4th and 5th strings counting from the top down ) so any monkey can calculate how much the mass density ( µ ) has to increase between strings for the tension ( T ) to remain constant. Even Al Gore can't get this one wrong. So here is the only math you need to know:
v = square root of (T/µ)
You then make individual notes by using the frets. The ratio of the widths of two frets is the 12th root of 2, about 1.059. The twelfth fret divides the string in two exact halves. Every twelve frets represents one octave. So that's how guitar makers know where to place the frets based on the size of the neck.
I guess that's enough gorey detail for Part I but for Part II we can talk about harmonics and the way they translate physics into the math in this whole 'music is math' concept.
*Also what I named my favorite sexual position of 2002.