A Brief Explanation of Math & Music

The connection between math and music is profound and pretty extensive, but here are some of the basics. It started with Pythagoras - yes, the same guy with the triangle theorem - a Greek man in the year 500 B.C. who had an entire philosophy, kind of a religion, with a large number of followers. His philosophies are considered among the most influential in history, and they believed, among other things, that everything in existence is composed of numbers.

So, probably also a fan of music, he tried to figure out the numbers in music. He experimented with a monochord, which is basically like a single guitar string, and found that by changing the length of the string (like holding down a guitar string) has a direct effect on the note that sounds.

For instance, putting your finger on the exact middle of a string on a guitar (the 12th fret), you are changing the length of the string to half the length that it is normally, the ratio for is 1/2 (or 1 to 2) since it is half. At this point, these two notes are the same, except one is an octave higher. Since sound is a wave in the air - like a wave in the water, except we cant see it - and the frequency determines what note it is, the frequencies of the notes behave in the same way that that the length of the string does. If you play the open (or full length) "A" string on a guitar, its frequency is 440 Hertz and is the "A" note - but then if you put your finger on the middle of the string, its still an "A" note, except an octave higher, and its frequency will be 880 Hertz. So, 2 times 440 = 880 and the ratio of the string lengths is 1/2!

The same can be said for the other notes in the scale. For instance, the ratio of string length from the same open "A" string to the "E" note (7th fret) is 3 to 2 (or 3/2) and the frequency of the "E" note is 660 Hertz. So, thats 660 to 440, or 660 divided by 440, which is equal to 3/2!!

For some reason our ears recognize these ratios without thinking about it, because simple ratios (such as 2/1 and 3/2 and 5/4) are the notes that sound pleasant to us - and our scales are built around these - whereas complicated ratios sound painful and grating to us.

So, many instruments after Pythagoras were built around his mathematical ratios. But there was a problem - it was hard to switch keys on these instruments (Like from the key of C to the key of E) because the relative ratios just didnt add up right. Along came Sebastian Bach who devised a way of dividing the frequencies into what he called the "well-tempered scale." Bach used this system to write songs in which he could easily switch between many different keys in the same song. But the sacrifice that he made for this ability was that the notes were no longer the pure mathematical ratios, but were instead close approximations.

This "well-tempered scale" was eventually developed into the "even-tempered scale," in which all the 12 half steps in the octave are the same distance apart. This "even-tempered scale" is the system that our modern "fretted" instruments use, like the piano and guitar, and this is what allows our music writers so much freedom in composition, though the notes you hear are only approximations of the pure notes that your ears want to naturally hear.