March 23, 2007
Understanding the Western Tempered Scale (Part 11 of 15):
The perfect 5th and the tempered 5th

Before you plunge into this tip, you might want to review what I told you back in November about the vibrational modes of a string.

The vibrational modes of an open tube are similar to these (i.e., think of a bugle playing Taps, each of those notes is a different overtone, but the first note in that song is actually the 2nd overtone).

The frequency of the first overtone is 2 times the fundamental frequency. The next one is 3 times the fundamental frequency. And so on, with 4, 5, 6, etc.

Overtone_Freq(n) = (n+1) * Fundamental_Freq

When you play a guitar string, the fundamental and all overtones will sound, but the overtones will have less amplitude than the fundamental. Our ears are accustomed to hearing these overtones played together. So, if you play the 2nd overtone and the 1st overtone together, you make what is called a perfect 5th. The ratio of the frequencies of these two pitches is exactly 1.5000000000.

We have seen how the tempered scale is built out of 12 equal half-step intervals, and the tempered 5th will end up being a little bit off from the perfect 5th (i.e., 1.498306 times the root note instead of 1.500000). Why do it that way?

IF we tuned one key to have a perfect 5th, i.e., to play perfectly in tune (we don't say what that means for the other notes), then every other key would actually be out of tune. The tempered scale was required to make pianos play "almost in tune" in all keys. Without it, some keys would sound perfectly in tune while other keys would sound way out of tune. So, the tempered tuning results in a piano in which all keys are equally out of tune (it is like a least squares fit) to minimize the out-of-tune-ness of all keys, so they all sound pretty good.

As soon as the tempered scale was figured out, that was the beginning of the end of all the local tunings in Europe. It took maybe a hundred years for people to accept that the tempered scale was usually the best way to do things, and another hundred years or so to figure out algorithms for how to tune a piano that way.

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