January 26, 2007
Understanding the Western Tempered Scale (Part 5 of 15):
Just What Does Equal Temperament Mean Mathematically?

In the late 1500's, people started talking about equal temperament in a scale. This is a theory about how exactly to space the notes in the Western scale. That theory didn't matter for violins so much, but it was very important for lutes or harpsichrds and later pianos. For a lute or a guitar, it is a theory about how to space the frets, each fret getting closer together in a uniform manner as you go up the neck, making the notes' pitches rise in a well-tempered manner. It took hundreds of years for that concept to take hold for all of Western music.

People started to realize that equal temperament between the half steps in the scale, which we talked about last Friday, meant that the ratio of the frequencies of adjacent notes a half step apart would have to be constant, no matter where in the scale you are.

Confused? Check it out. Consider "Do Re Mi Fa Sol La Ti Do". If we want to go from low Do to high Do, we have 12 half-step intervals we must go up (remember this from last Friday?). The rule of the game is that intervals are multiplicative. We know this from octaves. 110 Hz is an A, then to go up an octave we need to go to 220 Hz, then the next A is at 440 Hz, then 880 Hz, and so on until we are above 20,000 Hz, which only creatures like dogs and bats can hear.

So, to go up an octave means we multiply the note's frequency by 2.0000. To go up a half step, we need to multiply the frequency by some other number between 1.00000 and 2.0000, which I will call "m":

Freq_C = Freq_B * m (C is a half step above B)
Freq_C# = Freq_C * m (C# is a half step above C)
Freq_C# = Freq_B * m * m (C# is two half steps, or a whole step above B)

AND, if you go from A 220 Hz to A 440 Hz, you have to multiply by m 12 times:

440 = 220 * m * m * m * m * m * m * m * m * m * m * m * m

But, what does this mean???

If you can't wait until next week for the answer, you'd better get out your high school algebra book to figure it out yourself.

;