February 2, 2007
Understanding the Western Tempered Scale (Part 6 of 15):
How the Frequencies of Adjacent Notes Are Related

Get out your high school algebra book. We are looking for a frequency multiplier which separates two notes a half step apart, and we're calling it "m".

Recall from last week:
440 Hz =220 Hz * m * m * m * m * m * m * m * m * m * m * m * m

Or, simply:
440 = 220 * m^(12) (that's m raised to the 12th power)

Divide both sides by 220 and we see that 2 = m^(12) .

This tells us that m is a number such that when you multiply it by itself 12 times, you get the number 2. In other words, m is the 12th root of 2.

Again from your algebra book: the square root of 2 is the same as 2 raised to the 1/2 power. The cube root of 2 is 2 raised to the 1/3 power. So of course, the 12th root of 2 is 2 raised to the 1/12 power.

Now, get out your calculator: 1/12 is 0.0833333333333 (etc!) If your calculator has the y^x function (the x might be above the y a bit), press "2" "y^x" "0.08333333" "="
and, the answer is: 1.059463.

What is this number? It is m--the multiplicative factor to jump from one frequency to another frequency exactly half a step higher. Or, if we want to go a whole step higher, that would be two factors of m, or m^2 = 1.122462.

In other words, if A is 440 Hz, then B (a whole step higher than A) will be 440 Hz * 1.122462 = 493.883 Hz

If you are seeing this for the first time, and you are understanding it, that is cool - let me know.

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