February 16, 2007
Understanding the Western Tempered Scale (Part 8 of 15):
What is the meaning of 1.059463?

This is the half step interval. It comes out of the physical fact that when we double the frequency of a note, that new higher vibration tickles many of the same hairs in our inner ear, and we perceive it as being the same note in some way, even though it is higher - an octave higher, in fact. And it comes out of the fact that people in Europe towards the end of the Middle Ages started singing the 7-tone "Do Re Mi" scale, with two intervals closer than the others (i.e., 12 half steps in all), and also the artificial requirement that all half-step intervals be exactly the same size (multiplicatively, at least).

If we had 10 "half-step" intervals instead of 12, the number would be 1.071773 instead. But for 12, that number is 1.059463.

Now, back to the perfect 5th. The perfect 5th happens physically, with the vibration of strings or air columns. And the two notes in a 5th interval will share harmonics, tickling similar inner ear hairs, so our world is built to provide perfect 5ths, and our ears are built to appreciate perfect 5ths.

What does 1.059463 have to do with the perfect 5th? Let's go up the scale (remember that unplayed half step notes are represented by "^"):

Do	^	Re	^	Mi	Fa	^	So
1		2		3	4		5

So (so to speak) the 5th is 7 half steps above Do. In other words, to go from A 440 up to So, we need to multiply by 1.059463 7 times, which is 1.498061.

Several weeks ago, I said the "perfect 5th" above A 440 was E 660, but that the tempered E was a different number. Tempered E is 440 Hz X 1.498061 = 659.25

Now you understand (I hope!).

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