January 5, 2007
Understanding the Western Tempered Scale, Tuning, Hertz and all that--Part 2 of 15: The Perfect 5th and a Tempered 5th

An interval of one octave is made by doubling the frequency. An octave above A 440 Hz is A 880 Hz. An interval of a "perfect 5th" (which you may know from my books as the backbone of most chords) is made by multiplying the starting frequency by 1.5 - and you can understand where this comes from by going back to look at the vibrational modes of a string.

As you can see, the first harmonic has a wavelength half as long as the fundamental - as the wavelength and frequency are inversely related, cutting the wavelength in half multiplies the frequency by 2 - i.e., an octave. The 2nd harmonic has a wavelength 1/3 times the fundamental's wavelength, so its frequency is 3 times higher than the fundamental's frequency - or, its frequency is 3/2 (1.5) times higher than the 1st harmonic. These two harmonics make the interval of a perfect 5th.

If you know music a bit, you know if A is 1, B is 2, C# is 3, D is 4, and E is 5 (just going up the scale in the key of A). Hence, the frequency of E should be 1.5 x 440 Hz = 660 Hz. Actually, when I look up on my chart, I see that E is really 659.25. Why the difference? Because that is a "tempered 5th", not a "perfect 5th". Huh? Come back next Friday, we'll be on the road to understanding this.