What is a pure fifth?

Answer 1

If you are not interested in the process of 'tuning', go to the second paragraph. Fifths as tuned in equal temperament are tuned slightly "inside" of smoothe, so you would not find a pure fifth on a piano tuned in the standard fashion. You would construct a pure fifth by tuning all of the wavering out of the mutual harmonic of the tones, the same harmonic that is pulled "inside" to accommodate equal temperament. If the lower note is C, then the upper note of the fifth will be the first G above C. The harmonic to concentrate on will sound as one octave above the G. You should be able to hear the harmonic playing each note alone, and you must be able to hear the harmonic when playing the notes together. When all of the wavering is tuned out of this harmonic, you are left with a pure fifth. You can produce a pure fifth starting with a tone of any frequency, even one that is not a standard note. Your ability to hear the right harmonic is the only limiting factor.

It turns out that the mathematical relationship between the tones of a perfect fifth is a simple one, as is true for most of the basic intervals. The ratio is 3/2. The upper tone will have a frequency that is 3/2 times the lower frequency. Middle C has a frequency of 261.626 Hz, or cycles per second. The pure fifth above that is G, tuned at 392.438 Hz. You will see that this frequency is very slightly higher than the standard frequency of G in the system of equal temperament, which is 391.995 Hz. The difference is small, but when the notes are played together most humans can hear the slight wavering of the common overtone. All frequencies are rounded to 3 places.

Here's another way of understanding the 3/2 ratio. If you multiply the frequency of the lower note of the 5th by 3, and multiply the frequency of the of upper note by 2, the products will be equal if and only if the notes make a pure fifth. If the products are not equal, you will hear wavering that is directly related to the nonzero difference of the products.