Chris Jewell explains the Theory of Piano Tuning
There is a good chance that your eyes will gloss over when I explain a little theory, but never mind, for those who are interested here goes.
The basic rule states that when the pitch of a note goes up an octave, the frequency doubles. Are you with me so far? There are usually just over seven octaves on a piano, so although I know the frequency of bottom A isn't 10 Htz, lets just pretend that it is.
Therefore, by doubling 10Htz seven times we will arrive at the frequency of the top A. This will be 1280 Htz. (10Htz, 20Htz, 40Htz, 80Htz, 160 Htz, 320 Htz, 640Htz, 1280Htz.) Another way of getting from the bottom A to the top A would be to tune in 5ths. There are twelve 5ths in seven octaves. The frequency of the fifth of any given note can be found by multiplying the original note by three and then dividing that number by two. So for our example of bottom A, 10x3=30 30/2=15. So the frequency of the 5th up from the bottom A, which is E would be 15 Htz. If you do the same sum another 11 times, you will finish on the same top A note that we said should have a frequency of 1280 Htz. Unfortunately, it doesn't, which is a real nuisance. Work it our yourselves if you wish, but trust me, the figure you will arrive at will be slightly under 1297.5 which, as you can see, is higher than the 1280 Htz we know to be correct.
The difference between these two figures is known as the Pythagorean or Diatonic comma, and tuning your piano in equal temperament is the ability to distribute these 2.5 Htz equally throughout the range of your piano. So on a piano tuned in equal temperament, in order to make these two differing frequencies end up on the same note, every interval of a fifth has to be in fact slightly less than a fifth. This is what I am trained to do - knowing to what degree a fifth should be tuned flat to ensure that everything meets when you get to that top A.
Chris Jewell C&G Dis. NSC Tuning Dip