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Tuning Science
The tautness of a piano string determines its vibration rate, and the vibration rate of a piano string creates its pitch.  When two strings sound at the same time, the mathematical ratio of their vibration rates creates the following intervals:

1:1  unison
1:2  octave
2:3  perfect fifth
3:4  perfect fourth
4:5  major third
5:6  minor third

e.g.  If one string was vibrating at 400 cycles per second and another at 500, the two sounding together would create a perfectly tuned major third.

What does “in tune” sound like?  When two sounds form a unison and are exactly in tune, the sound waves in the air created by the vibrating source are reinforcing each other because their frequencies are exactly the same.  Thus two strings vibrating at 440 cycles per second, are creating waves in the air that reach our ears at the rate of 440 times per second.  The waves are in sync.

If one of the strings goes out of tune to 439 cycles per second, the waves from the two sources will go out of sync once every second.  They will also come back into sync once every second and reinforce each other for a moment before they go out of sync again.  At the moment they are in sync they reinforce each other and sound louder.  At the moment they are completely out of sync, they work against each other almost cancelling each other out, so they sound much weaker.  The resultant sound is perceived as a waver between loud and soft once every second; this waver we call beats.

If the 439 cycles per second string sinks further to 438, the beats between it and 440 will now occur twice per second.  The beats between 440 and 435 would sound five beats per second.  Therefore we can tell just how far out of tune two strings are by listening to the speed of the beats.

The explanation above also applies to two notes that are not unisons.  For instance, a perfect fifth above A-440 would be E-660 (since the ratio between the vibration rates of notes of any perfect fifth is 2:3).  If the E is out of tune at 659, the beat rate will be once per second


Equal Temperament
The fascinating thing about tuning a piano is that it won’t be in tune until all the intervals (except the unisons and octaves) are just the right amount out of tune.  Musicians with good ears could all tune their own pianos if it was simply a matter of putting everything exactly in tune.  The problem is that if you tune your piano exactly so it sounds good in the key of C major (for instance), it will sound pretty terrible when the music modulates to say Eb major. 

It was in the Baroque era leading up to the music of J.S.Bach, when musicians were experimenting with new types of tuning, and along with these experiments came new music with more extensive modulations.  Before that time, a musician had to retune the harpsichord in order to play a new piece in a different key.  The Well-tempered Clavier, Bach’s two books of twenty-four preludes and fugues, one in every major and minor key, were written to illustrate the fact that with proper tempering of the scale, you could play through the whole book without retuning.  Scholars argue about just how close Bach was to what we call “equal temperament,” but I think he must have been very close indeed or the preludes and fugues would not have sounded good to his discerning ears.

In order to achieve equal temperament, the tuner has to either stretch or squish certain intervals from their exact intonation, but please remember that this is not a lesson in how to tune a piano but a rather simple discussion of the kinds of things the tuner needs to consider as the tuning takes place.  Perfect fifths are slightly squished (a squished interval should probably not be called “perfect” but we are working with terminology from two different sources here).  Perfect fourths are stretched slightly more than the fifths are squished.  Thirds (both major and minor) are stretched considerably more than the fourths, and sixths are correspondingly squished.  You may wonder at this point just how the tuner knows how far to squish or stretch an interval.  The answer is two-fold.  As an interval is stretched or squished the beats between the two notes become faster, so the tuner needs to have a good idea of how fast or slow they should be.  Secondly, as the tuner stretches or squishes one interval, he/she is inevitably effecting other intervals involving other nearby notes that have already been tuned.  Those other intervals are used as checks on the interval being tuned at the moment.  Thus every interval has several other intervals that provide checks to its accuracy.


Tuning:  a Science, an Art, or a Magic
I think that the above description makes it clear that tuning a piano is a scientific activity.  Tuners are concerned with frequencies and beats which reveal scientifically whether the piano is in tune.  They do not need to be musicians in order to do this.  However, I have discovered over the years that at certain points in the process of tuning I inevitably stop, take off my scientist hat and don the hat of an artist.  There are three of these points:  when I have set the temperament in the middle octave, when I have tuned the bass section to that octave, and when I have finished tuning the treble.  At these points some kind of shift takes place in my brain, and I listen for a time as a musician, not a tuner.  I play chord sequences in every key and just listen as a musician.  Of course, these two ways of listening are not unrelated; if I have made a mistake scientifically, my artistic listening will reveal it.  It is like both are the opposite sides of the same coin.  Piano tuning is after all a science that supports an art.  My clients are always far more impressed with the tiny bit of Bach, Beethoven, Chopin, or Prokofiev that I give them after tuning than they are with the much longer time I spend getting their piano ready for the music of the great masters. 

Tuning is also magic.  This occurs when I tune my own piano at home, a piano with which I am very familiar.  After I have tuned my own piano, and I sit down to play it as a musician (not a tuner), it feels different!  Notice I said feels, not sounds.  Even though I have done nothing to regulate the piano which means it cannot feel different, it does!  I am always astounded at the new feel to my piano when it is newly tuned.  Of course the reality is that the piano sounds different.  Over the weeks when notes have gone ever so slightly out of tune, the unisons no longer reinforce one another like they used to, and are therefore not as loud.  Now, all of a tuning, they are much louder and my finger needs less weight to produce the same volume; it feels different!  Now that’s magic!


Staying in Tune 
After all these years of practise, I find it relatively easy to put a piano in tune.  However, making it stay in tune is another matter.  There are several different factors that determine how quickly a piano will go out of tune.  They include:
    -how well the tuner has equalized the tension on all sections of each string
Each piano string is divided into several sections between the various pins that it passes.  Only one of these sections sounds when the string is played.  A tuner employs various means to try to ensure that the tensions on all parts of the string are equal, but since only one section sounds, it in impossible to know accurately if this has been accomplished.
    -changes in humidity
An increase in humidity makes wood swell, and a decrease in humidity makes wood shrink.  This generally means that an increase in humidity will make a piano go sharp and a decrease will make it go flat.  I have often noticed that when I am called to tune a piano that has been moved from a less humid climate, it will have gone sharp as the wood of the piano took on the increased humidity of the new setting.  The opposite is also true.  In many areas there is a natural yearly cycle of dry and wet weather which will effect pianos that are not kept in a controlled environment.
    -changes in temperature
Higher and lower temperatures also cause a piano to go out of tune.  Part of this factor is related to humidity, because warmer air can hold more humidity than cooler air.
    -how the piano is played
In theory, whether a piano is played with great energy or very gently should not be factor in keeping it in tune if the tuner has managed to equalize the tension on all the sections of each string.  However, inevitably there will be imperfection in this (as explained above) and enthusiastic forte playing will tend to even out these tension differences, thus making the piano go out of tune.
    -whether or not the piano is moved around
Moving a piano will contribute to going out of tune.  Although pianos are built strongly, they are not absolutely rigid structures, and moving them will wrench things around sufficiently to effect tuning.  The larger the piano and the stronger it is built, the less moving will effect tuning.
    -the condition of the piano itself
If there is any looseness in any of the pins that a piano string touches, that string will not stay in tune well.  These are the tuning pins, bridge pins and hitch pins.