For new piano tuning customers or people wondering if their piano is out of tune. Very basic but useful information:

An out of tune piano can be easily identified by the harmonics or ‘beating’ of the intervals. In all but the bass notes of the piano, each note has three strings and getting these strings to exactly the same pitch each other is the essence of piano tuning. A piano tuner also has to tune a scale so that the notes are in tune relative to each other (he or she starts at middle C or C4, in the piano of the piano and then tunes in octaves). A badly out of tune piano will have a ‘honky tonk’ sound most people recognize from the out of tune pianos played in the salons in old western films. 

A piano tuner will tune the piano in fourths and fifths, trying to get the beats at one note per second for fourths and about half a note per second for fifths. This system of tuning is known as the equal tempered scale and is the most common musical scale used at present, used for the tuning of pianos and other instruments of relatively fixed scale. It divides the octave into 12 equal semitones. It is common practice to state musical intervals in cents, where 100 cents is defined as one equal tempered semitone. The cents notation provides a useful way to compare intervals in different temperaments and to decide whether those differences are musically significant. A useful parameter for comparison is the just noticeable difference in pitch which corresponds to about 5 cents.

– Richard, Piano Tuner Sheffield