Pianos suffer from an imperfection called inharmonicity and piano technicians have to make adjustments and compromises to account for this inharmonicity while tuning any piano. Piano strings vibrate according to mathematical principles, but in the real world of high tension piano wire, the strings do not vibrate exactly according to the mathematics. This is inharmonicity.

Pianos can be tuned either aurally (by ear) or by electronic device. Tuning by ear is a process of setting pitch of A above middle C at 440 Hz and then tuning a "temperament octave" using intervals and compromising those intervals to deal with inharmonicity and achieve equal temperament (which is defined below). The characteristics of equal temperament are then spread over the remainder of the piano by tuning octaves and other intervals out from the temperament.

Tuning by electronic device is using a machine which first measures the inharmonicity of the piano and then the machine calculates a tuning based on its measurements and provides a visual cue for when the note is at the proper pitch. Most piano technicians today use electronic devices, but good technicians will still do aural interval checks to be sure the machine is "correct" and will make adjustments if he or she does not like to machine's setting.

Pianos are tuned using partials. What are partials? They are subdivisions of the string vibration yielding higher pitches than the fundamental pitch of the string. All of these subvibrations occur in a single string and the human ear can pick them out with appropriate training.

The useful partials in piano tuning are:

The fundamental or first partial = the pitch of the note
The second partial = the pitch an octave above the fundamental
The third partial = the pitch an octave fifth above the fundamental
The fourth partial = the pitch two octaves above the fundamental
The fifth partial = the pitch two octaves and a third above the fundamental
All the way up to 12 partials.

When two strings of two different notes (an interval such as M3rd, P4th, P5th, M6th, Octave, etc.) are vibrating at the same time and some of their partials approximate each other in pitch, they create "beats". Beats are where the sound waves that are close to, but not at the same pitch, reinforce and cancel each other. The reinforcement is the "beat" we hear. This is same principle that causes the "wah-wah-wah" of out of tune unisons that you occasionally hear. Piano technicians use these beats to tune the piano by tuning intervals. The beats in M3rds, 6ths, 10ths give the piano its vibrato. We put beats into selected intervals, some fast and some slow and this is called tempering intervals.

The intervals used in piano tuning include thirds (fast beating), fourths (slow beating), fifths (slow beating), sixths (fast beating), tenths (fast beating), twelfths (slow beating), 17ths (fast beating), plus a few others. The interval beats are produced by partials of two different strings approximating each other.

Major Third: The 5th partial of the lower note against the 4th partial of the upper note (5:4 interval)
The partials approximate each other at a pitch 2 octaves above the upper note.
Perfect Fourth: The 4th partial of the lower note against the 3rd partial of the upper note (4:3 interval)
The partials approximate each other at a pitch 2 octaves above the lower note.
Perfect Fifth: The 3rd partial of the lower note against the 2nd partial of the upper note (3:2 interval)
The partials approximate each other at a pitch 1 octave above the upper note.
Octave: Many useful proximate partials here: 2:1, 4:2, 6:3, 8:4, etc.
There are many higher pitches at which the partials approximate each other.

These intervals are used to tune equal temperament. The essential characteristics of equal temperament are:

• A above middle C, termed A49, is at 440 Hz (cycles per second).
• In any section of the piano, playing M3rds, M10ths, or M17ths up the piano shows slowly increasing beat rates, but the beats are fast.
• In any section of the piano, 4ths, 5ths, and 12ths sound consonant with very slow beats at around 1 bps or less.
• Octaves sound beatless although they usually have very slow beats in order to tune the piano, on the order of 0.5 beats per second.
• Double octaves beat slowly and define the "stretch" of tuning depending on how fast we make them beat. This is the art of tuning pianos.

Various intervals are tempered or compromised to achieve these characteristics.

This information gives us the criteria to tell if a piano is in or out of tune. Play octaves and 5ths anywhere in the center section of the keyboard. If they both sound smooth and harmonious, the piano is in tune with itself or close to it. If octaves, fifths, or both sound noisy (beats), the piano needs tuning.

The fundamental question for your prospective piano technician should be "Is my piano at pitch and in tune?" since a piano can be in tune at some pitch other than A440.