• An octave is a doubleing or halveing of the frequency.

A₀=220 Hz,  A₁=440 Hz,  A₂=880 Hz osv.

• The octave is divided in 1200 cents, where each halftone is 100 cents. This division defines the equal tempered scale. Here is the mathematical formula to calculate the division of the frets on the fretboard:

L_n = Distance from the nut to fret n
n = Fret number
L = Distance between the nut and the bridge

• Two notes are unison (exactly the same) when no beats can be heard. Beats are the periodic and repeating fluctuations heard in the intensity of a sound when two sound waves of very similar frequencies interfere with one another. The sound waves alternately amplify and weaken each other. The closer the two notes come together, the beats become slower, to cease when the notes are the same.

A temperament is a way to define the notes within an octave. That definition is applied on all the octaves on the instrument. Before the invention of equal temperament, different temperaments were devised attempting to overcome the problem of playing in different keys, referred to as historical temperaments. Nowadays a temperament is a chosen divergence from the equal temperament to achieve tonal effects that will strengthen the musical character of a composition. Lots of different temperaments are defined. A temperament can be described by specifying the offsets in cents from equal temperament for each note within an octave.

While tempering implies a divergence from equal temperament (there is no longer 100 cents between all the half tones) the temperament will only have the intended effect in one or a few keys. You cannot play well sounding in all keys with any other temperament than the equal temperament.

The frequencies of harmonic overtones are even multiples of the main frequency. This is the case e.g. on wind instruments, but on stringed instruments the overtones are inharmonic.

Inharmonicity is caused by the fastening fault in the string endings. A string is fastened very hard in the nut and the bridge, and because of its stiffness it can't vibrate all the way to its ends. Close to the nut and bridge there are short parts of the string that does not vibrate properly. It is said that the effective length of a string is shorter than its geometrical lenght. The stiffer the string the longer the non vibrating parts.

These small, non vibrating parts closest to the ends of the string are of no importance for the note of the open string - you just tune the string to correct pitch. But apart from the main vibration, the sound of the string contains a series of overtones. The overtones divides the string in halves, thirds, fourths and so on. The wave lengths of the overtones are shortened by the non vibrating parts and the higher you get in the overtone series, the larger the affect. This makes the overtones pitch too high and the higher up in the series of overtones you get, the worse it becomes.

Here is an experiment you can perform on a regular guitar. Hit e.g. the A and D strings simultaneously and tune the fourth interval by listening to the beats. Raise the A string slowly. When the first beating disappear, another beating will appear - sort of hiding behind the first beating. Here you see how the overtones are displaced from one another! Without inharmonicity you only would have heard one beating.

Inharmonicity is causing great problems on pianos which have thick strings. The tone in an instrument is very much depending on the resonance of the overtones in the strings you don't play upon. If a piano was tuned to equal temperament it would be perfectly in tune, but it would have very little sustain because the too high overtones would resonate poorly. One way to improve the tone is to lengthen the strings, as on the grand piano. Then the strings become more slender in proportion to their lengths and the inharmonicity decreases. Another way is to stretch the octaves a little to make the too high overtones resonate easier, so called stretch tuning. (N.B. Don't confuse this with tempering where the octaves still are 1200 cents.) How much the tuning is stretched differs on different pianos. It depends on how inharmonic the piano is. In the middle register the octaves are typically stretched about three cents, while the deepest bass and the highest treble are stretched a lot more.

The tuning of e.g. guitars may very well be stretched, but unless you don't play a lot together with a piano there is hardly no reason to stretch the tuning. Guitars and other fretted instruments are indeed inharmonic (as we found in the experiment above), but not at all as much as pianos are because the strings are much slender. In wind instruments there is no inharmonicity - they have no strings.

There seems to be a common belief that the stretch tuning of the piano makes it impossible to bring it and e.g. a guitar in tune with each other. I dare to say that guitars intonate so poorly that they are not even close to equal temperament. If guitars would achieve equal temperament, the difference from the stretching of the middle register of the piano would hardly be noticeable.