Kirnberger temperament

[1] Kirnberger's tuning systems, or well temperaments are a way to artificially splice together two arcs on the "natural" spiral of fifths to turn it into an "unnatural" circle.

In Kirnberger's and his teacher Bach's time, keyboard musicians were experimenting with different unobtrusive ways to alter the spacing of notes around the spiral of fifths to close it into a circle, so that every note needed for every key was at hand, even if some rarely used key signatures might be very dissonant, but tolerable.

Some impractical but very consonant circular tuning systems exist, such as 31 tone equal temperament and 53 equal temperament, but the number of separate notes required to fill out any one octave on a keyboard far exceeds the space available on a playable keyboard (and the vast majority of the extra notes would probably never be played during the entire working life of the instrument).

For the most part, keyboardists insist that their pianos, harpsichords, and MIDI keyboards be limited to around 12 notes per octave, since no keyboard can be played that is so widened up with excess notes that a human hand cannot stretch across a whole chord, nor can the keys on the board be made so narrow – to fit more in the span of an ordinary player's hand – that even a skillful musician will often strike the wrong key among the tiny, closely packed notes.

This difference between the initial C and final C that is derived from performing a series of perfect tunings is generally referred to as the Pythagorean comma.

(Quarter comma meantone temperament has eight exactly pure thirds, but sacrifices four entire chords to achieve this end.)

Tempering any musical scale, however, is always a give-and-take situation: No temperament is a perfect solution to the fixed tuning problem.

But every temperament system is a mix of give-and-take compromises; each finds a way of dealing with the comma.