If you try to divide the musical domain by means of perfect fifths (the most consonant or agreeable of intervals), you rapidly run into a situation where what’s supposed to be a single note has several different possible values, depending on the pathway to that note. For stringed instruments without frets like the violin, where the strings are tuned to perfect fifths, this is resolved “by ear”, with the player making tiny adjustments which accord with the tonality of the piece being played. For brass and woodwinds, the tuning is “baked in”, which is why you have things like a “B-flat clarinet”, and composers call for instruments tuned in closest accord with the tonality of the composition.

As Andras Schiff laments, pianists never play in tune, because all intervals are pre-adjusted in order that no tone sounds out of tune regardless of what key the piece is played in. “Equal tempering” is a system whereby each note differs from its predecessor by a fixed proportion, resulting in a flat, colorless palette of tones. “Well tempering” is today a highly subjective art, in which some fifths are more perfect than others, with the result that each key signature bears a distinct tonal coloration.

From Book II of the Well-Tempered Clavier, here is the Prelude and Fugue in C-minor.