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Medieval church modes were replaced by the major and minor keys used in harmony. The technology that made this possible was fine tempering, without which a composer is stuck in one key, with a limited possibility of moving to a key up or down a fifth. Just tempering, which was used in the middle ages, was rational, which is easy to imagine as everything is based on integer divisions, but it falls apart because it is not possible to reach all notes that way. This left us with a number of ways to get to B flat, all of which produced different tones, and no way to get to F sharp.

By contrast, improved mathematical understanding of the Late Middle Ages lead to the realization that an irrational approach resolved the issues of just tempered system. In this system, there was a ratio between a note and the note a semitone lower that was the same as the ratio of 1 to the twelfth root of two. This was the well tempered system, in which some notes are very, very slightly out of tune, but it is possible to write and play music in any key. In celebration of this, J. S. Bach wrote the Well Tempered Clavier, which is a cycle in which pieces of every key are represented.

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