The problem of creating a fixed pitch scale for fretted instruments perplexed musicians for generations. In a sense, there is no actual solution since it is the mathematical equivalent of trying to divide 9 by 4 and have it equal 2; it simply isn't viable. It was Vincenzo Galilei, who first theorised that the problem could be alleviated by making all intervals equal divisions of the octave, eventually codifying the principle as the "rule of 18" in which the string is reduced by progressively by 1/18 of its length to produce a scale of 12 more or less equal semitones. This first "constant ratio" scale was not perfect but for the first time it allowed lutes to play tolerably in tune all over the fingerboard. Previously it had been necessary for lutenists to have an intimate knowledge of the parts of the fingerboard which could and couldn't be used and, in fact, tablature notation had been developed so that information could be conveyed not only about which notes were to be played but where on the fingerboard they had to be played.
In due course, and after much work by the musicologists and mathematicians of the renaissance and the enlightenment eras such as as Andreas Werckmeister, Marin Mersenne and Simon Stevin, the equal temperament system emerged.
Based on a single step ratio for the semitone derived from the 12th root of 2, equal temperament assembles larger intervals as multiples of the semitone. Unlike the harmonic series, which sacrifices the value of individual notes in favour of perfect harmonic consonance, equal temperament sacrifices harmonic perfection to achieve a fixed value for all twelve notes in the scale. It succeeds in this aim because the amounts by which the intervals differ from the harmonic intervals are very small and usually within the threshold of perception of most people.
