The history of musical temperament as we now it today already started with Pythagoras 500.B.C. Since then it went through several stages of development.
The Pythagorean tuning
Pythagoras created a diatonic system, which consists of 12 perfect fifths put on top of one another. But using only perfect fifths lead the “b-sharp” (which should enharmonical changed be the same note as C) on the 12th fifth to be a bit higher than the c which one would get from 7 octaves put on top of one another. The difference between those two notes can be given in the ratio 73:74, this numerical proportion is also called “Pythagorean comma”, it equals around a quarter of a chromatic semitone. Furthermore, the major thirds are a bit impure (higher than they should be). This kind of tuning can predominantly be used for music of the middle age for one voice, but becomes impure for polyphonic pieces.
The pure intonation is made from a row of partials, where, apart from the octave, the perfect 5th and major 3rd are adjusted to the tonic as well. Even tough the major thirds are harmonic for this kind of tuning, some problems still occurred. If one would start a major scale using pure intonation on c, for example, the first two notes (C and D) have the ratio 8:9 and one further (notes D and E) have the ratio 9:10. But, if one where to start a D-major scale the first two notes (also D and E) have to have the same ratio as the first two notes of the C-major scale (C and D). This means that the same notes from one scale have a different ratio when they’re being played in another key. The difference between that “big” and “small” whole-tone step is called “Syntonic comma” and has a ratio of 80:81, which equals about the fifth of a chromatic semitone. The pure intonation has the advantage of a nice sound, as long as the music stays in the same key.
In contrast to the Pythagorean Tuning the Meantone temperament doesn’t work with perfect fifths, but with major thirds. This way of tuning corresponded to the sound-ideal of the Renaissance. The Meantone temperament compensates the Syntonic comma and works with only one whole-tone, which lies in between the two different whole-tones of the Pure Intonation. Nevertheless, the fifths are a bit to small.
The well-tempered tuning
At around 1700 the German musician Andreas Werckmeister divided the octave into 12 equal intervals. Through that, every interval is slightly unexact, but this imperfection is barely noticeable, which allows musicians to perform in all of the 24 major and minor keys.
Johann Sebastian Bach wanted to make use of this new discovered way of tuning and therefore wrote two books (published 1722 and 1744), which he called “Das Wohltemperierte Klavier (=The well-tempered clavier). Each of those books include 24 preludes and fugues, one for all major and minor key-signatures in chromatic order.
- Knapp, W. and Peschl, W. (2005). Wege zur Musik-Oberstufe Band 1. Rum/ Innsbruck : Helbling, pp:46-47