Many ordinary organs are tuned to twelve tone equal temperament, where all twelve semitones are equal. But there are many twelve tone unequal temperaments. They arose out of dissatisfaction with
Many music practitioners have experimented with tunings in which the twelve semitones are not all equal, trying to find an arrangment in which many thirds are better than in twelve tone equal temperament, while organists can still play in all keys. A side effect is that such tunings lead to different 'sounds' for different keys.
The table below specifies only a small number of twelve tone unequal temperaments. In the top row, tones are ordered by fifths from left to right, starting at A♭ and ending at G♯. A just fifth has a pitch ratio 3/2. When we combine the fifths in the top row, we have to multiply their twelve pitch ratios. When we also subtract 7 octaves, we get a combined pitch ratio of (3/2)12 / 27 = 1.013643265, which is equivalent to 23.46001038 cents, almost a quarter of a semitone. That interval, called a Pythagorean comma, would be the interval between A♭ and G♯. But in any twelve tone tuning, these tones should have exactly equal pitches.
In order to make the pitches of A♭ and G♯ equal, some or all fifths must be slightly changed, where all changes (for one tuning) combine to minus one Pythagorean comma. One particular twelve tone unequal temperament is specified by one table row, giving the amounts by which fifths are changed, where p is shorthand for a Pythagorean comma. For example: -p/4 means that that particular fifth is narrower than 3/2 by one fourth of a Pythagorean comma. That is, by 23.46001038 / 4 = 5.865002595 cents. An empty cell indicates that that particular fifth has a pitch ratio of exactly 3/2. The data source used is Ro 2019.
| A♭ | E♭ | B♭ | F | C | G | D | A | E | B | F♯ | C♯ | G♯ | |||||||||||||
| a) | -p/5 | -p/5 | -p/5 | -p/5 | -p/5 | ||||||||||||||||||||
| b) | -p/4 | -p/4 | -p/4 | -p/4 | |||||||||||||||||||||
| c) | +p/4 | -p/4 | -p/4 | -p/4 | -p/4 | -p/4 | |||||||||||||||||||
| d) | -p/12 | -p/12 | -p/6 | -p/6 | -p/6 | -p/6 | -p/12 | -p/12 | |||||||||||||||||
| e) | -p/12 | -p/12 | -p/6 | -p/6 | -p/6 | -p/12 | -p/12 | -p/12 | -p/12 | ||||||||||||||||
a): Kellner. This tuning is claimed to be the one J.S. Bach wanted for his Wohltemperirte Clavier.
b): Werckmeister 1691.
c): Werckmeister IV.
d): Neidhardt II. Small city.
e): Neidhardt III. Big city.