Why did Bach only compose 15 inventions and not 24?

The Well-Tempered Clavier is dated 1722 and contains 24 preludes and fugues, one in each major and minor key.  This was possible because in that period there was a new way of tuning that allowed to play all 24 tonalities with the same temperament. 

So why are there only 15 keys in Bach’s Inventions and Sinfonias, which’s final autograph is dated 1723?

When Bach composed the Inventions and Sinfonias, he decided to use the keys that would be acceptable if played on an instrument tuned with the old tuning system, which was in that period the most used one. This is the system that we now call meantone temperament, but at the time, this was known to the musicians simply as “the method of tuning”.

This way of tuning however favoured some keys and made some other keys unusable. Sharps could not function enharmonically as flats, or vice versa. I, meantone temperament, as it was practiced in the 18th century and before, each key had its own special character. The usable keys were more harmonious than the same keys are in equal temperament!

Generally, the meantone temperament is a tuning that makes the following tones function as harmoniously as possible when combined in some intervals:

• C
• C# (and not Db)
• D
• Eb (and not D#)
• E
• F
• F# (and not Gb)
• A
• Bb (and not A#)
• B

Since this (meantone tuning) was the most common way of tuning, Bach only used these keys for the Inventions and Sinfonias. In fact, he even nearly went beyond the limits by composing in the F minor key. Since there was no true Ab in the meantone tuning (unless Bach used a modified mentone tuning, in which a few of the tones can sometimes be used enharmonically), the G# key had to be used as an Ab key. The augmented second F to G# could then be used as a very small minor third, making the character of the key quite more doleful than the same key would have sounded in equal temperament. It seems that the key of F minor has oten been used for music of similar character.

 Another consideration, is that the purpose of The Well-Tempered Clavier, was specifically to introduce pieces in all major and minor keys. This was made possible through the use of a system of tuning known as Well-Temperamnet, which not only made it possible to play in all keys, but alson preserved, to a remarkable extent, the characters of the keys. This system involve s a compromise tuning that allows enharmonic relationships betwen sharps and flats, and slight adjustments that would allow for example an F-natural to serve as an E-sharp.  Not that the tuning method used at that time called Well-temperament  is not exactly the same as our presently used system of equal temperament, which divides the octave into 12 equally out-of-tune semitones. In our modern equal temperament, the characters of the keys are destroyed, the only difference in the sounds of the keys is highness or lowness in pitch!