How many different notes are there in an octave? What about note names? The answer to the second is very interesting and this is part one of an exploration of that question.
Bach's Well-Tempered Clavier (henceforce WTC) consists of two books each containing prelude and fugue pairs in 24 different keys. Why 24? Well, if you look at a keyboard, you'll see there are 12 notes in the octave. Allowing for both major and minor keys we therefore have 24 major+minor keys to choose from and Bach wrote a prelude and fugue in each key in each book of the WTC.
But if we look at the key signature, it tells a different story. A key signature may consist of 1-7 sharps or 1-7 flats or nothing at all. Allowing for both major and minor keys that gives us 30 different keys.
Here are the 15 major keys that the key signature gives us (with the ones Bach uses in WTC in bold):
C# F# B E A D G C F Bb Eb Ab Db Gb Cb
The corresponding relative minors (with the ones Bach uses in WTC in bold again) are:
A# D# G# C# F# B E A D G C F Bb Eb Ab
Notice that the extra keys possible but unused by Bach are enharmonic with keys that are used. Db major is enharmonic with C# major, Gb major with F# major and Cb major with B major. Similarly A# minor is enharmonic with Bb minor, D# minor with Eb mintor and Ab minor with G# minor. That is not to say that Db major is the same as C# major—for one they have different key signatures and the names of each degree of the scale is different (more on that later). They may even sound different depending on the tuning system used.
But this explains why Bach wrote in 24 major+minor keys, even though notation provided him with 30—he avoided enharmonic duplicates.
But this isn't the whole story. Notice that:
The reason behind these two facts will be the subject of the next part.
The original post was in the category: music_theory but I'm still in the process of migrating categories over.
The original post had 4 comments I'm in the process of migrating over.