CIRCLE OF FIFTHS

If we order the major and minor keys by the number of sharps in their key signatures, we can conclude: each next key has the tonic that can be obtained by applying the ascending perfect fifth to the tonic of the previous one. The next figure shows this. We commonly use the key signatures with up to 7 sharps. 

Figure: The first four key signatures with sharps and the tonics of their keys 

If we order the major and minor keys by the number of flats in their key signatures, we can similarly conclude: each next key has the tonic that can be obtained by applying the descending perfect fifth to the tonic of the previous one. The next figure shows this. We commonly use the key signatures with up to 7 flats.

Figure: The first four key signatures with flats and the tonics of their keys

As we have already explained in chapter Inversion of intervals, instead of applying the descending perfect fifth, we can apply the ascending inverted perfect fifth, which is the ascending perfect fourth. We would get the same tonics. 

Similar rules hold for the sharps and flats that occur in the key signatures. In the key signatures with sharps, the following sharps appear: F#, C#, G#, D#, A#, etc. We can construct them by successively applying the ascending perfect fifth. This is illustrated in the next figure.

Figure: The sharps in the first four key signatures with sharps

In the key signatures with flats, the following flats appear: Bb, Eb, Ab, Db, Gb, etc. We can construct them by successively applying the descending perfect fifth. This is illustrated in the next figure.

Figure: The flats in the first four key signatures with flats

All this information (and more) can be seen in the so-called circle of fifths which is shown in the next figure. It shows 15 key signatures, ordered by the number of accidentals. The major and minor keys that share the key signatures are also annotated in the figure. There is a key signature without accidentals at the top of the circle (the C major/A minor key signature). There are 7 key signatures with sharps, starting from the top and moving clockwise, as well as 7 key signatures with flats, starting from the top and moving counterclockwise. 

Some key signatures can be shown in two enharmonically equivalent ways, using either sharps or flats. These are enharmonically equivalent key signatures which are shown as the pairs of key signatures at the bottom of the circle of fifths. Their keys are enharmonically equivalent keys. The tonics and the diatonic tones of enharmonically equivalent keys are enharmonically equivalent note names. For instance, the F# major key and the Gb major key are enharmonically equivalent and are used equally often because their key signatures both have 6 accidentals. The B major key and the Cb major key are enharmonically equivalent but the B major key is used more often because its key signature has one less accidental than the key signature of the Cb major key. For the same reason, the Db major key is used more often than the C# major key. The same holds for the relative minor keys.

Figure: The circle of fifths

It is interesting to note that the circle of fifths has 12 positions, just like clocks have. Both Bach and Chopin wrote a special set of compositions containing compositions in different keys that cover all the 12 positions in the circle of fifths. These are Bach’s Well-Tempered Clavier and Chopin’s Preludes. Bach’s Well-Tempered Clavier consists of one prelude and one fugue in the major key and one prelude and one fugue in the minor key for each of the 12 positions in the circle of fifths. Thus, it is a set of 48 compositions. It follows the circle of fifths by raising the tonic one half step so the keys are ordered in this way: the C major key, the C minor key, the C# major key, the C# minor key, … , the B major key, the B minor key. In fact, the Well-Tempered Clavier has two books and each one is structured as described above. Chopin’s Preludes consist of one prelude in the major key and one in the minor key for each of the 12 positions in the circle of fifths. Thus, it is a set of 24 compositions. It follows the circle of fifths in the clockwise direction so the keys are ordered in this way: the C major key, the A minor key, the G major key, the E minor key, … , the F# major key, the Eb minor key, the Db major key, the Bb minor key, … , the F major key, the D minor key.   

Closely related keys

The distance between two keys is defined as their distance in the circle of fifths. Keys which are at the same position or adjacent in the circle of fifths are called closely related keys. For instance, the C major key and A minor key are closely related keys since they are at the same position in the circle of fifths. The C major key and G major key are also closely related keys since they are adjacent in the circle of fifths. The same holds for the C major key and F major key. However, the C major key and Ab major key are not closely related keys. 

Closely related keys have key signatures that differ in no more than one accidental. For instance, the C major key and A minor key have the same key signature while the G major key has one more sharp in its key signature than the C major key. Similarly, the F major key has one more flat in its key signature than the C major key. 

Each key has five closely related keys: one of them is its relative key which occupies the same position in the circle of fifths and the other four are major and minor keys that are adjacent to it in the circle of fifths. 

For instance, the C major key has following closely related keys: 

• The A minor key (relative key)

• The G major key and E minor key (adjacent keys in the circle of fifths, clockwise direction) 

• The F major key and D minor key (adjacent keys in the circle of fifths, counterclockwise direction) 

The following figure illustrates this and also shows commonly used names for the closely related keys which are based on the names of the scale degrees (supertonic, mediant, subdominant and dominant).

Figure: Closely related keys of the C major key

The C minor key has following closely related keys: 

• The Eb major key (relative key)

• The G minor key and Bb major key (adjacent keys in the circle of fifths, clockwise direction) 

• The F minor key and Ab major key (adjacent keys in the circle of fifths, counterclockwise direction) 

The following figure illustrates this and also shows commonly used names for the closely related keys which are based on the names of the scale degrees (subdominant, dominant, submediant and subtonic).

Figure: Closely related keys of the C minor key