Circle of Fifths

The circle of fifths lays out the 12 tones in a circle, moving clockwise in fifths. If C is 12 o'clock, its fifth, G, is 1 o'clock. G's fifth, D, is two o'clock. And so on.

Inter-relation of Keys

So far we've really only talked about C Major and A minor, which are composed of the 7 white keys per octave on the piano. But scales can be built upon any of the 12 tones in the chromatic scale. We know the whole-half patterns of the major and minor scales, so we can figure out what notes any scale has, but what about the interrelation between scales? This is where the circle of fifths helps to keep things in perspective.

Let's look at the scales of F Major, C Major, and G Major:

F  - G  - A  - Bb - C  - D  - E  - F
C  - D  - E  - F  - G  - A  - B  - C
G  - A  - B  - C  - D  - E  - F# - G

F and C Major differ by only one note (C Major has a B while F Major has a Bb). Similarly, C and G also differ by one note (F vs. F#). Let's look at the whole-half patterns:

                  w    w    h    w    w    w    h
               F  - G  - A  - Bb - C  - D  - E  - F

   w    w    h    w    w    w    h    w    w    h    w    w    w    h              
C  - D  - E  - F  - G  - A  - B  - C  - D  - E  - F  - G  - A  - B  - C

                       w    w    h    w    w    w    h
                    G  - A  - B  - C  - D  - E  - F# - G

The whole-half pattern from F to F in the key of C Major is w-w-w-h-w-w-h. If we flatten the 4th note, the B, we get w-w-h-w-w-w-h - the major scale pattern.

The whole-half pattern from G to G in the key of C Major is w-w-h-w-w-h-w. If we sharp the 7th note, the F, we get w-w-h-w-w-w-h - the major scale pattern.

Remember that F is a fifth below C and G is a fifth above. We notice that if we change the key by moving in fifths, only one note changes. While no two major scales will have different tonic notes but identical sets of notes, some scales are more inter-related or harmonic than others.

The circle of fifths lists keys in 5ths, which is useful to determine how interrelated any two keys are.

The full circle only consists of the 12 tones, which can be arranged like the hours on an analog clock. The points opposite each other (12 vs. 6, 3 vs. 9, 11 vs. 5, etc.) are the least related, while the points nearest each other (12 and 1, 1 and 2, etc.) are most related. As we go right or clockwise, we sharp the 4th of the current key to get the next key. For example in C Major, sharping the 4th (F) gives us the scale of G Major. As we left or counter-clockwise, we flatten the 7th of the current key to get the next key. For C Major, we flat the B to get F Major.

So each movement is a single note difference. Note that the major scale has two half-steps, which means two notes right next to each other. Every other major or natural minor scale has to contain at least one of these notes, since the largest movement between neighboring notes is a whole step. So the least related scales will share 2 notes. For example, C and C# don't share any notes by name - every note in C is the same note in C# but sharped. However, E# = F and B# = C; so they still have two notes in common.

Full Circle

Let's find all 12 tones in order of fifths, going both ways from C:

                             C
                         F - C - G
                    Bb - F - C - G - D
               Eb - Bb - F - C - G - D - A
          Ab - Eb - Bb - F - C - G - D - A - E 
     Db - Ab - Eb - Bb - F - C - G - D - A - E - B
Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F#

F# is the same pitch as Gb, which means we've completed the circle. If we keep going, we still get unique key names, but the pitches will match other keys already in the circle. For example, continuing left (counter-clockwise), we reach Cb, which is the same as B. Going right (clockwise), we reach C#, which is the same as Db.

Key	Cb	Gb	Db	Ab	Eb	Bb	F	C	G	D	A	E	B	F#	C#
Flats/Sharps	7 b's	6 b's	5 b's	4 b's	3 b's	2 b's	1 b's	0 b/#'s	1 #'s	2 #'s	3 #'s	4 #'s	5 #'s	6 #'s	7 #'s
Unison	Cb	Gb	Db	Ab	Eb	Bb	F	C	G	D	A	E	B	F#	C#
2nd	Db	Ab	Eb	Bb	F	C	G	D	A	E	B	F#	C#	G#	D#
3rd	Eb	Bb	F	C	G	D	A	E	B	F#	C#	G#	D#	A#	E#
4th	Fb	Cb	Gb	Db	Ab	Eb	Bb	F	C	G	D	A	E	B	F#
5th	Gb	Db	Ab	Eb	Bb	F	C	G	D	A	E	B	F#	C#	G#
6th	Ab	Eb	Bb	F	C	G	D	A	E	B	F#	C#	G#	D#	A#
7th	Bb	F	C	G	D	A	E	B	F#	C#	G#	D#	A#	E#	B#
Oct	Cb	Gb	Db	Ab	Eb	Bb	F	C	G	D	A	E	B	F#	C#

Above is the list of the keys from Cb to C#. Note that this is actually 15 tones. Since C# = Db, Cb = B, and F# = Gb, we can eliminate them, leaving with only the 12 distinct tones on the circle of fifths.

Modulation

Modulation is the practice of changing keys. The source and destination keys are important. Many composers only modulate into destination keys based on the 4th or 5th of the source key, as these are the most harmonic keys.

Below is a table to show the most interrelated keys.

Common Notes	Diatonic Keys	Non-Diatonic Keys
6	P4, P5
5	M2	D7
4	M6	m3
3	M3	m6
2	M7	m2, d5

Minor Keys

So far I have always discussed the circle of fifth in terms of major keys. Since natural minor keys have the same notes as their relative major, the same rules apply. Other natural minor keys based on the 4th or 5th of the current natural minor key will have one note difference, and the 3 keys opposite the current key are the last related with 5 notes different.

Harmonic minor is a bit different. The M7 does not carry over to the next key in sequence. For example, in A Harmonic Minor, there is a G#. But in the key a fifth above, E Harmonic Minor, there is a G. Similarly, a fifth below A is D. D Harmonic minor has a G as well.