Circle of Fifths

The theory of everthing

The Circle of Numbers

An octave contains 12 unique notes a semitone apart (13 including the octave).

The Major (diatonic) scale consists of the eight notes (including the octave) separated by the intervals: Tone Tone Semitone Tone Tone Tone Semitone [TTS T TTS]. The fifth note in the scale is therefore 7 semitones above the root.

If we start at on the root note of a scale and step up 7 semitones we are on the 5th of that scale. If we then regard this as a new starting point and take another step up of 7 semitones we are on the 5th of this scale. By repeating this exercise 12 times we will visit each of the 12 notes in our original scale.

The reason is simply that 12 does not divide by 7. The remainder of 12/7 is 5 (12 % 7 = 5). So each time we step we are offset by 7 - 5 = 2 semitones. (Until the offset is divisible by 7, so really its: 2n % 7). Step 12 times and you will have visited every note and will be back at the beginning.

Hopefully, the table below illustrates the point.

123456789101112
123456789101112
123456789101112
123456789101112
123456789101112
123456789101112
123456789101112

The Circle of 5ths is just another way of writing this down. The reason the Circle of 5ths is so important is because of the properties that flow from presenting it in this way.

A final crucial point to notice is that if we step backwards from our starting point to the 5th our step will cover 5 semitones, not 7 ... because 12 - 7 = 5. Five semitones is a 4th, which is why the Circle of 5ths is often also called the Circle of 4ths. In the opposite direction the same arguments apply, its just that in this case 12 is not divisible by 5.

The Circle of 5ths

If we take the C Major scale as our starting point and do the 7 semitone forward hops as described above and arrange the notes in a circle we have the Circle of 5ths:
C
G
D
A
E
B
Gb/F#
Db
Ab
Eb
Bb
F
a
e
b
f#
c#
g#
eb/d#
bb
f
c
g
d

The Properties of the Circle of 5ths

Find the Key

Penatonic Scales

Chord Progressions

Tetra Chords

TBD