Lesson 24 - The Circle of Fifths

Intervals represented as two sine waves.

In western music, an octave is divided into twelve evenly sounding pitches as shown in the diagram below. A single semi-tone is divided into 100 cents, the human ear can discern around 5 to 10 cents in tonal difference. In other systems around the world, the tuning may be different, an octave many be divided into 72 pitches.

The blue sine wave represents a fixed bass note, and a black sine wave represents the high interval. We're going to stretch the black sine wave reducing it's frequency, in other words, we're starting from the high octave and moving down the keyboard. The right half of the each diagram is divided into twelve interval locations, which isn't represented as linear time, the time is selected for equal temperament tuning, meaning as we move down the keyboard in semi-tones, the tonal distance sounds even to our ears. By stretching the black sine wave like a spring, you can transverse all the intervals, a semi-tone at a time. Focus your attention on the octave diagram, the exact centre where the two sine waves intersect at zero amplitude. As you stretch the sine wave one semi-tone, diagram 7 results. Continue to stretch producing all the intervals, -7, +5, 5, -5, 4, 3, -3, 2, -1 and finally you will reach the bass note. 

NOTE: If you're going to count the semi-tones to find the interval, make sure you start counting from the right side of the diagram towards the middle,  not the middle towards the right.

The circle of fifths.

Now you have some understanding of the intervals, focus your attention on the fifth interval, has the smoothest sound of all the intervals and most similar to the octave and root note because both end points of the sine wave are all zero amplitude for both waves.

There are twelve notes in an octave, applying the western major scale formula to all twelve notes results in twelve major key signatures. There is a beautiful relationship between all key signatures illustrated in the circle of fifths diagram. All the notes are arranged around the circle, a perfect fifth apart, depending on the direction you move on the keyboard, up or down and the direction you move around the circle of fifths.  When we travel around the circle, each key signature differs from the previous key signature by a semi-tone. For example, the difference between the key of C and the key of G is the notes F and F#, F is the 4th interval in the key of C, F# is the major seventh interval in the key of G. To move from C to G, all you need to do is raise F a semi-tone to F#. Knowing what to do when moving from one key to another helps you with your improvisations.

So it turns out, by arranging the scales in fifths, there is a smooth transition from key to previous key, a semi-tone move, regardless of the direction you're travelling.

Observe the following diagram, if you're moving clockwise around the circle of fifths, up the keyboard, you're moving in fifths. Each key signature changes by a semi-tone from the previous key signature. Back in lesson 3, we learned how to construct each key signature, now we can see them around the circle of fifths. Starting on C, move up a fifth to G, and play all the white notes from G to high G, you need to sharpen the -7 into a major seven to construct the Ionian. When playing from G to G, all the white notes, this is a G Mixolydian, to make it G Ionian, you need to sharpen the -7 into a 7. So an Ionian has a modal equivalence ≃  Mixolydian #-7, G Ionian is the same scale as G Mixolydian #-7. 

X Ionian ≃ X Mixolydian #-7

X Mixolydian #-7 ≃ X Ionian

Remember the fifths rule.

If your right thumb starts on a white note, your pinky will end on a white note, except for the note B.

If you right thumb starts on a black note, your pinky will end on  a black note, except for the note Bb.

You can  use the fifth rule to remember the order at the piano. Start on the note C.

• Place your right hand thumb on C, the fifth is G.

• Place your right hand thumb on G, the fifth is D.

• Place your right hand thumb on D, the fifth is A.

• Place your right hand thumb on A, the fifth is E.

• Place your right hand thumb on E, the fifth is B.

• Place your right hand thumb on B, the fifth is Gb.  ( Fifth rule exception)

• Place your right hand thumb on Gb, the fifth is Db.

• Place your right hand thumb on Db, the fifth is Ab.

• Place your right hand thumb on Ab, the fifth is Eb.

• Place your right hand thumb on Eb, the fifth is Bb.

• Place your right hand thumb on Bb, the fifth is F. ( Fifth rule exception)

• Place your right hand thumb on F, the fifth is C.

Remember the note order around the circle of fifths.

Clockwise: C G D A E B and also B E A D G C

Counter Clockwise: C F Bb Eb Ab Db Gb and also Gb Db Ab Eb Bb F C

On the piano, apply the circle of fifths rule and you'll be able to see this order.

Remember the amount of sharps and flats for each key signature to help your visualisation.

• Key C has no black notes.

• Key G has one black note.

• Key D has two black notes.

• Key A has three black notes.

• Key E has four black notes.

• Key B has five black notes.

• Key F has one black notes.

• Key Bb has two black notes.

• Key Eb has three black notes.

• Key Ab has four black notes.

• Key Db has five black notes.

• Key Gb also has five black notes.

Remember all the black note names.

• Key C has no black notes.

• Key G has one black note, F#.

• Key D has two black notes, F# , C#.

• Key A has three black notes, F#, C#, G#.

• Key E has four black notes, F#, C#, G#, D#.

• Key B has five black notes, F#, C#, G#, D#, A#.

• Key F has one black notes, Bb.

• Key Bb has two black notes, Bb, Eb.

• Key Eb has three black notes, Bb, Eb, Ab.

• Key Ab has four black notes, Bb, Eb, Ab, Db.

• Key Db has five black notes, Bb, Eb, Ab, Db, Gb.

• Key Gb also has five black notes, Bb, Eb, Ab, Db, Gb, Cb (Cb is the note B).

Observe how the fifth rule is applied between F# and C#, C# and G#, G# and D#, D# and A# (Bb)

Observe how the fifth rule is applied between Cb (B) and Gb, Gb and Db, Db and Ab, Ab and Eb, Eb and Bb, Bb and F, F and C.

The circle of fourths.

The circle of fifths is also know as the circle of fourth because a fifth is the inversion of a fourth. Observe in the following diagram, from C to F going up the keyboard is a perfect 4th however if you start at C and go down the keyboard a fifth, you arrive at the same note F. F from C going up is a 4th interval, F from C going down is a 5th interval, this is best illustrated in the diagram. You call this an inversion, a fifth inverted is a fourth, a fourth inverted is a fifth, the same applies to all intervals. Every interval inverted is another interval, a minor 2nd inverted is a major 7th, a second inverted is a minor 7th, a minor 3rd inverted is a 6th, etc.

Since the circle of fifths is also referred to as the circle of fourths, there is often some confusion surrounding this, so let’s clarify. Some people believe going clockwise around the circle of fifths, each note is a 5th apart and going counter-clockwise is a 4th. This is not entirely correct, it’s not only the direction your transverse around the circle of fifths, it’s both the tonal direction you move on the instrument, up or down and the direction your transverse around the circle of fifths.

         Starting at the lowest C and moving up the keyboard, transverse clockwise around the circle of fifths, you move up in fifths.

         Starting at the highest C and moving down the keyboard, transverse clockwise around the circle of fifths, you move down in fourths.

         Starting at the lowest C and moving up the keyboard, transverse counter-clockwise around the circle of fifths, you move up in fourths.

         Starting at the highest C and moving down the keyboard, transverse counter-clockwise around the circle of fifths, you move down in fifths.