CIRCLE OF FOURTH AND CIRCLE OF FIFTH (things you must know about them) (part 5)
HERE IS THE MYSTERY SEE
MATHEMATICAL ILLUSTRATION OF THE CIRCLE OF FOURTH AND THE CIRCLE OF FIFTH.
How to Mathematically
Derive the Circle of Fifth
The circle
of fifths drawn within the chromatic circle as a star (dodecagram)
To explain
the circles, The circle of fifths, or
fourths, may be mapped from the chromatic scale by multiplication, and vice
versa. between the circle of fifths and the chromatic scale (in integer
notation ) multiply by 7 ( M7 ), and for the circle of fourths multiply by 5
(P5).
Here is a demonstration of this procedure. Start off with an
ordered 12-tuple (tone row ) of integers
(0, 1, 2,
3, 4, 5, 6, 7, 8,
9, 10, 11)
C D E F G A B
C♯or D♭ D♯or E♭ F♯or G♭ G♯or A♭ A♯or B♭
representing the notes of the chromatic scale: 0 = C, 2 = D,
4 = E, 5 = F, 7 = G, 9 = A, 11 = B, 1 = C♯, 3 = D♯, 6 = F♯, 8 = G♯, 10 = A♯.
Now multiply the entire 12-tuple by 7:
[0(7), 1(7), 2(7),
3(7), 4(7), 5(7), 6(7), 7(7), 8(7),
9(7), 10(7),
11(7)]
The ans:
(0, 7, 14, 21, 28, 35, 42, 49,
56, 63, 70, 77)
and then apply a modulo 12 reduction to each of the numbers
(subtract 12 from each number as many times as necessary until the number
becomes smaller than 12):
[0, 7, 14-12, 21-12, 28-2(12), 35-2(12), 42-3(12),
49-3(12), 56-4(12), 63-5(12), 70-5(12), 77-6(12)
(0, 7, 2, 9,
4,
11, 6, 1,
8 3, 10, 5)
which is equivalent to
(0, 7, 2, 9,
4, 11,
6, 1,
8, 3, 10, 5)
(C, G, D, A,
E, B, F♯, C♯, G♯,
D♯, A♯, F)
which is the circle of fifths. Note that this is enharmonically equivalent to:
(C, F, B♭, E♭, A♭, D♭, G♭, B, E, A, D, G,).
Circle Of Fourth
Enharmonic equivalents
These keys
are easily interchanged using enharmonic equivalents. Enharmonic means that the notes sound the same, but are written
differently. For example, the key signature of D♭ Major, with five flats,
contains the same sounding notes, enharmonically, as C♯ Major (seven sharps).
HOW CIRCLE OF FIFTHS
FORM THE CHROMATIC MOVEMENT
Chromatics movement: this is the movement of 12 steps in
the 12 keys. Many at times, it is called chromatic scale which goes thus in the key of C
C, C♯or D♭, D, D♯or E♭,
E, F, F♯or G♭, G, G♯or A♭, A, A♯or B♭, B
NOTE: The chromatic movement does not necessarily has to
start from the key of C.
This is how circle of fourth or fifth can be form with chromatic
movements
(C, G, D, A,
E, B, F♯, C♯, G♯,
D♯, A♯, F)
(C, C♯or D♭, D,
D♯or E♭, E, F, F♯or
G♭, G, G♯or A♭, A, A♯or B♭, B)
above- circle of fifth
below - chromatics
Here is the method to substitute circles of fifth to
chromatic movement in progressions
a. On chord substitutions
1. iii – vi – ii
( E, A, D)
|
Chord progressions
|
p
|
Chord substitutions
|
CM
|
Root
|
Chromatic movement
|
||
|
iii
|
E
|
G
maj
|
G
B D G
|
iii
|
E
|
G
maj
|
G
B D G
|
|
vi
|
A
|
G
maj7
|
G
B D G♭
|
biii
|
E♭
|
E♭min
dom
|
E♭
F♯ B♭ C♯
|
|
ii
|
D
|
F
maj
|
F
A C F
|
ii
|
D
|
F
maj
|
E
F A
C
|
Where CM – Chromatic Movement
2. v – I – IV
(G, C, F)
|
Chord progressions
|
p
|
Chords substitutions
|
CM
|
Root
|
Chromatic movement
|
||
|
v
|
G
|
B♭maj
|
B♭D F B♭
|
v
|
G
|
G dom
|
G B♭D F
|
|
I
|
C
|
C maj7
|
C E G B
|
I
|
F♯
|
F♯maj7
|
F♯B♭ C♯ F
|
|
IV
|
F
|
F maj
|
F A C F
|
IV
|
F
|
F maj7
|
F A C E
|
Where CM – Chromatic Movement
3. vii – iii –
vi (B, E, A)
|
Chord progressions
|
p
|
Chords substitutions
|
CM
|
Root
|
Chromatic movement
|
||
|
vii
|
B
|
B dim♭7
|
B D F A
|
vii
|
B
|
D min
|
A D F
|
|
iii
|
E
|
E dom
|
E A♭B D
|
bvii
|
B♭
|
B♭maj7
|
B♭D F A
|
|
vi
|
A
|
A minor
|
A C E A
|
Vi
|
A
|
A minor
|
A C E A
|
Where CM – Chromatic Movement
4. #iv – vii –
iii (F#, B, E)
|
Chord progressions
|
p
|
Chords substitutions
|
CM
|
Root
|
Chromatic movement
|
||
|
#iv
|
F♯
|
A
maj
|
A
C♯ E A
|
#iv
|
F♯
|
A
maj
|
A
C♯ E A
|
|
vii
|
B
|
A
maj7
|
A
C♯ E A♭
|
iv
|
F
|
F
dom
|
F
A♭ C E♭
|
|
iii
|
E
|
G
maj
|
G
B D G
|
iii
|
E
|
G
maj
|
G
B D G
|
Where CM – Chromatic Movement
Try this on the previous lessons (dia-tone
and tri-tone substitution) and see how it sounds.
Thanks I hope this helps.
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