Basically, the circle of fifths is a series of ascending perfect
fifths. It is called a circle because after twelve intervals in a
row, you end up on a note that is enharmonically equivalent to the
starting note, admittedly several octaves higher.
For example:
C, G, D, A, E, B , F#, C#, G#, D#, A#, E#, B#.
The B# is enharmonically equivalent (means the same thing as) a C.
It's only fair to note that if these were really 'perfect' fifths
with a 3:2 frequency ratio, then the B# would not be exactly in tune
with the C.
But back to the circle idea. If music theory people were
only interested in circles.... series that repeat.... why use the
fifths? Why not minor seconds. Those repeat after twelve notes
also..... a chromatic scale. Or, why not the circle of major
seconds.... those repeat after six notes.... the whole tone
scale. But they never hit the other six notes. Or, why not
a circle of minor thirds? That would repeat after four notes., like
this....
C E-flat, G-flat, A, C
Granted, the interval G-flat to A is not a third, spelling wise, but
enharmonically it is a minor third. And again, that series misses
the other eight available notes.
Or, why not a circle of major thirds, like this...
C E G# C
That repeats after just three notes, forms an augmented triad,
and misses the other nine notes.
Or, how about a circle of fourths? That amounts to going
backwards through the circle of fifths.
Perhaps people use the circle of fifths because it is the only pattern
besides a circle of minor seconds includes all twelve notes before
repeating.
But there is more to this than what I've said so far. It involves
'growing' the chromatic scale, or generating the need for all notes in
the chromatic scale, as a result of seeking to play in more than one
key.
To facilitate that concept, I think of the set of twelve chromatic
notes as showing on a round clock face. Both music notes and
hours on a clock repeat every twelve. The notes
positioned as follows:
Note
Hour
C
12
C# (D-flat)
1
D
2
D# (E-flat)
3
E
4
F
5
F# (G-flat)
6
G
7
G# (A-flat)
8
A
9
A# (B-flat)
10
B
11
After all, our twelve notes are periodic. their note names
repeat over and over as you ascend through the chromatic scale. My
concept is that as you move clockwise, the pitches ascend. It's
just an arbitrary concept but it helps me when I visualize it that
way. Here, I'm just going to try to explain how to evolve from
a
diatonic instrument that is useful for just in one key to a
fully chromatic instrument that is good to play in any key.
Let's begin with an
instrument set up to play only in the C major scale. I have a
chromatic guitar zither built for twelve notes per octave, but if I
were in the process of re-stringing it, I could just use seven notes
per octave like this: C, D, E, F, G, A and B. I have a diatonic
zither like that, but the partly strung chromatic zither makes a better
here. Anyway, a
set of those notes for playing in C (or a
minor, or D dorian, etc) would look like the one you see here....I
removed five of the
twelve notes.... C#, D#, F#, G% and A#, since they are not currently
installed on the zither. We don't need them for playing in C
major. I'm using roman numerals to indicate the positions in
the major scale: I = tonic, II = supertonic, III= mediant, IV=
subdominant, V = dominant, etc. So what you see here on the
clock face are five empty spaces where those sharp and flat notes would
have been.
Now, let's speculate
that someone picks up thispartly strung zither and objects to
using it in the key of C. To expedite matters, we
want to add only one more string per octave to add one more playable
key to the instrument. To find the new key, we rotate the
inner circle in the previous diagram, one 'click' at a time
clockwise. Its roman numerals showt the spacing of scale degrees
in a major scale. The first click shows a key requiring five extra
strings per octave Second click shows a key requiring two extra
strings per octave. Third click has a key requiring three
extra strings per octave: fourth click has a key requiring four extra
strings per octave. The fifth click shows F major, requiring just
one extra string per octave. But that is a perfect fifth down in
the circle of fifths, and I wanted to ascend first. Sixth click
shows F#, requiring five extra strings per octave. After seven
clicks we find the position where all but one of the scale
degrees on the inner circle match the notes currently on the zither....
ie, on the outer circle. Then we would install one more string
per octave to match the as yet unsatisfied scale degree in the
inner circle. So in this case, we would be adding an F# to each octave.
The arrow in the diagram shows the first increment in the circle
of fifths, from C to G. And this corresponds to adding one sharp
in the key signature, so that the key of G has one sharp.
Repeating this
process would take us farther up through the circle of fifths.
Now that we added the F#, we see that if we go seven more 'clicks'
clockwise, we have another point where we only need to add one more
string per octave to make another key available, namely the key of
D. Now the diagram has two arrows indicating ascent through the
circle of fifths, from C to G to D. This also adds a second sharp
sign to the key signature, so D major has two sharps in its key
signature. And our chromatic zither now has a C# in each octave.
Repeating this
process yet again would take us farther still up through the circle of
fifths. Now that we added the F# and C#, we see that if we go
seven more
'clicks' clockwise, we have another point where we only need to add
one more string per octave to make another key available, namely the
key of A. Now the diagram has three arrows indicating ascent
through the
circle of fifths, from C to G to D and then to A. This also adds
a third sharp sign
to the key signature, so A major has three sharps in its key
signature.
And our chromatic zither now has a G# in each octave.
Repeating this
process still another time would take us farther still up through the
circle of
fifths. Now that we added the F#, C# and G# to the zither
on each octave, we see that if we go seven more
'clicks' clockwise, we have another point where we only need to add
one more string per octave to make another key available, namely the
key of E. Now the diagram has four arrows indicating ascent
through the
circle of fifths, from C to G to D to A and then to E. This
also adds a fourth sharp sign
to the key signature, so E major has four sharps in its key
signature.
And our chromatic zither now has a D# in each octave.
Repeating this
process still another time would take us farther still up through the
circle of
fifths. Now that we added the F#, C#, G# and D# to
the zither on each octave, we see that if we go seven more
'clicks' clockwise, we have another point where we only need to add
one more string per octave to make another key available, namely the
key of B. Now the diagram has five arrows indicating ascent
through the
circle of fifths, from C to G to D to A to E and then to B.
This also adds a fifth sharp sign
to the key signature, so B major has five sharps in its key
signature.
And our chromatic zither now has an A# in each octave.
There is no more room to add any more strings to the chromatic guitar
zither. If we continued this process one more time, we would see
that having added the F#, C#, G#, D#, and A# strings to the zither
would allow us to start play in the key of F# major without
adding any more strings. We would have to refer to the seventh
degree of the F# major scale as an E#. This is often done in
music, but since it duplicates the F string, we will stop ascending the
circle of fifths at this point.
Thus far we have
not mentioned any flat keys. If we set the inner circle of our
diagram back at the C scale again, then we could have rotated the
inner circle seven clicks counterclockwise. That would have
positioned us at an F major scale. If our guitar zither only had
the C, D, E, F, G, A and B strings again, we would need to add just one
string, the B-flat string to each octave, to be able to play in
F. This is a descent in the circle of fifths, and adds one
flat to the key signature, so the key of F major has one flat in its
signature. Note the single arrow, indicating descent through the
circle of fifths from C to F.
And again, we could have
rotated the inner circle seven clicks counterclockwise. Since we have
added the B-flat strings to each octave, we find that we would
need to add just one
string, the E-flat string to each octave, to be able to play in
B-flat.
This descent in the circle of fifths, adds one more flat to the
key signature, so the key of B-flat major has two flat in its
signature.
Note the two arrows, in the diagram, indicating descent through the
circle of fifths
from C to F and then to B-flat.
And again, we could
have
rotated the inner circle seven clicks counterclockwise. Since we have
added the B-flat and E-flat strings to each octave, we find that
we would need to
add just one
string, the A-flat string to each octave, to be able to play in
E-flat.
This descent in the circle of fifths, adds one more flat to the
key signature, so the key of E-flat major has three flat in its
signature.
Note the three arrows, in the diagram, indicating descent through the
circle of fifths
from C to F to B-flat and then to E-flat.
And again, we could
have
rotated the inner circle seven clicks counterclockwise. Since we have
added the B-flat, E-flat and A-flat strings to each octave,
we find that we would need to
add just one
string, the D-flat string to each octave, to be able to play in
A-flat.
This descent in the circle of fifths, adds one more flat to the
key signature, so the key of A-flat major has four flat in its
signature.
Note the four arrows, in the diagram, indicating descent through the
circle of fifths
from C to F to B-flat to E-flat and then to A-flat.
And again, one
final time, (whew) we could have
rotated the inner circle seven clicks counterclockwise. Since we have
added the B-flat, E-flat , A-flat and D-flat strings to each
octave, we find that we would need to
add just one
string, the G-flat string to each octave, to be able to play in
D-flat.
This descent in the circle of fifths, adds one more flat to the
key signature, so the key of D-flat major has five flat in its
signature.
Note the five arrows, in the diagram, indicating descent through the
circle of fifths
from C to F to B-flat to E-flat to A-flat and then to D-flat.
And once again, we've filled up the string positions on the
chromatic guitar zither. If we turn the inner circle of our
diagram one more time, we find that we would be playing in G-flat major
(enharmonically equivalent to F# major) and the fourth degree of G-flat
major is a C-flat, which is enharmonically equivalent to a B, a string
that we already have and cannot add again. So we stop here.
Note that some pieces (or sections of them) are written in G-flat major
and do have a C-flat in the key signature.
Here is an interesting thing to notice. If you were keeping score
about what strings we added to the guitar zither, when ascending the
circle of fifths we added strings in this order:
F#, C#, G#, D#, A#
but while descending the circle of fifths, we added strings in this
order:
B-flat, E-flat, A-flat, D-flat, G-flat
and if you compare these enharmonically, you see that these two orders
of string addition are the reverse of each other.
To sum up, other than circle of minor seconds that produces a chromatic
scale, the circle of fifth is the only sequence of same type intervals
that takes twelve notes to come back to the beginning and that includes
all twelve notes of the chromatic scale. It is useful in
determining which keys use virtually the same notes, and in determining
the number of sharps or flats in a key signature.
begin with an
instrument set up to play only in the C major scale. I have a
chromatic guitar zither built for twelve notes per octave, but if I
were in the process of re-stringing it, I could just use seven notes
per octave like this: C, D, E, F, G, A and B. I have a diatonic
zither like that, but the partly strung chromatic zither makes a better
here. Anyway, a
set of those notes for playing in C (or a
minor, or D dorian, etc) would look like the one you see here....I
removed five of the
twelve notes.... C#, D#, F#, G% and A#, since they are not currently
installed on the zither. We don't need them for playing in C
major. I'm using roman numerals to indicate the positions in
the major scale: I = tonic, II = supertonic, III= mediant, IV=
subdominant, V = dominant, etc. So what you see here on the
clock face are five empty spaces where those sharp and flat notes would
have been.
Now, let's speculate
that someone picks up thispartly strung zither and objects to
using it in the key of C. To expedite matters, we
want to add only one more string per octave to add one more playable
key to the instrument. To find the new key, we rotate the
inner circle in the previous diagram, one 'click' at a time
clockwise. Its roman numerals showt the spacing of scale degrees
in a major scale. The first click shows a key requiring five extra
strings per octave Second click shows a key requiring two extra
strings per octave. Third click has a key requiring three
extra strings per octave: fourth click has a key requiring four extra
strings per octave. The fifth click shows F major, requiring just
one extra string per octave. But that is a perfect fifth down in
the circle of fifths, and I wanted to ascend first. Sixth click
shows F#, requiring five extra strings per octave. After seven
clicks we find the position where all but one of the scale
degrees on the inner circle match the notes currently on the zither....
ie, on the outer circle. Then we would install one more string
per octave to match the as yet unsatisfied scale degree in the
inner circle. So in this case, we would be adding an F# to each octave.
The arrow in the diagram shows the first increment in the circle
of fifths, from C to G. And this corresponds to adding one sharp
in the key signature, so that the key of G has one sharp.
Repeating this
process would take us farther up through the circle of fifths.
Now that we added the F#, we see that if we go seven more 'clicks'
clockwise, we have another point where we only need to add one more
string per octave to make another key available, namely the key of
D. Now the diagram has two arrows indicating ascent through the
circle of fifths, from C to G to D. This also adds a second sharp
sign to the key signature, so D major has two sharps in its key
signature. And our chromatic zither now has a C# in each octave.
Repeating this
process yet again would take us farther still up through the circle of
fifths. Now that we added the F# and C#, we see that if we go
seven more
'clicks' clockwise, we have another point where we only need to add
one more string per octave to make another key available, namely the
key of A. Now the diagram has three arrows indicating ascent
through the
circle of fifths, from C to G to D and then to A. This also adds
a third sharp sign
to the key signature, so A major has three sharps in its key
signature.
And our chromatic zither now has a G# in each octave.
Repeating this
process still another time would take us farther still up through the
circle of
fifths. Now that we added the F#, C# and G# to the zither
on each octave, we see that if we go seven more
'clicks' clockwise, we have another point where we only need to add
one more string per octave to make another key available, namely the
key of E. Now the diagram has four arrows indicating ascent
through the
circle of fifths, from C to G to D to A and then to E. This
also adds a fourth sharp sign
to the key signature, so E major has four sharps in its key
signature.
And our chromatic zither now has a D# in each octave.
Repeating this
process still another time would take us farther still up through the
circle of
fifths. Now that we added the F#, C#, G# and D# to
the zither on each octave, we see that if we go seven more
'clicks' clockwise, we have another point where we only need to add
one more string per octave to make another key available, namely the
key of B. Now the diagram has five arrows indicating ascent
through the
circle of fifths, from C to G to D to A to E and then to B.
This also adds a fifth sharp sign
to the key signature, so B major has five sharps in its key
signature.
And our chromatic zither now has an A# in each octave.
Thus far we have
not mentioned any flat keys. If we set the inner circle of our
diagram back at the C scale again, then we could have rotated the
inner circle seven clicks counterclockwise. That would have
positioned us at an F major scale. If our guitar zither only had
the C, D, E, F, G, A and B strings again, we would need to add just one
string, the B-flat string to each octave, to be able to play in
F. This is a descent in the circle of fifths, and adds one
flat to the key signature, so the key of F major has one flat in its
signature. Note the single arrow, indicating descent through the
circle of fifths from C to F.
And again, we could
have
rotated the inner circle seven clicks counterclockwise. Since we have
added the B-flat and E-flat strings to each octave, we find that
we would need to
add just one
string, the A-flat string to each octave, to be able to play in
E-flat.
This descent in the circle of fifths, adds one more flat to the
key signature, so the key of E-flat major has three flat in its
signature.
Note the three arrows, in the diagram, indicating descent through the
circle of fifths
from C to F to B-flat and then to E-flat.
And again, we could
have
rotated the inner circle seven clicks counterclockwise. Since we have
added the B-flat, E-flat and A-flat strings to each octave,
we find that we would need to
add just one
string, the D-flat string to each octave, to be able to play in
A-flat.
This descent in the circle of fifths, adds one more flat to the
key signature, so the key of A-flat major has four flat in its
signature.
Note the four arrows, in the diagram, indicating descent through the
circle of fifths
from C to F to B-flat to E-flat and then to A-flat.
And again, one
final time, (whew) we could have
rotated the inner circle seven clicks counterclockwise. Since we have
added the B-flat, E-flat , A-flat and D-flat strings to each
octave, we find that we would need to
add just one
string, the G-flat string to each octave, to be able to play in
D-flat.
This descent in the circle of fifths, adds one more flat to the
key signature, so the key of D-flat major has five flat in its
signature.
Note the five arrows, in the diagram, indicating descent through the
circle of fifths
from C to F to B-flat to E-flat to A-flat and then to D-flat.