Music Theory Basics:

The Circle of Fifths

COURSE WORKBOOK

Your Instructor: Ben Laude

Let’s return to “Happy Birthday,” this time with the knowledge that it begins on scale degree 4. Those first
notes, C-D-E-F, form the last four notes of the F major scale. Are they also a part of another major scale?

5̂ 6̂ 5̂ 1̂ 7̂

It turns out that these four notes are also the first four notes of a C major scale! This isn’t just a coincidence;
let’s examine why and how.

A four-note segment of a scale is also called a tetrachord. In a major scale, we talk about the first four notes
and last four notes as two different tetrachords, each spanning two whole steps and a half step. Look again
at the interval pattern of any major scale, and you’ll see that there actually is one repeating unit:

W W H W W W H
F G A B♭ C D E F
W W H W W W H
C DEF G A B C
Scale degrees 1-4 form the first tetrachord, while scale degrees 5-8 form the second one, and they are
separated by a whole step. The first part of “Happy Birthday” uses the top tetrachord of an F major scale,
which is the same as the bottom tetrachord of a C major scale.

Check out Invention No. 1 by J.S. Bach. He outlines the first tetrachord, and then catapults upward to the
second tetrachord on scale degree 5. In the second bar, he takes the whole melody up five notes (also
called a fifth), but leaves all the intervals intact! He essentially repeats the tune in the top tetrachord.

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Just as C-D-E-F is shared by both F and C major scales, the tetrachord G-A-B-C is shared by both C and G

major scales. This makes sense, because C is a fifth above F (five notes of a scale, or five spaces and lines

on a staff), and G is a fifth above C. Keys separated by a fifth will always share one tetrachord:

ˆ ˆ ˆ ˆ
1 2 3 4

G A B C D E F♯ G
ˆ ˆ ˆ ˆ
5 6 7 1

C D E F G A B C

If we extend this pattern and begin Bach’s Invention No. 1 in the key of G major, then D-E-F♯ is the top

tetrachord. In which key is it the bottom tetrachord? That would be D major – the key a fifth above G. Look

at the three tetrachords that we’ve examined thus far: C-D-E-F, G-A-B-C, and D-E-F♯-G. Each is separated

by a whole step, like the mortar between bricks. The F♯ is born from the repetition of this WWH pattern.

in G:

in D:

We cannot think of musical keys as separate islands that have nothing in common. They are in fact part of

the same body of water. Notice that these new accidentals fall in a specific place within each scale: F♯

appeared as scale degree 7 in G major. If we look at F major (the key a fifth below C), we find that B♭ appears

as scale degree 4. 7 is the half step below 1, while 4 is a half step above 3.

5th
G A B C D
5th
C D E F G A B C

F G A B♭ C D E F

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When we move key centers by a fifth, as we just did, we introduce two new terms. First is dominant,
meaning to move upward to the fifth scale degree of any key. Its mirror image is 1 down to 4 (covering the
interval of a fifth in the other direction,) which is called the subdominant. Next, we will shift keys by moving
in the dominant or subdominant direction with this motif from the Bach Invention.

If we continue to transpose this opening melody up a fifth, starting on C, we begin adding accidentals. G
major births F♯, D major creates C♯, and then G♯, D♯, A♯, E♯ (yes it’s a white key, the same as F!), and finally B♯
in C♯ major, exhausting all of the options for sharps. This makes sense, because C♯ major is just C major with
all the notes raised a half-step.

F♯ G♯ A♯ B♯
Accidentals:
C♯ D♯ E♯
Keys:
G D A E B F♯ C♯
Let’s take the first four notes of this melody (CDEF), ending on scale degree 4, and begin it on scale degree 1
of the key in the subdominant direction (which is F, the same as where we left off). Scale degree 4 in F is B♭.
In B♭ major we generate an E♭, continuing to A♭, D♭, G♭, C♭, and F♭, before we’ve exhausted all the seven
possible white keys that we can flat.

in F:

in B♭:

Accidentals: B♭ A♭ G♭ F♭
E♭ D♭ C♭
Keys:
C F B♭ E♭ A♭ D♭ G♭ C♭
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Through this process, we’ve created all the major and minor keys, treating C major as the center piece. As

we went, we added sharps or flats and retained them throughout the cycle. Once we added F♯, all the

remaining sharp keys contained an F♯. Similarly, once we added B♭, all the remaining flat keys had one.

The flats and sharps were also generated in a very specific order. The first sharp that we added, moving up a

fifth from C to G, was F♯, and so on, until we got to B♯. With flats, we began with B♭, moving down a fifth from

C to F, until we got to F♭. It’s important to remember this order, since composers use it in something called a

key signature to make it easy to read lots of sharps or flats in a piece.

F, C, G, D, A, E, B B, E, A, D, G, C, F

A shorcut for deducing key signatures: the last sharp in any key signature is also the leading tone of the

major key that you’re in. This should make sense, because scale degree 7 is always the most recent

accidental generated by moving up in fifths. For flats, the second-to-last flat is the key that you’re in. In B♭

major, the flats are B and E, B♭ being the second-to-last. Also note that most sharp keys begin on a natural

note and most flat keys begin on a flat note. We could’ve continued to F♭ major or G♯ major, but these aren’t

useful as keys since they require double-sharps and double-flats to spell out the related scales.

Essentially, we’ve constructed a number line with sharps on one side and flats on the other side. How can we

connect the ends of this line to create a circle?

subdominant direction (down a fifth) dominant direction (up a fifth)

C♭ G♭ D♭ A♭ E♭ B♭ F C G D A E B F♯ C♯
adding flats adding sharps

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Let’s introduce a concept called enharmonic equivalence. This says that F♯ and G♭ are fundamentally the
same sound, as are the sharp and flat spellings of any black key on the keyboard. This means that we can
connect the circle by saying that F♯ major (with 6 sharps) and G♭ major (with 6 flats) are really the same
key. Play these two scales on the keyboard and you’ll see they use the exact same pitches, just spelled
differently. What’s even more fascinating is that we derived these scales by going in different directions from
C: F♯ by moving toward the dominant, and G♭ by moving toward the subdominant.

the pitches in these two scales are equivalent!

F♯ Major Scale F♯ G♯ A ♯ B ♯ C ♯ D♯ E ♯ F ♯
G♭ Major Scale G♭ A♭ B ♭ C ♭ D ♭ E ♭ F G♭
Similarly, B major can also be called C♭ major, and C♯ major can also be D♭ major. Our number line has 15
keys, of which only 12 contain unique sets of pitches. The remaining 3 are enharmonically equivalent to
another key.

If we can derive the minor mode from a major scale, then surely all the key signatures will also apply to the
relative minors. Remember that the sixth scale degree of any major key is its relative minor (we can also
count two steps backward from the root to get to the same note!) Before going to the next page, try to
figure out the relative minor keys of all 12 major keys above on the now-completed circle of fifths.

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 a
d e
g b
c f
f c
f /g g /a
d /e

Every key has a set of 5 other closely related major and minor keys. Closely related keys are defined as

those with only one accidental different from a central key. We’ll explore this concept in further detail in a

later lesson.

Note that the same phenomenon of enharmonic equivalence applies to minor keys. E♭ and D♯ minor have

the same sound, and we can find examples from the repertoire in both keys. Seeing seven sharps or flats is

quite rare but it can happen, such as in Beethoven’s Op. 26 sonata in A♭ minor, or Bach’s Prelude and Fugue

in C♯ major.

The best way to internalize this circle of fifths is not by staring at it intently, but by playing music in all these

keys!

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Take Mozart’s “Eine Kleine Nachtmusik” and play the opening motif at a keyboard in the key of C. Notice that
the note you end on is the fifth of the key. Treat that as the new 1 and play the motif in that new key. Repeat
this in all sharp and flat keys. Shift octaves if you end up too low or too high.

For minor keys, let’s play the opening melody of Paganini’s Caprice No. 24 for Violin. This melody begins by
outlining scale degrees 1-2-3 of minor, and then uses the raised 6th and 7th from melodic minor. Play this
melody through all the sharp and flat keys.

If you’re looking for even more practice, try the major scale in all 12 keys, beginning in C, and then starting on
C♯, then D, D♯, etc. until you’re back at C. Rising by half step results in a zig-zag pattern across the circle of
fifths.

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