circle of fifths
The circle of fifths is a fundamental construct in music theory. There’s a damn good article about it on Wikipedia, so I intend here to explain several of the ways that it is useful to those who study music. I hope to make it relevant to a working musician (or one who aspires to be).
First, let’s look at the diagram. All 12 notes of the chromatic scale are arranged in a circle pattern, so that the interval formed between two adjacent pitches in the clockwise direction is an ascending perfect fifth.
circle of fifths
These are the same 12 pitches you know, but the arrangement is the secret here. There are several theoretical applications to this. We’ll begin with some basic ones, and progress to more advanced ones.
Key signatures
The first time most students of theory encounter this beast is in learning the key signatures associated with all keys. Generally speaking, the usual arrangement has C at the top of the circle. This key contains zero sharps or flats. As you progress clockwise, you add sharps. As you progress counterclockwise, you add flats. The numbers on this diagram indicate how many sharps or flats the key contains. The numbes aren’t really a part of the circle, just an aid for those of us trying to learn the key signatures.
Scale construction
You can build a scale by taking a group of adjacent pitches from the circle, and arranging them into consecutive order. If you pick a note and the six notes that follow it in the clockwise order, you obtain the major scale built on the second pitch from that group. For example, you start with Bb, and pick the group of pitches Bb F C G D A E. Arranged into a scale, this forms the F major scale: F G A Bb C D E.
This is one way to envision the construction of modes as well. That same seven-pitch group above is also the following modes:
Bb lydian
F ionian
C mixolydian
G dorian
D aeolian (natural minor)
A phrygian
E locrian
We can also build pentatonic scales by taking five adjacent pitches off the circle. In this case, selct a starting pitch and four adjacent pitches in the clockwise direction. This forms the major pentatonic scale associated to the first pitch of the group. For example, you start with G, and pick the group of pitches G D A E B. Arranged into a scale, this forms the G major pentatonic scale: G A B D E. Building a minor pentatonic scale is the same, except that the fourth pitch of the group is the name of the scale. The above grouping arranged E G A B D spells E minor pentatonic.
Building cadences
The harmonic formula of root motion by a descending perfect fifth is called an authentic cadence. It’s the strongest possible way for a chord to resolve to another chord. This relationship is fundamental to Western harmony. The notation V-I is often used to describe this, since the chords built from the V scale degree and I scale degree of the major scale provide a naturally occurring instance of this relationship. For example, in the key of D, the formula V-I is spelled as the chords A and D.
What should be fairly obvious is that these are adjacent pitches on the circle of fifths. In fact, any two adjacent pitches on the circle will have this same V-I relationship.
In jazz harmony there are several formulas for chord progressions that occur quite commonly. These are also built on root motion by fifths, which is couterclockwise adjacent notes on the circle.
ii-V-I is three adjacent pitches on the circle.
vi-ii-V-I is four adjacent pitches on the circle.
iii-vi-ii-V is four adjacent pitches on the circle.
In every case, the root motion associated with these formulas is what we’re interested in, not necessarily the qualitites of the chords. The major scale contains all the formulas listed above. It’s common enough in jazz to resort to chord qualities that lie outside a single major scale, but still adhere to the root motion by fifths relationship. Sequenced ii-V progressions a step apart are a commonly encountered example.
Dominant substitutions
It may take a minute or two to spot, but opposite positions on the circle are related by the interval of a tritone (three whole tones, or six semitones). You can view the tritone as the equal division of the octave into two parts, and this symmetry is reflected in the circle.
A commonly used substitution in jazz it the tritone sub for the dominant chord. If we take a ii-V-I progression (three adjacent pitches on the circle, remember?) and substitute the dominant chord with its counterpart whose root lies a tritone away, we form a root motion by descending semitones. For example, the key of A contains the ii-V-I progression Bm7->E7->Amaj7. The pitch directly opposite from E on the circle is Bb. So our substitute progression is Bm7->Bb7->Amaj7.
Other chord substitutions
Several common progressions above involved root motion by fifths. We can make tritone subs at any chord along these progressions, not just at the dominant, although dominant subs are more common. In minor blues it’s especially common for the chord a semitone above the V chord to precede the V chord. That would be written bVI-V-i in classical theory. They even have names like Italian sixth and German sixth for variations on it. But we jazz guys would tend to view it as a tritone sub for ii in a ii-V-i formula. No big deal.
Eventually, it gets useful to see root motion by fifths and root motion by descending semitones as equivalent. Functionally, they are. A lot of interesting chord progressions are derives from less interesting ones by use of this harmonic device alone.
Cross harp
It’s common for blues harmonica players to use the diatonic harmonica from the adjacent key on the circle of fifths, rather than the actual key of the song. So a blues harp player would pick the C harmonica to play a G blues. This relationship for all keys can be expressed as picking the harp from one step counterclockwise along the circle of fifths.
Slonimsky patterns
Okay, this is kind of far out, so if you’re a beginning theory student, please skip ahead. There’s a well-known book by Nicolas Slonimsky called Thesaurus of Scales and Melodic Patterns. The organizing principle of the book is the division of N octaves into M parts. (N and M represent some numbers here.) For example, we saw above how the tritone interval is the equal division of one octave into two parts.
There are other symmetries possible. Easy ones are the division of one octave into three and four equal parts. You can memorize that this yields a major third and minor third respectively. But if you just use the circle, you can pretty quickly spot these relationships graphically. One octave into six equal parts gives the major second.
Interestingly enough, we have to divide seven octaves into twelve equal parts, to get the perfect fifth. If you have a piano, this amounts to stacking perfect fifths up until you repat the pitch you began on. This will span a compass of seven octaves. And you will have played one trip around the circle of fifths.
