Key Signatures and the Circle of Fifths

Building the Circle: A Cycle of Fifths and Fourths

The Circle of Fifths is constructed from a single, powerful principle: repeatedly moving by the interval of a perfect fifth. As we discovered in previous lessons, the perfect fifth is the most consonant interval after the octave, possessing a simple frequency ratio of approximately 3:2. Starting from a note with no accidentals, C, and ascending by perfect fifths generates a sequence of keys.

This dual motion gives us our first practical rule:

• Moving clockwise by a perfect fifth adds a sharp to the key signature.
• Moving counter-clockwise by a perfect fourth adds a flat to the key signature.

This entire system is captured in the following diagram, which will be our central reference.

The Mathematical Engine: Why the Circle "Works"

A natural question arises from your mathematical background: why a circle of fifths? Why does this particular interval allow us to visit every one of the 12 chromatic notes exactly once before returning to the start, while a circle of, say, major thirds would not? The answer lies in the principles of group theory.

We can model the 12 pitches of our equal-tempered system as the cyclic group of integers modulo 12, denoted ${\mathbb{Z}}_{12}$, where C=0, C$♯$=1, and so on up to B=11. An interval is simply addition within this group. A perfect fifth corresponds to moving up 7 semitones, so it is the operation +7. Building the circle of fifths is equivalent to repeatedly applying this operation:
0 (C) $\stackrel{+7}{\to }$ 7 (G) $\stackrel{+7}{\to }$ 14 $\equiv$ 2 (D) $\stackrel{+7}{\to }$ 9 (A) ...

The sequence generated by repeatedly adding 7 mod 12 forms a cyclic subgroup $⟨7⟩$. The key insight comes from a theorem in group theory:

For a cyclic group ${\mathbb{Z}}_n$, the subgroup generated by an element $k$, written $⟨k⟩$, is equal to the entire group ${\mathbb{Z}}_n$ if and only if $k$ and $n$ are relatively prime, i.e., $\text{gcd}(k,n)=1$.

In our case, $n=12$ and $k=7$. Since $\text{gcd}(7,12)=1$, the element 7 is a generator of the group ${\mathbb{Z}}_{12}$. This mathematically guarantees that repeatedly adding 7 will visit every integer from 0 to 11 before repeating.

This mathematical property is why the circle of fifths is such a fundamental structure in Western music. There is no "circle of major thirds" because $\text{gcd}(4,12)=4\ne 1$. The subgroup $⟨4⟩$ only contains the notes {C, E, G$♯$}, forming an augmented triad.

The Physical Imperfection: The Pythagorean Comma

We have established the mathematical elegance of the circle in the idealized system of 12-tone equal temperament. However, if we look at the physics of pure intervals, a problem emerges. The pure perfect fifth has a frequency ratio of exactly 3:2. The octave is 2:1.

If we start at a pitch $f$ and ascend by 12 pure fifths, the final frequency will be $f\times (3/2{)}^{12}$. To compare this to our starting pitch, we must bring it down by some number of octaves. Seven octaves is $2^7$. Let's compare these two values:

They are not the same. The ratio between them, $(3/2{)}^{12}/2^7\approx 1.0136$, is known as the Pythagorean comma. This means that a circle built from pure, acoustically perfect fifths does not actually close. It forms a spiral.

The Circle of Fifths we use in modern music is a product of the 12-tone equal temperament system. This system slightly compromises the purity of the perfect fifth (making it $\approx 1.4983$ instead of $1.5$) in order to make 12 of them fit perfectly inside 7 octaves, thereby forcing the circle to close.

Finding Any Key Signature: The Practical Rules

With this deep understanding in place, let's return to the practical application of using the Circle of Fifths to identify key signatures.

Major Keys

1. Locate the Key: Find the major key on the outer ring of the circle.

2. Count the Steps from C: The number of sharps or flats is the number of steps (clockwise for sharps, counter-clockwise for flats) from the top position (C Major).

   • Example: E Major is 4 steps clockwise from C, so it has 4 sharps.
   • Example: A$♭$ Major is 4 steps counter-clockwise from C, so it has 4 flats.

3. Identify the Accidentals: The sharps or flats are added in a specific order.

   • Order of Sharps: F, C, G, D, A, E, B
   • Order of Flats: B, E, A, D, G, C, F (the reverse of the sharps)

   Applying this to our examples:

   • The 4 sharps for E Major are F$♯$, C$♯$, G$♯$, and D$♯$.
   • The 4 flats for A$♭$ Major are B$♭$, E$♭$, A$♭$, and D$♭$.

Minor Keys

As we learned in the last lesson, a minor key shares the same key signature as its relative major.

1. Find the Relative Major: Locate the minor key on the inner ring. Its relative major is directly outside it.
2. Use the Major Key's Signature: The key signature for the minor key is identical to that of its relative major.
   • Example: To find the key signature for F$♯$ minor, we find its relative major, which is A Major. A Major is 3 steps clockwise from C, so it has 3 sharps (F$♯$, C$♯$, G$♯$). Therefore, F$♯$ minor also has 3 sharps.

Conclusion

Today we have fully explored the Circle of Fifths, from its practical use to its deep structural origins. You now have a systematic method for identifying the key signature of any major or minor key.

Key Takeaways:

• The Circle of Fifths arranges keys in a cycle of perfect fifths (clockwise) and perfect fourths (counter-clockwise).
• Moving clockwise from C adds sharps in the order F, C, G, D, A, E, B.
• Moving counter-clockwise adds flats in the order B, E, A, D, G, C, F.
• This cyclic structure is a mathematical consequence of 7 and 12 being relatively prime, making the interval of a perfect fifth a generator of the group ${\mathbb{Z}}_{12}$.
• The closed circle is a feature of equal temperament, which compromises the purity of the fifth to resolve the Pythagorean comma.
• The key signature for any minor key is found by identifying the signature of its relative major.

You are now equipped with a complete map of the diatonic tonal system. In our next lesson, we will begin to explore scales that exist outside this strict major/minor framework, starting with the pentatonic and blues scales, which are fundamental building blocks for melodies and basslines in many forms of electronic music.