Circle of Fifths – The Science of Harmony
What it is
The circle of fifths is a map of the 12 musical keys, arranged like a clock. Each step around the circle moves up by a fifth, the distance from C to G. After 12 steps, you arrive back where you started.
Who created it, and when
- Around 500 BC: Pythagoras discovered the fifth. A string shortened to two thirds of its length sounds a fifth higher, a 3:2 ratio. Stacking fifths one after another became the basis of Pythagorean tuning.
- 1679: The Ukrainian composer Nikolai Diletsky drew the first known circle of fifths in his book Grammatika, written in Moscow.
- 1728: The German composer Johann David Heinichen published a circle of all 24 major and minor keys in Der General-Bass in der Composition. This is the circle musicians still use today.
How to read it
- C sits at the top, with no sharps or flats.
- Going clockwise, each key adds one sharp (♯): G, D, A, E…
- Going counter-clockwise, each key adds one flat (♭): F, B♭, E♭…
- Each major key has a relative minor that shares the same notes, like C major and A minor.
The ideas behind it
- Neighbours are family: keys side by side share 6 of their 7 notes. That is why music flows so easily between them.
- The fifth comes from nature: it is the 3rd harmonic of every note (3:2), the first new note in the harmonic series.
- Closing the circle: 12 pure fifths almost match 7 octaves, with a tiny gap called the Pythagorean comma. Tuning systems were invented to close it.
From one note to all 12
The 3rd harmonic of any note is its fifth. Now treat that fifth as a new first note and find its own 3rd harmonic. Repeat this 12 times, and you meet every one of the 12 notes of the chromatic scale, each one exactly once.
Step by step, from A:
Put them in order of pitch, and you have the full chromatic scale:
Here are the first five steps in 432 Hz and in 440 Hz tuning:
| Step | Note | 432 Hz | Sum | 440 Hz | Sum |
|---|---|---|---|---|---|
| 1 | A | 432 | 9 | 440 | 8 |
| 2 | E | 648 | 18 → 9 | 660 | 12 → 3 |
| 3 | B | 972 | 18 → 9 | 990 | 18 → 9 |
| 4 | F♯ | 1458 | 18 → 9 | 1485 | 18 → 9 |
| 5 | C♯ | 2187 | 18 → 9 | 2227.5 | – not whole |
432 Hz stays on 9 at every step. 440 Hz starts on 8, jumps to 3, and by the 5th step it is no longer a whole number.
After 12 steps, the circle comes back to A, just a tiny bit higher. That tiny gap is the Pythagorean comma.
The Camelot Wheel
DJs use a modern version of the circle called the Camelot Wheel. Each key gets a simple code: a number from 1 to 12 and a letter, A for minor and B for major. C major is 8B, and its relative minor, A minor, is 8A.
- Same number, other letter: 8B → 8A, from major to its relative minor.
- One number up or down: 8B → 9B or 7B, one step around the circle of fifths.
- These moves let two songs blend smoothly. This is called harmonic mixing.
The Camelot Wheel was created by DJ Mark Davis and is now used in DJ software such as Mixed In Key, Rekordbox, Serato and Traktor.
How Inner Dive uses it
In 432 Hz tuning, the fifth above A (432 Hz) is E (648 Hz), a pure 3:2 ratio. And 648 returns to 9 as well: 6 + 4 + 8 = 18 → 9.