## The Circle of Fifths

The circle of fifths is a beautiful thing, fundamental to music theory.

Sound is vibrations in air. Start with some note on the piano. Then play another note that vibrates 3/2 times as fast. Do this 12 times. Since

(3/2)¹² ≈ 128 = 2⁷

when you’re done your note vibrates about 2⁷ times as fast as when you started!

Notes have letter names, and two notes whose frequencies differ by a power of 2 have the same letter name. So the notes you played form a 12-pointed star:

Each time you increase the frequency by a factor of 3/2 you move around the points of this star: from C to G to D to A, and so on. Each time you move about 7/12 of the way around the star, since

log(3/2) / log(2) ≈ 7/12

This is another way of stating the approximate equation I wrote before!

It’s great! It’s called the circle of fifths, for reasons that don’t need to concern us here.

But this pattern is just approximate! In reality

(3/2)¹² = 129.746…

not 128, and

log(3/2) / log(2) = 0.58496…

not 7/12 = 0.58333… So the circle of fifths does not precisely close:

The failure of it to precisely close is called the Pythagorean comma, and you can hear the problem here:

This video plays you notes that increase in frequency by a factor of 3/2 each time, and finally two notes that differ by the Pythagorean comma: they’re somewhat out of tune.

People have dealt with this in many, many ways. No solution makes everyone happy.

For example, the equal-tempered 12-tone scale now used on most pianos doesn’t have ‘perfect fifths’—that is, frequency ratios of 3/2. It has frequency ratios of

$2^{7/12} \approx 1.4983$

I have tried in this blog article to be understandable by people who don’t know standard music theory terminology—basic stuff like ‘octaves’ and ‘fifths’, or the letter names for notes. But the circle of fifths is very important for people who do know this terminology. It’s a very practical thing for musicians, for example if you want to remember how many sharps or flats there are in any key. Here’s a gentle introduction to it by Gracie Terzian:

Here she explains some things you can do with it:

Here’s another version of the circle of fifths made by “Just plain Bill”>—full of information used by actual musicians:

If you watch Terzian’s videos you’ll learn what all this stuff is about.

### 3 Responses to The Circle of Fifths

1. OkCarl says:

Interesting. Has anyone ever applied this method to the resonant motions of populations of atoms and molecules?

2. If you’re interested in more exotic temperaments, I blogged a concept of mine.

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