Introducing one of the most amazing and beautiful ideas in musical theory: the Circle of Fifths.

We saw on our page Why are there only 7 note names? that the 2 closest harmonies in music scales are the octave and the fifth.

The octave is where the higher note is vibrating twice as fast as the lower note. The ratio between them is 2:1. The notes sound the same in a way one is higher than the other. We give them the same note name e.g. **A **and **A**; **B** and **B**, etc.

The fifth is where the higher note is vibrating 1.5 times or 3/2 times as fast as the lower note. The ratio between them is 3:2. E.g. the fifth of **A **is **E**.

It turns out:

- If we start with
**A**, then the note vibrating 3/2 times as fast is**E** - If we start with
**E**, then the note vibrating 3/2 times as fast is**B** - If we start with
**B**, then the note vibrating 3/2 times as fast is**F#/G♭** - If we start with
**F#/G♭**, then the note vibrating 3/2 times as fast is**C#/D♭** - If we start with
**C#/D♭**, then the note vibrating 3/2 times as fast is**G#/A♭** - If we start with
**G#/A♭**, then the note vibrating 3/2 times as fast is**D#/E♭** - If we start with
**D#/E♭**, then the note vibrating 3/2 times as fast is**A#/B♭** - If we start with
**A#/B♭**, then the note vibrating 3/2 times as fast is**F** - If we start with
**F**, then the note vibrating 3/2 times as fast is**C** - If we start with
**C**, then the note vibrating 3/2 times as fast is**G** - If we start with
**G**, then the note vibrating 3/2 times as fast is**D** - If we start with
**D**, then the note vibrating 3/2 times as fast is**A**

We have got back to **A **and, on the way, we have found all 12 of the notes on the piano keyboard, white and black!

And each time we have found the “fifth” note in the musical scale of the previous note! (Terms and conditions apply – see below).

This is called the “Circle of Fifths”:

Does your brain hurt yet? Best go and have a lie down while you absorb the wonder of the Circle of Fifths!

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**Terms and Conditions (the small print)**

*Note that in practice, applying the ratio 3:2 to define our 12 notes has limitations, because the circle does not quite get back to exactly the same place it started. You can show this on a calculator. If you multiply the ratio 3:2 or 1.5 by itself 12 times you get 129.75 (going round the circle of fifths from A back to A), but if you multiply 2 by itself 7 times you get 128 (all higher As vibrate 2 times the A below), and 128 does not equal 129.75! That means the A we end up with is vibrating slightly faster than it ought to, and the note is slightly higher or sharper than A! This is called the “comma of Pythagoras“.*

*To get round this problem on a piano, we use something called “equal temperament” which means slightly adjusting all the notes so that they sound ok together. It’s a compromise that has worked well for over 300 years!*

*For brass players, you can use your ears to make sure the notes sound good together. In practice, it means that any note we play might need to played slightly higher (sharper) or slightly lower (flatter) depending on what musical scale we are using, and what everyone else in the band is playing!*

*Also, if we keep multiplying by 1.5 we get a big number (we calculated it as 129.75 above). If we played a note vibrating that fast no one would hear it, not even your dog, because it would be so high in pitch! But that’s ok, because whenever the number gets too big we can just halve it to get the same note but an octave lower.*

*If we do this for the Circle of Fifths, we get the following ratios: Start with A as 1, then E is 1.5. B is 1.5 * 1.5 = 2.25, but divide it by 2, and we get B = 1.125 or 1 and 1/8th. In this way, we can keep all our notes between the starting A and the final A an octave higher*

*This subject is complicated even for an advanced music student, but if you want to understand more, look up “Musical Tuning” on the internet. E.g. Musical tuning on simple wikipedia.*

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