The Circle of Fifths and tuning

I wanted to return to the subject of tuning, following on from my previous post on the mathematics of guitar tuning and develop this subject further, because there’s another problem that’s not obvious, and which is rather counter-intuitive.  Professional musicians will already know this, but to most of us it’s a mystery and I’d like to explain it, succinctly but I hope clearly.

In common with a lot of my posts and articles, this is a topic which I’ve been ignorant about, and then have researched or worked out for myself.  I reckon that a lot of other people will be similarly ignorant, and so I hope this will be interesting (if a little geeky)

Most of you will probably already know that the standard tuning is A440.  This means that the A note above middle C is tuned to 440hz, and then all of the other notes are tuned around that.  How do you work out the frequencies of the other notes?  That’s where the Circle of Fifths comes in…or does it?.

As I mentioned in the earlier post, an interval of a fifth represents a 50% increase in frequency.  Hence a fifth above the reference A (i.e. E5, using scientific pitch notation) has a frequency of exactly 660hz.  Another fifth above that is B5, which logically has a frequency of 990hz.  If you keep going up by a fifth each time, you go through all 12 notes in the chromatic scale until you get back to an A after the 12th interval.  This gives us the following pitches, applying the 50% increase twelve time (I rounded all figures to 1 decimal):

Scale 1The last three notes are above the limits of human hearing and so they are outside the range of what’s useful.  If we bring each note down into the same octave by halving the frequency for each octave higher, we get the following frequencies for the chromatic scale from A4 to A5:

Scale 2a

Now compare the values for the two A notes: 440 and 892.  Clearly, 892 is more than twice 440, so we’ve ended up slightly sharp (by almost a quarter of a semitone).  Therefore, tuning by the circle of fifths doesn’t quite work

Let’s look at this another way.  What happens if, instead of starting from A, we start from the next note, A#?  Do we get the same results?  Well, let’s run the numbers.  Here are the results using the circle of fifths, but starting from A#4 at 469.9hz:

Scale 3The red notes all have different values from their equivalents in the previous table.   What do these notes have in common?  This comes back to the circle of fifths.  We started at a different point in the circle, and these notes are the ones that come after A (starting at A#, the circle goes: A#-F-C-G-D-A-E-B-F#-C#-Ab-Eb-A#), because the A is where we stopped our calculations for the A scale, and as we saw, the inaccuracies gradually increase as we go up by 50% each time.

In the table below, I have listed the note frequencies in the scale from A4 to A5, based on circle of fifth calculations starting at A, A# and also C# as an extra example.  Next to those, I have put the values for standard tuning, which is calculated mathematically with each subsequent note multiplying the previous by the 12th root of 2, so that when you have made 12 operations you have multiplied the original frequency by 2 (i.e. one octave).

Scale 4You can see that the standard frequencies are lower that those derived by our circle of fifth calculations.

That’s enough for now (phew!).  I will probably return to this subject again to explore its ramifications, but what conclusions can we draw from this?  Let’s see:

Conclusion 1: All tuning is an approximation

Conclusion 2: Tuning changes according to the key

The second conclusion is a slight leap from the results of this article, but I’ll probably come back to that in another article.  As a further complication, of course, when we are playing the guitar, the frequency of any given note will also depend on how hard you press the string, since that will affect its tension and pressing harder will sharpen the note (this is one reason why you will often hear professional guitarists saying that they never actually touch the fretboard; they are fretting very lightly to reduce this effect).  And once again, this is why it can be difficult to tune your instrument precisely.

As a starting point for further reading, here are a couple of related Wikipedia articles: