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So having tuned the open sixth, we're now going to look at the open major third G3 - B2.

This is the one where many guitarists are dissatisfied with the intonation, but often without being sure why, or what to do about it.

This open string pair is the odd one out, being tuned to a major third, rather than a perfect fourth. As you would have read about in the book, this means that when the guitar is correctly tuned to equal temperament, it is a fast beating, tempered interval. And that, really, is the root of the issue.

The correctly tuned, or tempered interval, sounds like this:

Whereas it may be that you would prefer it to sound like this:

Especially if it was part of this chord:

There is something very sumptuous and harmonious about a this kind of tuning (pure & beatless), but not everyone prefers it.

Let's have a look at the difference between the two. The beating in the first, correctly tuned one, sounds like this, when it is enhanced:

If you didn't hear the beating in the tempered interval, familiarise yourself with this, and then listen for it in the tempered interval (where it is much quieter):

Listen to the recording a few times, not just once. Your brain can literally learn to hear it, the more you practise.

Now listen to the second tuning again, which is tuned beatless or “pure”:

Compare it to the tempered interval:

There are two things that are different about the pure tuned interval. Firstly, the beating has disappeared. Secondly, the B is fractionally flatter. You may or may not hear that depending on your pitch sensitivity. But both factors generally have an influence on our perception of the intonation. As you can hear, the beating itself is quite a big influence on the sound of the interval.

Just to illustrate this influence of beating, let's have a listen to that same, pure tuning of the major third, but this time, with the beating put back into it, without altering the tuning between the two notes. Of course, in practice, this can't happen. But with sound engineering, we can do it. Here they are, side by side.

(There is still some slower beating present, which is quite usual, and is due to the false beating that we talked about in the book):

Pure third - no tempering, no beating:

Pure third - no tempering, but beating artificially added:

So you can see that the presence of the beating indeed makes a big difference, because in both these examples everything else is exactly the same. Whereas normally, in tuning the interval so as to introduce beating, you're actually changing the relative pitches of the notes, and the size of the interval.

Now the fact is, that because of the principle of temperament, because of the way the network of musical intervals works, as we talk about in the book, you can't have this open string third tuned pure, or beatless, without having to sacrifice the tuning of some of the other intervals.

So if you tune this pair of strings so that they don't beat, and then try to complete the rest of the tuning, you're going to run into unsatisfactory tunings in the other intervals.

Is is possible, on keyboard instruments, to do tunings that are not equal temperament and permit pure thirds in some keys, but the obstacle on the guitar is the fact that it has frets.

Unlike a keyboard, on the guitar, instead of having one string or course of strings for each note, which can be tuned independently, you have only 6 strings, and all the rest of the notes are then provided by the frets, whose positions are already fixed.

So different rules apply to the guitar, to those that apply to keyboards. Basically, the best way to tune guitar is based on equal temperament, for which it is designed in the first place.

However, that doesn't mean that this is an excuse for an open string G - B interval that sounds like this:

Rather than like this:

In that first, very wide tuning, the beating, enhanced, is like this:

This is a much faster beat rate than the proper equally tempered rate which should be no more than 8 beats per second, and usually, a little less. Not only is the beat rate here more than 2 times faster than it should be (It's about 17.5), but the relative pitch intonation of the two strings sounding very coarse.

It's actually quite common to hear this third mistuned as badly as this. It can result, very easily, from tweaking the tuining of perfect fifths and perfect fourths, to try to make them pure - beatless.

If you do this to the perfect fifths and fourths involving the notes G, D, A, E, and B - which you'll notice includes all the open strings, then this will happen. For example, it will happen just if you tune pure perfect fourths between the other open string pairs.

Here's the temperament circle:

You can see that major third G - B has the notes G, D, A, E, and B, on the arc over the major third line. This means that the tuning as perfect fifths or perfect fourths between these notes, determines the tuning in the major third G - B. And these notes happen to be the guitar's open strings.

We shouldn’t have more than the theoretical 8 beats per second in this open string third, and actually, we can often have less, if we know what we are doing, because of the business of strings stopping sharp, as we talk about in the book.