Coincidental HarmonicsWhy is it possible to tune two different notes to each other, e.g. E and A? This is possible because of coincidental harmonics. When you pluck an open string on a guitar, not only do you get the note produced by the open string as it vibrates along it's whole length, but also a series of upper partials. These are known as the Harmonic Series. You don't hear these upper partials because the open string, vibrating along it's entire length, displaces much more air than the upper partials and therefore is much louder. The volume of the open string drowns out the quieter upper partials. It is possible to tune an E to an A because both notes have a coincidental harmonic in their series of upper partials. That is to say, they both have an upper partial that vibrates at exactly the same frequency. If you rest your finger on the A string over the 12th fret and pluck the string, you get a harmonic that is an octave above the A produced by the open string. This harmonic is known as the first upper partial or the second harmonic (the first harmonic being the A produced by the open A string, also known as the fundamental). By resting your finger on the string at the 12th fret, you are stopping the whole of the string from vibrating and allowing the string to vibrate either side of your finger. This silences the fundamental and allows the first upper partial (second harmonic) to be heard. If you repeat this procedure at the 7th fret you will silence the first and second harmonics and hear the third harmonic. This harmonic sounds an E, which is an octave and a fifth above the fundamental A. By resting your finger above the 5th fret of the E string you hear the fourth harmonic of the E string, which is an E at exactly the same pitch as the one on the 7th fret of the A string. It is the beat rate between these coincidental harmonics that can be heard when tuning an E to A fourth. You do not need to pick out these harmonics to hear the beat rate. The beat rate can be heard just by playing the two open strings together. The upper partials themselves may be drowned out by the volume of the fundamentals but the beat rate between the two coincidental harmonics can be heard.
Using Intervals To Tune A GuitarIn standard tuning (E, A, D, G, B, e) you'll notice that four of the neighbouring pairs of strings are made up of fourths: E and A, A and D, D and G and B and e. The 2nd and 3rd strings (G and B) are made up of a major third. Firstly, set your A string using an A pitch fork (440Hz). The pitch created by the 440 fork corresponds to the 4th harmonic of the A string (above the 5th fret). Once again, you don't need to pick out the harmonic to hear the beat rate but you can if you want to hear it really clearly. NEVER bang a pitch fork on anything hard like a table. Strike it against some firm but with a bit of give in it; like the sole of a shoe or your knee if you're not too boney. Now play the E and A together and raise the E to just short of smooth. Check the accuracy of this by fingering the A string at the 2nd fret (B) and play the E and B together. Both the E A fourth and the E B fifth should be just off smooth. If you want to be really accurate, the fifth should be slightly smoother than the fourth, but there's not a lot in it. If the E A fourth is beating too rapidly then you need to raise the E a little more. If the E B fifth is beating too rapidly then you have raised the E too much and need to flatten it. Now tune the D to the A as a fourth. Bear in mind that whilst tuning the E to the A, the E was lower in pitch than the A and so you needed to stop short of smooth in order to make the fourth slightly wide. Because the D is higher than the A, you will need to go slightly past the smooth point to make the fourth wide. It's best in this case to tune it smooth and then check the the fifth by fingering the 2nd fret of the D string to get an A E fifth. If it is beating too much raise the D a little more and check both intervals again. Repeat the procedure used for tuning the A D fourth to tune the D G fourth. Now we come to the G B major third. Major thirds have a very rapid beat rate compared to fourths and fifths, which are almost smooth. This beat rate increases as you get higher in pitch. For example, the middle C (3rd string 5th fret) and the E (2nd string 5th fret) is a major third. This beats 10.38 times a second, almost a blur. I trained as a piano tuner in my early twenties and it was the part of the course that involved tuning this interval that separated the people who were going to complete the course from those who were not. If you have trouble hearing the beat rate between the G and B; tune the B string by fretting the G string at the 4th fret and tune the unison. Now pick out the harmonic that is just before the 4th fret of the G string and the one that is at the 5th fret of the B string. These are the coincidental harmonics for the G B third. You should now be able to clearly hear the rapid beat rate. To get a fourth and fifth for the G and B string: Finger the B string at the first fret and play it together with the open G for a fourth (G C) and the 4th fret for a fifth (G D). Then you can apply the method for tuning the fourths and fifths that was described above. Once the B string is done you can use the same method to tune the final fourth B e. You can also use octaves as a further way of checking for inaccuracies. Finger the D string at the 2nd fret and play it together with the open 6th E string as an octave. Remember, octaves should be smooth. Finger the G string at the 2nd fret and play it together with the open A string as an octave. Finger the B string at the 3rd fret and play it together with the open D string as an octave. Finger the 1st e string at the 3rd fret and play it together with the open G string as an octave. In fact, anywhere on the fret board where you can play an octave or a double octave along with an open string, you should do so; e.g. open G with the G at the 8th fret of the 2nd string or open D with the D at the 7th fret of the 3rd string. You can also apply this to fourths and fifths, or extensions of fourths and fifths i.e 11ths and 12ths or even 18ths and 19ths (these should sound just as sweet as the fourths and fifths). For instance, the open A string together with the open 1st e string as a 12th or the open D string together with the A at the 5th fret of the 1st string as a 12th. It's best to stick to double octaves, 12ths and 19ths when checking further up the fret board as these are the clearest to hear. You can of course check the 5th and 4th fret unisons too. If all your unisons and octaves are smooth and your fourths, fifths, 11ths and 12ths aren't beating too wildly, then you can be sure that your chords will sound sweet wherever you play them on the fret board. You could apply this method of tuning to achieve greater accuracy when setting the intonation on an electric guitar. I'm not going to go into the method for adjusting intonation as I'm sure someone else has already done so, but you could use double octaves, 12ths and 19ths to check the accuracy of your guitar's intonation.
StringsStrings can affect tuning and intonation. I find that about one out of every four or five packs of strings has a faulty one in there. I think it has something to do with the winding. Finally, if you find your guitar drifts out of tune often, (if you train your ears to hear interval beat rates you'll know about it alright) then it may be because you haven't set the string after a re-stringing.
String SettingAfter stringing your guitar and tuning it up to pitch. Be sure to set the strings by pulling them away from the fret board. Rounabout where the fret board meets the sound hole or pick up, take hold of the string with your first two fingers and with your thumb pressing against the front of your guitar, pull it away from the guitar. You'll find the pitch has dropped a lot because you are taking out the excess slack. retune and repeat several times until it no longer drops. You should feel the string becoming much more tense as you reapeat this proceedure. Be fairly gentle with the 1st ans 2nd strings. You should now find that your guitar stays in tune for much longer. If you don't know how to string a guitar the way they do in a guitar workshop then check around the internet or read a guitar maintainece handbook. Basically, it involves wrapping the string back under itself so that as you tighten it up it traps itself.