Tuning using Harmonics

Before we go any further, let's define some terms. When talking about harmonics on the guitar we are talking about the overtone series. I'll use the same terminology as the Harvard Dictionary of Music. The overtones series is a series of intervals relating to a specific pitch. Each pitch has an overtone series that consist of a set of frequencies that is related in a simple mathematically way to the original pitch. Often the pitch in question is called the the 1st harmonic and the rest of the notes in the overtones series are also numbered, i.e. 2nd harmonic, 3rd harmonic etc.. Sometimes the term "partials" is used instead of harmonic. Just to confuse things often the original pitch is called the fundamental and the notes of the overtone series are referred to as 1st overtone, 2nd overtone etc. . This can sometime be confusing because the term "1st overtone" in the one system is equivalent to "2nd harmonic or partial" in another system. Although one system is better for comparing the frequency relationship within the series, the terms "fundamental" AND "harmonic", are both commonly used in the discussion of the overtone series as it relates to the guitar. If all of this is confusing, forget I mentioned it. Really, the names aren't that important, the concept is the same either way.

Overtone series

fundamental/overtone                harmonic or partial
    terminology                        terminology

fundamental  =   freq          = 1st harmonic or 1st partial
1st overtone =   freq x 2      = 2nd harmonic or 2nd partial
2nd overtone =   freq x 3      = 3rd harmonic or 3rd partial
3rd overtone =   freq x 4      = 4th harmonic or 4th partial
4th overtone =   freq x 5      = 5th harmonic or 5th partial

     etc.

The pitch in question is called the fundamental (also called the first partial), in addition to the fundamental, their are several other frequencies added to create the sound we hear. One of the frequencies is double the original pitch (that's exactly one octave higher), another is three times the original pitch (one octave plus a perfect fifth higher), the next is four times the original pitch (two octaves higher) and then five times the original pitch (two octaves plus a major third higher). Each of the additional tones of the series is a little quieter than the previous. In theory, it continues indefinitely, in reality it doesn't. Steel string guitars have more of the upper harmonics than nylon string guitars. That is why steel string guitars generally sound "brighter" than nylons string guitars. Its all in the harmonic content, or the upper partials (I'm not talking dentistry here).

The existance of these overtones is easy to prove using a piano and another cool feature of nature: sympathetic vibrations. But since we not talking about the piano, we'll prove their existance on the guitar. Read on.

Here are locations of the harmonics for open strings.

• Fundamental (1st partial) = open string
• 1st overtone (2nd partial) = 12th fret (1/2 string length)
• 2nd overtone (3rd partial) = 7th fret (1/3 string length)
• 3rd overtone (4th partial) = 5th fret (1/4 string length)
• 4th overtone (5th partial) = approx. 4th (and 9th) fret (1/5 string length)

Instead of dealing with the terminology problem many guitarist refer to the harmonics by the fret at which they are located, i.e. the 12th fret harmonic, the 7th fret harmonic, the 5th fret harmonic and so on.

Before you can use harmonics in the tuning process you must first be able to consistly create them. Using your left hand lightly touch the 6th string at the 12th fret then pluck the string the string with a finger of your right hand then immediately remove your left hand from the string. You should touch the string exactly at the fret bar. Unlike when you normally play a note (you press slighty behind the fret bar), when playing harmonics you should place your fingers of the left hand directly over the fret bar. You should hear a chime-like sound. Try the same thing at the 7th fret, 5th fret or 4th fret. Each of those positions will isolate a different harmonic. You will be able to create the harmonics best when you pluck (right hand) near the bridge. Try it, several dozen times on all strings. Notice how much difference it makes when you pluck at various distances from the bridge.

Play the 12th fret harmonic on strings 2, 3 and 4 at the same time. Same thing at the 7th fret, now the 5th fret. Those are major triads in second inversion, first G then D and finally G again an octave higher. Chords in harmonics, it's a beautiful sound. Try it in minor now: 12 fret harmonic on string 1, 2 and 3. Oooo, does that remind you of something? Then play harmonics at the 7th fret on all three string and then the fifth fret. These are minor triads (in first inversion): Em , Bm then Em an octave higher.

The variety of techniques utilizing harmonics is beyond the scope of this page on tuning. I'll discuss the use of harmonics limited to the process of tuning the guitar.

Once the Low E in in-tune the process is simple (although the execution is a little tricky). It does require that you be able to easily hear the harmonics so this method is difficult to use in a noisy environment.

Here it the first step: The 5th fret harmonic of string 6 should sound the same as the 7th fret harmonic.

Why? Because both of those harmonics are the pitch "E". The 5th fret harmonic on string 6 is the 3rd overtone (4th partial, got it?). That note is two octave higher than the fundamental (open E string), hence it is also an E. The 7th fret harmonic on string 5 is the 2nd overtone (3rd partial). That note is one octave plus a perfect fifth higher than the fundamental (open A string), hence that note is also an E. The same E in the same octave range. It should be anyway, so you can use that fact to tune the A string.

Remember your E string is already in-tune, so if you can get both of the harmonics ringing at the same time you can turn the A string tuning peg to match the sound of the harmonic created on the E string. That's the tricky part, being able get both ringing at the same time. The most common way to do this is to pluck each string separately being careful when plucking the fifth string to not touch the 6th string and stopping the harmonic already ringing. It takes practice.

If it's nice and quiet, you'll notice that when you get close to being in-tune you well hear a vibration slow down (this is called a difference tone). When you come exactly in-tune the vibration will slow down to a stop. You are there.

The truth is that the we use a system called "equal temperament" and that presents a dilemma because the harmonic tuning method is slighty at odds with the equal temperament system. A short discussion of this issue comes later. For now, I fall back on the old phrase "close enough for government work".

Once you have tuned the A string within government spec, proceed.

Why? Same reason as before except this time both notes are A. I can simply copy and paste the previous paragraph and change letters and string numbers, the logic is the same. In fact here it is:

The 5th fret harmonic on string 5 is the 3rd overtone (4th partial). That note is two octave higher than the fundamental (open A string), hence it is an A. The 7th fret harmonic on string 4 is the 2nd overtone (3rd partial). That note is one octave plus a perfect fifth higher than the fundamental (open D string), hence that note is also an A. The same A in the same octave range.

If the harmonic on the 4th string is note the same sound as the harmonic on the 5th string then turn the 4th string's tuning peg until it does match. You might be tempted to tell some people "Hey!, can I have some quiet here. I'm trying to tune using harmonics!". I don't recommend doing that.

Once you get the 4th string in-tune continue.

There is a weak match between the 4th fret harmonic of string 3 and the 5th fret harmonic of string 2 but I don't recommend using those harmonics for reasons that are explained later in the section on tuning systems.

As it turns out the 7th fret harmonic of string 6 is the same pitch as the 2nd string (open). Play the harmonic at the 7th fret of string 6. Then play the open 2nd string (no harmonic) and adjust the string so that it matches the harmonic on string 6.

I usually do an addition tuning check when tuning string 2. After doing the above (7th fret harm. on string6 = string), I play the 12th fret harmonic on string 4 and that should match the regular note at the 3rd fret of string 2. First you pluck the harmonic at the 12th fret first and while it continues to ring out play a regular note (no harmonic) at the 3rd fret of string 2.

If you find that after tuning up the 2nd string using the 7th fret of string 6 that it doesn't seem in-tune with the second test with the 12 fret harmonic of string 4 then you might need to split the difference so that it is slightly (and intentionally) out-of-tune with both strings, hopefully not so much as to cause a big problem. This tuning compromise is inherent in our equal temperament system, discussed later. Notice that when tuning the second string, this step uses one harmonic note and one regular note. Any tuning adjustments should be made on string 2 only.

After you have tuned the first string, play the 1st string and the 6th string at the same time. They should be two octaves apart, in other words, they should sound good together. Try an E major triad, shown in chord diagram form below. It should sound nice and full, without any discordant sourness.

6 5 4 3 2 1 

o       o o
___________
|_|_|_o_|_|
|_o_o_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|

If it doesn't sound right, go through the whole process again. It is common to need to give a guitar a "twice over". Soon it will take you just a few seconds to go through the routine.

There are several types of tuning systems, most which divide the octave into several different smaller intervals. The equal temperament system divides the octave into 12 equal parts. Each octave up and down the pitch range is divided this way. The system, while a little out-of-tune with the musical forces in nature has the positive quality of allowing modulation to any key with acceptable intonation (close enough for government work). The need for a system of intonation such as equal temperament is summed up in this quote from the Harvard Dictionary of Music:

from the Harvard Dictionary of Music:

...The deviations (from the more "pure" pythagoreon and just intonation systems) are necessary because these two systems, although perfect within a small range of tones (mainly those of the C major scale) become increasingly inadequate with the successive introduction of the chromatic tones. For instance, the acoustically perfect fifth might well be used to obtain a succession of five or six fifths, c,g,d,a,e,b. However, if tones such as f#,c#,g#,d# are added in the same manner, the resulting tones cannot be satisfactorily used for melodies such as d,e,f#,g or d#,f,g,g# (meaning eb,f,g,ab). Moveover, the twelfth tone of the succession of fifths, b#, is noticeably higher than the tone c it would represent in our system of notation. Thus, it is necessary to devise methods that instead of being perfect in the simple keys and intolerably wrong in the others, spread the inevitable inaccuracy over all the tones and keys...

end quote

I don't deal with the raw mathematics of it all but when you measure the 3rd partial (which is one octave plus a fifth from its fundamental) against its corresponding note in equal temperament, you find they are not the same. The difference is small and barely perceptable but it exists. The equal temperament version is a little flat (lower) compared to the "pure" fifth from the overtone series.

A larger difference exist between the 5th partial and its corresponding note from equal temperament. The equal temperament version is noticable sharp compared to the overtone series. That is why I don't recomment using the 5th partial (found at the 4th fret) when tuning with harmonics. We are trying to get tuned to equal temperament and the 5th partial won't help us get there.

Although the 3rd partial (found at the 7th fret) is slightly at odds with equal temperament, it is close enough to be useful. Just be aware that the 7th fret harmonic will be ever so slightly higher that the corresponding equal temperament note. An itty bitty amount. For the most part you can ignore it. After you have tuned your guitar as well as possible and strummed a nice big E chord, there will still be some subtle differences (slow wavering vibrations) between the inherent conflicts between the various overtones of the different strings sounding together. When playing the E major chord one of the sources for the wavering is the conflict between the 5th partial of string 6 and the 2nd partial of the G# note on string 3 (fretted notes have harmonics also). Remember you don't have to isolate the harmonic (as we have been doing) for it to be present, it is already present, mixed in with all of the other harmonics to create the complete sound of the regular note. So the G# on string 3 has a harmonic G# one octave higher (its 2nd partial). This G# is in slight conflict with the G# generated by the 5th partial of string 6 (the low E string's 5th partial is G#). That's the way it is. Tuning is a compromise. You can notice these conflicts more on new strings because they have more overtones. When the strings age with corrosion and dirt build-up they first lose the high harmonics. You might not have as many conflicts in the overtone series(s) but you have other intonation problems

Wow, it's pretty involved... I just want to play guitar.