This lesson is divided into seven parts:
In order to gain a thorough understanding of tuning, it is necessary to learn a little scientific/historic background on the subject.
First, we'll explore a natural phonomenon known as the HARMONIC OVERTONE SERIES.
While there is evidence to suggest that certain ancient cultures may have plotted the harmonic overtone series, the credit for it's descovery is given to 19th century German physicist Hermann Helmholtz.
Helmholtz "discovered" that what we hear as a single pitch is actually a composite of many different pitches sounding together.
Let's have a look at a vibrating string:
This is the FUNDAMENTAL or overall vibrating pattern of the string.
Now, along this pattern of vibration are tiny dead spots called NODES, and if you touch one of these nodes while the string is vibrating, you will cause the vibrational pattern to change.
One such node is located exactly half-way along our vibrating string.
Try this experiment:
With your amp set to a clean sound, pick your low E-string, and then DAMPEN (stop it from vibrating) the string by laying your hand across it.
Now, play the string again, but, this time, lightly touch the string directly over the 12th fret. You should find that instead of dampening the entire string, you have only dampened the fundamental and are left with a higher bell-like note.
What you have done is to cause the string to vibrate like this:
This is but one of an entire series of nodal points along our vibrating string. (I have seen reference to at least 30 nodal points, but finding verification has been difficult.)
What follows is the vibrational patterns of the first four nodal points, their location along the string and the resultant pitch relative to the fundamental of E:
Fundamental - Open string - E
1st Harmonic Overtone - 12th Fret - E'
2nd Harmonic Overtone - 7th Fret, 19th Fret - B'
3rd Harmonic Overtone - 5th Fret, 24th Fret - E''
4th Harmonic Overtone - 4th Fret, 9th Fret, 16th Fret - G#'' (slightly flat)
Note: the ' marks along-side some of the pitches above represent an octave. One ' means that the note is in the next octave, while two ' ' means that the note is in the second higher octave.
We could continue to plot further overtones by adding one more vibrating section for each successive nodal point. For example, the 5th node yields this vibrational pattern:
It's all mathematically proportional, and therefore refered to as HARMONIC.
Now, all of these sectional vibrating patterns are refered to as OVERTONES, because, as Helmholtz discovered, they are all vibrating at the same time as the fundamental.
The fundamental vibration and the overtone vibrations combine to form what we percieve as sound.
The frequency (how fast or slow) of the fundamental vibration is responsible for what we perceive as the PITCH (how high or low) of a sound. Overtone vibrations, on the other hand, are responsible for what is called TIMBRE (pronounced tamber).
Timbre is not as easy to define as pitch. All of the nuances that help us distinguish the sound of, for example, a hammer hitting an anvil versus a jet plane passing overhead fall into the category of timbre. (ATTACK and DECAY are two other features that help us to distinguish one sound from another, but they are not within the scope of this particular discussion.)
Timbre is the direct result of the hierarchical structure (SERIES) of the harmonic overtones. In other words, which overtones are present + the relative strength or weakness of the individual overtones that are present = the sound characteristics of the instrument used to produce the sound.
So, what does all this have to do with tuning?
Simply put, the pitches that we use today are out of tune with the harmonic overtones.
This has been brought about by the desire of composers to write music that utilizes more than one key signature.
You see, originally the pitches used in music were defined by the overtone series. This is refered to as JUST INTONATION. (There is a lot of debate that rages regarding this subject. It's a lot like the "which came first, the chicken or the egg?" debate. This author (opens in new window) makes what I feel to be very valid points regarding the unconscious influence of the overtone series on the developement of music. Check it out.)
Now, the overtones are mathematically proportional to the fundamental. What that means to you and me is that no two fundamentals produce the exact same overtones. In other words, if an instrument were built to play in the key of C, and the overtones were used to generate the pitches for the rest of the scale, that instrument would not be capable of playing in tune in any key other than C.
This is how instruments were built for centuries.
Eventually, the desire for MODULATION (changing key) within a piece of music became great enough that instrument builders began to experiment with what is called TEMPERED TUNINGS.
In order to have an understanding of temperment, one must first come to terms with the CIRCLE OF FIFTHS.
Let's say that we are playing in the key of C, and we want to modulate to another. The smoothest transition will be found if we move to a new key that is very similar to the key we are in. In other words, the fewer notes that we have to alter in our original scale to accommodate the new scale, the easier it will be to modulate to the new key.
Now, take a look at the scale chart that you constructed in lesson 4. You will notice that, in both the "scales with sharps" column and the "scales with flats" column, each scale has one more sharp or flat than the previous scale. Also, notice that within the "scales with sharps" column that each successive scale is built off of the 5th note of the preceding scale, and that within the "scales with flats" column that each successive scale is built of the 4th tone of the preceding scale.
What this implies, is that the smoothest modulation from our original key of C will be to the key of G or the key of F, because we would only have to change one note (F to F# or B to Bb) to accommodate the key change.
Let's say that we modulate to G. Once there, we might want to modulate to another new key. If we choose the 4th tone of G for our modulation, that would take us back to the key of C (study your scale chart if this is not clear). Therefore,the only option for modulating somewhere new, is to go to the 5th tone (D).
If we were to continue modulating to the fifth tone of each scale, we would eventually arrive back on our original key of C (I guess the world is round after all).
This successive modulation can be expressed in a circle:
If you travel clockwise around the circle, each root is the 5th tone of the preceding root's scale. If you go counter-clockwise, each root is the 4th tone of the preceding root's scale.
Now, if you were to start on C and tune each 5th perfectly (no beats), by the time you got to the next C, You would find that it is out of tune with the C you started on.
Slight variances in pitch are measured in CENTS. One cent is equal to 1/1200 of an octave.
It so happens that our second C will be 24 cents sharp. That's nearly 1/4 of a half-step! (I verified this for myself by attempting to tune a piano and making the mistake of tuning the 5ths pure. I ended up with a piano extremely out of tune, and that's a lot of strings to have to re-tune!)
Also, take another look at the overtone chart. The pitches of the first four overtones define the MAJOR CHORD (just like the first 5 chords I showed you in lesson 1). From earliest times, the major chord has been one of the basic building blocks of music. In the case of the chart, that would be E G# B for an E Major chord (the 1st, 3rd and 5th notes of the E major scale).
Now, let's say that we tune a pure (matches the harmonic overtone) B from our E. Then, we tune a pure Gb from the B, a pure Db from the Gb, and a pure Ab (same as G#) from the Db (study the circle above). Were going to end up with a G# note that is 22 cents sharp as compared to the 4th harmonic overtone of E. That's enough of a discrepancy to cause our major chord to sound horribly out of tune! (Check out this page if you want to get into the mathematics of it all)
So, in order to arrive at a compromise between freedom of modulation and out of tune major chords, many different TEMPERMENTS (altering the pitches) have been devised throughout the centuries. The one thing they all have in common, though, is the need to take up the original slack of 24 cents and close the circle of 5ths. (This page has an easy to understand historical overview on the subject)
The tuning that we use today is called EQUAL TEMPERMENT. Equal temperment is arrived at by tuning each 5th in the circle 2 cents flat (2 cents x 12 keys = 24 cents). This leaves us with a major chord that is sharp by 14 cents (we can live with that) on the 3rd note of the scale, and 2 cents flat (barely noticable) on the 5th note of the scale, but allows us to modulate into any key with equal ease.
All the other notes of the scale are also slightly out of tune with the overtones.
This presents a bit of a problem, especially when combined with the distortion channel (lead channel, overdrive channel etc...) found on most guitar amplifiers. This type of sound is generated by over-emphasising the harmonic overtones until a saturation level is reached.
What this all means, is that every time you play a note, you are also setting into motion a whole slew of overtones. If your trying to tune one note to another, your hearing, not only the fundamental that you are trying to tune, but also the harmonics involved with both notes.
This is one of the reasons for most tuning methods using unisons (exact same note) or octaves. These notes are the only ones that are supposed to be in tune within equal temperment. Unisons and octaves also share the same pitches within the overtone series.
Now, recall that in the tuning section of lesson 4, I detailed two tuning methods that use harmonics, one good and one bad. The method that I labeled as good uses unison and octave harmonics in conjunction with fretted and open-string notes. This will yeild very accurate tuning.
The meathod that I labeled "bad", on the other hand, will leave your guitar out of tune.
Let's take a closer look:
When you play the harmonic at the 5th fret of the low E-string, the resultant pitch is E''. The harmonic at the 7th fret of the A-string is also an E. So far so good? Not quite.
If you look at our circle of fifths, you'll see that E is the 5th of A. Equal temperment dictates that the 5th is tuned 2 cents flat, but you just tuned them pure.
Now, the frets on your guitar are spaced to give accurate equal tempered pitches, so, in essence, you have forced the A-string to play 2 cents flat, which is backward from what you need. ( A is supposed to be 2 cents flat from D not E. Study the circle if this is not clear.)
Keep in mind that 2 cents is not really enough to notice, but you're about to do the same thing, again, to the D-string. You're going to be left with a D-string that is 2 cents flat compared to the A-string which was already out of tune!
By the time you get to the high E-string, you will find that it is noticably out of tune when compared to the low E.
There are a few more things to consider about the effects of overtones on tuning:
ELECTRONIC TUNERS - When using an electronic tuner, and trying to tune using an open string, you're asking the tuner to pick out the fundamental from within a barrage of overtones. Is it any wonder that the dial goes crazy? If instead, you tune by playing the harmonic at the 12th fret into the tuner, you will provide the tuner with a more pure tone to read. Switching to the neck pickup on your guitar and/or rolling back on the tone knob will also help.
DISTORTION - Distortion (the sound of your amps lead channel) accentuates everything, especially tuning errors.
POWER CHORDS - Power chords are made up of a root note and a 5th (this is covered more fully in the theory part II section of this lesson). They sound great with lots of distortion, but we just learned that the 5th is supposed to be out of tune, and that distortion is going to accentuate the problem. It is sometimes necessary to tune your power chords pure and "deal" with the out-of-tuneness of the rest of the notes.
MAJOR CHORDS - Major chords use the 3rd note of the scale, which, in equal temperment, is pretty far out of tune. They sound pretty bad if you just hang out on them (one of the reasons that Country music can sound so "sour"). Again, distortion makes the problem even worse (one of the reasons that power chords are so common in Rock music).
DEAD STRINGS - When your strings begin to suffer from too much "wear and tear", they begin to vibrate inconsistently which produces odd harmonics. This makes it almost impossible to play in tune.
HARMONIC CONVERGENCE - If the "New-Agers" are right, the next time the planets line up we're all doomed. In that event, I don't suppose it will matter whether your guitar is in tune or not :-)
This lesson is divided into seven parts:
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