Equal Temperament Guitar Tuning - the equal tempered scale
Equal Temperament for Guitarists
Guitarists, more than any other instrumentalists, need to understand Equal Temperament, or at least the basis of it.
Why? Because their instrument is fretted for Equal Temperament and the strings have to be tuned accordingly. Pianists don't need to worry about it because, although their instruments are tuned to Equal Temperament, the tuning is left to a professional Piano Tuner.
Tuning by Harmonics
Beginners usually do not know the technique of playing harmonics, but more advanced players should. Many are then lured by the idea of using the harmonics available on the 5th and 7th frets to tune their guitars. This inevitably results in a badly tuned instrument. I'll explain why later.
Tuning by Beats
Two notes, sounded together, form an 'interval'. Slight detuning of the interval results in audible 'beats', or a pulsating sound. Guitarists often try to eliminate all such beats, e.g. by playing a full chord of E then adjusting the 3rd string. But making one chord 'perfect' will only make other chords worse. In Equal Temperament, the only intervals that do not beat are the unison and the octave. Eliminating beats from any other interval is a big mistake, also to be explained later.
More about Beats
Unfortunately, there is another source of beats, even in a perfectly tuned unison. For example, an A on 6th string 5th fret might be in perfect unison with the open A on 5th string, yet still produce audible beats. The reason is that the strings are not fully independent, being coupled by the vibrating bridge (and to a lesser extent by the airborne sound wave). Vibrational energy is passed back and forward between the two strings. This exchange between the strings is the cause of the audible beats. Their character is different from detuning beats but it takes some experience to hear the distinction. To the physicist, it is an example of 'coupled harmonic oscillators'. The effect is far less obvious in solid electric guitars where the bridge does not vibrate.
Testing the Instrument
Guitarists sometimes complain that their instrument is impossible to tune correctly. Apart from cheap 'souvenir' guitars brought home from the Costa Brava, this is actually pretty rare. It takes a highly skilled luthier to make a truly fine guitar, but even the least competent should manage to get the basic dimensions correct. Far more likely is that the strings, not the guitar, are at fault. Check each string in turn by comparing the 12th fret harmonic with the 12th fret stopped note. They should be exactly the same. Very slight random variations are normal, but if every stopped string is equally sharp (or flat), you may indeed have an ill-made instrument. (But also check your stopping technique - you may be pressing too hard or at an angle to the fretboard). If the stopped notes are unacceptably different from the harmonics, buy some new strings. Good quality strings start life with a constant linear density (mass per unit length), but the effects of corrosion from left hand finger acids and wear from right hand plucking eventually causes unevenness along the length of the string. The 12th fret harmonic will always be correct, but the stopped note can be well out.
Equal Temperament is a musical tuning system that divides the Octave into 12 geometrically equal steps.
Each step is one semitone. When moving up one step, the frequency (or pitch) increases by 2^(1/12). Using this system, the Octave (12 semitones) is perfect, because 2^(12/12) = 2^1 = 2, an exact doubling of frequency. But the Fifth (seven semitones) is 2^(7/12) = 1.498 times the root. A perfect fifth, on the other hand is exactly 1.5 times. The difference is small, but it is significant. If you want to hear this, play the 6th string 7th fret harmonic. Now stop the string at the 7th fret and play the assisted harmonic at the 19th fret (effectively the new 12th fret). This note will sound very slightly flatter than the 7th fret harmonic. In Equal Temperament tuning, the slightly flat note is the correct pitch.
A common reaction is that Equal Temperament must be 'wrong'. Surely it's better to use perfect intervals instead?
- The theoretical problem with this is that you are then restricted to very simple music that remains mostly in one key. If you progress through the 'cycle of fifths' (C, G, D, A, etc) by perfect intervals, when you finally get back to C (after 12 steps) you'll find that it's a very different C from the one you started on! However, by slightly flattening each fifth, after 12 steps you'll arrive back exactly where you started. Thus Equal Temperament slightly compromises every interval (except unison and octave) so that all keys are equally acceptable. This brilliant invention is what made Western music so harmonically rich and varied.
- The practical problem, for a guitarist, is that your instrument has a fretboard layout mathematically designed for Equal Temperament. Unless your open strings are correctly tuned, all your 'cross-string' intervals become arbitrary.
The Wrong Way
Now we can explain why 7th fret harmonics should not be used to tune the guitar. Suppose you tune string 6 to a reference. If you then tune string 5 to string 6 by exactly matching the harmonics at the 5th and 7th frets, your 5th string ends up very slightly flat. You can then compound this error by tuning the 4th to the 5th in the same way, then the 3rd to the 4th. Progressively, each string gets slightly flatter. Finally, you can tune the open 2nd and 1st strings to the 7th fret harmonics on the 6th and 5th strings respectively, resulting in slightly sharp 2nd and 1st strings. The resulting mess sounds extremely unpleasant, especially intervals that use the very flat 3rd string with the sharp 2nd string. (The first position C major chord is particularly distasteful!) Many people 'tune' by harmonics in this way, convinced it's the 'correct' method, then make ad hoc adjustments (usually to the 2nd & 3rd strings) to try to salvage the result. This is doomed to failure.
The fact is, your instrument is built for equal temperament (by virtue of the mathematical positioning of the frets) and the strings have to be tuned accordingly. With a guitar, you can't opt out of equal temperament. If you want to employ 'Just Intonation' or 'Meantone Intonation' you need to take up the cello or violin instead.
The Right Way
There are many equally good ways to tune a guitar, but they all share a few do's & dont's:
- Don't ever use 7th fret harmonics when tuning
- Don't expect to eliminate all beats
- Do tune by unisons and/or octaves
Both of the following methods are good:
1. The Master A-String
- Use a concert A tuning fork to tune the 5th string to A. You may use the 5th or 12th fret harmonics if you like, as octaves (and double octaves) are perfect. When the 5th string is tuned, don't alter it again!
- Play the 7th fret E on the 5th string (the stopped note, not the harmonic!!)
- Tune the open 6th string to this note (one octave down). If you prefer, you can use the 12th fret harmonic for unison tuning.
- Again play the 7th fret E on the 5th string
- Tune the open 1st string to this note (one octave up). If you prefer, you can use the 19th fret assisted harmonic on Master String 5 for unison tuning.
- Play the 5th fret D on the 5th string
- Tune the open 4th string to this note (by unison)
- Play the 2nd fret B on the 5th string
- Tune the open 2nd string to this note (one octave up). If you prefer, you can use the 14th fret assisted harmonic on Master String 5 for unison tuning.
- Play the 10th fret G on the 5th string
- Tune the open 3rd string to this note (by unison)
The advantage of this method is that any slight detuning tendency is not passed cumulatively from string to string, as every string is tuned directly to the Master String. However, you should only use this method if your 5th string passed the 12th fret harmonic test described above. If your 4th or 6th string is more accurate, the method can easily be adapted, but as concert A tuning forks are the most readily available, the A-string is best.
2. The Electronic Tuner
With these, you get what you pay for. They are all guaranteed to be tuned to Equal Temperament. But some are more sensitive than others in detecting the accurate 'in tune' point. They are particularly useful for public performers who may have to tune in noisy environments. However, they are not particularly helpful in training your ear.
Whichever method you use, learn to trust it. If you tune by the Master A-String method, don't then play a major chord and start fiddling with individual strings. Remember that Equal Temperament is not perfect in any key, but it is the best compromise available, it is the tuning your guitar is designed for, and it will sound equally OK in every key.
Beware of these:
- Pitch pipes. It's OK to use the A-pipe only, then follow the Master A-String method, but don't use all six pipes, as they are rarely accurate enough. With cheaper ones, the pitch depends on how hard you blow!
- Pianos. Again, take one note from a piano, then tune your guitar. But don't expect the piano to be in tune. Most aren't. Electric pianos and electronic keyboards should be OK.
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