Tuning has always been a bugbear for guitarists. Every guitar player - and every guitar builder and repairer - is familiar with the problem. No matter how good the instrument, and how well tuned and adjusted, it never sounds perfectly in tune in all positions and keys.
This is not the fault of the guitar. It is not designed to play perfect intervals (except for octaves and unisons) in any position, or any key. It is designed to play the equal-tempered scale, and it is perfectly possible to adjust and intonate almost any well-made guitar so that it plays this scale pretty accurately. The problem with equal temperament, though, is that it is artificial, a mathematical construct, and it conflicts with the physical properties of real-world strings.
Real-world strings produce harmonics which are pure fractions of the speaking length of the string. The ancient Greeks and Chinese knew about the pure intervals, and constructed their musical scales around them. But Nature throws a spanner in the works by making the natural tone row irregular, so instruments tuned in this way cannot modulate to different key signatures without adding more intervals to the octave.
There is another problem in that 7 pure octaves and 12 pure fifths do not add upp the same:
7 octaves = (2/1) ^7 = 128
12 fifths = (3/2) ^12 = 129.74
The discrepancy works out to 24 cents (almost exactly a quarter-tone), and is known as the "Pythagorean Comma". Finding a way around these problems has been the cause of much controversy and many bitter arguments among music theorists for two and a half millenia.
To make a fixed-interval instrument with 12 notes in the octave useable in all the key signatures, the purity of the intervals has to be compromised. This is called "tempering". A temperament is a specific way of dividing the Pythagorean comma among the intervals of the octave. There many alternative ways to do this on keyboard instruments, and it is only in the last 150 years that equal temperament has taken over as the accepted standard.
As far as the guitar and other fretted instruments having 12 straight, unbroken frets to the octave are concerned, equal temperament is the only choice. Back in 1581, Vincenzo Galilei (Galileo's father), explained the need for equal semitones logically and correctly - "since the frets are placed straight across the six strings, the order of diatonic and chromatic semitones is the same on all strings. In chords, therefore, a C# might be sounded on one string, and a Db on another - this will be a very false octave unless the instrument is in equal temperament."
Equal temperament divides the octave into twelve exactly equal semitones. The resulting equal divisions are a logarithmic function of the speaking length of the string, rather than pure fractions, and thus are not a true analog of the natural harmonic series.
Equal temperament is the ultimate compromise. Tonal purity is sacrificed for ease of modulation. Depending on your viewpoint, equal temperament either a) makes every key equally in tune, or b) makes every key equally out of tune... The idea is to make it possible to play all intervals and chords, in all keys, with the same relative accuracy. Although every key is very slightly out of tune, every key is also useable. No key sounds worse than any other key. The same applies to all chords. Theoretically, that is. In practise certain intervals and chords can still sound dissonant. Thirds are especially troublesome, as the even-tempered minor third is 16 cents flat to the "pure" minor third and the even-tempered major third is 14 cents sharp of pure. The equal-tempered major sixth is 16 cents sharp of just, and the equal tempered major seventh is 12 cents sharp of just. The only interval which is identical in the two scales is the octave.
Those readers who are interested in the theory behind all this can check out my essay Tuning and Temperament, which goes into the history and development of tuning theory, from Pythagoras to the present.
The purpose of this article is to show how to get the best out of the equal tempered guitar.
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COMPENSATION
As we have seen, equal tempered fret spacing can be calculated mathematically to a high degree of accuracy. If this were all there was to it we would be laughing. However, if the bridge is placed at exactly the theoretical position (nut - 12th fret distance multiplied by 2), the fretted notes will get progressively sharper the further up the fingerboard one plays. This is because fretting the strings stretches them by a small amount, raising the tension and therefore the pitch of the notes produced. Action height is normally lowest at the nut and highest at the last fret, so the sharping effect increases with distance from the nut. To compensate for this, length is added to the string at the bridge end. The amount necessary varies from string to string, generally increasing from treble to bass. "Intonation" means adjusting this compensation until the open notes and the 12th fret notes of each string are exactly one octave apart.
Most modern electric guitars have a separate length-adjustable bridge saddle for each string, and the octaves can be intonated very precisely by the owner using an electronic tuner. Acoustic guitars usually have a fixed, narrow bridge saddle which gives little room for compensation.
Steel-string acoustic guitars usually have a straight slanted bridge saddle. Most nylon-strung guitars have either a straight bridge saddle with the same compensation for all six strings, sometimes with extra compensation on the G-string. This works fairly well due to the smaller differences in diameters in nylon string sets.
Most often the only way to improve the intonation of acoustic guitars is to install a wider saddle and file in the correct intonation points. Such work is best left in the hands of a professional with the appropriate equipment and experience.
INTONATION
Intonating most electric guitars is so simple that every guitarist with access to an electronic tuner should be able to do it himself. And since intonation is also affected by one's individual playing style - how hard one presses down the strings, for example - it makes sense that a guitar should be intonated by the person who is going to play it.
The nut, truss rod and action height should be adjusted to taste before you start intonating, otherwise you may as well not bother. It is also a waste of time (except in an emergency situation) to try to intonate with worn strings. For best results, restring and adjust the instrument, and then wait 24 hours to let the strings settle before fine-adjusting the intonation. (By all means give it another check and final adjustment 24 hours after that, too.) For most guitars you will only need a new set of strings, a screwdriver or key of the correct size for the bridge saddles' length adjusting screws, a good electronic tuner, and patience. Don't attempt to adjust your intonation by ear (unless you have perfect pitch) you'll only drive yourself crazy!
The goal of intonation is to adjust the length of each string individually until it plays pure octaves between the open string and the twelfth fret, between the first fret and the 13th fret, the 2nd and the 14th, and so on, as closely as possible. Start by tuning all six strings with the tuner (and keep checking the overall tuning throughout the procedure). The guitar should be held in playing position - the tuning will be noticeably affected by gravity, among other things, if the guitar is laid on its back.
Using the tuner, first compare the open string note to the note at the twelfth fret. The tuner should give exactly the same reading. If not, and the twelfth fret note is flat compared to the open note, the string length is slightly "too long", and the bridge saddle must be moved towards the neck. Conversely, if the twelfth fret note is sharp to the open note, the string is "too short", and the saddle must be moved away from the neck. Adjust the saddle, retune the open string and compare again. Repeat the procedure until the two notes agree. Do the same for the remaining strings.
Adding length to the string at the bridge end to correct the intonation at the 12th fret has an unfortunate side-effect, in that this also lengthens the distance from the higher frets to the bridge, which can throw the intonation off at the top end of the range.
To check for this, compare the 5th fret with the 17th fret, and the 7th fret with the 19th fret. If there is a problem, it may be necessary to compromise the 12th fret a tad to get acceptable intonation in the high register. If the guitar is seldom played above the 10th fret, though, it's obviously better to optimise the low end instead.
All the strings will end up slightly longer than the theoretical scale length, which is the distance from the nut to the twelfth fret x 2. The thicker the string, the more its tension increases when fretted. The lower strings therefore need more "compensation", as this small increase in length is called. A plain string needs more compensation than a wound string of the same diameter, so, in most cases, the high E string will be shortest, the B string a little longer, a plain G a little longer still, the D string a little shorter than the G, the A string a little longer than the D, and the low E longest of all.
Heavy gauge strings need less overall compensation than lighter gauges. This is because they are already at a higher tension than lighter gauges, and thus the percentage of tension added by fretting the strings is relatively less than for lighter gauges.
This is not the fault of the guitar. It is not designed to play perfect intervals (except for octaves and unisons) in any position, or any key. It is designed to play the equal-tempered scale, and it is perfectly possible to adjust and intonate almost any well-made guitar so that it plays this scale pretty accurately. The problem with equal temperament, though, is that it is artificial, a mathematical construct, and it conflicts with the physical properties of real-world strings.
Real-world strings produce harmonics which are pure fractions of the speaking length of the string. The ancient Greeks and Chinese knew about the pure intervals, and constructed their musical scales around them. But Nature throws a spanner in the works by making the natural tone row irregular, so instruments tuned in this way cannot modulate to different key signatures without adding more intervals to the octave.
There is another problem in that 7 pure octaves and 12 pure fifths do not add upp the same:
7 octaves = (2/1) ^7 = 128
12 fifths = (3/2) ^12 = 129.74
The discrepancy works out to 24 cents (almost exactly a quarter-tone), and is known as the "Pythagorean Comma". Finding a way around these problems has been the cause of much controversy and many bitter arguments among music theorists for two and a half millenia.
To make a fixed-interval instrument with 12 notes in the octave useable in all the key signatures, the purity of the intervals has to be compromised. This is called "tempering". A temperament is a specific way of dividing the Pythagorean comma among the intervals of the octave. There many alternative ways to do this on keyboard instruments, and it is only in the last 150 years that equal temperament has taken over as the accepted standard.
As far as the guitar and other fretted instruments having 12 straight, unbroken frets to the octave are concerned, equal temperament is the only choice. Back in 1581, Vincenzo Galilei (Galileo's father), explained the need for equal semitones logically and correctly - "since the frets are placed straight across the six strings, the order of diatonic and chromatic semitones is the same on all strings. In chords, therefore, a C# might be sounded on one string, and a Db on another - this will be a very false octave unless the instrument is in equal temperament."
Equal temperament divides the octave into twelve exactly equal semitones. The resulting equal divisions are a logarithmic function of the speaking length of the string, rather than pure fractions, and thus are not a true analog of the natural harmonic series.
Equal temperament is the ultimate compromise. Tonal purity is sacrificed for ease of modulation. Depending on your viewpoint, equal temperament either a) makes every key equally in tune, or b) makes every key equally out of tune... The idea is to make it possible to play all intervals and chords, in all keys, with the same relative accuracy. Although every key is very slightly out of tune, every key is also useable. No key sounds worse than any other key. The same applies to all chords. Theoretically, that is. In practise certain intervals and chords can still sound dissonant. Thirds are especially troublesome, as the even-tempered minor third is 16 cents flat to the "pure" minor third and the even-tempered major third is 14 cents sharp of pure. The equal-tempered major sixth is 16 cents sharp of just, and the equal tempered major seventh is 12 cents sharp of just. The only interval which is identical in the two scales is the octave.
Those readers who are interested in the theory behind all this can check out my essay Tuning and Temperament, which goes into the history and development of tuning theory, from Pythagoras to the present.
The purpose of this article is to show how to get the best out of the equal tempered guitar.
Back to top
COMPENSATION
As we have seen, equal tempered fret spacing can be calculated mathematically to a high degree of accuracy. If this were all there was to it we would be laughing. However, if the bridge is placed at exactly the theoretical position (nut - 12th fret distance multiplied by 2), the fretted notes will get progressively sharper the further up the fingerboard one plays. This is because fretting the strings stretches them by a small amount, raising the tension and therefore the pitch of the notes produced. Action height is normally lowest at the nut and highest at the last fret, so the sharping effect increases with distance from the nut. To compensate for this, length is added to the string at the bridge end. The amount necessary varies from string to string, generally increasing from treble to bass. "Intonation" means adjusting this compensation until the open notes and the 12th fret notes of each string are exactly one octave apart.
Most modern electric guitars have a separate length-adjustable bridge saddle for each string, and the octaves can be intonated very precisely by the owner using an electronic tuner. Acoustic guitars usually have a fixed, narrow bridge saddle which gives little room for compensation.
Steel-string acoustic guitars usually have a straight slanted bridge saddle. Most nylon-strung guitars have either a straight bridge saddle with the same compensation for all six strings, sometimes with extra compensation on the G-string. This works fairly well due to the smaller differences in diameters in nylon string sets.
Most often the only way to improve the intonation of acoustic guitars is to install a wider saddle and file in the correct intonation points. Such work is best left in the hands of a professional with the appropriate equipment and experience.
INTONATION
Intonating most electric guitars is so simple that every guitarist with access to an electronic tuner should be able to do it himself. And since intonation is also affected by one's individual playing style - how hard one presses down the strings, for example - it makes sense that a guitar should be intonated by the person who is going to play it.
The nut, truss rod and action height should be adjusted to taste before you start intonating, otherwise you may as well not bother. It is also a waste of time (except in an emergency situation) to try to intonate with worn strings. For best results, restring and adjust the instrument, and then wait 24 hours to let the strings settle before fine-adjusting the intonation. (By all means give it another check and final adjustment 24 hours after that, too.) For most guitars you will only need a new set of strings, a screwdriver or key of the correct size for the bridge saddles' length adjusting screws, a good electronic tuner, and patience. Don't attempt to adjust your intonation by ear (unless you have perfect pitch) you'll only drive yourself crazy!
The goal of intonation is to adjust the length of each string individually until it plays pure octaves between the open string and the twelfth fret, between the first fret and the 13th fret, the 2nd and the 14th, and so on, as closely as possible. Start by tuning all six strings with the tuner (and keep checking the overall tuning throughout the procedure). The guitar should be held in playing position - the tuning will be noticeably affected by gravity, among other things, if the guitar is laid on its back.
Using the tuner, first compare the open string note to the note at the twelfth fret. The tuner should give exactly the same reading. If not, and the twelfth fret note is flat compared to the open note, the string length is slightly "too long", and the bridge saddle must be moved towards the neck. Conversely, if the twelfth fret note is sharp to the open note, the string is "too short", and the saddle must be moved away from the neck. Adjust the saddle, retune the open string and compare again. Repeat the procedure until the two notes agree. Do the same for the remaining strings.
Adding length to the string at the bridge end to correct the intonation at the 12th fret has an unfortunate side-effect, in that this also lengthens the distance from the higher frets to the bridge, which can throw the intonation off at the top end of the range.
To check for this, compare the 5th fret with the 17th fret, and the 7th fret with the 19th fret. If there is a problem, it may be necessary to compromise the 12th fret a tad to get acceptable intonation in the high register. If the guitar is seldom played above the 10th fret, though, it's obviously better to optimise the low end instead.
All the strings will end up slightly longer than the theoretical scale length, which is the distance from the nut to the twelfth fret x 2. The thicker the string, the more its tension increases when fretted. The lower strings therefore need more "compensation", as this small increase in length is called. A plain string needs more compensation than a wound string of the same diameter, so, in most cases, the high E string will be shortest, the B string a little longer, a plain G a little longer still, the D string a little shorter than the G, the A string a little longer than the D, and the low E longest of all.
Heavy gauge strings need less overall compensation than lighter gauges. This is because they are already at a higher tension than lighter gauges, and thus the percentage of tension added by fretting the strings is relatively less than for lighter gauges.
