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Guitar Geometry And Intonation Error

This page will help you to understand guitar geometry as related to guitar setup and intonation.   It explains the causes of intonation error that necessitate both nut compensation and saddle compensation.

The guitar is fretted to produce musical notes in a scale called equal temperamentEqual temperament means that the 12 notes within each octave are evenly spaced.  Well, not exactly - the frets are placed closer together as the notes get higher, such that the frequency doubles in the span of an octave.  But this happens evenly - each fret space is .94387 as wide as the previous fret space, as you go up the fretboard.  

The auditory perception is that the notes sound evenly spaced.  There is no inherent problem with intonation due to the fretboard design.  Intonation problems are primarily the result of the additional tension created from the strings being fretted, and other lesser effects.  These need to be compensated for at both ends of the strings.

When the frets are laid out properly on the fretboard, which they almost always are, then the fretted notes can all be made to play in tune with each other.  That will happen when intonation corrections are made at both the nut and the saddle.   This is because the fretboard layout (distance between frets) is mathematically analogous to the 12-tone equal themerament musical scale.

Some think that equal temperament is the inherent problem with intonation, because it does not produce the most perfect natural intervals.  They might make tuning adjustments in order to “sweeten” individual chords.  But, when a guitar is properly compensated, it is amazed how good equal temperament can sound, with or without additional sweetening or string bending!  Actually, sweetening that players do can be partially to get some natural harmonies, but are mostly to overcome their guitar's poor intonation.

Use of equal temperament is almost universal in western music.  Other temperaments are possible, but not ordinarily practical on a guitar, since they make some intervals more pleasing at the expense of others.  An additional problem is that other temperaments are much more difficult to lay out on the fretboard.

People differ in their sensitivity to pitch error.  Sometimes I hear people say that intonation is not a problem for them, but then they are the first to complain, “these strings just don’t stay in tune”.  If your intonation is marginal to begin with, then even a slight change will throw it over the edge, and will be noticeably out of tune.

If we compensate at the bridge, but not at the nut, then notes near the 12th fret will be in tune with the open strings, but notes nearer to the nut will be sharp.  This sharpness may be further aggravated by the use of a capo.

Where on the neck do you usually play?  The answer to that should inform you as to where you most need to be in tune.  A lot of acoustic players play mostly at the lower end of the fretboard, while others, especially electric players, play over a wide range.   Some playing styles use mostly open chords, while other styles use only closed chords and melodies.

As you might know, the higher the action, the more compensation is needed, because when the action is high, fretting a note will add even more tension to the string.

Your fretboard needs to be accurately laid out.  If you have any doubts about the accuracy of you frets, you can Check Out Your Fretboard.

It should be understood that, even with the best compensation techniques, results will be imperfect to some degree.  Every piece of wood is different and the shape of every neck is different.  Our geometric models are very good approximations, but every guitar body vibrates differently, perhaps resulting in resonances which can pull a note or two slightly off pitch.  Results of these affects are, however, often exaggerated, and  take the blame for lack of nut compensation.

Your adjustments don't need to be perfect - look at it this way: If a do-it-yourself-er makes a 10% error in correcting his intonation, that means he has improved his intonation error by 90%!  Who would complain about that!

Sources of Intonation Error Requiring Bridge Compensation

Intonation correction should be made at the saddle - not to make the 12th fret note in tune with the  open string, but to make, as much as possible, all of the fretted notes in tune with each other.  To do this, we need to move the point of suspension of the string on the saddle toward the butt of the guitar, to increase the vibrating length of the string until the note fretted at 12 or 14 is exactly in tune with the note fretted at fret 2.  Then, assuming that your fretboard is accurate, all of the notes from fret 1 to fret 12 and beyond will be in good tune.  

Why are we not using the open strings to adjust the saddle?   Because, if the nut has not been compensated, the open strings are out of tune with the fretboard!  (I know, this conflicts with what you have always been taught.)

We have noticed that some strings require more compensation than others.  Why do we need to compensate at the saddle?   The following are main elements of intonation error, requiring compensation at the saddle:

Saddle compensation takes care of string stiffness and other effects that vary over the length of the fretboard.

Intonation error from pulling a string to the top of a fret

When we fret a string, we add tension to the string, raising the pitch.  Lengthening the string at the saddle compensates for that added tension by resulting in lower pitch for any fretted note, compared to the open string.  We increase the string length by changing the point of suspension on the saddle.

Generally, as we move up the fretboard, the tension becomes greater, because the strings are higher off the fretboard.   Compensation at the bridge takes care of this nicely because the added distance to the saddle becomes a greater proportion of the vibrating length as you move up the fretboard.

Intonation error from the stiffness of a string

Because of the stiffness of a string, the string does not bend perfectly at the edge of the saddle.  The effective point of suspension, as far as vibration is concerned, is beyond the actual point of suspension.  The effective point of suspension at the bridge is just a bit in front of the saddle.  Also, the effective point of suspension of any fretted note will be ahead of the top of the fret.  Both of these contribute to the need of saddle compensation.

We may not have thought of the stiffness effect before, because we compensate by trial and error, and make the required adjustments irrespective of the specific causes of intonation error.

We can easily see now, why the smallest wound string needs less compensation than the biggest unwound string: the stiffness of a string is mostly determined by the string’s core size, not its overall size, and the core size of the smallest wound string is small compared to its overall size, and it is also smaller than the biggest unwound string.

Sources of Intonation Error Requiring Nut Compensation

Compensation at the nut is needed in a similar pattern as that of the saddle - strings with a bigger core need more compensation.  Compensation is done by effectively moving the nut toward the first fret, either by shortening the fretboard and moving the nut forward, by extending portions of the nut over the fretboard, or by a combination of the two. 

Why do we need to compensate at the nut? 

We need compensation at the nut to make the open string in tune with the note at fret 1 (and at subsequent lower frets).

Intonation error from pulling the string to its fretted depth

For bridge compensation, I cited (above) pulling the string to the top of the fret, however, the added tension from pressing the string on to the depth of your pull is compensated at the nut. That added tension is approximately the same at any fret.  (If there is any variable component of the added tension, it will be included in the saddle adjustment.) 

We don’t normally pull the string all the way to the fretboard, and we don’t all pull to the same depth, so it’s necessary to be mindful in testing, to approximate the force that the guitar owner will use.

Intonation error from pulling extra height from the nut

We often set the nut slots slightly higher than the level of the frets.  This allows for wear and tear and a safety margin.  However, some people set the nut a lot higher than the first fret.  In any case, pulling from the added height of the nut requires some nut compensation. 

Intonation error from the stiffness of a string

Because of the stiffness of a string, the string does not bend perfectly at the edge of the nut.  The effective point of suspension, as far as string vibration is concerned, is beyond the actual point of suspension on the nut.  However, the same thing happens at the first fret and each subsequent fret: the effective point of suspension is beyond the actual crown.  Therefore, the effective distance from the nut to the effective point of suspension of the first fret is the same as the actual distance from the nut to the  first fret, as far as string stiffness is concerned.  Therefore, string stiffness is not a significant part of nut compensation.

(Note: A luthier has advised me that the string actually moves more freely at the fret crowns than at the nut.  This would mean that nut compensation would be affected by string stiffness, and that it would be in the reverse direction as other components, ie, tending to lengthen the distance to the first fret.)

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