Intonation

Making a classical guitar play in tune is an issue guitarists struggle with constantly.  A major reason for this is the inherent instability of nylon.   Frequent tuning of classical guitars will always be necessary because of conditions beyond the control of players or luthiers.

However, many of the problems guitarists encounter when trying to get their guitars to play in tune are built right into their instruments.  Although there are some practical limitations on the level of perfection which can be achieved in intonation, due to innate features of real-life classical guitars, almost any guitar in otherwise playable condition can be made to play with very good intonation.  The purpose of this article is to shed some light on the physical intonation variables–the variables under the control of luthiers–in classical guitars in a way which is accessible to guitarists and luthiers alike.

First, it may be helpful to show how fret spacing is determined.    The equal-tempered scale, which is the standard scale for traditional western music and therefore for the guitar, is based on a mathematically derived constant, 17.817.   For any scale length (nut to saddle), the first fret is placed a distance from the nut which is equal to 1/17.817 of the total scale length.  The distance between the first and second fret is 1/17.817 of the distance between the first fret and the saddle; the distance between the second and third fret is 1/17.817 of the distance between the second fret and the saddle;  and so on for each succeeding fret.  Using this calculation, the distance between each succeeding fret becomes shorter, and the distance from the nut to the twelfth (octave) fret ends up being exactly one-half of the total scale length.

Obviously it is necessary for the spacing of frets to be accurate if good intonation is to be achieved.  Yet it is amazing how many guitars, especially luthier-made instruments, have inaccurately spaced frets.  Whenever a guitar refuses to play in tune, fret spacing is the first thing that needs to be checked.

Assuming fret spacing is accurate, the second important variable affecting intonation is the stretching of the string which occurs when a note is fretted.   This stretching increases the total length of the string, which increases the tension on the string, just as if we had tuned up with the tuner, and therefore causes the note to play sharp.

Things are not quite this simple, however.  Each string behaves differently with regard to sharping tendency when fretted.  There are three related rules which apply here: sharping from fretting is inversely proportional to pitch (the pitch rule); pitch is proportional to string tension (the tension rule), and; string tension is proportional to string mass (the string mass rule).

The pitch rule tells us that a guitar will display a global tendency to more sharpness as the open-string pitch goes down, and the Eb string does in fact go sharp more than the Et string.  However, the tension rule and the string mass rule also come into play, and we see this especially when we compare the sharping behavior of the G and D strings.  If we were to apply the pitch rule only to G and D, we would expect more sharping from D than from G.  But D in fact sharps less than G. This is because of the metal windings on D which add mass.  Even though D is lower in pitch than G, it has higher tension than G and therefore sharps less.  If you want to test this, you can tune your monofilament G-string down to D; the string will now sharp more than when it was tuned to G.

We can rectify the tendency to sharpness from fretting by adjusting the total length of the string (referred to as “compensation” in luthiery terminology).  But where do we make the adjustment (nut and/or saddle), do we add to or subtract from string length, and how much adjustment do we make?

To answer these questions, we need to first examine in more detail what happens when a note is fretted.  The first component of stretching occurs when the string travels to the crown of the fret (“travel stretch”).   However, when a guitarist frets a note, the finger isn’t pressed directly down on the fret itself; instead, the finger is pressed down behind the fret, and more pressure is applied to make sure a firm string-to-fret contact is established in order to avoid buzzing.  This causes additional stretching (“fretting stretch”).   Each of these string stretch components must be addressed at the appropriate end of the string for best intonation results.

The correct place to compensate for travel stretch is at the saddle, by setting the saddle itself back and/or by moving the string breakoff point back if there is adequate working room on the saddle, thereby adding to string length.   The reason for this has to do with the relationship between the amount of stretch and the ratio of compensation to active string length.  As we fret the string on succeeding higher frets, the total string length when fretted, and thus the string tension, increases because of the slope of the fingerboard with reference to the open string.  Why, then, doesn’t each succeeding note get sharper?  Because at the same time the ratio of the compensation at the saddle to the active string length is increasing proportionally.

In practical terms, it is possible to fret a note with travel stretch only by pressing down on the string with a small piece of wooden dowel so that the string is sandwiched between the dowel and the fret crown.  By playing the note thus fretted into a stroboscope, we can check if the amount of saddle setback is producing a note which is in tune.  The compensation at the saddle is correct when the note at the twelfth fret is exactly one octave higher than the fretted note.  You can also check the result by ear: the compensation at the saddle is correct when the harmonic at the twelfth fret is exactly equal to the fretted note.  Why the twelfth fret?  Because the harmonic and fretted notes are theoretically equal at the twelfth fret but not at other frets.

The correct place to compensate for fretting stretch is at the nut, by setting the string breakoff point forward, which reduces the distance from the nut to the fret and thereby takes away sharpness caused by fretting stretch.  The reason for this has to do with the relationship between the ratio of fretting stretch to total stretch and the ratio of nut compensation to inactive string length.  As notes are fretted higher on the fingerboard, the ratio of fretting stretch to total stretch decreases because of the fingerboard slope which increases the string-to-fret distance as the notes get higher.  At the same time, however, the ratio of nut compensation to inactive string length is decreasing proportionally.

The most practical way to compensate the nut is to set the nut forward by removing an amount from the nut end of the fingerboard equal to the compensation required by the G-string, which, as it turns out, requires the most compensation at the nut.  Setforward can then be adjusted (decreased) for the other strings by cutting facets into the leading face of the nut to move the breakoff points back as required.   Compensation at the nut is correct when any note on the fingerboard fretted normally, as when playing the guitar, plays in tune.  In practice, it is more difficult to accurately compensate a nut than a saddle.  Fortunately, the accuracy requirements of nut compensation are also less, because fretting stretch itself, which is largely under the control of the player, varies more than travel stretch, which is determined by the physical properties of the guitar.

When establishing compensation values, the nut compensation required for each string must be determined after saddle compensation values have been established.   This is because fretting stretch always follows travel stretch when notes are fretted by a guitarist.

If one starts with a guitar to which nothing has been done to correct intonation, adding compensation at the saddle as described above will produce a dramatic improvement in intonation, especially as playing moves to higher positions on the fingerboard.  Many guitar makers are aware of this, even if they are not acquainted with the details, such as the need for individual compensation for each string.  Less well known, however, is the benefit of adding complementary compensation at the nut, which distributes the intonation improvement over the entire fingerboard.

Applying all the above measures, I have developed empirically a set of compensation values for each string on a classical guitar as follows:

Intonation Setbacks/Setforwards

StringSaddle SetbackNut Setforward
Et1.00mm<1/64"
B1.50mm1/64"
G2.25mm1/32"
D1.75mm1/64"
A2.00mm1/64"
Eb2.50mm1/64"

These values have been tested with a stroboscope on a number of my hand-crafted guitars strung with D’Addario J45 strings and have consistently produced very accurate intonation results. Notice that saddle setback values are in 0.25mm increments; this is the maximum accuracy practically achievable when carving bone, but it is more accuracy than the most skillful player can match with playing technique.

You may be asking, why are the nut setforward values shown in inches while the saddle setback values are in millimeters?  The reason is that the exact measurement values that have proven to produce the best results for the nut are most effectively implementable when denominated that way, nothing more.

It is important to note that any compensation values must be tested on fully stretched-out strings. If a guitar is adjusted for intonation with fully stretched-out strings and then played immediately after re-stringing with brand new strings, it will sound a little flat, particularly when played in higher positions.

The measures described above can enable a guitar to sound any note in tune anywhere on the fingerboard; assuming the open strings are properly tuned, the guitar will function accurately as an equal-tempered instrument. Fine-tuning the guitar to achieve interval harmonicity, however, requires the player’s skills in tuning the instrument.

It is well known that equal temperament is deficient because of anomalies in harmonicity which show up especially when chords are played. By careful tuning, the guitarist can restore harmonicity by tuning in small compromises in equal temperament. In this respect, the guitar is different from, say, a piano, where the technician can tune each note individually and thus fine-tune the piano’s interval harmonicity independent of the player. The technical measures described in this article for adjusting a guitar’s intonation, however, will ensure that the guitarist’s tuning optimizes the guitar’s harmonicity at any position on the fingerboard, not just at the positions where chords are played to tune the guitar.

I have not attempted to describe all the possible conditions in a guitar which can affect intonation. For example, my guitars are built with what I refer to as a double-twist fingerboard to optimize string action. My compensation values may need to be adjusted for instruments which do not employ this feature. There are any number of potential features or problems or defects which could be present in a guitar and affect intonation in such a way as to require attention or even prior remediation before any attempt is made to upgrade intonation. When adjusting intonation, each instrument must be looked at individually.

(This is a revised & updated version of an article which first appeared in Guitar Review, Summer 1990.)

3 thoughts on “Intonation”

  1. I really liked reading this. I have a new 7-string multi-scale classical guitar made in Parana, Brazil, and I am having some problems with the tuning. The 2nd and 3rd strigs are playing a little sharp moving up the scale, so adjustment at the saddle and nut are in order. But the 4th string is playing a little flat up the scale — that is something I don’t quite understand. I’ve thought about sanding about 1.5 mm off the nut. Ideas? Anyway, thanks for the article!

    1. The 2nd & 3rd strings could possibly be improved by moving the saddle break-off points back according to my specs. The 4th string is a bit more problematic: what you report (flatness) suggests excessive saddle setback. I would strongly discourage tampering with the nut; a properly compensated nut requires shortening the fingerboard and creating a new, wider nut with proper break-off points, something for an expert only.

  2. Thanks a lot for your very interesting article. I’m really struggling with intonation and I’ve come to realize that most luthiers underestimate the importance of nut compensation. Not easy to tell them what they ought to do, though… 🙂

    Francis

    PS: there are 2 D strings in your chart.

Leave a Reply

Your email address will not be published. Required fields are marked *