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 to the fret and thereby takes away sharpness caused by
stretching. 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. 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:
| String | Saddle Setback | Nut Setforward | |
| Et | 1.00mm | 0.25mm | |
| B | 1.50 | 0.50 | |
| G | 2.25 | 1.00 | |
| D | 1.75 | 0.50 | |
| A | 2.00 | 0.50 | |
| Eb | 2.50 | 0.50 |
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
setback/–forward 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. 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. For the guitar to function as a
well-tempered instrument, 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 turn the piano into a
well-tempered instrument 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 a treble-to-bass fingerboard surface twist 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.)