When a guitar refuses to play in tune, fret
spacing is the first thing that needs to be checked. Assuming
fret spacing is correct, 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
increase the tension on the string, and therefore causes the note
to play sharp.
Each string behaves differently with regard to sharpening tendency
when fretted.
There are three related rules which apply here:
1) sharpening from fretting is inversely proportional to pitch
(The Pitch Rule).
2) Pitch is proportional to string tension (The Tension Rule);
and 3) 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
low E string does in fact go sharp more then the high E string.
However, the tension rule and the string mass rule also come into
play, and we see this especially when we compare the sharpening
behavior of the G and D strings. If we were to apply the pitch
rule only to G and D, we would expect more sharpening for D then
from G. But D in fact sharpens less then G. This is because of
the metal winding on D which adds mass.Even though D is lower
in pitch than G, it has a higher tension than G and therefore
sharpens less.
If you want to test this, you can tune your unwound G-string down
to D, the string will now sharpen more when fretted at the nut
than when it was tuned to G. We can rectify the tendency to sharpness
from fretting by adjusting the total length of the string 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 break-off point back, thereby adding
to string length. The reason for this has to do with the relationship
between the amount of stretch and the ratio of compression to
active string length. As we fret the string on succeeding higher
frets, the total string length when fretted, and this 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.
The correct place to compensate for fretting stretch is at the
nut, by setting the string break-off 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 ration of fretting stretch and the ration of the nut
compensation to inactive string length. As notes are fretted higher
on the fingerboard,the ratio of fretting stretch decreases because
of the fingerboard slope. At the same time, however, the ratio
of nut compensation to inactive string length is decreasing proportionally.
Compensation at the nut is correct when any note on the fingerboard
fretted normally, as when playing the guitar, plays in tune. Adding
complementary compensation at the nut, distributes the intonation
improvement over the entire fingerboard.
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