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HOME / BASIC GUITAR SETUP

                     Basic Guitar Set-up 101
                               by
                 Charles Tauber, Copyright 1996


"Things should be made as simple as possible, but no simpler."
Albert Einstein


1. INTRODUCTION

Many of the techniques and much of the theory which I have outlined
below is based upon what I have learned from those generous enough
to share with me what they knew. The techniques I have described
are only one way of doing things; there is no one right method,
though some methods are quicker and easier than others. What is
more important than rigidly adhering to a set of techniques is to
understand the theory behind the techniques. Understanding the
theory provides you with a basis for separating the facts in guitar
set-up from the abundant mythology. Once you understand what you
are trying to accomplish (the theory) you can adopt, or adapt,
specific techniques for accomplishing each task.

While the following discussion strictly applies to steel string
acoustic guitars, the basic theory is the same for classical and
electric guitars, as well as a wide variety of other fretted string
instruments.


2. THE FOUR ADJUSTMENTS

There are four, and only four, basic, universal adjustments that
affect the playability of every guitar. These are as follows:
adjusting the amount of relief (or "bow") in the neck, adjusting
the string height at the saddle, adjusting the string height at the
nut and adjusting the intonation. These four adjustments are what
I refer to as "basic guitar set-up". 

In contrast with "repair" work, which is work that may need to be
performed on a specific instrument in order to maximize the
playability of that particular instrument, set-up work is a series
of adjustments that need to be performed at least once on every
guitar. Very few manufactures take adequate time to properly
perform these adjustments. Even when manufacturers do, the set-up
is very general and aimed at the "average" player, rather than the
specific preferences of an individual player.


3. NECK RELIEF

When reduced to its simplest, the mechanics of the guitar is little
more than the mechanical amplification of vibrating strings which
are stretched across a (semi) rigid structure. To understand guitar
set-up, it is helpful to know a little about how the strings
vibrate and the structure which maintains their tension. The
tension on the strings is maintained by fixing and supporting each
end of the strings, one end at the head and the other at the
bridge.  While at rest, due to the tension imposed upon each
string, each string forms a nearly perfect straight line between
its end-supports - the nut at the head and the saddle at the
bridge. For a guitar, the vibration of a string is initiated by
first displacing the string from its rest position - the straight-
line position it assumes while it is at rest and under tension -
and then releasing it. The elasticity of the string, and the
tension imposed upon the string, causes the string to overshoot its
natural rest position until it reaches nearly the same displacement
in the opposite direction. Due to a "loss" of energy with each
overshoot, the amplitude of the string diminishes ("decays") until
it eventually returns to its rest position.

The general shape assumed by the vibrating string is that of a
shallow curve that begins at one end of the string, is a maximum at
the string's mid-span, and ends at the other end of the string.
Thus, there is no displacement of the string at the nut or saddle,
and there is a maximum displacement near the 12th fret, the
string's theoretical mid-point. The amplitude of a vibrating string
depends upon several factors including string tension, string
material and the initial displacement of the string (i.e. how
heavily the string is plucked or struck). The same set of strings
when tuned below concert pitch will have a greater amplitude than
when at pitch. Similarly, a lower tension string (such as "silk and
steel", or light gage), when tuned to pitch, will have a greater
vibrating amplitude than a higher tension string (such as phosphor
bronze, or medium gage) tuned to the same pitch. Any string type
will have a greater amplitude if struck harder (ie. given a greater
initial displacement). Thus, the amplitude of the vibrating strings
will vary depending upon the type and gage of strings used and the
player's "style" of playing.

To accommodate the amplitude of the vibrating string, there are two
options: either raise the height of the strings (the "action")
sufficiently that the bottom of the vibrating strings do not touch
the tops of the frets, or make the top of successive frets assume
the shape of the vibrating string. The string height can be
minimized by doing the latter, which reduces the distance that the
strings must be depressed for fretting. This, in turn, makes the
guitar easier to play. 

Practically, the way in which the tops of successive frets are made
to conform to the amplitude of the vibrating strings is to
introduce a slight curvature to the neck. This curvature, which is
a slight upwards concavity, is usually referred to as neck
"relief". The amount of neck relief required depends upon several
factors, including string height, and, of course, string amplitude.
If the string height is sufficiently great, no neck relief is
required; the vibrating strings will clear the tops of the frets
regardless. This, however, increases the distance that the strings
must be depressed, which makes the guitar harder to play.
(Depressing a string against a fret stretches the string,
increasing the tension imposed on that string. The more the string
is stretched, the greater the tension; the greater the tension the
greater the force required to depress the string further.)

Since the amplitude of the vibrating string depends upon the type
and tension of strings used and the "attack" used to displace the
strings, the amount of neck relief must also depend upon these same
factors as well as the individual preferences of the player. While
there is no one universally correct setting for neck relief which
accommodates all the variations of these factors, as a general
guideline, approximately 1/64" or 1/32" of relief is typical. This
is generally measured at a fret that is at the mid-span of the neck
(typically the 7th fret) and is the distance from the top of the
fret to the bottom of a string when the string is fretted at the
first fret and at a fret where the neck joins the guitar body,
typically the 14th fret. By simultaneously depressing a fully-
tensioned string against the first and 14th fret, the string forms
a straight edge spanning the first and 14th frets. Alternatively,
a metal straight-edge can be laid along the fingerboard (on top of
the frets) and the distance measured from the top of the 7th fret
to the bottom of the straight edge. 

For a guitar equipped with an adjustable truss rod, adjusting the
amount of neck bow is quite simple. While there are a number of
designs of adjustable truss rods, each shares the same basic
principles of operation. Specifically, when a threaded nut is
tightened on a threaded metal rod, the resulting tension in the rod
alters the curvature of the neck in which the rod is embedded.
Adjustment of the rod involves tightening or loosening the nut on
the rod. Tightening the nut increases the tension in the rod, and
consequently increases the amount which the rod counteracts the
pull of the strings, thereby reducing the bow in the neck.

The truss rod nut may be located at either end of the rod, at
either the guitar's head or from inside the soundhole. At the
guitar's head, the nut is often concealed under a small plastic or
wooden plate, fastened with small screws. From the soundhole, the
truss rod nut may be directly accessible through a hole in the
cross brace, or may be located at the heel block, often accessible
only by completely loosening the strings. To tighten or loosed the
nut, you will require either an allan key or specialized socket
wrench. (Standard allan keys are available at most hardware stores
and specialized socket wrenches for this purpose are available for
several dollars from luthier supply houses. Some manufacturers will
include an appropriate wrench or key, particularly if they use a
non-standard arrangement (eg. Larrivee).)

Since the amount of string tension imposed upon the neck changes
the curvature (bow) of the neck, whenever possible, the adjustment
should be done while the guitar strings are at full tension. Where
this is not possible (eg. some Fender electric guitars) it is an
iterative process in which an adjustment is made with no string
tension and then the measurement is taken after the strings are
returned to full tension. This is repeated until the adjustment is
correct. The required number of turns of the nut depends upon the
amount of bow in the neck, the truss rod design and its
installation. Regardless, the adjustment required rarely exceeds
one or two full turns of the nut, and is often less than one.
Usually, the truss rod nut is tightened by turning clockwise and
loosened by turning counter-clockwise.

In addition to this adjustment, it is often helpful to sight down
the neck - from nut to soundhole - to observe the "trueness" of the
fingerboard and identify any frets which may not be fully seated.
A true fingerboard is one which has no bumps or hollows along its
length (ie. is "flat"). For guitars with a joint between the neck
and the body, it is a common fault to have a hump in the
fingerboard just beyond the joint, typically beyond the 14th fret. 
The truer the fingerboard, the lower the string height can be set
prior to the strings buzzing against the frets.

For guitars with either no adjustable truss rod, or that have
sufficiently untrue fingerboards, the remedy is the same; remove
the frets and dress (true) the fingerboard, followed by refretting.
This procedure is "involved" and takes a skilled repairperson
several hours to complete; adjusting the truss rod is a trivial
adjustment that can be performed by the layperson in minutes.
Hence, truing of the fingerboard will not be discussed in this
article.

It is very important to understand how changing the amount of bow
in the neck affects the height of the strings relative to the tops
of the frets. Ideally, from the point of view of ease of playing,
the guitarist desires the fingerboard surface to be straight
("true") along its length and parallel (or nearly parallel) to the
bottom of the guitar's strings. This would result in the strings
maintaining a constant distance above the fingerboard along the
fingerboard's entire length. This, in turn, would require a uniform
effort to depress the strings and would provide the greatest ease
of playing. 

If a curvature (bow) is introduced into the fingerboard (and neck),
the strings will no longer be a uniform distance from the tops of
the frets; some places along the string will be closer to the
frets, while others will be farther away. If the curvature is
concave upwards ("bowed"), the strings will be at their maximum
height above the tops of the frets at the mid-span of the curve.
Conversely, if the curvature is concave downwards ("back-bowed"),
the strings will be at their minimum height above the tops of the
frets at the mid-span of the curve. The characteristic symptoms of
an overly bowed neck is a high action at the nut that becomes
higher around the 7th fret. The characteristic symptoms of a back-
bowed neck is fret buzz in the middle frets, around the 7th fret,
and buzzing of the open strings against the first fret. Often,
action exhibiting either of these characteristics is a result of an
incorrect neck bow, and is "fixed" by adjusting the bow in the
neck.

It is important to note that the curvature of the neck affects the
string height at both the nut and at the middle frets. It is,
therefore, essential that the correct amount of neck bow be set
prior to any adjustment of the string height at the nut or saddle.
While the amount of bow affects the string height, it should never
be used to specifically attempt to adjust the string height at
either the nut or saddle. Adjusting the amount of neck bow is a
separate adjustment that must be made prior to and independent of
adjusting the string height at the nut and saddle. This cannot be
over emphasized. First set the correct amount of bow in the neck,
then, once it has been set, leave it at that setting and then
adjust the string height at the nut or saddle, if necessary.


4. ADJUSTING THE STRING HEIGHT AT THE SADDLE

The saddle height can be adjusted either before or after adjusting
the string height at the nut. My preference is to adjust the string
height at the saddle prior to adjusting the nut.

Begin by measuring the distance from the top of the twelfth fret to
the bottom of the sixth string while all of the guitar's strings
are at full tension. I prefer to make this measurement by laying a
6 inch ruler, on edge, adjacent to and parallel to the string. The
ruler is supported at one end at the twelfth fret and along its
length by the adjacent frets, eleven, ten, nine, etc. The ruler I
use is calibrated along its end - as well as along its length - and
conveniently measures the distance from the top of the twelfth fret
to the bottom of a string. Any similar method which measures the
distance from the top of the twelfth fret to the bottom of the
string can be used. Repeat for each remaining string.

Although the ideal string height depends upon the preferences of
the player and the type and construction of the guitar, a typical
"good" playing string height ("action") for a steel string acoustic
guitar is about 3/32" at the sixth (bass E) string and about 5/64"
at the first (treble E) string, as measured from the top of the
twelfth fret to the bottom of the strings. The intermediate strings
increase in string height gradually from the first to sixth
strings. The increase in string height from one string to the next
accommodates the increase in vibrating amplitude that accompanies
the increase in string diameter. Due to the type of music played,
the materials from which the strings are manufactured, the string
tensions used and the type of guitar construction, classical
guitars have a higher action, while many electric guitars have a
lower action. 

There are only three possibilities for the measured string height:
the strings are at the desired height, the strings are higher than
desired or the strings are lower than desired. In the first case,
no adjustment is required. Each of the other two possibilities are
discussed below.

STRINGS TOO HIGH

Using elementary geometry, it can be shown that a change in the
string height at the twelfth fret requires about twice the amount
of change at the saddle. For example, if a string height measured
at the twelfth fret is 4/32", and the desired measurement is 3/32",
the change in height at the saddle necessary to lower the string by
1/32" at the twelfth fret is about 2/32".

Using the measurements taken, calculate the amount that each string
needs to be lowered at the saddle. Measure the amount of saddle
height that is projecting above the top surface of the bridge. The
saddle must project at least 1/16"  from the top of the bridge.
This ensures that the strings exert a sufficient downward force on
the saddle to prevent the strings from vibrating side-to-side on
the top surface of the saddle. The side-to-side vibration often
causes a string to rattle or be muted. If the 1/16" projection can
not be maintained, a neck reset or shaving of the bridge may be
necessary, both of which are jobs for either the professional
repairperson or the skilled amateur.

As an optional next step, you may wish to mark, using a pencil, the
location along the saddle at which the strings cross the saddle.
Completely loosen all of the strings and remove the saddle from the
bridge, noting or marking at which end of the saddle is the first
string. Although the saddle should only fit snugly, but not
tightly, its removal can often be facilitated with a pair of
pliers. (Saddles should never be glued to the bridge.)

Most steel string acoustic guitars, as well as most electric and
some classical guitars, have the top surface of the fingerboard
domed or arched across its width. (Many players find a domed
fingerboard easier and more comfortable to play.) To achieve the
correct height of each string, the contour of the top (bearing)
surface of the saddle will generally follow the same curvature as
the surface of the fingerboard. However, to accommodate the slight 
increase in string height towards the bass strings, the saddle
contour deviates somewhat from the contour of the fingerboard.

>From the top surface of the saddle, measure, and mark with a
pencil, the calculated reduction in saddle height required for each
string. A smooth curve can be drawn through the pencil marks on the
face of the saddle. Clamp the saddle in a vise and remove the
excess saddle height with a file, filing to the drawn line. When
this step is correctly completed, the top (string bearing) surface
of the saddle will be flat across its width and curved along its
length.

A string supported by a relatively large, flat bearing surface will
tend to vibrate from side to side over the width of the supporting
surface. This causes the string to vibrate against the supporting
surface, resulting in either a buzzing sound or a muted string. To
prevent this, the width of the top surface of the saddle must be
reduced by chamfering the top edges of the saddle and rounding the
remaining surface. Premature string breakage at the point of
contact of the saddle is the result of rounding the saddle to too
sharp a point (too small a radius). Creating too extreme a point
also results in premature wear of the saddle; a string will quickly
wear a notch into the top of the saddle. Except in very certain
circumstances, a guitar saddle should not be notched to accommodate
the strings. A notched saddle is often the cause of poor
intonation, buzzing and muted strings. 

In addition to the importance of correctly shaping the top of the
saddle, it is also very important to accurately locate where each
string is supported, or "breaks", across the width of the saddle.
Accurately locating where each string breaks over the saddle and
shaping of the top surface of the saddle are discussed below in the
section entitled "Setting Intonation". Upon completing the shaping
of the saddle, the saddle is returned to the bridge and the guitar
retuned.

STRINGS TOO LOW

If the strings are too low, either a new, taller saddle can be made
or the height of the existing saddle can be raised using shims. In
making a new saddle one begins with an oversize saddle-blank, and
the process of completing the saddle is very similar to reducing
the height of an overly tall saddle, as described above. 

Materials for shims can be plastic, wood, metal or paper, although
I generally use wood veneers of maple or rosewood. Shims are cut to
the width of the saddle, and are stacked underneath the saddle in
the slot in the bridge. Placing shims beneath a saddle reduces the
amount of the saddle that sits within the slot in the bridge.
Effectively, the depth of the slot is reduced by the thickness of
the shims, which places a practical limit on the amount that a
saddle can be shimmed. Ideally, a minimum of half of the total
height of the saddle should be within the slot, with the remainder
projecting above the top of the bridge. Provided that the saddle is
the correct width for the slot, this ensures that the saddle is
adequately supported and prevents the saddle from leaning under the
pressure applied by the strings. 

Some guitars are fitted with a saddle design which readily allows
the saddle height to be adjusted. Generally, these arrangements
consist of a saddle suspended between two threaded posts, the
height of which can be adjusted using a screw driver. While this
arrangement makes it relatively easy to adjust the saddle height,
it has a number of significant disadvantages, which explains why it
is not more commonly used.


5. ADJUSTING THE STRING HEIGHT AT THE NUT

Once the neck relief and the string height at the saddle have been
correctly set, the string height at the nut can be adjusted, if
necessary. At the nut, each string sits in a slot cut into the top
surface of the nut. The purpose of the slots is to maintain the
spacing of the guitar's strings and, in guitars without a "zero
fret", to maintain the height of the strings at the nut.

The first thing to determine is whether or not the string height at
the nut requires adjustment. There are a number of ways of doing
this, some qualitative and some quantitative; I prefer to use
qualitative methods. Quantitative methods, which I will not discuss
in detail here, involve using feeler gages similar to the way in
which they are used in setting the string slot depth, as described
below. 

Qualitative methods, by definition, involve determining by feel,
rather than measurement, whether or not the strings are too high or
if they are too low. To begin with, check that the string height at
the nut is not too high. The criteria used to determine if the
string height at the nut is too high is that the effort required to
fret any or all strings at the first fret should not be greater
than the effort required to do so at any other fret. One test of
this is to play a barre chord at the first fret and at, say, the
seventh fret. If it requires more effort to apply the barre at the
first fret, then the nut is too high. (Recall that since you have
already adjusted the bow in the neck and the string height at the
saddle, the string height beyond about the third fret should
already be as low as you prefer it.) Another test is to compare the
amount of effort required to depress a single string against the
first fret with the amount of effort necessary to depress the same
string at the second fret while the string is still depressed at
the first fret. This can be done by fretting a string at the first
fret with the first finger of the left hand then, without lifting
the first finger, depress the same string at the second fret with
the second finger of the left hand. Compare the effort required in
each case. If more effort is required to fret the first fret, the
string height at the nut is too high.

Next, provided that the string height at the nut was not found to
be too high, it is necessary to ensure that the string height at
the nut is not too low. The usual symptom of a nut being too low is
that an unfretted string will vibrate against the first fret when
the string is plucked. This results in a buzzing sound.
Qualitatively, the ideal string height at the nut is one in which
if you pluck each open string slightly harder than you would during
normal playing, the open string will just begin to vibrate against
the first fret.  To check that the strings are not too low,
individually pluck each string quite firmly; the string should just
begin to buzz when plucked slightly harder than it would be during
normal playing. If it won't buzz at all, the strings are probably
higher than they need to be; if it buzzes at less than normal
plucking force, the string height at the nut is too low.


LOWERING THE STRING HEIGHT AT THE NUT

If you found that the string height at the nut was too high, the
following method can be used to reduce that height. The string
height at the nut is usually measured adjacent to the nut as the
vertical distance from the surface of the fingerboard to the bottom
of the string. Quantitatively, the ideal string height at the nut
for each string is the minimum height above the fingerboard that
just allows that vibrating string to clear the first fret.
Typically, this is several thousandths of an inch greater than the
height of the first fret. Any higher than this and the string is
unnecessarily difficult to depress; any lower and the string
vibrates ("buzzes") against the first fret. Thus, the ideal string
height at the nut depends upon the height of the first fret and the
amplitude of the vibrating string (at the first fret). Since the
height of the fret wire varies from one manufacturer to the next,
and the amplitude of the vibrating string depends upon the player's
attack and the type and tension of strings used, there is no
universally correct dimension for string height at the nut.
Instead, it is necessary to set the string height at the nut based
upon theory, the particular preferences of the player and the
particulars of the individual instrument.

The method described below for adjusting string height at the nut
is very similar to that given by Dan Erlewine in his book "Guitar
Player Repair Guide". The tools necessary for this adjustment are
a short straight edge (ruler), a standard set of feeler gauges and
either a set of calibrated nut files or an X-acto saw and a tear-
drop needle file. Feeler gauges are thin, accurately calibrated
metal strips that are used for gauging the size of a gap. The
feeler gauges need not be a special set; they can be obtained from
any auto-supply or hardware store or purchased from a luthier
supply house. Nut files are special files that are manufactured to
cut a round-bottomed slot of a particular width. Although a
complete set of nut files includes widths ranging from .010" to
.058", a "starter" set consisting of .016", .025" and .035" widths
is quite adequate. While nut files are easy to work with and remove
much of the guess work from nut slot filing, they are expensive,
and not essential. An excellent alternative is to purchase an X-
Acto saw blade which attaches to a standard X-Acto knife handle and
one or more needle files. The saw and handle is available at hobby
shops, is inexpensive and is very useful in cutting nut and saddle
materials. Small tear-drop shaped needle files are available at
hardware stores and hobby shops in at least two different sizes and
can be used very effectively for cutting the slots in guitar nuts. 

Begin by measuring the height of the first fret. One way of
measuring this is to place a straight edge on the top of the first
two frets, so that it straddles the first and second frets, and
then slide feeler gauges - individually, or stacked - between the
fingerboard and the straight edge, until the gages just fill the
space between the fingerboard and the straight edge. (The straight
edge runs parallel to the strings, while the feeler gages are
inserted from the edge of the fingerboard parallel to the frets,
midway between frets one and two.) A typical fret height is about
.040". To the fret height you measure, add about .008", which is
about the amount by which the string will vertically clear the
first fret. This total is the approximate height above the
fingerboard that each string should be. As a subtle adjustment, I
often vary the nut height by about two or three thousandths of an
inch from first to sixth strings, with the first string being lower
than the sixth. Stack the appropriate combination of feeler gauges
to obtain a combined thickness of the correct value.

Reduce the tension on the first string sufficiently that you can
lift the string out of its slot and slide it towards the second
string, letting the first string rest on the top of the nut. With
all but the first string at full tension, place the end of the
stacked gauges on top of the fingerboard so that the edge of the
gages are touching the nut. The gages are inserted under the first
string from the treble side of the fingerboard so that the length
of the gauges runs parallel to the frets and all but the first half
of an inch or so overhangs the edge of the fingerboard.



                         |         | stacked gauges
                         |         |    
                         |         |          edge of fingerboard
                 .........         |............................. 
                 .       .         |                     |
  1st string slot.-------.         |                     |
             ----------------------------------------first string 
                 .       .         |                     |
                 .       .         |                     |
                 .       .----------                     |   
                 .       .                               |  
                 .       .                            first fret
                    nut

                           Enlarged Top View of Nut Area


Using either a nut file or the X-acto saw, deepen the slot until
you just begin to contact the top surface of the stack of feeler
gauges with the file or saw. The slot should be filed (or cut) so
that the string breaks over the nut at the leading edge of the slot
- at the face of the nut nearest the bridge. This is accomplished
by filing or sawing the slot at a downward angle from the
fingerboard towards the head. Failure to have the string break at
the leading edge of the nut can result in poor intonation and
string buzz. The width of the slot should be several thousandths of
an inch larger than the diameter of the string. If the slot is too
narrow, the string will bind in the slot, often causing premature
string breakage. One symptom of a binding nut slot, in addition to
frequent string breakage, is that the change in tension in the
vibrating portion of the string does not occur smoothly when
adjusting the tuning pegs; the string's pitch changes suddenly,
lagging behind the adjustment in the tuning peg. Once you have
filed to the appropriate depth, remove the stack of gages,
reposition the first string and return the string to full tension. 

Repeat the process for the second string, sliding the stack of
gages beneath the first string and extending under the third
string. When completed, repeat the process for the third string,
using either a nut file or a needle file to deepen the slot; the
saw kerf is not sufficiently wide to accommodate the diameter of
strings larger than the second. To adjust the height of the fourth
string, insert the gages from the bass side of the fingerboard and
repeat the same process. Repeat for the fifth and sixth strings.


RAISING THE STRING HEIGHT AT THE NUT

If you found that the string height at the nut was too low, or you
cut the string slots too deep, the string height at the nut can be
increased. One common way of increasing the string height at the
nut is to place a shim beneath the nut. To do this, remove the nut,
cut a shim to the same width as the nut, glue the shim to the
bottom of the nut and reglue the nut to the neck. Guitar necks are
commonly made from mahogany or maple, and shims can easily be made
from a matching veneer, and if necessary, the visible edge of the
veneer can be stained with an appropriate color of felt tip pen.
Once the nut has been shimmed sufficiently, the string height can
then be lowered, if necessary, using the same procedure as
described above. 

6. INTONATION

Generally, intonation refers to the ability of an instrument to
play in tune. Specifically, intonation refers to how closely the
notes produced by an instrument (or voice) conform to desired
pitches. Hidden within this one simple statement are two quite
complex questions: "what are the desired pitches?" and, once those
pitches are determined, "how can an instrument be adjusted so that
it accurately produces the desired pitches?" Since both of these
issues are quite complex, and involve such diverse areas of study
as instrument making, acoustics, music theory, and human
perception, my intention in this section is to address these
questions by giving an overview of the most basic theory relevant
to setting-up the intonation of a guitar. 

In the process of answering these questions, we must first
understand what pitch is, and determine which pitches are desired.
To do so, we must understand some of the basics of musical
acoustics and the vibration of strings. Once this has been
introduced, we can then examine the practical aspects of adjusting
the guitar so that it will produce these pitches.


BASICS OF MUSICAL ACOUSTICS

Sound is produced as a result of the movement of an object. When an
object which has been at rest is disturbed, the object vibrates.
The vibration of that object is transmitted through a medium,
usually air, by similarly disturbing the medium, until the
disturbance reaches our ear drums. The disturbance of our ear drums
is what our brain interprets, and what we call sound. In
interpreting sound, we can differentiate between "noise" and other
sounds, such as speech, the wind rustling through the trees or the
sound of a guitar string. In contrast with noise, sounds which we
perceive to be musical in nature are produced by an object
undergoing a repetitive oscillating motion that is sustained over
some period of time. The vibration of a guitar string is an example
of a this type of sustained motion, which is often referred to as
"periodic vibration". 

Physicists have rigorously characterized periodic vibration in
terms of several parameters, of which the most important to this
discussion is the number of times the object oscillates during a
given time period. This quantity is referred to as the frequency of
vibration, and can be measured in repetitions ("cycles") per
second, which is often called a Hertz (Hz), in honour of the
physicist Heinrich Hertz. From the musician's perspective, the
"pitch" of a musical sound and the frequency of vibration of that
sound are synonymous: the higher the frequency of vibration, the
higher the pitch of the sound.

Although an object can be forced to vibrate at virtually any
frequency, the practical range of frequencies that most people can
hear is limited to the range from about 15 Hz to 15,000 Hz. Given
that musical notes of any frequency within this range can be
produced and heard, which frequencies should be used to make music?
If you were to play a single line melody comprised of individual
notes played successively one after the other, you would be free to
choose whatever frequencies you like. However, as soon as a second
line, or harmony, is played with the melody, you would find that by
virtue of the relationship of the sounds, many frequencies sound
very harsh and very undesirable.

Historically, there have been a variety of schemes for determining
the precise relationships which yield pleasing ("consonant")
musical sounds. The earliest method is attributed to the ancient
Greek scholar, Pythagoras, who is most famous for his theorem
related to right angled triangles. Pythagoras is credited with
being the first to define the relationship between two consonant
notes or pitches. For this purpose, he devised a simple instrument
called a monochord, which consisted of a single string stretched
across two fixed end-supports with an intermediary moveable third
support. The third support divided the single string into two
separate segments so that, when plucked, one pitch could be
produced from each segment. By moving the third support along the
length of the string, the ratio of the lengths of the two segments
could be varied. He found that when the lengths of the segments are
in the ratios of 1:1, 1:2, 2:3 or 3:4, plucking the two segments
produces pitches that are consonant. (The musical intervals
represented by these ratios are the unison, octave, fifth and
fourth, respectively.)

Based solely upon the consonant ratios 1:2, 2:3 and 3:4, a musical
scale can be established in which the ratios of the string lengths
are entirely comprised of whole numbers. If we start on a note
called C, the note C', an octave higher, will be in the ratio 1:2.
The first note between C and C', a fourth from C to F, is the ratio
of 3:4. The second note, a fifth above C, gives the note G at a
ratio of 2:3. A fourth below G gives the ratio of 3/2 of 3/4, or
9/8, for the note D. When this process is continued, the major
scale, with its ratios, is obtained as follows:

C      D      E      F      G      A      B        C'
1:1    8:9    64:81  3:4    2:3    16:27  128:243  1:2  

This scale is referred to as the Pythagorean diatonic scale.
Missing from the diatonic scale are the chromatic notes - the
sharps and flats. If one continues with the same process of
obtaining the ratios of the string lengths until all of the
chromatic notes are found, one observes the curious result that two
sizes of semitones exist, and that the size of the semitone depends
upon where it occurs and from what starting point it was
calculated. (One size of semitone, for example, from F to F#, is
2,048:2,187, while the other size of semitone, say, from E to F, is
243:256.) This gives rise to a number of difficulties inherent in
this scheme, the result of which is that how consonant a particular
note sounds depends upon the reference being used. For example, one
expects the note B#, which should be the same pitch as C', to be in
the ratio of 1:2. However, if one starts on the note C and reaches
B# in 12 steps of a fifth at a time (i.e. C, G, D, A, E, B, F#, C#,
G#, D#, A#, E#, B#) the note obtained for B# is actually
262,144:531,441 rather than 1:2. The amount of this "error", which
is also exactly the difference in the two sizes of semitones, is
referred to as the Pythagorean Comma. Its magnitude is
524288:531441, as seen, for example, in the ratio between B#:C'.

Practically, this unavoidable consequence of the Pythagorean scheme
is of great importance. For example, if you were to play a piece of
music in the key of C, if any time the octave C' was played it
produced the pitch of B#, it would sound quite dissonant - it would
sound sharp. Similarly, if in playing a piece of music you require
B# and actually get C', it too will sound quite dissonant - it will
sound flat. For centuries, the means of dealing with this
deficiency was to alter the tuning of an instrument to suit the
particular key in which a piece of music was played, while also
avoiding changing keys into one which would encounter particularly
dissonant intervals that resulted from the Pythagorean Comma.

In attempts to overcome, or "temper", the deficiencies of the
Pythagorean system, a number of other schemes have been devised,
each with varying degrees of acceptance and success. One scheme for
doing so is equal temperament. In this scheme, the problems that
arose in the scheme of Pythagorean temperament, from having two
different sizes of semitones, are eliminated by dividing the octave
into 12 equal parts, one for each of the 12 semitones that comprise
the octave. The only commonalities between Pythagorean and equal
temperament are that the frequency of the octave is exactly twice
that of the starting note and that notes which are in unison have
the same frequency: that is, only the consonant ratios 1:2 and 1:1
are retained. All other notes differ between the two schemes.

In order to maintain the frequency of the octave at twice the
frequency of the starting note, and for each of the 12 semitones to
be of equal size, each semitone must be equal to 2 to the exponent
1/12 times the frequency of the previous note. (21/12 is equal to
1.05946..., not a whole number, and not the ratio of two whole
numbers.) Thus, the frequency of the note C# is 21/12 times the
frequency of C, the frequency of D is 21/12 times the frequency of
C#, or 22/12 times the frequency of C, D# is 21/12 times the frequency
of D, or 23/12 times the frequency of C, and so on. In general, the
"nth" semitone above a starting note is 2n/12 times the vibrating
frequency of the starting note. The advantage of this scheme is its
consistency: each and every semitone is exactly the same size.  The
consequence of this is that, for example, an interval between two
notes an octave apart will always sound the same regardless of the
two notes being compared: the interval of C to C' is exactly the
same as F# to F#' or C to B#. Similarly, the interval of a perfect
fifth from C to G will sound exactly the same as the same interval
from, say, B to F#. The disadvantage of this scheme is that the
"purity" or "sweetness" of some of the Pythagorean intervals is
lost as a result of the tempering. Thus, equal temperament is a
compromise between the ability to play equally in tune in any key
and the loss of purity of some of the intervals. 

Equal temperament began to be widely used in the latter half of the
18th century. The Well Tempered Klavier, by J.S. Bach, was written
as a testament to the versatility of equal temperament, with one
prelude and fugue written in each and every key. Today, equal
temperament is universally used in music of the Western culture,
with the exception of "Early Music" instruments which strive to
authentically recreate the sound of music which predated the use of
equal temperament. 

After centuries of debate, the note "A" has been standardized to be
440 Hz: this standard is referred to as "concert pitch". With A 440
Hz as the starting point, and equal temperament as the mathematical
scheme, the vibrating frequency of every musical note can be
determined by calculation.

Having determined which frequencies (pitches) we require an
instrument to be capable of making, it is helpful to first
understand a little of how the object undergoing vibration produces
these notes before turning to the practical aspects of setting up
an instrument. Thus, we next turn our attention to the vibrating
string.


VIBRATING STRINGS

Consider a "simple" string of uniform diameter and cross section
supported at each of its ends. The string is of vibrating length L,
subject to a tension T and made of a material of density per unit
length d. Then, when the string is excited, the fundamental
frequency, f, at which it will vibrate is theoretically given by

f = 1/(2L) x (T/d).      (1)

Of particular interest to this discussion is the fact that the
vibrating frequency of a string is inversely proportional to its
length: as the vibrating length of the string becomes shorter, its
frequency of vibration (pitch) increases. While maintaining the
same tension on a string (i.e. not retuning the string), more than
one pitch can be obtained from that string by "stopping" it
anywhere along its length. Stopping a string divides the string
into segments, often resulting in one segment vibrating while the
others effectively do not. Since each vibrating segment of the
string is shorter than the unstopped string, the frequency of
vibration of each segment is higher than that of the unstopped
string. Thus, an infinite number of vibrating frequencies - all
higher in pitch than the unstopped (or "open") string - can be
obtained by stopping the string.

The strings of "string instruments" - guitars, lutes, banjos,
violins, violas, etc. - are generally stopped in one of three ways.
The first of these is to touch the string at a specific location
along its length, which causes the point of contact with the string
to remain stationary while the entire string vibrates. This
stationary point, called a node, forces the entire string to
vibrate in whole-number divisions of the unstopped string length.
Thus, a node can occur at the exact midpoint of the string, which
divides the string precisely in two, at exactly a third of the
string, which divides the string precisely in three, or at
locations that divide the string in 4, 5, 6, etc. The notes
produced this way are often referred to as "harmonics". Since these
are exact whole-number divisions of the string, the pitches of the
notes produced belong to Pythagorean temperament, rather than equal
temperament. 

The second method of stopping a string is by using the fingers to
depress a string directly against the fingerboard, as is done with
the violin family of instruments. The placement of the player's
finger on the string stops the string and defines the vibrating
length (and pitch) of that string. Thus, producing a desired pitch
largely involves placing the fingers (stopping) in the correct
location along the length of the string. The third way of stopping
is by depressing the string against a fixed bar or chord - a "fret"
- which generally spans the width of the fingerboard. (Depressing
the string against a fret is usually referred to as "fretting" the
string.) The use of one or more frets allows the musician to
accurately, and repeatably, shorten the vibrating length of a
string by depressing the string against a fret. The frets finalize
where along the length of the strings they are stopped. Thus, the
task of obtaining the desired pitches involves obtaining the
correct vibrating string length to produce the desired pitch. 

While there are a variety of methods for calculating the placement
of frets, they are nearly universally based on equal temperament.
Conceptually, the simplest of these determines what shortened
string length is required to produce the next note one semi-tone
higher in pitch. This is calculated by dividing the unstopped
vibrating string length by 21/12. For example, for a 650 mm long
guitar string supported at one end by a nut and at the other end by
a saddle, the (theoretical) distance from the saddle to the first
fret is 650/21/12 = 613.5 mm. Each successive fret location can be
determined recursively using the vibrating string length of the
previous fret to determine the required string length for the next
fret. To continue the example, theoretically, the second fret
location is 613.5/21/12= 579.1 mm when measured from the saddle. (In
practice, for reasons that will become apparent from the discussion
below, it is generally easier to measure the fret locations based
upon the remaining non-vibrating portion of the string, from the
nut location to the fret, rather than the vibrating portion of the
string, from the saddle to the fret.)

Unfortunately, the vibration of a string is considerably more
complex than was assumed for the "simple" string described by
equation (1). Practically, the primary condition not accounted for
in equation (1) is related to the geometry of the fretted string.
When the string is unfretted and in its rest position, it assumes
the shortest distance between its two end supports - namely, a
straight line. As the string is depressed against a fret (or
fingerboard), it no longer forms a straight line between its end
supports. Instead, the string is stretched to form two straight
segments, extending first from one end support down to the fret,
and then from the fret up to the other end support. The stretching
of the string increases the tension on that string. As can be seen
from equation (1), this has the effect of increasing the pitch of
the vibrating string. For frets which are accurately placed in
accordance with equal temperament, the fretted notes will sound
sharp. In order to compensate for this increase in tension, an
adjustment is made to the instrument which increases the length of
the vibrating portion of the string. Increasing the string length
by the correct amount flattens the pitch of the slightly sharpened
fretted note, thereby producing the correct pitch. This slight
increase in vibrating string length is referred to as
"compensation". 

The amount of compensation required is directly proportional to the
amount of the increase in string tension that results from fretting
a particular string. The amount of this increase in tension depends
upon the amount that a string is stretched as a result of fretting,
the tension on that string, the mechanical properties of the string
and the vibrating string length. The amount that the string is
stretched depends largely upon how far it must be depressed until
it contacts the fret, which depends upon how high the strings are
above the tops of the frets. Thus, for the same guitar, a high
action will require greater compensation than a relatively low
action. 

The tension on a string, the mechanical properties of the string
and the vibrating string length are all inter-related, as suggested
in equation (1). If one were to tune a typical high E (first)
guitar string down two octaves, until it is the same pitch as the
low E (sixth) string, the tension on that string would be very low
- in fact, it would be reduced to 1/4 of its original value. As a
consequence of the very low tension, the string would not sound
very well: it would lack volume and sustain. This highlights the
fact that the best sound is obtained from a string that is at a
tension relatively near its breaking strength. In this example, the
string tension is nowhere near its breaking strength. One of the
ways to increase the tension on this string is to increase its
density per unit length. This is accomplished by increasing the
effective diameter of the string, either by using a larger diameter
wire, or by winding a thick string around a thinner "core" string.
Thus, while maintaining the same string length, the tension
necessary to tune the string to pitch increases as the diameter
increases. Conversely, while maintaining the string length and
tension, strings of descending pitches can be obtained by
progressively increasing the diameter of successive strings. This
is essentially what is done on guitars and related string
instruments. However, a consequence of this is that the increase in
density per unit length is accompanied by an increase in the
stiffness of the string, which increases its resistance to being
fretted. The result is that the greater the diameter of the string,
the greater is the compensation required. Furthermore, since each
of the strings on a guitar is of a different diameter and slightly
different tension, each string requires its own unique amount of
compensation.

ADJUSTING THE INTONATION OF A GUITAR

>From the previous discussion, we have seen that the guitar's frets
are placed according to equal temperament and that the action of
fretting a string increases the tension, and consequently pitch, of
the fretted notes. To compensate for this increase in pitch, the
length of each string is increased slightly by an amount dependant
upon each individual string. In this section, we will examine the
practical aspects of how to adjust the guitar so that the notes
produced by the fretting of the strings are in accordance with the
frequencies required for equal temperament.

Fret positions, as briefly described above, are determined without
regard for the compensation that the strings will require. This
means that the 12th fret, the octave of the open string, is located
at precisely half of the theoretical - non-compensated - vibrating
string length: the distance from the face of the nut closest to the
bridge to the 12th fret is exactly half of the theoretical
vibrating string length. The theoretical vibrating string length is
often referred to as the "scale length" to distinguish it from the
actual final vibrating string length, or the "compensated length".
On a compensated string, since the distance from the nut to the
saddle is increased by the amount of the necessary compensation,
the distance from the nut to the 12th fret is shorter than the
distance from the 12th fret to the saddle by the amount of the
compensation. Typically, the amount of compensation required is
between about 3 mm (1/8") and 5 mm (3/16"), depending upon the
string.

The procedure used to adjust the intonation is to alter the precise
string length of each string until the pitch of two notes produced
on the same string are identical. The two notes that are compared
are the note produced by fretting a string at the 12th fret and the
note produced by the harmonic at the 12th fret of the same string.
These notes are both octaves of the open string and are in unison
with each other. As previously noted, harmonics are produced when
a string is divided into a whole number of equal vibrating segments
and, consequently, belong to the scheme of Pythagorean temperament.
Recall, however, that the frets are placed using equal temperament
and that the only places where the two schemes produce notes of the
same frequency are at the unison (1:1) and the octave (1:2).
Comparing the pitches of any notes other than the unison or octave
is like comparing apples and oranges; they will not, and should
not, be the same. (An example of this is that, with the exception
of octaves of the open string, the correct position (i.e. the one
that produces the clearest sound) for stopping a string to obtain
a harmonic is not directly above the center of the corresponding
fret, but is slightly offset.) This is particularly relevant during
routine tuning of the instrument.

The two pitches - the 12th fret note and the 12th fret harmonic -
can be compared with the unaided ear, although more accurate
results can be obtained by using an electronic tuner. The
intonation is correctly adjusted when the two pitches are
identical. Practically, however, there is limit to the ability of
the human ear to differentiate between differences in pitch. This
varies depending upon the frequency and intensity of the note as
well as one's own physical limitations and musical training. To aid
in quantifying deviations in frequency, a semitone has been divided
into 100 parts, each part being called a "cent". A well trained
musical ear can generally distinguish between frequencies 3 or 4
cents apart. When setting the intonation, it is preferable, though
not essential, to use an electronic tuner which is calibrated in
cents.

Prior to performing any adjustment of the intonation, the
intonation of the instrument to be set-up should be accessed. This
should be done using a new set of strings that are all tuned to the
pitch at which you will usually play them. (All subsequent
intonation adjustment should be done with the same new strings all
tuned to pitch.) In addition, the string height at the nut and
saddle should be correctly set-up, since the compensation required
is, in part, dependant upon string height. Beginning with the sixth
string, compare the 12th fret note with the 12th fret harmonic:
record the discrepancy and repeat for each of the remaining
strings. If, for any string, it is found that the discrepancy
between the two notes is greater than 3 or 4 cents - or audibly
different - the vibrating string length for that string will
require lengthening or shortening. Notes which sound sharp require
the string to be lengthened, while notes which should flat require
the string to be shortened. If the notes are of the same pitch, no
adjustment is required: the intonation of the guitar is properly
set-up.

If an adjustment is required, it is usually helpful to first ensure
that the saddle is located so that it can support the strings at
their correctly compensated lengths. If the saddle is not correctly
located, the strings cannot be supported at their correct length
and the instrument cannot be properly intonated. On most
commercially manufactured guitars, a narrow, 3/32" or 1/8" wide
saddle provides the end-support for the strings at the bridge. The
saddle is inserted in a slot in the bridge and is usually slanted
to provide greater compensation as the diameter of the strings
increases. Where each string breaks over the thickness of the
saddle determines the precise vibrating length of each string. The
primary factor in adjusting the guitar to play in tune is obtaining
the precise, correctly compensated string length.

To determine if the saddle is in the correct location, first
determine the guitar's scale length, add an approximation of the
necessary compensation and then measure the location of the saddle
relative to the nut (or zero fret, if the guitar has one). To
determine the guitar's scale length - the theoretical string length
used to calculate the placement of the frets - double the measured
distance from the face of the nut (or zero fret) to the middle of
the 12th fret. (A 36" long, straight edge that is calibrated in
either or both inches or millimeters is helpful.) To this
measurement, add an approximation of the compensation necessary:
about 3 mm to the length of the first (treble E) string and about
5 mm to the length of the sixth (bass E) string. For example, if
you measured 320 mm from the nut to the 12th fret, doubling it
gives a 640 mm scale length. Ideally, the first and sixth strings
should break over the center of the saddle's thickness at about 643
mm and 645 mm, respectively, when measured from the face of the
nut. While Guitar scale lengths are often calculated in inches, I
find millimeters are easier to work with, regardless of which units
were originally used to place the frets. 

A relatively frequent impediment to correctly setting the
intonation is that the saddle is located too close to the nut, not
allowing a sufficient increase in string length. The most common
case of this, due to an insufficient slant of the saddle, or simply
due to insufficient compensation being given, is not being able to
sufficiently lengthen the sixth string, with the consequence that
it plays sharp. Another frequently seen impediment is the saddle
not being of sufficient width to accommodate the range of
compensation required from one string to the next. The very common
example of this is that the second ("B") string cannot be
sufficiently lengthened, and consequently plays sharp. This is due
to the fact that the second string on steel string acoustic guitars
generally requires greater compensation than its immediate
neighbours, and requires a string length for which the string
should break behind (i.e. longer than) the thickness of the saddle.
(For electric guitars and classical guitars, typically it is the
third string, rather than the second, where this occurs.) It is
quite common to find steel string acoustic guitars who's second
string is insufficiently compensated. One simple remedy to this
problem is to use a thicker saddle. If the saddle is either
incorrectly located, or insufficiently thick, the saddle must
either be moved, replaced with a thicker one or a compromise made,
with the string length adjusted within the physical limitations of
the saddle location.

The exact amount of compensation required is determined by trial
and error. The correct amount results in the 12th fret note and 12
fret harmonic being exactly the same pitch. Practically, a simple
way of adjusting the string length is to cut one inch segments from
the excess length of a set of guitar strings. By bending a short
length of string into a "V" shape, it can be slid under the string
- between the bottom of the string and the top of the saddle - and
re-positioned on the top of the saddle, as necessary, to change the
vibrating string length until the pitches of the notes match. (To
aid in inserting the "V" under one of the guitar strings, it is
often necessary to loosen the string somewhat, insert the "V", and
then return the string to pitch. Once under a string, a pair of
pliers is often helpful in re-positioning the "V".) The gage of the
string to use to make the "V" depends upon the gage of the guitar
string it will support and the shape of the top of the saddle: feel
free to experiment and choose what works best for you. It is
somewhat easier if the top of the saddle is flat across its width,
as is the case with a new saddle or in some circumstances when
lowering the saddle height. Once the "V" has been correctly
located, its center can be marked with a sharp pencil on the top of
the saddle. When this has been repeated for each of the strings,
the saddle is removed from the guitar and the top of the saddle is
shaped. 

Shaping of the top of the saddle involves two components. The first
is to shape the top surface of the saddle so that the string breaks
at a precise location on the width of the saddle. This determines
the exact vibrating string length of each string. The second
component of shaping the top of the saddle is to provide a "clean"
support for the string. This involves bringing the top of the
saddle to a rounded point so as to support the string over a narrow
area, while not being so sharp as to be easily grooved by the
string or cause pre-mature string breakage. The length of the
saddle can be divided into one segment for each string, and the top
of each segment can be individually bevelled and rounded with a
file to correctly position the point created at precisely the
pencil marking determined using the "V"s. When completed, the
shaped saddle can then be returned to the bridge and the instrument
brought to pitch and tuned.

TUNING A GUITAR

As a prerequisite to playing in tune, a guitar's intonation must be
correctly adjusted. However, even after the intonation has been
correctly adjusted for each individual string, the guitar as a
whole will only sound in tune if each of the strings is properly
tuned relative to each of the others. There are numerous correct
methods for tuning a guitar, as well as numerous incorrect methods.
What differentiates the correct methods from the incorrect methods
is the adherence to, or violation of, one simple principle. Rather
than get caught up in the specific aspects of each of these
methods, we will examine the general principle. Once the general
principle is understood, you are then free to create your own
method.

Virtually all tuning methods involve altering the tension of a
string until a note produced on that string exactly matches a
standard to which the note is being compared. The standard can be
a note produced by a pitch pipe, a tuning fork, or an indicator on
an electronic tuner. Of these, an electronic tuner likely provides
the easiest means of initial tuning since each string is compared
to an independent standard, rather than to the other strings, and
most tuners provide a visual indication of when the string is in
tune. 

If an electronic tuner is used to tune only one string, or not used
at all, then once one string is tuned to a standard, the guitar is
tuned to itself using notes produced by the guitar. In any method
which tunes the guitar to itself, there is one basic principle that
must be honored. This principle is that - except in very specific
ways - one cannot mix equal temperament and Pythagorean
temperament. Recall that harmonics belong to Pythagorean
temperament and all fretted notes belong to equal temperament, and
that the only notes which are the same between the two are octaves
and unisons. Practically, this means that a guitar cannot be
correctly tuned using any method that compares any fretted note
with any harmonic that is not an octave (or multiple of an octave)
of an open string. For example, the harmonic produced at the fifth
fret is a fourth higher than the open string. The harmonic at the
seventh fret is two octaves above the open string. A common, and
incorrect, method of tuning is to compare the pitch of the harmonic
on the fifth fret of, say, the sixth string with the seventh fret
harmonic on the fifth string. This compares an octave (of the fifth
string) to a fourth (of the sixth string). The fourth belongs to
Pythagorean tuning while the octave is common to both Pythagorean
and equal temperaments. If the fifth string is tuned this way, the
fretted notes of the fifth string will be out of tune with the
fretted notes on the sixth string. Conversely, it is consistent,
and correct, to compare the fifth fret note on the first string -
an equal temperament note "A" - with the seventh fret harmonic of
the fifth string - a note two octaves above the open A string
common to both equal and Pythagorean temperament. Any method based
upon not mixing equal and Pythagorean temperament is correct.

Once the instrument has been correctly tuned using only equal
temperament notes, many more experienced players will often tweak
the tuning slightly for certain notes. As was mentioned earlier,
certain notes loose a quality of "purity" as a result of the equal
temperament. As a compromise, many players will alter the tuning -
not the intonation - of the instrument so that notes specific to a
key in which they play regain some of the lost purity. This is a
question of personal preference and, to some degree, musical
training. Since electronic tuners give notes belonging to equal
temperament, they are of no use to the average player in tweaking
the tuning.


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