Fretted Instrument Bridge Saddle Compensation Calculator

The bridge saddle location of the guitar, bass guitar, mandolin, ukulele and of most fretted instruments is moved a bit back from its nominal location. This change in location is called bridge saddle compensation. Without it, all fretted notes will play sharp to some extent, due to stretching of the string during fretting and also bending stiffness. Compensation effectively lengthens the string length of fretted notes. The longer string lengths make the pitch of the fretted notes flatter, thus compensating for the sharping effects. This page contains an online calculator which can be used to calculate compensation. The calculator provides individual string compensation values and also straight saddle compensation for instruments that feature a straight angled saddle. It contains data for a number of types of instruments and can be easily used to calculate compensation values for those instruments. The calculator can also be used to calculate compensation for non-standard instruments.

Looking for a calculator that will tell you how much to move the string contact point of the saddle of an existing instrument to achieve accurate intonation? Check out the Adjusting Bridge Saddle for Proper Intonation page.

About Compensation

The scale length of an instrument is the distance between the edge of the nut and the nominal bridge saddle position. For guitars and other fretted instruments the placement of the frets is based on the scale length. See the page entitled Calculating Fret Positions on this site for more info on fret placement. For fretless instruments the bridge is generally located at the nominal bridge position. If the bridge was located here for fretted instruments though, the pitch of each note would be sharp and this sharpness would increase the farther up the neck you went. There are two major factors involved with this phenomenon. The first is that the strings of an instrument are positioned at some distance over the frets. The distance is small, but pressing a string down to a fret stretches it a little, which raises its tension a little, which raises its pitch. The other factor is that strings exhibit bending stiffness, and this increases the vibrating frequency of the string a bit, which also sharpens the pitch.

How much pitch is sharpened depends primarily on the stiffness of the string, or for wound strings, of the string's core. This stiffness is a function of the diameter of the string and also of the material it is made of. The solid nylon of the upper classical guitar strings and the nylon floss of the cores of the lower strings is not very stiff, so stretching the strings while fretting doesn't raise the pitch all that much. Consequently not much compensation is required. Contrast that with the solid steel of steel guitar strings, which is quite stiff. Fretting steel strings raises their pitch significantly and they require more compensation. No matter what the material, the bigger the diameter of the string (core), the stiffer it is, so large diameter solid strings will need more compensation than thinner ones.

The generally accepted solution to this sharping problem is to move the bridge saddle a bit farther away from the nut. This effectively increases the length of the string from fret to bridge saddle, which flattens the pitch and thus provides some compensation for the sharping effects described. Bridge-only compensation is by no means a mathematically perfect solution, but in practical terms it may very well provide a solution which is well within the limits of variability which is not under the builder's control (string mass variability, note-to-note fretting pressure variability¹, etc.), and which is also within the limits of pitch perception for musicians in real musical contexts. Its persistent and widespread use may attest to its practical utility. Likewise the limited use of systems and devices claimed to improve intonation beyond what is provided by bridge saddle only compensation, even though such systems and devices have been available for quite some time, may indicate that bridge saddle only compensation is in fact functionally optimal. Recent research suggests that all popular compensation strategies are perceptually equally effective.²

Just to define the term: Bridge compensation is the distance the bridge (saddle) must be moved from the nominal bridge saddle position to achieve the desired compensatory results. For the most part compensation has historically been calculated empirically. An instrument was set up with the bridge (saddle) located at the nominal bridge position, and then it was moved back by increments until both the open string and the fretted octave were in tune. This is still the way it is done for instruments like electric guitars that have intonation adjustment available. For instruments with fixed bridges and saddles, like typical acoustic guitars, once compensation was figured out, the compensation distance(s) was recorded and future instruments were simply built with the bridge saddle(s) located at the derived position. It should be noted that each string will usually have its own compensation value. Most acoustic instruments will make use of a straight angled saddle though. Because a straight saddle rarely can be located to provide ideal compensation for all strings, the use of a straight saddle represents a mathematical compensation compromise. But here again, considering the high popularity of the straight saddle, this compromise may not be of (much) practical significance.

About This Calculator

Given some information about the instrument and the strings that will be on it, it is possible to do an accurate job of estimating compensation. This is generally not something you'd want to do with pencil and paper as there is a lot of math. But it is an ideal application for an online calculator such as this, which will do all of that math for you.

This calculator makes use of compensation equations that are variants of those developed by industrial polymer physicist Sjaak Elmendorp. His equations are described in his American Lutherie article entitled "It's All About the Core, or How to Estimate Compensation".³ I highly recommend this article for background about compensation. More advanced users of this calculator will probably need to reference it, although basic users will not. The article fully describes the derivation of his compensation equations; it describes validation testing of the model and resultant accuracy specifications; and it describes measurement limitations which should be considered when making use of those equations. Most of this information will not be repeated here. If you are interested, see the article. By the way, even though this article is short, appeared in a popular press journal, and describes a very informal-appearing investigation, the author is a true scientist and the methodological rigor of the experiments and the reporting of same is exemplary.

The calculator outputs offsets from the nominal bridge saddle location to the compensated bridge saddle location for each string, and also for a straight bridge saddle. Refer to the following diagram.

Using the Calculator (Basic)

The calculator has been designed so that it can be used for simple compensation estimation for a number of standard instruments, and also for much more advanced use. Basic use is pretty straight forward:

1. Select an instrument from the drop down menu at the top of the calculator and click on the "Fill" button. This will fill the subsequent forms with data for a typical configuration of that kind of instrument.

2. (optional) Modify common parameter values for scale length and action as needed. Note that all fields accept numeric data only. Values are in inches. Note also that average action values for the instrument should be specified. If you make any modifications here, click on the "Propagate" button. This will update the subsequent forms with your changes.

3. Scroll down past the tabs containing the data for each string to the section entitled "Calculate Results". Click on the "Calculate" button.

A compensation value for each string will appear in the red results fields below. Each of these values is an offset from the nominal bridge saddle location to the compensated bridge saddle location for that string. Also included are straight saddle offsets for the first and last strings. These are also offsets from the nominal bridge saddle location. Connecting these two with a straight line represents the compensated straight saddle. If you are using the calculator to derive compensation for a classical guitar or other instrument that has a straight saddle that is set perpendicular to the centerline of the fretboard, take the difference between the two straight saddle offset values, divide that in half, and add that result to the smaller of the two straight saddle offset values. The result is the compensation offset for the saddle.

The Calculator

Using the Calculator (Advanced)

Selecting an instrument from the menu fills in the forms with data for a typical instrument of that type - typical scale length, typical action and typical string set. Adjusting any of the common parameters and propagating them modifies the form data. But it is the string data forms that the calculator uses to estimate compensation. The data in the forms can be modified directly to further refine the compensation estimation. Describing a new instrument from scratch is detailed in a subsequent section. This section contains information on typical modifications to the string data forms you may want to make.

Excluding Strings from the Straight Saddle Calculations

Excluding strings from the straight saddle compensation calculations will make straight saddle compensation more accurate for the remaining strings. The two cases in which you might want to do this are when you intend to individually compensate one string, and when there are strings that vibrate at a pitch at which humans don't have particularly good pitch perception.

A number of steel string guitars with straight saddles provide separate compensation for the B string. Likewise a number of classical guitars that make use of a straight saddle will provide separate compensation for the B and/or G string. This is because these strings need enough compensation to make them way out of line of a straight line saddle. If you intend to use such a saddle on your instrument, you can specify that the calculator should not consider certain strings in its straight saddle calculations. On the tab for those strings, un-check the box labeled "Include this string in straight saddle compensation calculations?" This will result in more accurate straight saddle calculations for the other strings. That increase in accuracy may be perceptible.

People don't have very good pitch perception in the bass range (and it is even worse in the double bass range) so it may be advantageous to exclude the lowest string from guitars and basses from straight saddle compensation calculations. The resulting compensation for those strings probably won't result in perceivable differences, and compensation accuracy of the remaining strings will be improved, possibly perceptibly.

Multiple Scale Length Instruments

You can see on the tab for each string that scale length is specified on a per string basis. If you are designing a multiple scale length instrument you can set the scale length for each string individually. Note that if your instrument will make use of a custom string set then you'll also need to specify some critical dimensions for the strings. This process is covered in detail in the next section.

Specifying an Instrument from Scratch

As you can see from the parameters in the tabs, most of the information needed to estimate compensation relates to the instrument's strings. So if you want to use the calculator to estimate compensation for an instrument that does not appear in the drop down menu, most of the work involved in doing that will be in describing the instrument's strings. In an ideal world all of this information would be readily available from string manufacturers, but in fact little of it actually is. The very first thing you'll want to do if at all possible is to get or compose a set of strings from D'Addario. Why? Because D'Addario is the only string manufacturer that provides unit weight and tension data for most of the strings in their product line.

Specifying an instrument involves filling out the parameter fields in the forms as described below. If you are going to spend the time to do this for a common instrument you would be doing the world a great favor if you send your completed forms to me so that I can include this instrument in the drop down menu. Thanks. Be sure to include some information on the instrument and string set, and please provide your email address so I can contact you with any questions. Here are the parameters and how to fill these fields in.

Number of strings (Common Parameters Form)

This is pretty straight forward. Just fill in the number. Note that on the strings tabs the first string is always number 1. By convention this should be the highest pitched string.

Number of courses (Common Parameters Form)

For instruments like guitars and ukuleles where the strings are not in courses, this should be the same as the number of strings. For instruments like mandolins (which have 4 courses of two strings each) or 12 string guitars (which have 6 courses of two strings each) enter the number of courses. This value is used to calculate straight saddle compensation values. Note that the first course is always number 1, and should contain the highest pitched strings.

Course

The string course that this string belongs to. Course are sequentially numbered from 1-n, where 1 is the highest pitched course. String #1 is always in course #1. If the instrument does not have strings in courses then set this value to be the same as the string number.

Scale length

The scale length for this string, in inches. Note that for steel strings it is critical to be accurate about this. For nylon strings, not so much.

Action @ 12th fret

Action for this string, measured from the top of the crown of the 12th fret to the underside of the string, in inches. Note that for steel strings it is critical to be accurate about this, although you really can't get very accurate with ruler measurements.

Unit weight

The unit weight of the string in pounds per inch. To get this data, look up your string set on the D'Addario website. The site will identify the part number of each string in the set. Go to D'Addario's Catalog Supplement/String Tension Specifications document, and look up the string part number. Copy the unit weight into this field. If this data isn't available or if you are not using D'Addario strings, you can calculate unit weight if you know the scale length, frequency the open string is tuned to, and string tension. The equation for this calculation can also be found in the D'Addario catalog supplement document. It may also be possible to directly measure unit weight if you have access to a high resolution laboratory scale. Note that this is a critical parameter, so if you attempt to weigh strings you may want to weigh a small pile of them.

Pitch

The pitch of the open string, in Hz. You can look this up in the Table of Musical Notes and Their Frequencies and Wavelengths on this site.

Diameter

For unwound strings, this is the diameter of the (unstretched) string, in inches. For D'Addario strings you can look this up on their website. For wound strings, this is the diameter of just the core of the string, not the whole string. Unfortunately you can't simply look this up anywhere, so you'll have to measure this using digital calipers. This measurement is simple enough to do for steel strings. Just measure the core where it sticks out near the end of the string. For wound nylon strings things are a little more involved. Since the string core of these strings is made of nylon floss, the diameter can't be measured directly. You'll have to unravel and straighten a bit of the winding wire, measure its diameter with the calipers, multiply that measurement by 2, then subtract that result from the diameter of the whole string to get the diameter of the core. Note that for all strings this measurement is critical for compensation estimation. It is a good idea to take a number of caliper measurements and then take the lowest one you get.

Hex core?

The core of some steel strings is hexagonal in shape. If this is the case with this string, check this box. When you take the diameter measurement above, be sure to measure across the flats of the hexagon shape.

Material

This field in the form is a menu of materials. Chose the one that describes what a plain string is made of, or what the core of a wound string is made of. Steel strings are made of steel. The core of all steel strings is made of steel, no matter what they are wound with. Plain nylon strings are made of nylon. For the purposes of compensation estimation you can just use nylon as the material for fluoropolymer (so-called carbon fiber and so-called titanium) strings as well. Plain gut strings are made of gut. The wound strings in nylon, fluoropolymer, gut and so-called silk and steel string sets have cores made of nylon floss. If the strings you have are made of (or have cores made of) some other material, select "-other-" for the material. You'll have to look up or figure out the modulus of elasticity for that material though. See below.

Modulus of elasticity

If you set the material of the string/core to "-other-" you'll have to enter a value for the modulus of elasticity (MOE) of that material, in GPa. You can generally look such things up, but you may have to resort to figuring this out yourself. Check out Dr. Elmendorp's article for information on how to do the latter. If you have set material to something other than "-other-", this field is ignored.

Include this string in straight saddle compensation calculations?

Un-check this box if you don't want this string to be included in the calculation for the location of a straight saddle. Do this only if you intend to individually compensate this string. Un-checking this box for strings with compensation that is unusually out of line (like the B string of steel string guitars) can improve the accuracy of the estimation of straight saddle location. Note that the calculator won't calculate straight saddle compensation unless at least three strings in the set are included.

Extra Information Provided by the Calculator

In addition to compensation values, the calculator results include some information that may be useful when considering alternate tunings for conventional instruments, or when designing new instruments or string sets. This information is included here just because the calculator is already calculating most of this internally anyway.

String Tension

The tension each string is subjected to is a function of its vibrating length, its mass per unit of length, and the pitch of the note it is tuned to. For the exact equation modeling this, see the page on this site entitled String Tension. The calculator presents string tension in pounds, which is typically the way it is presented by string manufacturers on their string packages.

There are a number of reasons why string tension is of interest to designers and builders of plucked string instruments. If tension is too high it can damage the instrument. Even if an instrument can withstand high tension strings, their use may make it more difficult for players to fret the strings. Low tension causes other problems. Strings that are too slack require higher action, because they vibrate in a broader arc when they are plucked. Lowering string tension makes the resulting note sound less pure in tone. Although this may not be noticeable or objectionable with small changes in tension, greatly lowering tension may make for unacceptable tone.

The string sets for plucked instruments are usually designed for approximately uniform tension across the strings of a set. This is an ergonomic issue, but is also helps to provide uniform loudness for each string when chords are played.

Some general trends can be observed by looking at the tensions of the strings for various types of instruments. Steel strings are always of considerably higher tension than nylon or gut strings. Higher pitched instruments tend to have lower string tensions than lower pitched instruments, given the same string (core) material.

% of Maximum String Tension

Commercial string sets tend to have string tensions that are well below the tensile strength of the strings (or for wound strings, of their cores). As mentioned above, the higher the tension of the string, the more pure the resulting notes will sound to the listener. This is due to the fact that there are two restoring forces that cause the string to vibrate after it is plucked. These are string tension, and the bending stiffness of the string (core) material. If a string didn't have any bending stiffness it would vibrate in a harmonic fashion, with a fundamental frequency of vibration and a series of harmonic vibrations at frequencies of integer multiples above the fundamental. Harmonic vibration results in what humans perceive to be pure-sounding notes. But all real strings exhibit bending stiffness. The restoring force contributed by bending stiffness is constant and independent of string tension. What this means is that the higher the string tension, the bigger the percentage of the restoring force of vibration is contributed by string tension, and the purer the notes will sound.

From this it would seem that the higher the string tension the better. In fact, for non-fretted string instruments such as harps, the general design rule for string sets is to maintain tension at about 80% of the tension at which they would break. This percentage results in notes that are as pure-sounding as possible, while providing a reasonable margin of safety against string breakage. It is possible to use such high tensions because the strings of harps need not be fretted. It is generally the case that maintaining such high tensions for fretted instruments would result in difficulty in fretting of the strings, so lower tensions are generally used.

Data in this column is presented as the percent of the maximum tension that the string could withstand before breaking. This is calculated as a function of the ultimate tensile strength (UTS) of the material that the string (core) is made of. UTS data used in the calculations is generally quite approximate. Designers of string sets should be aware that values here above 80% should be considered to be problematic.

Inharmonicity

It was mentioned above that lowering the tension of a string makes the resulting note sound less pure, because doing so increases the percentage of the restoring force of vibration that comes from the bending stiffness of the string, and this in turn causes the string to vibrate in a less than harmonic fashion. The property of vibrating in a less than harmonic fashion is called inharmonicity. Lowering string tension increases inharmonicity. Shortening the vibrating length of the string also increases inharmonicity. Values in this column compare the frequency of the (perfect) second harmonic of the open string with that of the calculated frequency of the second partial, and present this difference in cents.

Musicians and instrument builders often wonder just why certain instruments must have scale lengths that fall within relatively narrow ranges. Wouldn't it be a lot easier to play, say, a steel string electric bass if it had a scale length as short as a guitar? And if so, why are there not steel string electric basses with these short scale lengths? The reason is because doing so would result in an instrument the notes of which sounded so impure that it would be objectionable. Oh, the open strings might sound good enough, but fretting up the neck further shortens the vibrating length of the string, which increases inharmonicity. It is the case that the typical scale lengths used for instruments represent a compromise of acceptable inharmonicity on the bass string of the instrument. Most players understand that the fretted notes way up the neck on the lowest string of a guitar don't sound that good.

Looking at the inharmonicity values presented in this column for typical string sets for a number of different instruments reveals some interesting patterns. The lower strings for extended range instruments such as the five string bass exhibit high inharmonicity. Higher pitched instruments exhibit high inharmonicity in general, yet we rarely have complaints about the purity of the notes of instruments like the mandolin and the ukulele. Nylon strung instruments avoid high inharmonicity of the bass strings by using floss cores, which offer very low bending stiffness.

Some Technical Details

The compensation equations used by this calculator are variations of those specified in [3]. The article also shows how to derive material MOE and gives examples for nylon and nylon floss. These materials and other polymers will exhibit nonlinear stress/strain curves and thus their elastic moduli will be functions of tension. The calculator makes use of the variable elasticity data for nylon and nylon floss from the article. Gut also exhibits variable elasticity, and values are calculated from my own experiments. Straight saddle compensation estimation is done by simple linear regression.

Occasionally someone will assert that string stretch associated with fretting involves the entire length of the string from anchor to post, not just the part from nut to bridge saddle. I have performed limited experimentation on this and from that I am currently satisfied that this is not the case - normal fretting simply does not overcome string/saddle or string/nut friction and so stretch during normal fretting is limited to that part of the string from saddle to nut. If anyone can reference definitive experiments showing different results I would be interested in the citations.

The percentage of breaking tension data is based on material UTS data. That for gut is based on my own limited experiments. The UTS for steel strings and cores varies with core diameter, and is a function of draw down. Data is generally available from manufacturers in tabular form, but here I make use of a polynomial and fitted coefficients.

Calculating bending stiffness of floss cores appears to be an extremely difficult problem. Here I am treating these as solid nylon, which in practice turns out to be a reasonable approximation.

References and Suggestions for Further Reading

1. Mottola, R.M. “Same-Fretted-Note Intonation Variability of the Steel String Acoustic Guitar” OSFPreprints, 2018

2. Mottola, R.M. “Blind Listening Evaluation of Steel String Acoustic Guitar Compensation Strategies” OSFPreprints, 2017

3. Elmendorp, Sjaak “It's All About the Core, or How to Estimate Compensation” American Lutherie, #104

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