Liutaio Mottola Stringed Instrument Design



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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.

Initial appearance: July 17, 2014
Last updated: July 27, 2017

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 variability, etc.), and which is also within the limits of pitch perception (and the desirability of pitch accuracy) 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.

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 based on 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".1 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. 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

Bridge Saddle Compensation Calculator

Instrument
Selecting an instrument from the menu and then clicking on the Fill button below will fill in the form with typical values for that instrument. You can modify any of these if you wish or just leave them alone to calculate typical compensation values for the selected instrument. Select -none- to clear the entire form.
Instrument:
Click this button to fill the form with values for the selected instrument.
Status of fill:

Common Parameters
Enter or modify common parameters in the fields below. If you make entries or changes here, click on the Propagate button below to propagate these values to the parameters for the individual strings. All values should be in decimal inches. Nonzero values are required in all fields.
Scale length:
Action @ 12th fret on 1st (highest pitched) string:
Action @ 12th fret on last (lowest pitched) string:
Number of strings:
Click this button to propagate these values to the parameters for the individual strings.
Status of propagation:

String Parameters
Enter or modify string parameters in the fields below. There is a tab for each string (up to 12). Note that except in special cases or if you are an advanced user, you shouldn't need to make any modifications here. All values should be in decimal inches except as noted. Nonzero values are required in all fields.

Scale length:
Action @ 12th fret:
Unit weight (lb/in):
Pitch (Hz):
Diameter (core diameter for wound strings):
Hex core?
Material (core material for wound strings):
Modulus of elasticity (Core MOE for wound strings) Leave blank unless material = other. (GPa):
Include this string in straight saddle compensation calculations?
Scale length:
Action @ 12th fret:
Unit weight (lb/in):
Pitch (Hz):
Diameter (core diameter for wound strings):
Hex core?
Material (core material for wound strings):
Modulus of elasticity (Core MOE for wound strings) Leave blank unless material = other. (GPa):
Include this string in straight saddle compensation calculations?
Scale length:
Action @ 12th fret:
Unit weight (lb/in):
Pitch (Hz):
Diameter (core diameter for wound strings):
Hex core?
Material (core material for wound strings):
Modulus of elasticity (Core MOE for wound strings) Leave blank unless material = other. (GPa):
Include this string in straight saddle compensation calculations?
Scale length:
Action @ 12th fret:
Unit weight (lb/in):
Pitch (Hz):
Diameter (core diameter for wound strings):
Hex core?
Material (core material for wound strings):
Modulus of elasticity (Core MOE for wound strings) Leave blank unless material = other. (GPa):
Include this string in straight saddle compensation calculations?
Scale length:
Action @ 12th fret:
Unit weight (lb/in):
Pitch (Hz):
Diameter (core diameter for wound strings):
Hex core?
Material (core material for wound strings):
Modulus of elasticity (Core MOE for wound strings) Leave blank unless material = other. (GPa):
Include this string in straight saddle compensation calculations?
Scale length:
Action @ 12th fret:
Unit weight (lb/in):
Pitch (Hz):
Diameter (core diameter for wound strings):
Hex core?
Material (core material for wound strings):
Modulus of elasticity (Core MOE for wound strings) Leave blank unless material = other. (GPa):
Include this string in straight saddle compensation calculations?
Scale length:
Action @ 12th fret:
Unit weight (lb/in):
Pitch (Hz):
Diameter (core diameter for wound strings):
Hex core?
Material (core material for wound strings):
Modulus of elasticity (Core MOE for wound strings) Leave blank unless material = other. (GPa):
Include this string in straight saddle compensation calculations?
Scale length:
Action @ 12th fret:
Unit weight (lb/in):
Pitch (Hz):
Diameter (core diameter for wound strings):
Hex core?
Material (core material for wound strings):
Modulus of elasticity (Core MOE for wound strings) Leave blank unless material = other. (GPa):
Include this string in straight saddle compensation calculations?
Scale length:
Action @ 12th fret:
Unit weight (lb/in):
Pitch (Hz):
Diameter (core diameter for wound strings):
Hex core?
Material (core material for wound strings):
Modulus of elasticity (Core MOE for wound strings) Leave blank unless material = other. (GPa):
Include this string in straight saddle compensation calculations?
Scale length:
Action @ 12th fret:
Unit weight (lb/in):
Pitch (Hz):
Diameter (core diameter for wound strings):
Hex core?
Material (core material for wound strings):
Modulus of elasticity (Core MOE for wound strings) Leave blank unless material = other. (GPa):
Include this string in straight saddle compensation calculations?
Scale length:
Action @ 12th fret:
Unit weight (lb/in):
Pitch (Hz):
Diameter (core diameter for wound strings):
Hex core?
Material (core material for wound strings):
Modulus of elasticity (Core MOE for wound strings) Leave blank unless material = other. (GPa):
Include this string in straight saddle compensation calculations?
Scale length:
Action @ 12th fret:
Unit weight (lb/in):
Pitch (Hz):
Diameter (core diameter for wound strings):
Hex core?
Material (core material for wound strings):
Modulus of elasticity (Core MOE for wound strings) Leave blank unless material = other. (GPa):
Include this string in straight saddle compensation calculations?

Calculate Results
Clicking the Calculate button will perform the calculations and put the results in the fields below. All compensation values are positive offsets from the scale length(s) provided. In addition to the individual compensation values, straight saddle compensation values are provided for the first and last strings. All values are in decimal inches. Need SI units? Copy and paste values to the units calculator in the right column of this website, above.
Click this button to calculate results.
Status of calculations:
String# Compensation Straight Saddle Compensation
1    
2    
3    
4    
5    
6    
7    
8    
9    
10    
11    
12    

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 the following section. This section contains information on typical modifications to the string data forms you may want to make.

Instruments with Strings in Courses

For our purposes here there are two classes of multi-string courses, those where each string in each course is identical to the other string(s) in the course, and those where the strings are not the same. An example of the first class is the mandolin, which has four courses, each with two identical strings. For the purposes of compensation estimation with the calculator, this class of instrument is best considered just like it has individual strings. You can see that the mandolin instrument in the drop down menu is a four string instrument. An example of the second class of instrument is the twelve string guitar. In this instrument the lower four courses each contain two strings tuned an octave apart from each other. If you do not want to use a straight saddle for an instrument of this type but only want the compensation values for each individual string, then there is no problem with setting up the calculator from scratch for a twelve string instrument. But if you do want to make use of straight saddle compensation, the calculator will not give you the best results. It has no notion of strings in courses and will calculate straight saddle compensation as if the instrument has twelve equally spaced strings. Probably the best thing to do for instruments such as this is to calculate compensation for just the "regular" 6 strings, and then to separately calculate compensation for the octave strings. With these results in hand you can construct a straight saddle that has separate compensation for those octave strings. Or of course you can do what most manufacturers do and simply use a straight saddle compensated for the non-octave strings only.

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

This is pretty straight forward. Just fill in the number. Note that on the strings tabs the first string is always number 1.

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 be too 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?

Uncheck 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.

Some Technical Details

The compensation equations used by this calculator are variations of those specified in [1]. 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, using a simple incremented x-value per string.

References and Suggestions for Further Reading

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