# Measuring Scale Length of Stringed Instruments

Elementary texts state that the scale length of a stringed instrument is the vibrating length of the (open) string. But this is often not the case, depending on the instrument and its construction details. Understanding the relationship between scale length and string length is important to folks attempting to duplicate or draw up plans for existing instruments. This page discusses the relationship between scale length and string length for different instruments and describes methods for measuring scale length of existing instruments. It includes a calculator for determining the scale length of fretted instruments. The calculator is also useful to determine which fret layout system was used, and for evaluating how accurately the frets were placed.

Last updated: December 14, 2022

## Contents

The relationship between scale length and string length is best discussed in the context of the type of the instrument. The three major classes of instruments in this matter are those with unstopped strings such as pianos and harps, those with stopped strings but do not have frets, and fretted instruments.

## About Instruments with Unstopped Strings

The scale length for each string of an instrument with unstopped strings like the piano and the harp is the same as the vibrating length of the string. As such, scale length can be measured directly from the vibrating string length for these instruments. Note that by unstopped strings, I mean strings the vibrating length of which are never shortened in the process of playing the instrument. Note also that since each string in such an instrument is generally a different length, the whole scale length abstraction is not too meaningful when discussing instruments of this type.

The actual scale length of instruments of this class (violin family and other fretless instruments where fingering positions are unmarked) can also be measured directly from the vibrating length of the open string, that is, from the nut to the bridge saddle. Note that most instruments of this class exhibit some string divergence, that is, the string spacing is wider at the bridge than it is at the nut, but scale length is always measured straight down the center of the fingerboard from the nut to the bridge saddle. A distinction should be made between the actual scale length of an instrument and the scale length intended by the builder. Because the bridge on this class of instrument is movable. it is often the case that an instrument is set up with the bridge moved from its nominal position. This can be done by carelessness during setup, but it is often the case that the bridge is relocated purposely to improve tone. Small variations in scale length are undetectable by players, and this moving about of the bridge can in no way be considered erroneous. But for someone attempting to document dimensions of an existing instrument, it is important to locate the bridge exactly where the maker intended it to be before measuring the scale length. Violin makers file small notches in the walls of the f-holes to indicate nominal bridge placement. For documentation efforts, the bridge should be positioned so that the saddle is directly in line with a line drawn between the notches of the two f-holes. Care should be taken when making scale length measurements of this class of instrument, to be sure the bridge is not bent forward. This invariably happens with age of the bridge, due to down force exerted by the strings.

Also in this class of instruments are those with fret analogs, such as lined fretless bass guitars, or instruments that are otherwise permanently marked with fretting position indicators. Instruments of this class almost always have the bridge saddle moved a small amount from its nominal position, so as to slightly lengthen the vibrating length of the strings. This is done to improve intonation of the fretted notes. The lengthening effectively flattens the notes a bit, and it is done to provide some compensation for the sharping of the fretted notes that happens during fretting. Fretting a string bends it down toward the fingerboard and this bending raises its pitch a bit. Pitch is also raised a bit by the bending stiffness of the string, which is higher the more the vibrating part of the string is shortened by fretting.

Since a majority of fretted instrument have some bridge saddle compensation, and since compensation varies from string to string, it is not possible to directly measure scale length on this class of instrument as the vibrating length of the open string. However, on modern instruments that do not also add some compensation at the nut end, it is possible to measure scale length by measuring the distance from the nut to the 12th fret and then multiplying that value by 2.

Note that the above specifies modern instruments, specifically instruments that used the 12th root of 2 fret placement system. Many older instruments have frets located using the Rule of 18, and in these instruments the distance between the nut and the 12th fret is a little bit shorter than half of the nominal scale length. So, if the frets of an instrument were laid out using the Rule of 18, you cannot calculate scale length by measuring the distance from the nut to the 12th fret and then multiplying that value by 2. While on the subject of older instruments I should also point out that many older instruments were built with fairly loose tolerances as far as fret placement goes.

Nut end compensation is becoming increasingly popular, as it has the potential to further improve intonation on fretted instruments. You can often identify an instrument that employs individual string nut compensation by examining the nut carefully. On such instruments, the front face of the nut will not be parallel to the first fret, or the front face will be different for each string, cut back by filing or notching or moved forward with the addition of extensions. But be aware that there are many instruments that have intentional or unintentional nut compensation that is not so apparent. In these instruments the distance between the nut and the first fret has been shortened. Although it is unlikely to find such nut compensation intentionally applied to older instruments, it is often the case that builders of new or old instruments unintentionally added a bit of nut compensation simply by sawing the nut end of the fretboard off using the same saw as was used to cut the fret slots. Unless the location of the nut end was positioned to compensate for the saw kerf thickness, the fretboard ends up with half the kerf thickness of nut end compensation.

If a modern instrument with 12th root of 2 fret placement does not have a compensated nut or any errors in the placement of the nut, then you can determine scale length by measuring the distance from the nut to the 12th fret and then multiplying that value by 2 as described above. One practical problem is that it is difficult to ascertain if there are issues with nut placement. If it does have a compensated nut (or if you suspect it has one or just are not sure) then the scale length can be derived by measuring the distance between the 1st and the 13th frets and multiplying this value by 2.1189. This technique will work for all fretted instruments with equal temperament fret spacing (12th root of 2, virtually all modern instruments) and so is probably the most reliable. The accuracy of the result will depend upon the accuracy with which those frets were installed, and also the accuracy of your measurement.

As should be apparent from the preceding text, obtaining accurate scale length data is dependent on a number of things that are not under your control, and some of these are difficult to determine. I wrote an article in American Lutherie explaining all the hairy details of attempting to identify the actual scale length of fretted instruments.1 See the article for more information on this subject. Folks needing to accurately identify scale length of an instrument may want to make use of the calculator below.

## Fretted Instrument Scale Length Calculator

The calculator takes as input fret-to-fret measurements you make using digital calipers. You can make a single measurement and the calculator will figure out the scale length of the instrument in either the modern $\sqrt[12]{2}$ (Twelfth Root of Two) fret layout system and the historical Rule of 18 system. Advanced users can enter up to ten different fret-to-fret measurements. If you enter at least two, the calculator will figure out the following:

• the fret layout system used
• the scale length
• the amount each fret-to-fret distance deviates from that of the calculated scale length

That the calculator will figure out the fret layout system used is very useful, because the historical transition from Rule of 18 to the Twelfth Root of Two took place over the course of decades. This makes it difficult to speculate which system was used for any one instrument, without definitive historical information on the construction of that instrument.

The last item in the list above can give you an idea of how accurate the frets were laid out, and can also help to identify single frets which are way out of position.

Advanced users can also enter the measured nut-to-first-fret distance, and the calculator will report on how accurate the nut was placed relative to the frets.

### Taking Fret-to-Fret measurements

As is the general case with software of any kind, the results the calculator return can be no more accurate than the measurement data provided to it. Fret-to-fret measurements should be done using digital calipers, measuring between frets along the centerline of the fretboard. It is a good idea to measure between frets that are as far apart as can be measured using the calipers you have. Doing so reduces the effect of measurement errors. So for example, it is better to measure the distance between frets 1 and 4 (assuming your calipers can open wide enough to do this) than between frets 1 and 2. When entering multiple measurements, it is also a wise idea to measure from different fret pairs. So for example, rather than taking three measurements between frets 1 and 4, 1 and 5, and 1 and 6, it is generally better to take measurements between frets 1 and 4, 2 and 5, and 3 and 6.

What is needed are measurements from the apex of the crown of one fret to the apex of the crown of another. Since the location of the crown apex cannot be exactly ascertained, a good way to simulate this is to measure the distance between the bridge side of the lower fret and the nut side of the upper fret, and then add to this the measured width of one fret. When measuring from the nut to the 1st fret, the measurement is taken from the fretboard surface of the nut, to the nut side of the fret, and then half the width of the fret is added to that.

### Simple Use of the Calculator

The calculator will calculate the scale length of an instrument from a single fret-to-fret measurement. This is the easiest way to use it. To do this, first take a fret-to-fret measurement as per the instructions above. Then:

• Select the units of measurement (inch or mm);
• Select the fret layout system. Use $\sqrt[12]{2}$ (12th root of 2) for a modern instrument, or Rule of 18 for an antique instrument. Note: For simple use do not select the option labeled calc.;
• Enter the number of the first (lowest) fret that you took the measurement between, in the box on the first line labeled F1#;
• Enter the number of the second (highest) fret that you took the measurement between, in the box on the first line labeled F2#;
• Enter the distance you measured between the crowns of these two frets in the box on the first line labeled F1-F2 Distance;
• Check the check box that is next to the distance box on the line. This indicates to the calculator that the data on this line should be used in its calculations;
• Click on the Calculate button. The calculated scale length will appear in the Scale length box. Whatever Fret Layout System you selected will also appear in the results.

### Scale Length Calculator

 Units inch mm Fret Layout System calc. $\sqrt[12]{2}$ rule of 18 F1# F2# F1-F2 Distance ✔ Error Optional Status: Javascript must be enabled in your browser to use the calculator. Results Scale length: Fret layout system:

If you are not absolutely sure of the units or fret layout system used in the instrument, you may want to select different values for the Units and Fret Layout System and then recalculate and compare the results.

### Advanced Use of the Calculator

The calculator may yield more accurate scale length results and will display other useful information if more than one fret-to-fret distance measurement is entered. Doing so is highly recommended. Instructions for advanced use are similar to those for simple use, only additional measurements are added on additional lines of the calculator. Entering at least three measurements is a good idea. As well as entering additional measurements, you can also do the following:

• If you are not absolutely sure of the fret layout system used for the instrument, and you have entered and checked the check box for more than one fret-to-fret distance, you can (and should) select calc. for the Fret Layout System. The calculator will calculate which Fret Layout System was most likely used, and report that in the results, proceeded by the '?' character to indicate that this is estimated;
• The check box on each line containing measurement data should be checked. This lets the calculator know to use that data in its calculations. Un-checking a line may be useful if you suspect that line may involve a badly placed fret. More information on this case is available below;
• You can optionally enter the measurement from the nut to the first fret, in the line just above the Calculate button. If you do this, that measurement will not be included in the scale length calculation, but the calculator will analyze the accuracy of that measurement in terms of the calculated scale length. More info on that below.

### Interpreting Results

Scale Length- After the Calculate button is clicked, the calculator will calculate a scale length based on the measurement data on each line. It will then average all of these calculated scale lengths together. Data on a line where the check box is not checked will not be used in the calculated average scale length. The calculated scale length will appear in the results box near the bottom of the calculator.

Errors/Deviations- The calculator will also calculate an error value for each line on which data have been entered. This is the difference between the entered measurement and what this measurement should be for the calculated scale length. The calculator also color codes this error information. If it appears in green, the deviation is unlikely to have an audible effect on intonation. If it appears in orange, the deviation is probably within acceptable tolerances for hand-measured and hand-cut fret slotting. If it appears in red, it is likely that one or both of the frets involved in the measurement have been badly located, to the extent that intonation problems will occur.

Eliminating Outlier Measurements- There are two uses of the error values and their color coding. The first is to help attain a more accurate scale length calculation by eliminating outliers. If most of the errors are insignificant (colored coded in green) but some are coded in orange or red, un-checking the lines where the errors are significant will remove them from use in subsequent scale length calculations. Doing so and clicking again on the Calculate button will change the calculated scale length value.

Identifying a Badly Placed Fret- The second use of the error values is to help identify a single badly placed fret. This is best described by example. If there is a significant error in the measurement between, say, frets 2 and 6, it is possible that one of those frets has been badly positioned. Including an additional measurement from another fret to fret 6 would help to identify if fret 6 was badly placed. If both of the measurements involving fret 6 have significant errors, then it is likely that fret 6 is badly placed.

Identifying Nut Placement Anomalies- As mentioned, one reason that scale length measurements taken from the nut cannot be assumed to be reliable is because the nut end of the fretboard is often cut short, either accidentally or intentionally. If nut-to-first-fret measurement data is entered, the error value for that line can help identify what is going on. Note that the most common unintentional shortening at the nut end happens when the same saw that is used to saw the fret slots is used at the 0th fret position to saw off the end of the nut, without compensating for the width of the saw kerf. Since the fret slotting saw is nominally 0.022" thick, an error value here of -0.011" would indicate that this was done. Note that I have seen cases where it was apparent that the builder knew that compensation for the width of the saw kerf was necessary, but unfortunately got the math wrong. It is also increasingly common for the nut to be intentionally placed closer to the 1st fret, to provide some amount of nut-end intonation compensation.

### Rationalizing the Results

It is useful when performing forensic analysis to determine scale length to consider all the clues at your disposal. I've already mentioned that the Fret Layout System used for antique instruments (for our purposes here, any instrument built before 1920) can be counted on to be Rule of 18, and that the 12th Root of 2 is the system used for all modern instruments (again, for our purposes here, any instrument built after 1990). Also already implied is that errors in nut placement are very often the result of erroneous compensation for the fretting saw kerf width, and these tend to be multiples of half that width. The measurement system used when the instrument was designed is also a useful clue if you have that information. Instruments built anywhere in the world other than the USA almost always would have had their scale lengths specified in SI (metric) units, almost assuredly in whole millimeters and probably in five millimeter chunks. Instruments built in the USA (with the exception of classical guitars) will almost always use USCS units (also called English or Imperial) and will likely be expressed in chunks no smaller than sixteenths of an inch.

1. Mottola, R.M. "Measuring Scale Length of Fretted Instruments" American Lutherie #136, 2019, p. 48.

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