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.
Last updated: September 11, 2018
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.
The scale length for each string of an instrument with unstopped strings like pianos and harps 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. Since 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 bridge 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. 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 to the nut, 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. Many older instruments have frets located using the rule of 18 and in these instruments the 12th fret is a little bit shorter than half of the nominal scale length. It turns out that for most practical purposes this will not make any difference to measurements though. But 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. Scale length measurements may yield only approximate results.
Nut end compensation is becoming increasingly popular, as it has the potential to further improve intonation on fretted instruments. You can 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 or 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 located to compensate for the saw kerf thickness the fretboard ends up with half the kerf thickness of nut end compensation.
If a modern instrument does not have a compensated 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. 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 (virtually all modern instruments) and so is probably the most reliable.