In an article on the subject appearing in American Lutherie1, editor Tim Olsen asserted that optimal action (i.e., minimal string height over the frets) could not be achieved if the fret tops described a cylindrical section. Since the strings themselves of most modern instruments diverged in distance from the nut end of the neck to the bridge saddle and since the strings are essentially straight, there would necessarily be greater distance than is optimal between the outside strings and the frets under them at the bridge end of the neck. He went on to opine that the strings probably describe something closer to a conical section and that, for optimum action, the tops of the frets must also describe a similar section.
Last updated: January 22, 2019
A number of other authors picked up and expanded upon this theme including Hartstein2, who provided a formula which can be used to determine the parameters of a cone described by the strings of an instrument. As presented in the article it solves for the radius of the cone at some distance from the nut of the instrument:
Rx = (Rn * (1 + Dx * (Sf – Sn) / (Df * Sn))
Dx = an arbitrary distance from the nut;
Rx = the radius of the cone at that distance from the nut;
Sn = the distance between the centers of the outside strings at the nut;
Df = the distance from the nut to a reference location where the distance between the centers of the outside strings is known or can be measured;
Sf = the distance between the centers of the outside strings at Df;
Rn = the radius of the arc formed by the bottoms of the nut slots;
Please note that I have taken the liberty of changing the descriptions of the parameters somewhat to more precisely describe a conical section which would be described by the strings of an instrument. This formula is very useful in the task of determining if in fact the strings of an instrument describe a conical section. Using the distance from the nut to the bridge saddle(s) for Dx and taking all other parameter values from measurements made of an instrument, the formula can be used to solve for Rx, which in this case will be the radius of the arc of the bridge saddle (or described by the individual saddles on instruments so equipped).
I've done this for a few dozen instruments of different types, both in preparation for and as follow up research to an article related to this subject3 for American Lutherie. Comparing the calculated radius of the bridge saddle(s) with the measured radius, I have yet to find an instrument where they are the same, within a reasonable degree of accuracy. This indicates that, at least in these cases, the strings do not describe a conical section but some other surface.
Note that these findings have some practical implications for those wishing to optimize playing surface geometry. Does this finding mean that the action is sub-optimal on all of these instruments? Not at all. Probably the most important rule to consider is that the optimal relationship between the strings of an instrument and the playing surface should be considered on a string by string basis, without worrying too much about the geometric shape described by all the strings together. All that matters is that the relationship between each string and the tops of the frets directly underneath it is optimal. Although a perfectly conical section described by the strings can be mated with a perfectly conical playing surface to yield optimal action, optimal action can also be had by, say, strings that describe a more flattened, elliptical cone if the playing surface also describes such a cone. Considering this, my appreciation for the ad hoc fret leveling jobs done by some of the most respected repair people has increased dramatically. These folks manage to take instruments with cylindrical fingerboards, level the frets so the fret tops are optimally related to each string over them, set the appropriate saddle curve, and then optimally adjust the action, all in an ad hoc manner and without worrying too much about the name given to the geometric shape of the fret top surface they have established. Also considering this, the effort and tooling involved to make conical surface fingerboards hardly seems worth it, as the same action can be had by far less tedious methods.
American Lutherie #101 featured what I consider to be the definitive article on this subject4. Written by F.A. Jaén, the article sums up this issue and covers all of the geometry involved, and concludes that the difference between an optimal shaping of the fret tops and a perfect cylindrical shape for same is so small as to be insignificant. My experience indicates that, as a consequence of the method most often used to level and dress frets, even this difference disappears in actual practice, even if that is not the explicit intention of the luthier.
The practical ramifications of all this for luthiers attempting to achieve lowest action for fretted instruments are clear. So called compound radius fretboards are not worth the additional effort required to make them. The same action and tolerances are achieved by using cylindrical fretboards and normal fret leveling and dressing techniques.
Olsen, T. “Cylinders Don't Make It”
American Lutherie #8, 1986.
Hartstein, E. “Conical Radius Fretboard Formula”
American Lutherie #34, 1993, p. 46.
Mottola, R.M. “Optimizing Playing Surface Geometry”
American Lutherie #89, 2007, p. 56.
Jaén, F.A. “Not Only Cones Make It - And Cylinders Almost Do”
American Lutherie #101, 2010, p. 52.