Optimization of the Geometric Relationship between the Strings and the Playing Surface of Fretted Instruments

Copyright © 2007 R.M. Mottola

[This article originally appeared in American Lutherie #89.]

Post Publication Notes - I no longer use either the sandpaper-on-glass method of fret leveling described here, nor do I use the fret leveling bar/beam described here either. I do use the short mill file method described here, and have been using only that method for the last fifteen years. I've found using the mill file for fret leveling yields results that are just as accurate as any other method and generally a lot quicker as well, shaping the fret tops to describe a quasi-conical surface. I find that beginners readily understand what they need to do using the mill file, filing in the direction of the string lines, and achieve accurate results right away. For the latter reason this is the fret leveling method that I recommend for beginners in the book Building the Steel String Acoustic Guitar. Note that the work described here and additional research led to another article appearing on this website about a related subject. See "Lutherie Myth/Science: The Strings of a Stringed Musical Instrument Describe a Conical Section".

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For optimum action a string must be positioned as close to the playing surface (fret tops, or for fretless instruments the fingerboard surface) as possible. Since a vibrating string exhibits greater displacement in the center of its length than at any other point, first order optimization typically angles the string in relation to the playing surface so that it is very close at the nut and widens out as it approaches the bridge, leaving enough clearance at the center so that the string can vibrate without buzzing on the playing surface at any point. Further optimization is possible by adding relief to the playing surface, an upward curve that ideally optimizes string-to-playing surface proximity at all fretting positions. Philip Mayes did a nice article on relief geometry which appears in the Guild of American Luthiers book Lutherie Woods and Steel String Guitars which I highly recommend. I am going to ignore the issue of relief in the current discussion simply for the sake of clarity. Please consider that any of the string-to-playing surface relationships discussed here could be further optimized by the application of appropriate neck relief.

The angled relationship (sans relief) between a string and the playing surface under it can be easily represented in a two dimensional drawing. A 2D representation is also suitable for the relationship between multiple strings and the playing surface when the playing surface (and nut and bridge saddle) are flat. But both the representation and the optimization can get a little hairy if the playing surface is cambered. Guitar fingerboards are narrower at the nut end than they are at the bridge end, and so the strings diverge as they approach the bridge. Electric guitars and acoustic steel string guitars typically have fingerboards with cambered playing surfaces, and in order to keep the strings close to the tops of the frets both the nut and the bridge saddle tops are also cambered. The surface described by the strings is a function of the divergence of the strings and the cambering of the nut and bridge saddle, along with the fact that the strings are essentially straight.

Tim Olsen described the geometry involved here very well in his article entitled "Cylinders Don’t Make It" (American Lutherie #8, Big Red Book of American Lutherie 1) and opined that the strings describe the surface of a conical section. Although there are some combinations of nut camber, bridge saddle camber and string divergence that will indeed result in the strings describing a conical surface, this is not the case with all combinations. Consider the truncated cone shown in figure 1. If a section of a cone is taken by intersecting the cone with a plane, the outside edges of that section will only be straight if the plane intersects the cone on two of its generation lines (straight lines on the surface of the cone that run from the apex to the base). Another way to say this is that the outside edges of the intersection of a cone and a plane will only be straight if the plane goes through the apex of the cone. Any other intersection results in the edges forming a parabola (figure 2).

The restriction that the edges of a conical section described by instrument strings must be straight lines (because the strings are straight) means that there is a fixed relationship between the divergence of the strings, the radius of the curve at the bottoms of the nut slots, and the radius of the curve at the bridge saddle. One could arbitrarily specify any two of these and the value for the third would have to follow. It was interesting that, when I measured nut camber, saddle camber and divergence of a number of instruments I could find none where the strings described a conical surface. Those interested in the formula which shows these relationships can find it in the informative but inaccurately titled article "Conical Radius Fretboard Formula" by Elaine Hartstein which appeared in American Lutherie #34 (Big Red Book of American Lutherie 3). The formula there is for conical sections described by strings (not for the fretboard as the title states) and if you substitute the radius of the curve described by the bottoms of the nut slots for that of the fingerboard itself at the nut as the article states then you can accurately derive values for the attributes of that string surface using the formula presented.

Assuming that one were to dimension the components of an instrument so that the strings did describe a conical surface it would be possible to optimize the relationship between the strings and the playing surface for best action if the playing surface also described a conical surface. The two cones would be axially oriented to each other and that of the playing surface would be more acute, such that an optimum angled relationship would exist between the playing surface and a string that was located anywhere on the string cone. Figure 3 shows the relationship between these two cones.

Tim Olsen’s article pointed out that for optimally low string action for all strings of the instrument (and assuming the strings describe a conical section), the tops of the frets of the instrument must also describe a conical section. It should be pointed out that this is not the only geometric relationship that would result in optimally low string action. Optimal action only requires that the relationship between each string and the playing surface beneath it is optimal. Depending on radius values at the nut and bridge and on string and fingerboard divergence this could result in quite complex surfaces. These seem to appear more frequently on fretless instruments of the violin family than on fretted instruments. Consider figure 4, which shows a model of the string surface of the double bass, looking from the nut end. The radius of the bridge of the bass is not much different than that of the nut, and given the string divergence the resulting surface is not conical. In the model the scale length has been reduced to only 1" which makes it much easier to see the twist in the surface, clearly visible between the lowest two (rightmost) strings. But it is possible for a bass to exhibit optimal action if its fingerboard is shaped to a similar surface. In the remainder of the article I am going to consider only the conical string surface and conical playing surface relationship, but I did want to point out that this is not the only relationship that can yield optimum action.

Since most guitar fingerboards are cambered in a cylindrical fashion, and since in theory the fret crowns are at a fixed distance above the fingerboard surface, the action of the outboard strings will be unnecessarily high above the frets near the bridge end of such fingerboards. Tim’s article proposed a simple thought experiment to demonstrate this, imagining the surface of a large diameter pipe representing a cylindrical playing surface and two rods representing the outboard strings. If the rods are positioned above the surface of the pipe and they remain parallel (that is, they do not diverge) then optimum action can be attained. If the rods are positioned so that they diverge it is not possible to maintain optimum action along the entire lengths of the rods. If the geometry here is not readily obvious, actually trying this out with a couple of rods or dowels and a hunk of 4" PVC pipe or a coffee can will make it clear.

So if the strings of an instrument diverge and the tops of the frets describe the surface of a cylindrical section then string action can never be optimal. And optimal action is a desirable feature, especially in the realm of electric guitars. Two more desirable features are generally discussed in the context of optimizing guitar playing surface geometry. The first is the desirability expressed by some players for a relatively short radius camber to the playing surface. The general theory here is that the shorter radius makes for a more comfortable playing surface, one that better fits the natural curve of the fingers. The second desirable feature is the ability to make large displacement string bends. These two features are quite at odds, since the smaller the playing surface radius the shorter the possible string bends. This is due to the fact that bending a string on a cambered playing surface can pull it "downhill" of the next higher fret enough so that the string touches that fret. This doesn’t happen on a flat playing surface and for all practical purposes it doesn’t happen on larger radius playing surfaces either. Even though these features are at odds it is possible to provide short radius playing surface near the nut end and larger radius playing surface further up the neck, where all the really long string bends are performed. This compromise is realized in a conically shaped playing surface, and this is why these two features are mentioned in the context of this discussion.

Although the above referenced article discussed the subject of conical section playing surfaces in the abstract it did not provide data on the distances involved in real instruments. I'll fill in some of that here. The question to be answered is just how much lower can the action be if a conical section playing surface is used instead of one describing a cylindrical section. Figure 5 shows the geometry involved in schematic form. Figure 5A shows the curve of both cylindrical and conical surfaces at the nut end. Figure 5B adds the larger radius curve of a conical section surface at the bridge end. Figure 5C adds the curve of the bridge end of a cylindrical section surface. The distance marked d represents the maximum action optimization possible with the use of a conical section playing surface.

I'll put some real numbers to this in a bit but first let me note that d will be larger when comparing conical surfaces with short radius cylindrical surfaces, and smaller when comparing conical surfaces with longer radius surfaces. Extending this and putting it in practical terms, there will exist some cylindrical radius big enough that the advantage of a conical section playing surface will be small enough not to matter.

Calculating values for d requires formulae for deriving arc parameters. In the following, I am assuming a typical electric guitar fingerboard 1.625" wide at the nut and 2.207" wide at the 24^th fret. Using a 24 fret fretboard and the full width of the board in my calculations means the values calculated for d will be worst case – in shorter boards d will be smaller, as it will be if the E to E string span is used in the calculations rather than the full width of the board.

Comparing a cylindrical 16" radius section to an optimal conical section that has a 16" radius at the nut end, the value of d is 0.018" and the radius at the 24^th fret is 29.5". Comparing a cylindrical 10" radius section to an optimal conical section that has a 10" radius at the nut end, the value of d is 0.028" and the radius at the 24^th fret is 18.43". Although apparently small, these values of d are only meaningful when they are examined in the context of the entire guitar neck structure and the tolerances to which it is built. For comparison, consider that the accuracy of the straight edge I use when doing fret work is 0.0015" over 12". Consider also that the 0.028" value for d translates (at the 24^th fret) to approximately 1/2 turn of the 6-32 action adjustment screw typically used on bridges for Fender style electrics. Most guitarists would consider this to be a meaningful difference.

A typical electric guitar neck is hardly a rock solid structure, and it bends somewhat as the instrument is played. I wanted to get some idea of how much it bends during playing, to see if the calculated values for d were substantial when compared to the active dimensional range. To this end a number of solid body electrics were clamped to the top of the table saw, and a dial gage was positioned and zeroed over the center of the 12^th fret. A small weight (an arching plane weighing 12.25 oz) was placed at the 10^th fret and the deflection was read on the dial gage (Photo 1). Typical deflection was only 0.001", indicating that the neck structure is likely rigid enough during playing so that the calculated values of d are probably significant.

While on the subject of how significant the calculated values for d are, I should point out that probably the most meaningful evaluation should be whether or not guitarists can actually sense the difference in string action. I did no test to attempt to quantify that and could find no research on the subject but will assume for the sake of the remainder of this article that such differences in action are perceptible, but allow that they may not be.

Having established values for string action improvements that can be had by the use of conical section playing surfaces and ascertaining (or at least asserting) that these values are probably significant in practical application, let us consider some methods by which such surfaces can be implemented. By far the most common method is to use a conically shaped fingerboard. If the surface of the fingerboard describes a conical section and the fret wire is of uniform height, then the surface described by the fret crowns will also describe a conical section. This approach was pioneered commercially by Warmoth Guitar Products (warmoth.com) and is available in what they call their compound radius fingerboard option. Based on a cursory survey, as far as I can tell no large commercial instrument manufacturer offers necks using conical section fingerboards, but this approach seems to be popular among small shops.

There are some issues associated with this approach. Tooling used to shape the fingerboard to a conical section is a bit more involved than that needed to shape cylindrically, but not very much so (see Brian Woods' elegant method in American Lutherie #87). But by far the biggest issue is fret installation. Since the radius of the fingerboard surface is different at each fret position the frets must be hammered in or, if pressed in, a separate caul is needed for each fret (or each few frets). This may make this construction option prohibitively expensive for larger manufacturers and may partially explain why its use is not more widespread.

Another approach to producing a conical section playing surface is to use a cylindrical fingerboard and just level the frets to describe a conical section. Remember, for optimal action it is the shape described by the fret crowns that ultimately matters. Since so little material is removed using this approach there is plenty of metal in typical large fret wire to do this. And the use of a cylindrical fretboard makes it possible to press in frets using a single caul. This is the approach I use to achieve an approximate conical section playing surface in my own work and in fact is the only way I ever level frets. I'll describe two methods I use to do an ad hoc conical fret leveling job, and also the way someone else does it. All of these approaches start with a fingerboard with a relatively small cylindrical radius. None of these approaches produce accurate cones, but all of them produce approximations that are well within the tolerances of normal fretwork.

Before describing my ad hoc conical fret level approach it would be a good idea to see what it is we are trying to achieve. It became obvious that it would not be possible to take a meaningful photograph of the results of such a fret leveling job due to the great size ratio between the width of fret wire and the length of a guitar fingerboard. So instead of a photograph I’m using a computer generated solid model with similar dimensions to that of a guitar fingerboard, save the scale length, which is 8". This makes for a severely shortened fingerboard, which better shows how the fret tops look after approximate conical leveling. See figure 6.

As mentioned I use two techniques for doing ad hoc conical fret leveling jobs. The first is used for necks that can be removed from the guitar or that are fretted before the neck is installed. A long piece of stick-on abrasive paper is put down on a flat surface. I use the slab of 0.5" thick glass I have for most hand sharpening operations around the shop. The abrasive strip should be at least 4" wide and longer than the fingerboard. The neck is inverted, fret side down on the abrasive and rubbed back and forth in a rolling motion that approximately describes the surface of a cone. I start with the side of the fingerboard that is farthest from me parallel with the edge of the abrasive that is nearest to me (photo 2, figure 7 bottom). The neck can be rocked end to end, indicating that this portion of the fret top surface is not yet level. I rock it so the frets on the nut end are in contact with the abrasive and then push the neck away from me with a rolling motion so that it ends up on the other side of the abrasive strip with the near side of the fingerboard parallel to the far side of the strip (photo 3, figure 7 top). At the same time as I'm rubbing and rolling the neck back and forth I'm pressing down harder on the nut end of the neck during the ends of the stroke, and on the bridge end of the neck during the middle of the stroke. This sounds more difficult to do than it really is. Keeping in mind the picture of how the fret tops are supposed to end up when this operation is done, it is quite a simple matter to do an excellent job the very first time you try it. Progress is tested by attempting to rock the neck end to end on the abrasive. When you are done there will be no rocking on either side or in the middle.

The end result should be as pictured in figure 6. The top of the first fret at its center should be pretty much untouched, while the top of the last fret should be well ground in the middle to a long radius curve. Likewise the top of the first fret will be somewhat ground down at the ends but the top of the last fret will be pretty much untouched at the ends. Checking your progress with a straight edge positioned down the tops of the frets at each side and in the middle will help to gauge how much more grinding is needed. A final check of the radius of the top of the last fret and that of the first using radius gages is done. My ad hoc conical fret leveling jobs for Fender style guitar necks (starting with a 10" radius fingerboard) go from less than 10" at the first fret to about 16" at the last fret, more or less the same dimensions established by Warmoth. Notice that these are not exact numbers. This is an ad hoc technique so I never really know in advance the exact dimensions of the surface I’ll end up with. This turns out not to be a problem in practice. The nut will be marked with a half pencil resting on the first few frets, and the saddle (for acoustic instruments) will be marked using a half pencil taped to a stick which is slid across the last few frets, so all components will share the same approximate conical relationship.

One thing that may be helpful when first attempting this technique is pre-cambering the tops of the first and last frets using concave sanding blocks and then working with the first and last frets completely off the ends of the abrasive so that they are guaranteed not to be touched during the initial leveling process. I put a piece of duct tape on the glass under where the first and last frets are, as it is the same thickness as the abrasive strip I use. After all the grinding is done and the frets are level and their tops describe an approximate conical surface, a few quick passes taken with the fingerboard shifted up and then down so that the first and last frets get abraded too will blend them in with the rest of the frets perfectly. Advanced fret levelers (i.e., those that have performed this operation once) can probably forgo this special treatment. You can also add a kind of small circular scrubbing motion to your back and forth rubbing, which will make fast work of this operation. It takes me about a minute to do fret leveling in this manner.

As stated I use two ad hoc conical fret leveling techniques. The second is for use on an instrument with the neck attached. Here I use a flat length of rectangular metal tubing or a steel level longer than the fretboard and with one surface covered with stick-on abrasive. The leveling operation is the same as that described above only upside down, with the abrasive covered surface rubbed over the tops of the frets (photo 4). I find that, using the level or the short side of the rectangular steel tubing, it takes forever to perform this operation by simply rubbing back and forth. The circular scrubbing motion described above is pretty much mandatory. Progress is checked with the straight edge and radius gauges as described above (photo 5).

Jim Mouradian of Mouradian Guitars in Cambridge MA keeps the action optimally low on some of the best guitars in the Boston area. He shared with me his technique for approximate conical fret leveling. His technique is even more ad hoc than mine. Jim uses a short mill file for the job and begins by filing at the bridge end of the fingerboard, passing the file along the frets in this area in line with the desired conical surface, and leaning more heavily on the file when passing over the centerline of the fingerboard since more metal must be removed in this area to effect a longer radius camber here. He takes no radius measurements but instead relies on the look of the flats filed onto the frets. He also doesn’t take explicit straight edge readings of the fret tops here. Since the file he uses is short and a good deal of it is overhanging the fingerboard when working on this end, he can feel frets that need more file work by rocking the file over them. Once he is satisfied with the leveling job at the bridge end of the fingerboard he works his way up the neck, blending the surface established at the end into the rest of the frets, rocking the file to test for flatness as he goes. The first fret is barely touched by the time he gets there. Jim says that working on the bridge end of the fingerboard first also allows him to deal with the effects of any fingerboard end rise that may be exhibited on the instrument.

The techniques described here may sound quite imprecise and in fact they are. Still, they work well and they are no more imprecise than the methods generally used for leveling frets no matter what surface geometry is sought. Consider how difficult it is to try to level frets to describe a cylindrical surface using a small mill file, particularly when working at the fret ends. The file must be kept parallel to the surface of the cylinder, which means it hangs over the ends of the frets at an angle. It was just the difficulty of this process that led me to the idea of doing an ad hoc conical surface fret leveling job on the very first instrument I built years ago. Working in isolation and with an explicit understanding of the underlying geometry, I just assumed that an approximate conical leveling job must be the way everyone did it.

Considering again the difficulty in doing an accurate cylindrical leveling job, I wondered if a number of folks didn’t end up doing quasi conical fret leveling inadvertently, simply by filing along the natural "string lines" of the fingerboard. While I was at Jim Mouradian's shop I took the opportunity to measure the fingerboard and fret radii of a number of the instruments in for repair. These measurements were taken with a shop built radius gauge (photo 6).

Only instruments that had low action, clean-looking fretwork and not appreciably worn fingerboards were measured. Measurements were taken of the fingerboard in a number of locations, and of the first, 12^th and last frets. If the radius of the last fret was way off (as it often was) I backed off and measured the next to last fret. I didn’t take enough measurements to come to any sorts of conclusions, but some of the data are intriguing (Table 1).

The fret tops of the two new instruments and three of the used instruments did describe conical (more or less) sections although they were all pretty shallow. I don’t know how frets are leveled at the shops that produced these (or in the case of the used instruments, the shops that did any subsequent repair work) but the data do suggest at least the possibility that folks may inadvertently level to approximate conical sections. Since the fret work on all the instruments measured looked fine on careful visual inspection and none had obvious wear, the variance among the frets of most of the other instruments (Danelectro, Martin, Fender, Wal) may indicate something about the accuracy to which fret leveling can be done in general. I wonder if the Washburn wasn’t leveled using a long cylindrical sanding bar. And finally it is intriguing, especially in the context of whether or not guitarists can actually feel the differences in string action we are talking about here, to consider the Gibson. The frets on this one were leveled to approximately a reverse conical section, but the instrument played just fine, with comfortably low action (my subjective assessment). Clearly this is an area that could benefit from some formal research.

There is a lot of room for additional study on this topic, particularly aimed at ascertaining whether the lower action afforded by a conical playing surface is really detectable and desirable by players. But given that it is possible to perform an approximate conical fret leveling job with no more effort than it takes to level frets in a cylindrical fashion, it seems to me at least to be worth doing.