Here's a copy of a post I made in 2012, covering this subject. The subject was the geometry of leveling fretless fingerboards, and the conversation got into the differences between cylindrical and conical (compound) radiuses.
I think this discussion is getting confusing because you guys are missing one element. I'll try my hand at explaining the geometry here:
First, a normal, typical bass fingerboard is cylindrical. That is, it has the same radius at the nut as at the heel.
However, as described above, the strings aren't parallel. They are closer together at the nut than they are at the bridge. Even though the radius of the undersides of the strings may be the same at the nut as at the bridge, the underside of the strings actually describe a slightly different shape than a cylinder.
The strings going right down the center of the fingerboard will be true. That is, if you lay a straightedge down the centerline of a normal cylindrical fingerboard, it will lay flat. However, as you move the straightedge off to the side at an angle to the centerline, following the actual paths of the outboard strings, you'll find that the center will be high. The straightedge will rock on a high spot in the center, as if the neck were backbowed. The further off-angle to the centerline you go, the worse this condition will be. Also, the smaller the fingerboard radius (rounder) is, the worse it will be. If you cut the fingerboard to a pure cylinder, you'll end up with high-spot buzzing on the outboard strings, but not the center strings.
So how do you trim the surface of a cylindrical fingerboard to correct for this problem? The process is just like Musiclogic described above; you level-file right along the actual string paths, blending in between them. This process is often called "conical filing", but it isn't really forming a cone shape. That's what gets confusing. It's actually forming a slight "hourglass" shape. The radiuses at either end of the fingerboard are untouched. You are trimming away wood at the middle of the fingerboard (lengthwise), but only on either side of the centerline. So, the radiuses at the nut and the heel may be 12", but the radius at the 7th fret will be slightly less, like 11 3/4". I prefer to call this "hourglass filing" to minimize the confusion.
On a fingerboard that has been properly "hourglass filed" like this, a straightedge placed along all of the string paths will be dead flat. When the straightedge is placed parallel to the centerline, but off to either side of center, there will be a slight gap under the middle, a "relief". But, down the center, there's no gap. This is where owners often get really confused when trying to check the relief and adjust the truss rod. A high quality, hourglass-filed fingerboard can give you confusing relief readings if you don't understand what you are looking for.
If you want to get really technical, the real mathematical description of this hourglass-filed fingerboard is.....wait for it.....an offset hyperbolic paraboloid! I'm sure that brings back some frightening memories from your school days. Picture an hourglass, where you take the narrow waist and push it off to the side enough that one side becomes straight. That's what an "hourglass-filed" fingerboard looks like. The centerline of the fingerboard is right on that straight side. The further off to either side you go, the more hourglass-shaped it becomes. Try not to get a headache.
So, how does this relate to compound radius fingerboards? A "compound radius" fingerboard is just another term for a conical-shaped fingerboard. That is, the radius is larger at the heel than it is at the nut. The surface is a section of a cone. Basically, the more conical you make the fingerboard shape, the less need there will be for the "hourglass filing" correction. There is a point where the surface becomes conical enough that no hourglass filing is needed. However, most commercial compound radius fingerboards aren't that radical, and still need a little bit of correction to make them flat along the string paths.
What confuses things even more is that instruments built with compound radius fingerboards often also use a flatter radius on the bridge than on the nut. So, the underside of the strings are also on a conical shape, but they still aren't parallel, so the fingerboard surface still ends up needing some hourglass-filing correction. Don't hurt yourself picturing that one.
How does this relate to frets? Exactly the same way. A really good fretjob has this hourglass-filing correction as part of it. The typical process involves using a straight file or diamond stick or oilstone, working along the string paths and blending in between. Leveling frets with sandpaper on a radius block is only a way of roughing them in. It takes an extra step to make them really true for the strings.
Let me repeat: All of this hourglass-filing stuff becomes less noticeable and necessary as the fingerboard radius gets flatter (larger radius). A flat fingerboard doesn't have this effect at all. That's one of the reasons why manufacturers and builders like to go with flatter fingerboards; it's less complicated. I personally like to build my basses with very round fingerboards, from 7 1/4" to 4" radius. At a 4" radius, all of these geometry issues become much more obvious. That's why I've spent the time working with them and trying to understand them.
I hope this helps clarify the discussion?
Here's another post I wrote, a little further down, explaining more:
Yeah, upright bass fingerboards really exaggerate all of this geometry. I do a lot with hot rodded Baby Bass necks. Dramatic conical fingerboards, combined with a wide taper in the string spacing and relief cuts that have to rotate around the cone! Shaping out an upright bass fingerboard is a lot harder than most people understand.
You've got to keep the terminology straight though:
Doing an "hourglass" trim does not, by itself, change a cylindrical fingerboard into a conical fingerboard. You can do an hourglass trim on a cylindrical fingerboard, or you can do an hourglass trim on a conical fingerboard.
If you're changing a cylindrical fingerboard into a conical (or "compound radius") fingerboard, that would correctly be called a "conical trim". You are changing it into a cone, so that the radiuses are now different at the two ends.
Upright bass fingerboards start out as a conical shape, around 2 1/2" radius at the nut and 4" radius at the heel. To true it up for low action, it needs to be "hourglass filed", which makes the middle a little skinnier, but doesn't change the radius at either end. That would correctly be called an "hourglassed conical" fingerboard.
Now, most electric basses have cylindrical fingerboards, with the same radius on both ends. To trick it out for low action, you can "hourglass file" it. But that doesn't make it into a conical (compound radius) fingerboard. It just makes it into an "hourglassed cylindrical" fingerboard.
Of course if you want to, you can do both: First "conical file" it to convert it from a cylinder into a cone (that is, making it into a compound radius fingerboard), then "hourglass file" it to correct for the taper in string width.
Is that more understandable?
I know this all sounds complicated, but it comes down to the same basic thing: In terms of playability, the only thing that matters is the 1/4" wide strip of wood directly underneath each string. The rest of the fingerboard is just there for decoration. That's what Big B is doing with his technique with the jointer: Get the string paths straight, and then blend off the excess between them.
A few years back, I worked on some really radical instruments that had reverse compound fingerboards! They were 7 1/4" radius at the nut and 2 3/4" radius at the heel, if you can imagine that. Six strings, short scale length. The fingerboard geometry was crazy. They required a significant amount of hourglass filing, close to 1/8" off in the middle of the fingerboard. As I mentioned above, the more radical the radius, the more hourglassing is required. For those necks, I ended up making a special router fixture that cut six round-bottom flutes directly along the string paths. Once those paths were created, I would round off the spines between the flutes. It was surprising how hourglassed the final shape of the fingerboard was.
Here's the whole thread:
http://www.talkbass.com/threads/even-fretless-fingerboard-levelling-tips.905673/
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