In summary, to obtain lowest possible action, the strings must be set to a radius somewhat flatter than the radius of a (cylindrical) neck because:
- Two circles of the same radius cannot comfortably “nest” within each other, therefore the string radius should be somewhat greater than that of the neck to begin with
- Because there is a difference (spread) between string spacing at nut and bridge, outer strings must effectively "wrap" around the surface of a cylinder to reach from bridge to nut
- Assuming a cylindrical neck of a single radius, the more you twist/wrap a string around the surface of a cylinder, the nearer that string gets to contacting the cylinder’s surface. Because strings closer to the centerline of the neck are less affected by this and thus can have lower action than outside strings, the pattern of the ideal string radius as measured from the bridge is even flatter/larger than a simple concentric circle surrounding the neck radius
- Compound radius necks with true conical grade won't suffer from this issue
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Imagine stacking two shin guards on top of each other; each has the same radius, so when stacked they create a crescent-moon shape. The lower shin guard is your guitar neck, and the upper guard is the radius of your strings. Look between the guards. You will see a crescent moon shape, meaning strings toward the center of the neck are higher than those on the edges. In other words; once you’ve raised the high and low E where they won’t buzz, the G & D are too high, if you are strictly adjusting everything based on the radius of the neck/shin guard.
To set each string at an equal height, radius should be a concentric circle around the neck; like toilet paper around the cardboard tube, your strings should be set at a greater radius than the radius of the neck itself.
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Easy enough; but further considerations cause the ideal string radius to be even flatter than that. The problem is, the above action/radius measurement method only holds true if the nut and the bridge of the guitar are the same width with all strings running parallel. For the purposes of this thought experiment, imagine a lead pipe is your neck. Create a bridge and nut for your lead pipe guitar by wrapping two metal bands around the lead pipe at some distance from each other. Place strings across these bands at equal spacing, each parallel to the pipe.
Then remember, on most guitars the bridge has wider spacing than the nut. To accurately depict what’s going on with a guitar on your lead pipe instrument, you need to move the strings farther apart on the metal band that represents your bridge. You’re now twisting the strings around the pipe, and the strings on the 'outside' (your low/high E) are the ones most in need of twisting. After a certain amount of twisting, the string will make contact with the pipe itself somewhere close to the “nut” of your instrument. To compensate for this, you need to raise the action at the bridge on the outside strings until they have adequate clearance above the surface of the lead pipe.
[size=0]*(If this part of the thought experiment is too difficult to you, try this in practice- put two rubber bands around a soda can and put a straight edge across them, representing your string. Now try to hold the straight edge in one place on the top rubber band, while moving it along the bottom rubber band. At a certain point the straight edge will contact the can.)[/size]
So the simplest way to raise the action is by increasing the radius of the metal band that represents the “bridge." The issue is by increasing the radius overall, now you’ve altered the action of the strings toward centerline of the pipe, which didn’t get as twisted around the pipe and were therefore already near the correct action.
To get the ideal action, you really want to lower the inner “D and G” strings back down near where they were, while holding the outer strings at the further distance above the pipe as measured from the “bridge” where they will not buzz. Doing so causes the radius pattern followed by the strings to be even flatter.
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Recap- String radius is flatter because a.)It should be concentric to the neck to begin with, b.)wrapping a string around a cylinder will cause it to become closer to the cylinder's surface, and c.)outside strings are most affected by 'wrapping.'
Considerations & Implications
- A truly conical neck radius is ideal, with a bridge radius somewhat larger than the neck radius at the high frets. This negates the greater action the outside strings require over high frets
- The above example is simplified and doesn't factor in any differences required in action height between larger/smaller guage strings, or irregular string tensions in different tunings etc... It's unlikely the final action settings will follow anything like a true arc.
I just made myself sound like a freak about action; and I was once, but I'm not now. I'm fine with just setting all my guitars to a medium-low action of ~2mm ballpark on all strings at the 12th fret and being done with it... Even my Soloist. But it does irk me when people say "string action should be matched to the neck radius" without quantifying further. Taken on the surface, that statement is not true if you want to get the most out of any neck, whether cylindrical or compound radius. In general, it should always be somewhat flatter than the radius.
In reality I think being pragmatic is the best approach (especially considering the latter points on string gauge/tension etc.) Do you like your action low? Simply tweak each saddle until it's as low as it can go without buzzing. Don't think you need to give a **** about using a radius gauge, there's more to it than that anyway. There.
...Still. I dislike the idea of bridges with fixed radii, even if I don't really think it matters
