The Eminence bass
An occasional discomfort in my left-hand reminds me that this bass has too many strings, resulting in the top end of the neck being a little fatter than one might expect. I have a hunch that the fingerboard is not as conical as it could be; i.e. there should be a change of radius of the board's surface which is proportional to the spacing of the strings.
So there is scope for reducing the bulk at the top-end of the neck by reducing the top radius of the fingerboard, better aligned with the change in string-spacing.
This requires a bit of wood removal.
For reliability, in the hands of a fool, it requires a tool to control how the wood should be removed to ensure that the result will be a useful conical surface. Such a tool would have a cutter (router) travelling in a straight line resembling the line of each of the strings. It would perform a length of cutting, then traverse by an increment to repeat the longitudinal cut. This traverse would be dictated by the required radius at each end of the fingerboard.
A conic is described as a ruled surface, formed by straight lines joining the apex with a circular base. The bass strings are good examples of such lines, and it can be imagined that their extended paths will converge at a point above the instrument. As long as the tool tracks a set of paths from the base (bridge) radius to the apex, as described by the strings, a conic will form. For now I'll include some ball-park figures to get a feel for the task.
It is possible that there is more to be gained from such a router tool than a simple conic surface. It might also be allowed to hollow out the conic a little, to make more room for string vibration.
A suitable introduction for this adventure would be to describe the shape known as a hyperboloid. Perhaps Google can do it better than me.
Ambitious perhaps, but the hyperboloid is still a ruled surface, therefore could be achieved with a router tool running in straight lines. Only now, the lines are not perpendicular to the circumference of the base; as the angle moves away from a right angle, so a hollow starts to appear. In bass terms, if a tool path between the bridge radius and the top-nut radius was allowed to drift from the perpendicular, conical path, more wood would be removed from somewhere not quite in the middle.
Which all sounds a bit vague.
Where from and how much?
To explain "where from", I will assume that the tool is performing well, and has already formed the desired conical fingerboard, correct ratio and radii. It is at this point I will decide whether to call it a day, or start offsetting the tool path to remove more wood. I need to know where the wood will disappear, relative to the scale-length. Some Excel number juggling reveals that the focal point of any further gouging, that is, the point of maximum hollowing depth, is dependent on the ratio between bridge and top-nut radii. If the scale-length is chopped into 100ths, a percentage describes the distance of this focal point from the top-nut. Starting with the ratio of
- 1: 1 = 50%
- 1.5: 1 = 45%
- 2: 1 = 41%
- 2.5: 1 = 39%
- 3: 1 = 37%
- 5: 1 = 32%
- 10: 1 = 25%
For my purposes, 39% puts the biggest dip at 402mm from the top-nut, just an inch shy of the middle of the fingerboard. Good news and worth a try, I reckon. Alternatively I could reduce the ratio of my conic (but not after I've cut it!). Start again. I could PLAN to settle for a lower ratio conic of, say, 2: 1, with a view to gaining a hollow 2% closer to the centre of scale, but I feel the conic is the primary goal here , so I'll stay with 2.5: 1.
As for "how much", this is very much "wet a finger and test the breeze" kind of definition. It may be apparent by now that I could hollow out the board by different amounts from one side to the other, sympathetic to string pitch, simply by progressively reducing/increasing the offset angle of the tool-path.
From here onwards, the offset angle of the tool-path refers to the relative position of the tool-path at the bridge position and the top-nut position; i.e. the ends of the scale length of the strings. The figures show a 45° offset with dotted lines. So the tool crosses the top-nut position at a point 45° clock-wise with respect to the point at which it crosses the bridge.
- If I hollow out to a nominal depth of 1 mm for the 1st string,
- 1 x 4/3 = 4/3 mm for the 2nd string
- 4/3 x 4/3 = 16/9 mm for the 3rd string
- 16/9 x 4/3 = 2.4 mm for the 4th string
- 2.37 x 4/3 = 3.2 mm for the 5th string
This says that the fat string has over three times more displacement than the thin string.
On my bass, concern is mainly for the two fat strings, which are rattling on first few positions. But to consider leaving the treble side alone, and start hollowing from the crest of the board to the bass side, would be more technically challenging than beneficial.
A further thought is to bend the path of the tool to the depth required for the thin string. This allows the maximum depth to be wherever I want. Either central to the scale-length, or up to 55% of the scale-length to compensate for the one-side hyperboloid effect. More sums needed...
I would be interested to hear other opinions on this; as far as I know it's not been a precise topic before now, obscured within the art of the luthier.
For instance, allowing for string displacement is just one thought; there is also an apparent benefit to string height on the upper reaches. However, I don't need a lower action on the thin string because I still need to get my fat finger under it for pizzicato playing.