why not a cone?
I will make some drawings, to try to graphically illustrate why this is a complicated problem.
The four strings together are called the "string band". The strings are, of course, just about perfectly straight; they sag a little bit from gravity, an effect which matters almost not at all; and as they vibrate, the middle portion swings through a much wider area than the ends, an effect which matters quite a lot. But in this study, I am not (yet) trying to compensate for such effects: the strings will be assumed to be straight lines. Similarly, in real basses, it is often desirable to have more distance between the lowest string and the fingerboard (aka action height) than for the upper strings. Here, I will assume a fixed action height for all four strings, at least to get started.
The neck shape is characterized by two radius values: the radius at the nut, and the radius at the bridge. If the radius at the points in between changed smoothly, at a constant rate (i.e., linear), then the shape of the neck would be conical. But in actuality, a conical shape will not work, as I'll try to show.
The issue is that the string band subtends a different angle on the two circles. At the nut, the radius is larger (e.g., 9 inches), but the strings are close together (e.g., 1.75 inches), and so a very narrow angle of the large 9" radius circle, is subtended (or "covered"). Down at the bridge, the radius is smaller (e.g., 3 inches), but rather than getting correspondingly closer together, the strings actually fan out (e.g., to a width of 2.5 inches). Thus, on the smaller 3" radius circle, a much wider angle is subtended.
The trick is, this means that the strings are not parallel to longitude lines, on the theoretical cone connecting the two circles. The pathway on the surface of the cone, connecting the endpoints of a given string, is actually a curve. The straight path of the string between those endpoints goes "underground", through the body of the cone, not on the surface. In a practical sense, this implies that the string will buzz at certain fingering positions. A cone will not work.
If this result is not obvious, perhaps a thought-experiment, "reductio ad absurdum", will help. Suppose that the strings go from a realistically-small angle on the 9-inch circle, such as perhaps 10 degrees, to an extremely-exaggerated angle on the 3-inch circle: let's go for something above 180 degrees (fig 3-C). The pattern of the strings at the bridge end is thus almost a square or rectangle. The geometrical problem remains the same. Picturing the cone connecting the two circles, you can imagine that the outer two strings of the four, now start out on "top" of the cone at the 9-inch end, but then move down around to "underneath" by the time they get to the 3-inch end. Clearly, the pathway along the surface of the cone connecting the endpoints of one of these strings, is a long spirally-curved path, a bit like the stripe on a barber's pole. The straight line connecting the endpoints goes through the interior of the cone, nowhere near the surface except at the ends.
The only way a cone shape could work, is if the strings followed longitude lines of the cone. Then, the string band would subtend the same angle on both circles. With the larger radius being at the nut, this would mean that the strings would have to start out very widely spaced, getting much closer together towards the bridge. This would be just about unplayable. Or alternatively, if one wanted to preserve the correct arrangement where the strings fan out slightly towards the bridge, then the radius at the nut would have to be slightly smaller than the bridge radius, rather than considerably larger. The strings would be closer together at the nut, but they'd be in a significantly curved shape, just like they are at the bridge. This would be awkward to play as well.
In order to have the compound radius that we want, with nice flat strings near the nut for fingering convenience, but with tightly-radiused strings near the bridge, for bowing convenience: in order to have that, we cannot use a fingerboard which is a conic section.
The four strings together are called the "string band". The strings are, of course, just about perfectly straight; they sag a little bit from gravity, an effect which matters almost not at all; and as they vibrate, the middle portion swings through a much wider area than the ends, an effect which matters quite a lot. But in this study, I am not (yet) trying to compensate for such effects: the strings will be assumed to be straight lines. Similarly, in real basses, it is often desirable to have more distance between the lowest string and the fingerboard (aka action height) than for the upper strings. Here, I will assume a fixed action height for all four strings, at least to get started.
The neck shape is characterized by two radius values: the radius at the nut, and the radius at the bridge. If the radius at the points in between changed smoothly, at a constant rate (i.e., linear), then the shape of the neck would be conical. But in actuality, a conical shape will not work, as I'll try to show.
The issue is that the string band subtends a different angle on the two circles. At the nut, the radius is larger (e.g., 9 inches), but the strings are close together (e.g., 1.75 inches), and so a very narrow angle of the large 9" radius circle, is subtended (or "covered"). Down at the bridge, the radius is smaller (e.g., 3 inches), but rather than getting correspondingly closer together, the strings actually fan out (e.g., to a width of 2.5 inches). Thus, on the smaller 3" radius circle, a much wider angle is subtended.
The trick is, this means that the strings are not parallel to longitude lines, on the theoretical cone connecting the two circles. The pathway on the surface of the cone, connecting the endpoints of a given string, is actually a curve. The straight path of the string between those endpoints goes "underground", through the body of the cone, not on the surface. In a practical sense, this implies that the string will buzz at certain fingering positions. A cone will not work.
If this result is not obvious, perhaps a thought-experiment, "reductio ad absurdum", will help. Suppose that the strings go from a realistically-small angle on the 9-inch circle, such as perhaps 10 degrees, to an extremely-exaggerated angle on the 3-inch circle: let's go for something above 180 degrees (fig 3-C). The pattern of the strings at the bridge end is thus almost a square or rectangle. The geometrical problem remains the same. Picturing the cone connecting the two circles, you can imagine that the outer two strings of the four, now start out on "top" of the cone at the 9-inch end, but then move down around to "underneath" by the time they get to the 3-inch end. Clearly, the pathway along the surface of the cone connecting the endpoints of one of these strings, is a long spirally-curved path, a bit like the stripe on a barber's pole. The straight line connecting the endpoints goes through the interior of the cone, nowhere near the surface except at the ends.
The only way a cone shape could work, is if the strings followed longitude lines of the cone. Then, the string band would subtend the same angle on both circles. With the larger radius being at the nut, this would mean that the strings would have to start out very widely spaced, getting much closer together towards the bridge. This would be just about unplayable. Or alternatively, if one wanted to preserve the correct arrangement where the strings fan out slightly towards the bridge, then the radius at the nut would have to be slightly smaller than the bridge radius, rather than considerably larger. The strings would be closer together at the nut, but they'd be in a significantly curved shape, just like they are at the bridge. This would be awkward to play as well.
In order to have the compound radius that we want, with nice flat strings near the nut for fingering convenience, but with tightly-radiused strings near the bridge, for bowing convenience: in order to have that, we cannot use a fingerboard which is a conic section.
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