first numerical results...
Well, I guess maybe the code is OK; I'm not certain of that, but looking more closely at the values and the context, I think they might be correct. Anyway, here's a graph of what I'm getting:
(This plot was produced using GNU Octave, from data produced by the C function I showed in the last post.)
This is depicting the Y-position of the circle centers (the X-position is always 0), and the radius values of the circles. Y values are negative because everything is measured relative to the top surface of the fingerboard.
Dimensions are in mm (not that it matters). The horizontal axis (this would be the Z axis, the way we've been naming things), however, is just an abstract position number: 0 for the nut end, 100 for the bridge end. In reality, this distance will be something more like 1000mm.
Initial string positions (nut end):
(-10,0) (10,0)
(-30,-3) (30,-3)
Final string positions (bridge end):
(-44,0) (44,0)
(-45,-35) (45,-35)
These values are arbitrary, for testing the code. They are of the basic form similar to the real values (i.e., the radius starts out large and gets smaller), but the shape is somewhat exaggerated in order to help convince myself that the code is working. I.e., the shapes of the curves in the graph are also of the right general form, but they don't correspond to any realistic design.
As you can see, neither function is linear. We already knew this had to be the case.
More surprisingly, the radius value is not continually-decreasing: the lowest radius value occurs somewhere in the middle of the fingerboard length, not at the bridge end as I might have expected. This is the benefit of graphing things!
These results shed interesting light on the art of the master instrument-builders. If my hypothesis is correct, they essentially start with only the beginning and end points of the two curves, numerically-speaking. Draw a line connecting the two points, and the difference between that, and the final curve as shown in the graph, is the material they carve away, sand away, or otherwise shape in accordance with their art and proprietary templates. This shaping process is called "relief".
This is depicting the Y-position of the circle centers (the X-position is always 0), and the radius values of the circles. Y values are negative because everything is measured relative to the top surface of the fingerboard.
Dimensions are in mm (not that it matters). The horizontal axis (this would be the Z axis, the way we've been naming things), however, is just an abstract position number: 0 for the nut end, 100 for the bridge end. In reality, this distance will be something more like 1000mm.
Initial string positions (nut end):
(-10,0) (10,0)
(-30,-3) (30,-3)
Final string positions (bridge end):
(-44,0) (44,0)
(-45,-35) (45,-35)
These values are arbitrary, for testing the code. They are of the basic form similar to the real values (i.e., the radius starts out large and gets smaller), but the shape is somewhat exaggerated in order to help convince myself that the code is working. I.e., the shapes of the curves in the graph are also of the right general form, but they don't correspond to any realistic design.
As you can see, neither function is linear. We already knew this had to be the case.
More surprisingly, the radius value is not continually-decreasing: the lowest radius value occurs somewhere in the middle of the fingerboard length, not at the bridge end as I might have expected. This is the benefit of graphing things!
These results shed interesting light on the art of the master instrument-builders. If my hypothesis is correct, they essentially start with only the beginning and end points of the two curves, numerically-speaking. Draw a line connecting the two points, and the difference between that, and the final curve as shown in the graph, is the material they carve away, sand away, or otherwise shape in accordance with their art and proprietary templates. This shaping process is called "relief".
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