The facing curve
There is a lot to discuss about the formula used to define the facing curve. An obvious methode is to use a curve along a circle. It results in a curve with an unnoticable transition into the flat part of the table.
Theoretical determination of the curve of a bend reed is complex. There is a theory about bending of beams using the Euler–Bernoulli beam equation.
But it is complex to make a realistic equivalent of the bending reed.
At first the reed is not a beam with an even thickness over the whole length. The reed becomes thinner near the tip, so that would result in a more curved reed
near the tip and also a need of a more curved facing near the tip. However that is based on the idea of one single force evenly distrubuted over the whole length of the beam. The experience however of players with the shape of the baffle shows that that affects the distribution of the force over the length of the reed. A high baffle could result in more resistance. So the shape of the baffle tip could have effect on the ideal curve of the facing.
So it seems that we have to rely on practicle formulas to define the facing curve:
- The radial curve
- for x> x_facing=(x_tip-l_facing)
- f(x) = r-sqrt(r*r - (x_tip-x)*(x_tip-x))
- r = l_facing /sin(2*atan((h_tip/l_facing))
- This is a nice curve with a smooth transition to the flat part of the table. But it is not adjustable. It is fixed and defined by the tip heigth x_tip and the facing length l_facing.
- references: http://www.corrybros.com/facings/4585851806
- The elliptical curve
- The elliptical curve is in fact a modified radial curve. It uses a circle that is resized in x or y direction: radius_X = N* radius_Y.
- The idea is maybe that the N factor creates some adjustability of the facing curve. But that is limited. The facing can be made more curved by using a high value of N. But it is hardly possible to make it less curved than the radial curve when the tip opening is less than 10% of the facing length.
- The power curve
- f(x) = h_tip * (1 – x/l_facing) ^ p
- practicle values for p are 1,4 to 2 (= radial)
- The advantage of this curve is that is is adjustable with the power factor. Disadvantage is maybe that for some values the transition to the table is not as smooth as the radial curve. For the value of 2.007 the curve is almost equal to the radial curve.
I measured the facing curve of one of my bass clarinet mouthpieces. A Buffet Crampon EPE JM.
Then I compared the curve with three different curves. The power curve, the circle curve and the elliptical curve.
The power curve was almost a perfect fit using a power of 1,91. The elliptical curve can only be more curved than the circle or radial curve.
So that means that I have to modify my design an apply the power formula.