I apologize for the length of this post.
Right, how I made my extension piece.
This can be broken down into a few steps: measurement, calculation/design, building and tuning.
The first step was to measure all of the tone holes on my instrument: the measurements were frequency, diameter, depth and distance from the top of the instrument to the top of the tone hole (not the with the mouthpiece or neck but where the neck connects to the body). In hindsight I really only needed to measure the notes in the table below. All of my measurements were in Inches as that was the caliper that I had. All measurements should be to the hundredth of an inch or smaller.
All math modified from Acoustical Aspects of Woodwind Instruments, Revised Edition by C.J. Nederveen
After the measurements were taken and recorded in a data table, i created a ti-89 titanium graphing calculator program to do the math for me (program available if you have an 89, just ask). To start out I calculated the acoustical length from the equation Acoustic length=speed of sound/(4*frequency). If using inches the speed of sound is 13622.0472 in/sec and if mm it is 346000 mm/sec.
I then calculated the length to hole with the mouthpiece attached
D1=.325*di of hole*(di hole/(2*g)^.39-1)
g=2^12-1
lah=(.25+di hole+D1)*(di bore/di hole)^2-.45*di bore
z=.5*g*sqrt(1+4*lah/(g*acoustic length))-.5*g
D2=(-3.3*10^-4)*frequency/(speed of sound)
LL=acoustic length-z*acoustic length
To calculate to length of the mouthpiece
Create this data table in Microsoft excel or similar
Numbers
note frequency number Actual Length LL Mouthpiece Length
A 196 1 (=LL-Actual Length)
G# 185 2
Thumb F 155.56 5
E 146.83 6
low A 98 13
low G 87.31 15
low F 77.78 17
Lowest E 73.42 18
Run a linear regression for frequency vs. mouthpiece length
Run a linear regression for number vs. mouthpiece length
Write the equations down and weight them so they are more closely related to the lowest E
Calculate the Mouthpiece length for the next notes, table below
note frequency number LL Mouthpiece Length Actual Length
extension D# 69.3 19 (=LL-Mouthpiece Length)
extension D 65.41 20
extension C# 61.74 21
extension C 58.27 22
note B 55 23
Calculate the relative lengths of the pieces
change distance
E flat to D# (=Actual length first-Actual length second)
D# to D
D to C#
C# to C
C to B
Design
I created my extension piece in five parts, one for each tone hole and another because i made a female joint where i should have made a male joint. All of the pieces will have the same interior and exterior diameter as your instrument and it is advisable to make all the tone holes the same size to facilitate finding keywork and pads (all of mine are .85 in. diameter). Skills on the lathe are highly recommended in this next part, else you can have someone else make the part for you and you can tune it.
The lengths that were calculated earlier are to the top of the tone hole. The first piece should start as the length from E to D# holes, minus the length of the top of the E tone hole to the start of the bottom tenon. Then that piece needs to have designed into it a female joint that is slightly larger in diameter(by .001-.002 inches) than the diameter of the bottom tenon cork, slightly compressed by the caliper. Then add the radius of your tone hole length to the bottom of this piece and that is the center of the tone hole. draw in the tone hole, add about an inch and a half to the piece and create a male joint. continue with previous steps, until the last piece is reached. Add the remaining length to hole where B should be and create a male joint to fit into the bell. (I made a female joint and it took me a week to fix it).
Building: I had access to a 3D printer, so i used that for the majority of my model. A lathe and drill press would work fine, or sending plans off for someone else to make it.
Tuning: This piece will be slightly out of tune, it is a fact of life. To sharpen the note sand the top of the male joint down as well as the tenon in question to fit. To flatten-pull out
Final Shaping and such
there needs to be an eighth inch deep, quarter inch wide groove running around the middle of each tenon to attach cork. This is best done with a hand file or lathe. The tone holes need to be leveled then cut to the proper shape. The outside and inside need sanding, any sharp corners need a slight bevel and if it is made of abs plastic it needs painted, proper clarinet wood needs oiled and other woods, oiled, waxed, or otherwise waterproofed, not sure on the best method.
*most mistakes can be fixed with super glue
Pictures: