Hi there,

I've been a member of this forum for a while but I don't think I've
posted before.  I come from a science/engineering background and was
thinking about facing curve formulations the other day (circular,
eliptical, quadratic, cubic, etc.) and it occurred to me that there is
a simpler formulation for a facing curve that may be useful to people.

I've posted two files -- the first is a pdf which contains the maths
of the formulation (quite long, not necessarily very interesting, but
there nonetheless!), and the second is a spreadsheet allowing you to
calculate facings.

The first formulation is a quadratic equation where the facing is
forced to be a tangent to the table at the breakpoint (just like the
circular and eliptical models). It turns out that this produces almost
exactly the same facing as a circular arc (to with 1e-5) but the
equation is much simpler and you don't have to fit to find the
parameters (you still only need the tip opening and facing length to
define the whole curve).  I've called this the "restricted-quadratic"
facing.

The second formulation is a generalisation of the first -- it allows
you to change the exponent to anything you like. This can produce a
wide variety of facings, all of which also are forced to be a tangent
to the table at the breakpoint. I've called this the
"variable-exponent" facing.  I've included an example where I use it
to fit to a set of 7* Link measurements from Theo Wanne's website --
the fit is very good. Once you've found an exponent you like, you can
produce facing curves for different tip openings and facing lengths
just as with the restricted-quadratic facing described above (the
spreadsheet lets you do this).  This might well make it easier to
reverse-engineer facings from old mouthpieces for a variety of
different tip openings and facing lengths (rather like Theo Wanne's
"Ring" facing).

I should also mention that both of these formulations prevent you from
having any bumps or humps in the facing, unlike a generic cubic fit.

I don't think that any of this is going to change the world but it
might be useful to someone! What I'm not saying is that what people
have done before is wrong -- I'm just saying that there is a simpler
way of going about things that still makes sure that the facing meets
the table smoothly.

Anyway I hope someone finds it interesting...

Paul.

PS.  The solve routine in OpenOffice seems to have trouble fitting the
variable-exponent model to a given facing. The Excel solver works
fine, though!

Should have said - the files are under "Misc"

[ Attachment content not displayed ]

[ Attachment content not displayed ]

Thank you for the contribution Paul!  Your presentation is top-notch.

My gut says that it may be possible to show analytically that the radial formulation and the restricted-quadratic (parabolic) formulation are essentially the same in the region that we are interested.  It would probably take an order-of-magnitude analysis to drop some insignificant terms.  Mathematicians hate this but engineers do it a lot (tossing away small terms).

I took a quick look at the variable-exponent curves vs the elliptical curves I use.  The main difference I see is that radius of curvature of the elliptical curve gets shorter as the curve nears the tip of the facing.  On the exponent curve it does the opposite.  It gets flatter near the tip.  

In my thinking, the reed is thinner near the tip so it might like to flex more... and the thicker part of the reed near the table would prefer to flex less.  So I have been trying various elliptical facing curves.  I find they have the advantage of short facings for the high notes without resorting to short facing lengths.  But they are more resistant to low note response than radial facings of the same facing length.  

I have worked with a few players who squeak on elliptical facing curves.  So there is a potential disadvantage for some.  But these players also squeak on other curves with some resistance built in, like short facings.

I have read that some good clarinet facing curves are near parabolic, but a perfect parabolic curve is not the best.  I can see where exploring the variable-exponent might be the key to finding some good clarinet (and sax) curves.


      

Yes you can show indeed show that the restricted-quadratic and radial
curves are mathematically similar by a simple analysis of the two
equations (I didn't include it because I thought people might not be
interested!).  Here it is:

Radial:

l = M - (2Rf-f^2)^(0.5)
  = M - (2Rf)^(0.5) * (1 - f/2R)^(0.5)

Then use the binomial expansion (1+x)^n = 1 + nx + ...

l = M - (2Rf)^(0.5) * (1 - f/4R)
  = M - (2Rf)^(0.5) + (f^(1.5))/((2R)^(0.5))

If f is small (as it is in our case), you can ignore the last term so

l = M - (2Rf)^(0.5)

which is the same as 

l = M - (f/a)^(0.5),

if a = 1/(2R).

If you check the analysis sheet on the spreadsheet you'll notice that
the calculated R (E4) and the calculated a (G4) do indeed satisfy this
equation.

I'll have to think about the radius of curvature at the tip. It should
be possible to do something similar to reproduce that...

[ Attachment content not displayed ]

--- On Wed, 11/12/08, John Delia <bzalto@...> wrote:

>After looking at this, Tom, I am not sure it is worth the trouble.  j

Who is "Tom" you are referring to?

For guys with good math chops, this work is no trouble at all.  It is actually fun to apply it in areas you have not seen it used before.  Keep at it Paul!




      

[ Attachment content not displayed ]

Hi Paul,

Thanks for your posts.  I explored this myself some time back, and
came to similar conclusions, ie:

1. A radial is virtually identical to a parabolic curve within the
region applicable to a mouthpiece facing.  But obviously the
mathematical meaning of the two curves is different.

2. The parabolic formula is simpler than a radial formula, but this
doesn't matter much when the computer can instantaneously calculate
all the curve co-ordinates you could ever want.

3. A quadratic equation always yields either a parabola or a straight
line.  So Theo Wanne's "Ring" facing is really just a parabolic curve.
But correct me if I'm wrong!

4. The variable exponential curve looks interesting, perhaps someone
will find an application for this?

5. I've found elliptic curves good on alto and especially soprano sax.
 The tighter curve near the tip seems to make high notes a bit easier
and in tune.  I experienced no advantage over radial curves on tenor sax.

6. Clarinet facings are still a bit of mystery to me - I've never
found a magic formula for these, and published curves seem to be all
over the show.  But someone out there must of solved it!

Regards,
Andrew


--- In MouthpieceWork@yahoogroups.com, "paul_brimicombe"
<paul.brimicombe@...> wrote:
>
> Yes you can show indeed show that the restricted-quadratic and radial
> curves are mathematically similar by a simple analysis of the two
> equations (I didn't include it because I thought people might not be
> interested!).  Here it is:
> 
> Radial:
> 
> l = M - (2Rf-f^2)^(0.5)
>   = M - (2Rf)^(0.5) * (1 - f/2R)^(0.5)
> 
> Then use the binomial expansion (1+x)^n = 1 + nx + ...
> 
> l = M - (2Rf)^(0.5) * (1 - f/4R)
>   = M - (2Rf)^(0.5) + (f^(1.5))/((2R)^(0.5))
> 
> If f is small (as it is in our case), you can ignore the last term so
> 
> l = M - (2Rf)^(0.5)
> 
> which is the same as 
> 
> l = M - (f/a)^(0.5),
> 
> if a = 1/(2R).
> 
> If you check the analysis sheet on the spreadsheet you'll notice that
> the calculated R (E4) and the calculated a (G4) do indeed satisfy this
> equation.
> 
> I'll have to think about the radius of curvature at the tip. It should
> be possible to do something similar to reproduce that...
>

Friends:
(sorry by my poor english language)

Ok. all respect formulas; but will be interensting to see draws or 
graphical pictures (in millimeters - tenth an cententh of mms)
and to do more understanding the flanks of curve rails of mouthpiece, 
be saxophone mouthpiece or clarinet mouthpiece.

Some radial is not similar to oval curve, absolutelly - for more 
teeny that whether it-; because the radial diagram have one only 
point of origin and the parabolic or oval curve have two (or more) 
points of origin. The hit or goal, is encounter - to succes and 
error - the right place that points, in both instances.

I am graphical designer and i know the minimal of geometry elements, 
i not handling the mathematics formulas, how the majority of repair 
mans or luthiers; for the less in my country. But, will be very 
interenting to combine both theoricals solutions, isn't that so?

And, principally, can to see the diagrams or draws exactly as that.

For example: i have beautiful Excel diagrams, but all those are not 
exactly and visually proportional of the real facings (the curve 
rails of mouthpieces) or mouthpiece. Are correct the points in the 
curve; but is incorrect the real visual proportion and that can 
confuse us.

Hence, i am treating to encounter one way by to combine both system 
of measurement the curve rails in relation to the plane of the table;
but visually easy to understand. I wait that you understand me |:)=)

How tell Andrew, correct me if I'm wrong!

Fraternally yours

Silverio
(from the norwest of Argentine Patagonian portal)


--- In MouthpieceWork@yahoogroups.com, "andrewhdonaldson" 
<andrewhdonaldson@...> wrote:
>
> Hi Paul,
> 
> Thanks for your posts.  I explored this myself some time back, and
> came to similar conclusions, ie:
> 
> 1. A radial is virtually identical to a parabolic curve within the
> region applicable to a mouthpiece facing.  But obviously the
> mathematical meaning of the two curves is different.
> 
> 2. The parabolic formula is simpler than a radial formula, but this
> doesn't matter much when the computer can instantaneously calculate
> all the curve co-ordinates you could ever want.
> 
> 3. A quadratic equation always yields either a parabola or a 
straight
> line.  So Theo Wanne's "Ring" facing is really just a parabolic 
curve.
> But correct me if I'm wrong!
> 
> 4. The variable exponential curve looks interesting, perhaps someone
> will find an application for this?
> 
> 5. I've found elliptic curves good on alto and especially soprano 
sax.
>  The tighter curve near the tip seems to make high notes a bit 
easier
> and in tune.  I experienced no advantage over radial curves on 
tenor sax.
> 
> 6. Clarinet facings are still a bit of mystery to me - I've never
> found a magic formula for these, and published curves seem to be all
> over the show.  But someone out there must of solved it!
> 
> Regards,
> Andrew
> 
> 
> --- In MouthpieceWork@yahoogroups.com, "paul_brimicombe"
> <paul.brimicombe@> wrote:
> >
> > Yes you can show indeed show that the restricted-quadratic and 
radial
> > curves are mathematically similar by a simple analysis of the two
> > equations (I didn't include it because I thought people might not 
be
> > interested!).  Here it is:
> > 
> > Radial:
> > 
> > l = M - (2Rf-f^2)^(0.5)
> >   = M - (2Rf)^(0.5) * (1 - f/2R)^(0.5)
> > 
> > Then use the binomial expansion (1+x)^n = 1 + nx + ...
> > 
> > l = M - (2Rf)^(0.5) * (1 - f/4R)
> >   = M - (2Rf)^(0.5) + (f^(1.5))/((2R)^(0.5))
> > 
> > If f is small (as it is in our case), you can ignore the last 
term so
> > 
> > l = M - (2Rf)^(0.5)
> > 
> > which is the same as 
> > 
> > l = M - (f/a)^(0.5),
> > 
> > if a = 1/(2R).
> > 
> > If you check the analysis sheet on the spreadsheet you'll notice 
that
> > the calculated R (E4) and the calculated a (G4) do indeed satisfy 
this
> > equation.
> > 
> > I'll have to think about the radius of curvature at the tip. It 
should
> > be possible to do something similar to reproduce that...
> >
>

--- On Thu, 11/13/08, andrewhdonaldson <andrewhdonaldson@...> wrote:

3. A quadratic equation always yields either a parabola or a straight
line. So Theo Wanne's "Ring" facing is really just a parabolic curve.
But correct me if I'm wrong!

It has a linear and parabolic terms, so it can be a straight line, parabola, or anything in-between.  It is not constrained or restricted to be tangent to the table.  Your data set has to be good enough to ensure this will happen when using this type of general-purpose equation.  Theo's data set looks pretty good.  But few of us have those resources.  

I think it is better to work with mathematical curves that only require a few data points or inputs to generate an entire facing curve that is tangent to the table and has a good shape.




      

That's right -- the quadratic formulation I suggested forces the
facing to meet the table smoothly whereas a generic quadratic curve
does not necessarily do this.

I also agree that it's best to use as few parameters as possible ---
it makes things easier to work with and you have a better idea of what
each of the parameters actually does!


--- In MouthpieceWork@yahoogroups.com, Keith Bradbury <kwbradbury@...>
wrote:
>
> 
> --- On Thu, 11/13/08, andrewhdonaldson <andrewhdonaldson@...> wrote:
> 
> 3. A quadratic equation always yields either a parabola or a straight
> line. So Theo Wanne's "Ring" facing is really just a parabolic curve.
> But correct me if I'm wrong!
> 
> It has a linear and parabolic terms, so it can be a straight line,
parabola, or anything in-between.  It is not constrained or restricted
to be tangent to the table.  Your data set has to be good enough to
ensure this will happen when using this type of general-purpose
equation.  Theo's data set looks pretty good.  But few of us have
those resources.  
> 
> I think it is better to work with mathematical curves that only
require a few data points or inputs to generate an entire facing curve
that is tangent to the table and has a good shape.
>

Hi Keith, I don't understand what you mean by "anything-in-between". 
All parabola are geometrically similar - there are no curves in
between a parabola and a straight line that result from a quadratic
equation, just different sized parabola.

In other words, if a parabola begins at a tangent to the table, there
is only one parabola that will pass through the tip . . .

Regards,
Andrew


--- In MouthpieceWork@yahoogroups.com, Keith Bradbury <kwbradbury@...>
wrote:
>
> 
> --- On Thu, 11/13/08, andrewhdonaldson <andrewhdonaldson@...> wrote:
> 
> 3. A quadratic equation always yields either a parabola or a straight
> line. So Theo Wanne's "Ring" facing is really just a parabolic curve.
> But correct me if I'm wrong!
> 
> It has a linear and parabolic terms, so it can be a straight line,
parabola, or anything in-between.  It is not constrained or restricted
to be tangent to the table.  Your data set has to be good enough to
ensure this will happen when using this type of general-purpose
equation.  Theo's data set looks pretty good.  But few of us have
those resources.  
> 
> I think it is better to work with mathematical curves that only
require a few data points or inputs to generate an entire facing curve
that is tangent to the table and has a good shape.
>

--- On Fri, 11/14/08, andrewhdonaldson <andrewhdonaldson@...> wrote:

>>>Hi Keith, I don't understand what you mean by "anything in-between". 

A quadratic equation is Y=aX^2+bX+c.   If a=0 you get a pure straight line, if b=0 you get a pure parabolic curve.  But if a=.5 and b=.5 you get half a line and half a parabola.  This is what I mean by an "in-between" curve.  "c" just determines where it crosses the Y axis when X=0.

For a facing curve, a, b, c are actually determined by the data set and a least squares analysis.  They are difficult to tweek on your own to make a facing curve that is a little more open/closed.  Theo actually came up with a linear fit for facing length based on tip opening and also a, b, c based on tip opening (I going from memory here but you can check his site).  So he can get different curves based on tip opening.  But only one type of curve for each tip opening and facing length the way he uses it.

I like elliptical curves since I can deal in just 3 terms to describe a target curve.  The tip opening, facing length and the ratio of the major/minor axis of the ellipse that passes through the tip and facing length.  After a while, you get a feel for what different ratios can do.  Also, a ratio of "1" is the same as a radial curve.  

The same can be done with the variable exponent curve, though I have not worked with this curve.  


      

--- In MouthpieceWork@yahoogroups.com, Keith Bradbury <kwbradbury@...>
wrote:


> A quadratic equation is Y=aX^2+bX+c.   If a=0 you get a pure
straight line, if b=0 you get a pure parabolic curve.  But if a=.5 and
b=.5 you get half a line and half a parabola.  This is what I mean by
an "in-between" curve.  "c" just determines where it crosses the Y
axis when X=0.

Doesn't that just give you a different sized parabola?

Regards,
Andrew

I think a quadratic is only a parabolic if b=0.  Then, different values of a will give you different sized parabolas.  Otherwize, it is different sized and shaped quadratics.

--- On Sun, 11/16/08, andrewhdonaldson <andrewhdonaldson@...> wrote:

--- In MouthpieceWork@ yahoogroups. com, Keith Bradbury <kwbradbury@ ...>
wrote:

> A quadratic equation is Y=aX^2+bX+c. If a=0 you get a pure
straight line, if b=0 you get a pure parabolic curve. But if a=.5 and
b=.5 you get half a line and half a parabola. This is what I mean by
an "in-between" curve. "c" just determines where it crosses the Y
axis when X=0.

Doesn't that just give you a different sized parabola?

Regards,
Andrew




      

> 1. A radial is virtually identical to a parabolic curve within the
> region applicable to a mouthpiece facing.  But obviously the
> mathematical meaning of the two curves is different.

This is true.  The distinction is that while all radial curves are
approximately quadratic in shape, not all parabolic curves satisfy the
restrictions placed on the radial curve.  In the general formulation
of a quadratic y = ax^2 + bx + c the facing can meet the table sharply
(i.e. with a sudden change in gradient).  This is not true for the
radial facing, which is restricted to meet the table smoothly.  The
restricted-quadratic facing curve I suggested also meets the table
smoothly.

Paul.

Vintage ebonite alto saxophone and tenor saxophone blanks for sale on eBay!

--- On Tue, 11/18/08, paul_brimicombe <paul.brimicombe@...> wrote:
From: paul_brimicombe <paul.brimicombe@...>
Subject: [MouthpieceWork] Re: New facing curve formulations: restricted-quadratic and variable-exponent
To: MouthpieceWork@yahoogroups.com
Date: Tuesday, November 18, 2008, 3:36 AM










    
            > 1. A radial is virtually identical to a parabolic curve within the

> region applicable to a mouthpiece facing.  But obviously the

> mathematical meaning of the two curves is different.



This is true.  The distinction is that while all radial curves are

approximately quadratic in shape, not all parabolic curves satisfy the

restrictions placed on the radial curve.  In the general formulation

of a quadratic y = ax^2 + bx + c the facing can meet the table sharply

(i.e. with a sudden change in gradient).  This is not true for the

radial facing, which is restricted to meet the table smoothly.  The

restricted-quadrati c facing curve I suggested also meets the table

smoothly.



Paul.




      

    
    
	
	 
	
	








	


	
	


      

Yeah, I always assume that a useful facing curve will meet the table
at a tangent.  So when I consider parabolic facings, I'm thinking
about the one and one only parabola that will satisfy this condition -
which turns out to be almost exactly the same as a radial curve.  Is
that what you mean by a restricted quadratic equation?

Regards,
Andrew


--- In MouthpieceWork@yahoogroups.com, "paul_brimicombe"
<paul.brimicombe@...> wrote:
>
> > 1. A radial is virtually identical to a parabolic curve within the
> > region applicable to a mouthpiece facing.  But obviously the
> > mathematical meaning of the two curves is different.
> 
> This is true.  The distinction is that while all radial curves are
> approximately quadratic in shape, not all parabolic curves satisfy the
> restrictions placed on the radial curve.  In the general formulation
> of a quadratic y = ax^2 + bx + c the facing can meet the table sharply
> (i.e. with a sudden change in gradient).  This is not true for the
> radial facing, which is restricted to meet the table smoothly.  The
> restricted-quadratic facing curve I suggested also meets the table
> smoothly.
> 
> Paul.
>

One last word...

--- In MouthpieceWork@yahoogroups.com, "paul_brimicombe" 
<paul.brimicombe@...> wrote:

>  ...In the general formulation
> of a quadratic y = ax^2 + bx + c the facing can meet the table 
sharply...

However, it should be noted that a mild kink in this curve, where it 
meets the table, can easily be blended in during refacing.