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Catenary curve in instrument backs
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sdantonio
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Joined: 09 Apr 2007
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Location: Bellingham, Massachusetts, USA

PostPosted: Mon Apr 09, 2007 10:03 am    Post subject: Catenary curve in instrument backs Reply with quote

Hi All,

I have been looking at old instruments in photos, technical drawings, as well as those comming in my shop recently.

On the top plate is seems that the catenary curve is reasonable symmetrical along the length on the instrument.

On the back it seems that the catenary is "to short for the back" and "skewed towards the heal". Another way of thinking of this is that the flat spots at the ends of the catenary where the scoup goes is shorter on the heal end of the back and considerable longer, 2 to 2.5 times, on the neck end.

I have noticed the same things appears to be the case on the archtop guitar vamily of instruments.

Is there an acoustic reason for this? Is there any fixed ratio that has been working out on acoustic principles for the flats on the two ends on the back catenary?
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Michael Darnton
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PostPosted: Mon Apr 09, 2007 10:24 am    Post subject: Reply with quote

A catenary is a very specific shape, not just any old symmetrical curve, and I can't imagine that you should find too many tops with a catenary long arch, except in the very lowest class of factory instruments from the late 1800s. I haven't noticed many on back long arches, either. Perhaps you're doing something wrong in your analysis?
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sdantonio
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Location: Bellingham, Massachusetts, USA

PostPosted: Mon Apr 09, 2007 11:00 am    Post subject: Reply with quote

OK, so when I posted a while ago refering to the long arch as a spline (which was definitely wrong) and I go jumped all over by everyone (apropriately so in this case) saying the long arch wasn't a spline it was a catenary. Now it's not a catenary any more. Ok, what geometric shape is the long arch? How should I refer to it to be accurate?

And getting back to my question, the scoop area on the neck end of the back seems to be considerable wider than at the heel end of the long arch. Why and is there a proper geometric ratio between the two widths?
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Michael Darnton
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PostPosted: Mon Apr 09, 2007 1:10 pm    Post subject: Reply with quote

Well, I don't know what you're looking at, so I don't know what the answer is, but I do know that shapes are specific--that is, an egg isn't spherical, even though it's generally rounded, and to call it spherical is an error. Catenary is the shape you get when you suspend a chain between two points. A catenary of a specific height and width is only one shape. Since the top and back aren't the same shape, even though they have about the same height and length, they can't both be catenaries. As to what they are. . . . that's for you to describe.
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moronsreign
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Joined: 26 Mar 2007
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Location: Idaho

PostPosted: Mon Apr 09, 2007 1:52 pm    Post subject: Top & back arch Reply with quote

If I hold a chain suspended, it makes a catenary, by definition. If I suspend it over a flat surface and allow a portion of the center to lie on that surface, I get an approximation of the long arch of the top of a violin.You could also make an argument that the curves at the ends are a cycloid, as Michael has presented.The center is not quite flat, but very slightly arched also, between thos two other curves at the ends.
The back is either a single catenary or cycloid curve, but I'm not aware of any consistent assymetry at the ends. I'm just an amateur so this is a personal observation.
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sdantonio
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PostPosted: Mon Apr 09, 2007 2:25 pm    Post subject: Reply with quote

Ok, lets look at a typical Amait
http://www.usd.edu/smm/Violins/Amati3366/3366AmatiViolin.html
You may have to blow the picture up a bit to see it more clearly.

If you look carefully at the back there is a convexed area to the back arch, at the ends there are 2 concaved areas, one at either end, i.e. the scoops.

The same is true with the top and is easier to see in the picture.

Lets not use any names for these curves except convexed area and concaved areas, or regions, of the back and top lengthwise arches respectively, if you want me to I can scan one in and work out the equation for the curve in polar coordinates, but it won't add anything to the question. But phrased this way we are not using any known named curves and so we will get completely past that.

On the top, the two convexed areas appear approximately to have the same length while on the back, the one at the button end appears to be longer than the one at the heel (or endpin end on cellos). Another way to phrase this is that the convexed area of the back does not appear to have it's maxima corresponding to the midpoint of the back but appears to be offset towards the heel end of the instrument.

My question is, is there a fixed relationship between the length of the button and heel concaved areas of the back and can those fixed relationships be related to the total length of the instrument back. i.e., is the button end concaved area 2 times that of the heel end concaved area? Or 2.5 times, etc.

What I'm getting at in the long run is; If I work out such an equation that defines the back arch of a “typical" instrument, say the Amait violin, and then I am asked to make up an instrument of a size that I do not have a set of quinta for (say a 1/4 size), can I run the equation for that lengthwise curve on a plotter to generate that curve and will it always follow that (and I'm pulling these numbers out of the air right now) the heel end concaved area is 1/5 the total length of the back, the concaved region is 3/5 and the button end convexed region is 2/5.

Secondly, will this equation defining the back or top lengthwise arches be directly scalable between different members of the violin family (violin, viola, cello, etc.). To the eye the lengthwise arch of the violin looks very similar in shape to that of, say, the cello with the obvious difference of a scale factor.

Or am I running into the old Praetorius argument that you should be able to take one perfect instrument, say the violin, and by scaling it, obtain the bass, tenor and alto members of the family. At this time however, I am interested in looking only at the top and back lengthwise arches.

Sorry for the confusion earlier. English is very inaccurate, but if I came out here with a equation with a dozen term of more, which is a pretty accurate description of the actual lengthwise curve for a Amati back I probably wouldn't get an answer at all.
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sdantonio
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PostPosted: Mon Apr 09, 2007 2:29 pm    Post subject: Reply with quote

the heel end concaved area is 1/5 the total length of the back, the concaved region is 3/5 and the button end convexed region is 2/5.

Of course I screwed up a line. Ment to say

the heel end concaved area is 1/5 the total length of the back, the convexed region is 3/5 and the button end concaved region is 2/5.
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sdantonio
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PostPosted: Mon Apr 09, 2007 5:11 pm    Post subject: Reply with quote

I know how to phrase the question now.

Lets define terms, minima, maxima (no definition needed), x intercept, the point where the curve passes through the x axis.

Take a violin back and scribe a straight line along the center from the heel through the botton, this by definition is the x axis.

There are exactly 2 x intercepts to the lengthwise arch, exactly 2 minima and one maxima.

Travel along the curve from the heel you pass through a minima, and then back to the first x intercept (the heal end recurve or scoop). Region I

Continue along and you pass through the maxima and back down to the second x intercept, Region II. Disregard this section, it it unimportant to the question.

You then travel from the second x intercept, through the secomd minima and out to the beginning of the button. Region III (button end recurve)

The chord for region III> chord for region I, simetimes considerable longer.

knowing that Amati set the position of his f holes according to an approximation of the golden proportion and that Strad did the same thing, is there an exact of approximate proportion between the chords of these two regiona and the total length of the instrument bach plate?


Is there a fixed ration that describes this relationship.
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Mikes
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PostPosted: Mon Apr 09, 2007 5:39 pm    Post subject: Reply with quote

sdantonio, if you go look at a bunch of long arches front or back you will quickly see that there is a lot of variation out there, even just within one makers work like Stradivari. Maybe some of the more experienced people could shed some light on how different arches effect the sound.

Mike S
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JWH
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PostPosted: Mon Apr 09, 2007 7:12 pm    Post subject: Reply with quote

sdantonio:

Two things: What you are looking at is distorted by virtue of camera angles as well as age related problems with tops and backs under stress. Your use of the term 'typical' rarely explains anything in the violin business and especially in this instance. Smile
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sdantonio
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PostPosted: Mon Apr 09, 2007 11:32 pm    Post subject: Reply with quote

A little technical note of curves, catenary and otherwise. Since some people were not overly happy with my use of the term catenary I will explain my use of the term in detail.

The name catenary curve describes a family of curves based on the cosh function, not just a single curve. It has also been described as the curve formed by allowing a chain to hang freely.

Lets first mention curtate cycloids since everyone is familiar with them here. A cycloid in general is made by placing a point of the circumfrence of a circle and rotating that circle through 360 degrees. The curve that the point traces out is a cycloid. The curtate cycloid is made by placing that point on the circles radius at some point p such that r>p>0 (i.e. between the radius of the circle and a radius of zero). As such, there are an infinite number of curves that make up the family of curtate cycloids. Only one curve of this family describes the violin crossarchings.

Catenaries likewise are curves that obey a cosh function and describe the chain as previously mentioned. The classical catenary is described as a chain where all the links are of identical mass. This is the only one described in the engineering and mathematics books because it is the curve that describes the cables in suspension bridges and other enigneering applications.

This however is only one member of the catenary family of curves. The others are derived by changing the weighting constants of the equation. In a real life model, imagine a chain where the links are not all the same mass. This will shift the minima of the resulting curve. To approximate the top of the convexed part of a violin lengthwise arch, arrange the heavier masses so that the chain minima (or the curve maxima when you flip it over) is at the bridge position. This shifts the curve slightly so that it is no longer symmetrical. To approximate the back, arrange the chain link masses so that the chain minima is slightly more shifted towards the heel of the violin, i.e. at the sound post position. For those of you with small chains, try it yourself. Hang one or more small paperclips on the chain to vary one of the liks masses and vary the tension (pull the endpoints apart or bring them closer together).

Using this, you can approximate the shape of any violin top to a great degree of accuracy by using smaller links and numerous masses. But the important thing to remember is these are all catenary curves. All members of the catenary curve family.
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Michael Darnton
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PostPosted: Mon Apr 09, 2007 11:41 pm    Post subject: Reply with quote

I post a respose, he replies:

sdantonio wrote:

The name catenary curve describes a family of curves based on the cosh function, not just a single curve. It has also been described as the curve formed by allowing a chain to hang freely.

Lets first mention curtate cycloids since everyone is familiar with them here. A cycloid in general is made by placing a point of the circumfrence of a circle and rotating that circle through 360 degrees. The curve that the point traces out is a cycloid. The curtate cycloid is made by placing that point on the circles radius at some point p such that r>p>0 (i.e. between the radius of the circle and a radius of zero). As such, there are an infinite number of curves that make up the family of curtate cycloids. Only one curve of this family describes the violin crossarchings.

Catenaries likewise are curves that obey a cosh function and describe the chain as previously mentioned. The classical catenary is described as a chain where all the links are of identical mass. This is the only one described in the engineering and mathematics books because it is the curve that describes the cables in suspension bridges and other enigneering applications.

This however is only one member of the catenary family of curves. The others are derived by changing the weighting constants of the equation. In a real life model, imagine a chain where the links are not all the same mass. This will shift the minima of the resulting curve. To approximate the top of the convexed part of a violin lengthwise arch, arrange the heavier masses so that the chain minima (or the curve maxima when you flip it over) is at the bridge position. This shifts the curve slightly so that it is no longer symmetrical. To approximate the back, arrange the chain link masses so that the chain minima is slightly more shifted towards the heel of the violin, i.e. at the sound post position. For those of you with small chains, try it yourself. Hang one or more small paperclips on the chain to vary one of the liks masses and vary the tension (pull the endpoints apart or bring them closer together).

Using this, you can approximate the shape of any violin top to a great degree of accuracy by using smaller links and numerous masses. But the important thing to remember is these are all catenary curves. All members of the catenary curve family.

So, I have corrected my repsonse, above. Screw it, then. If a catenary can be anything you want it to be, then there's nothing to discuss. [edit], why do I even bother to try..... I am SO out of here......
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sdantonio
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PostPosted: Mon Apr 09, 2007 11:57 pm    Post subject: Reply with quote

Hi Michael,

That is what I'm getting at. BTW, I read your article on circles. Excellent. I have actually used some of it in some design work on other instruments. What I'm looking to see if we can get at is what the violin top and back looked like after it was distorted.

The reason I tend towards catenaries and cycloids, rather than circles, was that both curves were approximated in the time of Amati and solved analytically (at least in the cast of the cycloid) during Strads lifetime. Both, being pretty astute and knowlegable about anything effecting their science (I consider them to be the first true violin scientists rather than just simple craftsmen) I assume they had a working knowlege of the curves and would apply anything that generated a beautiful symmetry (beautiful symmetry typically being equated to beautiful sound in those days as it still is today).

So, now, how do we proceed in deriving an analytical solution to the "perfect" lengthwise curve. I believe we have the best widthwise curves in the curtate cycloids. Another collegue of mine has proposed stacking up curtate cycloids to derive the lengthwise arch. Draw one, then draw another by rolling the circle along the first cycloid (instead of rolling along the straight line), superposition the two, then add in others as necessary. As necessary, varying the radius and displacement of the point. I think there must be a simpler solution using catenaries.

What I was looking for initially is the endpoints of the catenary as a function of the total body length. A ratio if you will, and then to fit catenaries from there.

You know, I should write more of these posts at home and fewer posts from work so I don't feel rushed and can think more clearly. It may help to make them more understandable.

And to respond to a previous reply, yes the shape of the arch mattes. Using the chain analogy again. Less tension in the chain and you get a steeper catenary which will give you a violin with more projection, but will be very diffacult to control. More tension gives you a flatter curve more like the baroque instruments. A softer sweeter sound, less projection, but easier to control.
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sdantonio
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PostPosted: Tue Apr 10, 2007 12:09 am    Post subject: Reply with quote

Michael Darnton wrote:


So, I have corrected my repsonse, above. Screw it, then. If a catenary can be anything you want it to be, then there's nothing to discuss. ----, why do I even bother to try..... I am SO out of here......


No, I have valued your feedback here for a long time.

Yes, a curve of the catenary family can be very flexable. But the trick is establishing the rules by which workable equation can be developed.

One of my professors used to tell us jokingly, there are an infinite number of equations that have not yet been discovered. If you want to become famous, pick one and name it after yourself. The trick is to pick one that actually describes something important (the Shroedinger equation, the Dirac equation, the Einstein field equations etc.).

The catenaries can be anything... within reason... the trick is to pick the right set of rules to transform it into something that matters. I'm trying to do that using mathematical modeling, but I also need the feedback from people who have built a whole lot more instruments than I have (and have built better ones).
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JWH
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PostPosted: Tue Apr 10, 2007 11:06 am    Post subject: Reply with quote

sdantonio wrote:
The name catenary curve describes a family of curves based on the cosh function, not just a single curve. It has also been described as the curve formed by allowing a chain to hang freely.


Contrary to what you've written, a catenary curve is not a family of curves but is a single curve only varied by scale with coordinants that don't change in relationship to themselves. The formula stays the same whether I 'pull your chain' or someone else does, rendering it slack or taut.
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