Cremonese arching explained


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I was planning that I would save this for a letter to the editor of Strad magazine for this fall when I would have more time and make it complete with fancy equations, etc. There seems to be a considerable amount of interest in the subject right now, so I will go ahead and cast this pearl. Let me tell you the history of what lies ahead (poets can tune out right here with no loss of vital information). As a young graduate student in physics I was a theoretician, and quite naturally took a number of graduate math courses. The only math course I took in which I topped the PhD math students was The Calculus of Variations. I topped that course because I was the only student to get one of the two problems on the final exam, which was to calculate the shape of a catenary of a particular inhomogeneous chain, a chain in which the mass /length of the chain increased linearly along its length rather than being homogeneous. IIRC (this was 42 years ago!), the professor gave us the physics, which I already knew, and asked us to set up and solve the equations as a minimization problem. I still have had a fondness for that curve, although I couldn’t come close to solving that problem today.

When I started studying the violin, Torbjörn Zethelius was kind enough to send me his article on inside-first arching which I found very persuasive as to the method of construction. As I further studied arching, however, I became convinced that the crossed catenary approach leads to incorrect arch shapes. (Torbjörn, please do not take offense at what follows, I greatly admire your knowledge and I view what follows as an extension of your inside-first insight) Admittedly, I have never held a Strad, but the arching profiles resulting from the crossed catenaries just don’t look right to me. Imagine for a moment that one is in St. Louis, walking in a (very large) circle around the Gateway Arch. If you are looking in a direction perpendicular to the plane of the arch, what the arch appears to be is a catenary. As you walk around the circle, the arch you see no longer has a catenary shape, but appears as a more pointed curve, and that is the problem I have with the crossed-catenary approach. Looking at my Strad posters and the photos in the Johnson and Courtnall book, it is obvious that the top arch (generally) has a relatively flat section in the center. The crossed catenary approach leads to a more pointed top longitudinal arch profile than even that of the ordinary catenary, which also does not have the proper profile for the shape of the longitudinal top arch.

The only thing I have found in any of my books as to the shape of the top arch is that Heron-Allen describes it as the arc of a circle having a radius of three times the length of the violin body. I have also seen it described as a cubic spline curve. It is neither. I have examined everything I can find online as to the proper curve to describe the longitudinal top arch and have found nothing. As I looked at the posters and photos, I recognized the curve. It took only a trip to the jewelry department of a craft store to confirm my intuition. If what I am about to describe for the top longitudinal arch has been written before, it is in something I cannot find or some $800 book which I am not about to buy. I therefore claim to have discovered this all by myself!

The shape of the longitudinal top arch of the Cremonese violins (for which I have good pictures and curves) is that of a catenary of an inhomogeneous chain. The shape of longitudinal bottom arch for these same violins is an ordinary catenary, i.e. that of a chain having uniform mass/length. I am persuaded that the cross arches are the result of simple catenaries carved inside first, then transferred to the exterior. The edge flute carving leads to a curve that is for all reasonable purposes a curtate cycloid.

The inhomogeneous chain used to create the longitudinal top arch is composed (generally) of two sections of heavy chain attached to either end of a section of much lighter chain having a length of 100 to 175 mm, or 4-7”. The lighter the center chain, the flatter the curve of the center section of the arch. Having one uniform chain and two different sectional chains I can match exactly the longitudinal arches for about ten violins for which I have good pictures. Further, all the cross arches on my Strad and Guarneri posters have a long section of excellent match between a catenary and the actual curve.

The method of construction generally follows that laid out by Torbjörn. The arch shape is decided using the appropriate chain and a horizontal line on the wall the length of the distance between the end blocks. The high point is decided upon and that point marked on the chain (for most cases, this isn’t necessary for the back curve). The high point is located on the plate and drilled to a depth equal to the desired (exterior) height of the arch minus the maximum thickness of the finished (graduated) plate. The longitudinal arch is carved using the appropriate chain between the end-blocks as template. The interior cross arching is then carved using the plain chain as a template. The depth of the longitudinal arch at any point defines the cross arch catenary. Once the interior is carved, the arching is transferred to the exterior using the graduation punch set to the maximum desired thickness of the plate. The punch is set up with a support post having a rounded top. After the top shape has been carved and scraped down to the punch marks the plate will be of uniform thickness. The support post on the graduation punch is next replaced with one having a hollowed center to allow the convex top surface to fit securely and rigidly into the hollow as the punch forces the wood onto the post. The interior is then punched and scraped to the final graduation pattern.

A couple of observations: the marks on the top of the cello in the picture given by MD are totally consistent with the above procedure, as are the inside punch marks on the Strad violin. The punch is used twice, first on the outside, then on the inside. Secondly, there is no need to use a drill with a special end-point to begin the carving of a plate. Guarneri may have used one simply to save time. Thirdly, it is much quicker to make a few chains than to make templates for every desired arching shape.

Finally, there are some pathological violins that have an out of the ordinary shape. If you have the Hill book on the Guarneri family, examine the Andrea Guarneri violin of 1676, which has flat center sections for both top and bottom, only one chain required. Also, examine the Pietro Guarneri violins of 1708 and 1686 in which he places the high points of both top and bottom above the upper corners. You do this by having more of the heavy chain at the end of the violin where you want the high point to be located. For a truly flat center section, use the heaviest jewelry chain you can find with a center section of thread. You will find great entertainment in pondering arching profiles. For all of you pros who adopt this, my royalty fee is 10% payable immediately upon sale of any violin that uses the TJR copyrighted arching profiles. TJR is, of course, short for “Tony, Joe and Roger.” Have fun with this, I’m going fishing……….

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I would be very careful about drawing conclusions from photos that haven't been taken with such analysis in mind. In particular, it's impossible to make a useful photo of the back and top simultaneously, and most photographs are not consistent even regarding whether the top or back is favored, or neither, and by how much.

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I understand all the limitations of photos. I also understand that I can find nothing else that comes even close to reproducing the curves and is consistent with all the facts that Torbjörn has given us. When you have several Strad posters tacked to the wall and can move from one to the next reproducing the measured curves, I think you will not find my conclusions unwarranted. Chains are cheap, try it yourself.

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One other point I meant to make and forgot: The point of the ordinary catenary arch is that for uniform loading, the catenary arch converts stresses into purely compressional forces, i.e. no shear forces or bending moments within the arch. The inhomogeneous catenary arch should try to do the same thing, although it has to struggle a bit more. The inhomogeneous catenary is, in fact, made up of sections of ordinary catenaries which have matching slopes at the join points, but uniform loading would have to be replaced by a particular non-uniform loading to produce purely compressional forces.

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Roger. This is nice stuff you are doing. ...It's real good food for thought to say the least!

However...& I am not sure how it relates to your thoughts..... but almost certainly long arches on Old Cremona's have distorted a bit to form their present shapes over time compared to their original form. .....So what we see on Strad posters for instance does require a degree of interpretation....You will be aware of the war of forces involved.

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Hey Roger,

I wonder if this is not a case of overanalyzing....

Did you try to make an arch (preferably a whole strung up violin) following my method before you embarked on your mission to improve it?

There is a reason that I use two catenaries diagonally across the longitudinal arch, and not a single one along the center. Think about it.

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The inside arching method was first described by Euro Pelluzzi in a book called "Tecnica Costrutiva degli Antichi Liutai Italiani" (sold out, just in Italian).

Pelluzzi mentions a Cremonese manuscript in support to his theory that is called "LIBRUM SEGRETI DE BUTTEGHA - Regule et Formule Phoniche per Liutaro et Violinaro", once in the library of the Schollar Patetta Federico and now in the Vatican Library. This book is mentioned also in the last VSA Journal.

This manuscript is dated 1795 (when the golden period was over... ...), I think a more serious study about this manuscript should be done. For the measures it uses the "braccio Cremonese", mentioned by Sacconi.

The manuscript gives convex radius, which prompted Peluzzi to the conclusion that they were meant to be used on the inside. Pelluzzi's ideas are also based on the reflection of the light in concave mirrors. I find it a bit too complex to be feasible, but I may be wrong.

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Hi Torbjörn:

If someone is one of the usual suspects for over analyzing, well c'est moi! I did think about it a lot. And that thought process led me to conclude that I didn't like the shape of the curve when viewed from the side. Another thing that led me to the above is that there are distinct axes of strength in the wood. It seemed to me more understandable to have catenaries along directions of constant strength rather than cutting across them. I can't come up with any physical intuition for the behavior of a shell with directions of maximum rigidity to bending running diagonally across an axis of symmetry. One benefit I can see is that the diagonal catenaries should suppress torsional thick-plate modes of vibration, but I think that there are graduation patterns that will do the same thing while still having arching shapes that (to my eye) are preferable.

No, I haven't even finished the first violin. I can be challenged on any aspect of violin making on that one. I am finding that juggling fly fishing, bicycling and violin making has at least one too many activities for this old retired guy. Please understand that my whole approach to violin making is, in fact, one of over analysis. This is just entertainment for me. I have a whole series of ideas I am going to try in this very first one. My next wild idea is how to adjust the tone of a finished violin to be what a player wants. If my first violin is POS, it really doesn't matter. I'll give it to some kid and try to improve the next one. The real fun for me lies in trying to understand the whole system and the sensitivities to the various approaches. In my mind, arches and graduation are the foundation of the acoustics and that is why I found the catenary and your method so intriguing. But I just have to try my own ideas.........I do hope that you will understand that I am not being critical of you or your violins. I just find a great deal of satisfaction that I understand the arching approach when my chain perfectly covers the line of the longitudinal arches of a Strad poster

Hi Luis:

Well, I was sure I had never found what I was looking for in the small number of books available to me. I'll add one more to the "unavailable" list. When I see anything related to concave mirrors, I immediately suspect that parabolas will come into the discussion somewhere, and the parabola was at first thought to be the shape of the hanging chain.

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The violin was created during the renaissance in Italy, a time of alchemy and discovery, and I believe that the original violin designers were using their creative intuition to create a "form that follows function". They didn't use scientific thinking like people do now. That's why they were so successful. I think intuition is the most important tool in violin making, not scientific method.

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I believe that the original violin designers were using their creative intuition [...]

The problem for us is that where the old makers' intuition was programmed by the sorts of things that were known then, a modern maker's intuition is full of the flotsam of modern life.

We do not have the luxury of just 'doing what comes naturally', we have to choose what to put into the soup of our "maker's intuition".

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Well, Jonathan, that is certainly a common view. Does it also explain the shape of the violin, based upon the golden section, that is so prevalent? Is that simply intuition? I find little difference in using the golden section plus a pair of dividers to define in two dimensions the shape of the body, location of ff holes, etc. and using a simple two or three piece chain to describe a third dimension. Particularly so when we already have information which shows that the catenary is a curve very similar to the curtate cycloid which MD's laser photos show that one of the Amati's was using for cross arches. Search for posts by John Masters on this subject. While there is no doubt that a highly skilled craftsman makes a big difference, there is similarly little point in denying an unmistakable similarity to a simple, common and very strong shape, one that was discovered long before the violin.

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Did you know that the father of Strad's first wife was a mathematician?

Do you think they discussed proportions, curves and arches over a Chianti?

Recall that these might have taken place with Galileo's polemics still fresh on people's minds.

-----------------------

Later

What was I thinking, the above is impossible.

They would not be drinking Chianti.

------------------------

Even later...

I thought this was amusing considering the secretiveness of lutherie:

In 1671, Hooke announced to the Royal Society that he had solved the problem of the optimal shape of an arch, and in 1675 published an encrypted solution as a Latin anagram in an appendix to his Description of Helioscopes, where he wrote that he had found "a true mathematical and mechanical form of all manner of Arches for Building," He did not publish the solution of this anagram in his lifetime, but in 1705 his executor provided it as:

“ Ut pendet continuum flexile, sic stabit contiguum rigidum inversum. ”

meaning

“ As hangs a flexible cable, so inverted, stand the touching pieces of an arch. ”

---------------------------

CORRECTION

The mathematician was the father of Strad's first wife's husband!! See later post by MANFIO (#21)

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I just think that people in the renaissance didn't think the way we do today. Of course they were using sacred geometry, but if you look at the evolution of rebecs to viola da bracchio and to violins you can see the trial and error of the creative mind making small changes to finally reaching a perfect creation. I think that involved a lot of intuition about what the next change in their design would be. I think sacred geometry is inside all of our minds.

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Hi Torbjörn:

If someone is one of the usual suspects for over analyzing, well c'est moi! I did think about it a lot. And that thought process led me to conclude that I didn't like the shape of the curve when viewed from the side.

Hi Luis:

When I see anything related to concave mirrors, I immediately suspect that parabolas will come into the discussion somewhere, and the parabola was at first thought to be the shape of the hanging chain.

Hi Roger,

I do appreciate that you're finding inspiration in my method. Certainly, diagonal catenaries by themselves doesn't give a genuine Cremonese appearance when viewed from the side. I think that you forgot what the cross catenaries along the upper and lower bouts do to the side view. That's why I asked if you had tried my method. If you had, you'd find that the catenaries across the bouts changes the side appearance into a flatter longtudinal curve. In fact, you'd end up with a sideview that you might actually recognize from the posters!

I'm glad that the article makes people think. That's maybe the most important thing about it.

Roger/Manfio,

I was aware of the book that you've mentioned, and that it indicates that the inside was made first. But as you say, it needs more research. It may well be that Andrea Amati thought that he was making parabolas when he was working with catenaries. I should look into the meaning of parabolas in Andrea Amati's time.

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Yes, it's a pity that we can't study the old manuscript... I asked some friends in Italy and they just know the same as I: it's on the Vatican Library...

About Alessandro Capra: I'll quote Tom King and Andrew Dipper in their article on the last VSA journal:

"Galileo Galilei is recorded in history as the inventor of a proportional compass. But this claim was disputed in coutr by a Cremonese architect named Alessandro Capra (fig. 30). This very same

Cpra was the father of Giacomo Capra, the first husband of Stradivari's first wife Francesca Ferraboschi. He was also the grandfather of Antonio Stradivari's two stepchildren. So you can see that Stradivari had a direct association with Capra, who was a mathematician and geometer. Capra wrote a book called La Nuova Architettura Famigliare that contains a section on mathematics and proportion. This lin teween master is important. It wasn't that Stradivari was just a violinmaker working at the bench; the grandfather of his stepchildren was also a mathematician.l He would have had plenty of opportunities to discuss geometries with the man and certainly would have been aware of his book" (Page 79).

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