LEARNING CURVES

How did the old Cremonese makers work out their archings? Gareth Ballard presents a detailed method for analysing the various curves – and creating your own from the same principles

Using a dial calliper on an instrument back

ALL PHOTOS, GRAPHS AND DIAGRAMS GARETH BALLARD

Many violin makers, myself included, obsess about the methods of describing the outline of their instruments, but strangely glaze over when asked about the arching. I can only imagine that this is largely because they think it is unfathomable – surely not dull. In my opinion, it is the shaping of the arching, and the subsequent thicknessing of the front and back plates, that really distinguish a maker’s ability to produce a good-sounding instrument. These are the factors that govern the way the wood will flex. In this pursuit we look to the Cremonese masters for inspiration, although the method they used was never recorded or passed on to later generations. Some believe the Italian plague of 1629–31 was largely responsible for this break in the passing-on of knowledge, and there is some fascinating debate on the subject; suffice to say, violin makers have been working blind ever since.

In this article I would like to share with you a technique I have developed over the past 15 years or more, that highlights the true structure of the Cremonese instruments and, critically, then allows you to copy the instrument’s structure authentically (which I will argue is not possible with current approaches).

When creating the arch of a new instrument, the standard method is to make arching templates from plans such as The Strad ’s posters. External templates are made for five cross-sections across the plate (front or back) and one for the long arch. The wood is then worked until a perfect fit is made when a template is held to the arching exterior.

The issue I have with this approach is that you will, inevitably, have to make your plate either thicker or thinner than the original to achieve the same flexibility, as every piece of wood is unique. Once you have created a perfect copy of the outside, the plate is turned over and wood is removed from what will be the interior until you reach the desired flexibility. So you cannot simply copy the graduations (thicknesses) of another instrument as this will not result in the same degree of flexibility. What’s more, its propensity to flex will even vary within one piece of wood. If the outside remains constant and you have different graduations (thicknesses) from the instrument you are copying, you will have inadvertently changed the constructional shape, and your instrument will therefore not vibrate the same way.

Figure 1 shows the cross-section of two instruments superimposed on each another, one red and the other black. The dotted line describes the midpoint between the interior and the exterior and represents the constructional shape of each. Copying the exterior of an instrument, as in the standard approach adopted since the Cremonese, does not mean you are copying the constructional shape if your thicknesses are different, as seen by the discrepancy between the two dotted lines.

The technique I have developed identifies the constructional shape of the original and allows you to maintain it, even though the graduations on your instrument are different. This reveals some remarkable insights as to how these cross-sections relate to one another, and how these shapes can actually be created from scratch.

THE MIDLINE

The important point to realise is that the constructional shape of an arch is not that of the exterior or interior, but the line that bisects these two surfaces – the midline (figure 2). If you copy the outside of an instrument and then remove wood from the inside to your desired flexibility, you will not have the same thickness as the original and therefore not the same structure. Violin maker Nigel Harris introduced the concept of the midline in his work on ‘End Arch Ratios’ (EAR), for the purpose of examining the relationship between the height of the arch at the upper and lower bouts relative to the C-bout, and expresses this as a ratio (bit.ly/3zDxNHo; PDF).

To identify the midline for the whole arching of an instrument from plans such as a poster, I enter the height measurements of an arch cross-section (interior and exterior) into an Excel spreadsheet, which instantly provides me with a midline value (using the formula: SUM then divide by 2). The data can then be displayed in chart form. Once this constructional line is identified, I estimate the thickness of my finished wood so I can draw in the external heights for my new instrument. This will not be the same as the heights of the copied instrument because mine will have different graduations, but it will keep the same midline.

FIGURE 1 Cross-section of two instruments superimposed on each another

FIGURE 2 Cross-section of a violin top at its narrowest point, indicating the midline (red)

THE CATENARY

I also plot a computer-generated catenary curve over the midline to highlight any distortions, and to provide a point of reference to compare one instrument to another. It also allows me to draw perfect curves for the midline of an instrument I wish to build from scratch. The catenary (from Latin catena meaning chain) is the natural curve made by a hanging chain and has remarkable structural properties.

In Stradivari’s time, architects would have been well aware of these qualities. For example, stone buttress supports were carved upside down, guided by the shape of a hanging chain. The idea that they may well have been used by the Cremonese in violin making has been presented by Torbjörn Zethelius (The Strad, August 2006 and November 2017). He suggests that a hanging chain may have been employed to guide the shaping of the interior of the plate. What is unique about my own approach is that I generate the catenary in the computer to do this rather than using an actual chain. This then allows me to compare the catenary curve to the midline rather than the interior.

Zethelius puts forward an insightful technique but other than the use of the catenary, my approach is quite different. I could not find evidence of catenaries in the diagonals of the convex section. Maybe a chain was used here but subsequent blending of the hollowing changed the result. I prefer to use Harris’s EAR analysis to examine and determine the bout heights, which would otherwise have been determined by the diagonal catenaries. What I like about the Harris approach is that, like the method I am proposing, it allows you to compare classic models and then apply your own informed suggestions. When I examine the long arch, it is purely to check that the midline flows smoothly.

WHAT’S IMPORTANT IS UNDERSTANDING THE SHAPES AND THE STRUCTURES WE ARE DEALING WITH

Once the curve is generated, you can make very precise adjustments to the length and height of the catenary. I do this 0.1mm at a time for the height and 1mm for the length, to achieve the closest possible match. Of course, this precision would not be possible holding an actual chain.

For the majority of the instruments I have examined, I have found the midline matches the computer-generated catenary very well, for both front and back (figure 4). I cannot safely say that this was a deliberate intention but I believe so, as in some cases it is so precise. I am not sure it even matters, as I am not trying to say this is how the Cremonese made their instruments, only that the catenary matches extremely well in many cases. I would also propose that, if the catenary indeed has such magical qualities, the place it will be employed the best is along the line of construction.

FIGURE 3 Cross-sections at the narrow point of (a) 1735 ‘Plowden’ Giuseppe Guarneri ‘del Gesù’ back, (b) 1649 ‘Alard’ Nicolò Amati back, (c) 1718 Antonio Stradivari ‘Maurin’ front, (d) 1676 Andrea Guarneri ‘Conte Vitale’ viola back

CREDIT

FIGURE 4 Comparing a hanging chain to the ‘Sauret’ Guarneri ‘del Gesù’ plans

The front and back have a remarkably similar constructional shape. Essentially the extra thickness in the middle of the back makes it appear more pointed. There is a beauty in the simplicity the catenary reveals: it seems to follow the midline so perfectly at the narrow point for its entire length, and gives a real sense to what is happening in the other sections.

It’s true that the adjustments made when working this way will often provide quite small corrections, but this advantage is by no means my principal motivation for its use. We have to understand that the violin in particular only just works, and most violins do not sound exceptional. If small differences did not matter, we would all be making superbsounding instruments every time. But when everything is just right, there is a perfect storm and a wonderful sound ensues. It seems clear to me that the creation of the soundbox is by far the most important factor, and this is why we have to be so careful to get it right.

I would not wish to imply that you cannot make a good instrument using the template method, as that is patently not true. What I hope this technique will offer is a clearer way to see and understand your work. What is important to me is understanding the shapes and the structures we are dealing with, otherwise we will always be playing catch-up with the great makers, copying without understanding or learning – and potentially not copying accurately at that. In the modern age we have technologies that allow us to go even further than they could. With this method, and simply using a spreadsheet to work out the maths, you develop a clearer understanding of the arching design and what great archings look like in their construction, whether flat, high, full or scooped. You can then go on to create your own design, confident that you are perfectly in tune with the greats of the past.

FIGURE 5 Spreadsheet for the 1743 ‘Sauret’ by Giuseppe Guarneri ‘del Gesù’, with data for the original and new model both represented. The catenary varies with the midline no more than 0.1mm. Also note the slight variation in the exterior of the original to the model.

CREDIT

FIGURE 6 Taking the measurements from the ‘Viotti’ poster

IN PRACTICE

With older plans, such as that of the ‘Viotti’ in figure 6, you may only have the external profile. The internal has to be calculated by subtracting the graduations given in a separate plan. More frequently these days, we are given actual CT scans so this is unnecessary. I make a measurement along the crosssection every 10mm and every 5mm for the narrow point. I lay a ruler along the bottom edge of the arch section and measure the heights using vernier callipers at the given points.

As the arch may not be symmetrical I find the average of the two sides. Often a depression on one side will create a bulge on the other, and this process cancels out such distortions. I enter the heights into the spreadsheet and using the Excel function (SUM then divide by 2) I determine the midpoint between the internal and external surfaces and show this in a chart as the midline.

I then plot the computer-generated catenary curve over the midline curve, trying to find the best match. I do this by fixing the height of the catenary to that of the midline, and adjusting the catenary length. At the narrow point, the midline adheres to the catenary curve for its entire length, turning up sharply just for the recurve at the edge. The other sections tend to follow the catenary for about two thirds of their length before blending into the recurve.

Having a visual representation of essentially the convex catenary areas of the arch (i.e. the actual constructional shape, not just the exterior) gives a remarkable insight into the design and how this varies between models.

In the spreadsheet I estimate my final graduations, and mark them on each side of the original’s midline to give me my new external heights. A thickness of 3mm in the wood is marked as 1.5mm above the midline and 1.5mm below for the interior. Once you have some experience, your estimations should bring you within around 0.1mm. If you need to make your graduations thinner than you estimated, you can turn the plate over and finish off from the outside to maintain the midline. For this reason, I recommend that you err on the generous side with your graduation estimations until you feel more confident.

Once the measurements for the five cross-sections and long arch have been represented in Excel charts and the midline indicated, I transfer the exterior heights of my new model to the wood. By new model I mean ‘the same midline as the original’ but with my own graduations.

Then I draw the cross-section lines on the back of the plate and mark the heights at 10mm intervals, 5mm for the narrow point.

YOU CAN NOW CREATE YOUR OWN DESIGN, CONFIDENT THAT YOU ARE IN TUNE WITH THE GREATS OF THE PA ST

FIGURE 7 Five crosssections taken from the front of the 1735 ‘Plowden’ showing both the original and new models (not to scale)

FIGURE 8 Violin back with external height measurements marked, taken from the Excel chart

Now the plate is ready to be worked. I clamp the dial calliper in the vice and hold the wood horizontally, to avoid a false reading, and upside down in order to see the numbers.

With the data stored in an Excel spreadsheet your efforts are nicely preserved for a repeat of the same instrument. What’s more, the spreadsheet can easily be used as a framework for a different model, by simply replacing the data.

When I have built up an idea of how these cross-sections should look, in terms of the relative heights and lengths of the catenaries, I can create my own instrument from scratch. This means I will need to determine the height and the length of the catenaries for the second and fourth cross-sections. I also look at the midline of the long arch to make sure the graduations I have estimated don’t lead to any buckling and that it runs smoothly.

Once I have decided the height I want the catenaries, I experiment with their length to manipulate the fullness, or scoopiness, they create in the arch. If you take a given height in the midline (not the external heights marked on the plate) of cross-sections 1,3 and 5, say 8mm, and sketch in a relief line around the plate through these points, as in Sacconi’s drawings, you can adjust the plans until that 8mm point appears in the right place in cross-sections 2 and 4.

I hope this method reveals some neglected issues and helps other makers improve their own decision making. I also hope this will encourage others to develop their own instruments, confident that they are genuinely informed by the centuries of experimentation that led to the ‘golden period’ in Cremona.