# The “Four Circles” of the Violin: Generalising the System

**Part 1**

The shape of the violin is a beautiful synthesis of visual and acoustic art. Its form has evolved with its music throughout its five-century history: first curvaceous and full, then made more slender and toned by musical virtuosity. At its core though, the shape of the violin has always been governed by geometry — the morphological changes only slight tweaks to a few ratios.

In this series, I derive a generalised series of equations to describe the shape of a violin body, building on the work of luthier Kevin Kelly and his “Four Circles” system of violin construction. Further, I will apply my generalised model to historical violins and investigate trends in the works of Amati, Stradivari, and Guaneri.

**Why Do We Want Generalised Equations?**

Geometric construction relies only on a straight-edge and a compass to create a shape. When using geometric construction, everything follows from the first line I draw. The beauty is that I don’t need to know any dimensions other than my initial one because using my straight-edge and compass to construct ratios will define all the subsequent dimensions. If I draw a line of 10cm and use geometric construction to divide it into seven segments, I know each one will be 1.43cm without ever having to measure 1.43cm on ruler.

The problem is, geometric construction isn’t much good when I want to define a fraction that is irrational, or even one that has a large denominator. It’s fine splitting a line into 7 segments to get a ratio of 3/7, but what if I want to express 33/77? It’s completely impractical to use purely geometric construction to draw a ratio like this.

At this point, I need to break out my calculator and ruler to define a distance for my compass. This fine if for something simple like a straight line, but what if the thing I want to draw is more complex? To know what distance I need to set my compass to, I need to know the *relationship *between the different elements I’m drawing. I need equations that describe these relationships.

But why do I need to *generalise* the equations? After all, a Stradivari-type violin is defined by a very specific set of ratios. Why can’t I just create a specific set of equations?

A specific series of equations is fine, but the clue to their limitation is in their name: they’re* specific*. If I wanted to draw an Amati Grand Pattern, I’d need to work my equations out all over again. The same is true of any other violin shape. I have no *flexibility*. What I want is a *generalised* series of equations that means I can put *any* ratios in I want into.

Furthermore, as I’m going to use my model to “reconstruct” historic violins, it pays to have a system that’s very flexible. It turns out that there is a lot of variation even among violins of the same luthier. More on that later.

**Generalising “Four Circles” System**

Kelly’s “Four Circles” system is a method of geometrically constructing a violin based off traditional Cremonese design. His method has many similarities to those used by other luthiers, historic and contemporary, and has the advantage of relatively simple corner-construction. In his lecture, Kelly discusses his system using ratios specific to Amati and Stradivari violins. Here, I will ignore specificity to qualitatively generalise his method, while adhering to the same basic construction principles.

Kelly’s system derives its name from the use of four primary circles to define the shape of the violin body: the lower, upper, and centre bout circles.

The “reference” circle is the lower-bout circle — the other body circles are defined in terms of the lower bout radius being 1 unit. All of these circles are freely variable as functions of the lower-bout radius. Though the correspondence is rough, we can see this defines the “shoulders”, “waist”, and “hips” of the violin.

I think it would actually be more appropriate to call this the “five circle” system as there is a “missing” circle not apparent in Kelly’s construction: the circle defining the distance between the two centre-bout circles, or the “waist” circle. Drawing the fifth circle isn’t strictly necessary as you can just use a line (as I have below), but it is important for generalisation that this distance is variable as a function of the lower-bout radius.

The four/five circles define the horizontal bounds of the violin body — none of the subsequent circles added will affect the width of the instrument. The vertical boundaries, however, are only established after the addition of further construction circles. One could see these circles as existing only to “flatten” the upper and lower bouts, as they serve no subsequent function in locating the corners. The positioning and radii of these circles will be discussed in much more detail in Part 2.

In acoustic terms, the violin shape is essentially complete at this point. The corners of a violin are primarily structural elements with aesthetic styling — they are not a resonant part of the soundboard. Changing the shape of central area will affect the acoustic profile of the instrument, all else being equal.

As an aside though, I do not think it is correct to attribute the increased sound projection and brighter timbre of Stradivari-type instruments over older violins to the changes to the soundboard shape alone. Changes to other design elements such as the arching, and advancements in materials and tools are also likely to have had an impact.

The construction of the corners is somewhat more complex than the body owing to the need to maintain as much flexibility as possible in the equations. While Kelly uses some of the intersections of the body circles to define the position of the corner construction lines, I have left these freely variable distances as a function of the bout radii. This is because while the body circles are freely variable, using them to position the corners results in too few degrees of freedom. Moreover, the corner geometry is somewhat independent of the body, as will be discussed fully in Part 3.

**Conclusion**

The above method generalises Kelly’s method to make it applicable to *any* violin body shape. The exact functionality and behaviour of this generalisation is likely opaque without the mathematics, but this will (hopefully) become clear is the subsequent sections.

For now, here is the resultant mathematical model in action:

**Sections**

Part 2: Body Equations

Part 3: Corner Equations

Part 4: Analysing Historical Violins

**References**

Kelly, K. (2019, January 14). *The ‘Four Circles’ System of Violin Making*. Retrieved from The Strad: https://www.thestrad.com/lutherie/the-four-circles-system-of-violin-making/8545.article