The “Four Circles” of the Violin: The Body Equations

Part 2

In Part 1, I introduced a qualitative generalisation of Kevin Kelly’s Four Circle system of violin design. In this section, I will derive the general mathematical relationships that define the body of a violin.

Terminology

A note on the use of the word “body”. From hereon, I will use the term “body” to refer to the section of the soundboard excluding the corners. I will use “violin” to refer to the acoustic soundbox of the instrument, which includes both the body and corners. This is to avoid a confusing equivocation.

A further note is regarding anything with a [subscript] “c”. Unicode doesn’t permit the use of the letter c as a subscript. Therefore, I will simply append “c” to a variable — ex. “hc” instead of “h_c”. This does not mean h×c.

The Four Circles

Recall that the violin body is defined by the four circles of the lower, centre, and upper bouts.The centre bout circles are separated by the waist-line/circle and are both tangent to the upper and lower bout circles.

Fig. 1: The four bout circles.

Fig. 2: The body waist dimension.

Let the radii of these four distances be defined as follows:

Fig. 13 The basic dimensions of the violin body.

The lower-bout radius (rₗ) is the reference radius. All of the other radii are functions of this radius.

Where kₓ is the body scaling-constant for the corresponding circle and is 0<kₓ≤1.

The Secondary Body Circles

Four circles (which I will call “minor” circles) of radius rₘ are then added in pairs to each of the upper and lower bouts. The origins of the minor circles lie on the horizontal diameter of the bout circles, and their edges are tangent to the edges of the bout circles.

Fig. 4: Minor circles shown in green.

rₘ has the following relationship to rₗ:

To complete the construction, a circle originating from edge of the bout circle and tangent to the two corresponding minor circles is added to the upper and lower bouts. Let these be the “major” circles. Note that the radii of the upper and lower major circles is different, being R₂ and R₁ respectively.

Fig. 5: The upper and lower major circles.

This completes the construction of the body, with the maximum lateral and vertical boundaries now established. Below is an illustration of the body outline once the circles have been trimmed, with the arcs retaining their corresponding colours.

Fig. 6: The complete body outline.

Relating R₁ and R₂ to the Body Constants

From a given choice of body constants (kₘ=1/2, kᵤ=2/3, etc.) the radii of R₁ and R₂ are fully defined — their size cannot be adjusted to maintain the same tangency relations. If I were to plot these circles, I can do so without using a calculator or ruler.

This is fine, but the radii R₁ and R₂ are of fundamental importance to the overall vertical dimensions of the violin — they define the height of the instrument. Violins are typically built to a height (~356mm for a 4/4 violin). At the moment, there isn’t a clear relationship between height and my choice of body constants that define R₁ and R₂. We therefore want to find a relationship between R₁ and R₂ and the body constants, and eventually use that to determine the height of the violin.

Fig. 7: The relationship between R₁ and R₂ and the height of the body.

Let’s start with R₁:

Fig. 8: Detail view of the lower-bout construction

We know that the the major circle is tangent to the two minor circles, and therefore the radius R₁ must pass through their origins. We also know that the origin of the lower major circle is coincident with the edge of the lower-bout circle. Hence, we can form the construction shown in Fig. 6, and use trigonometry to derive the following relation:

This gives us an exact relationship between R₁ and the choice of body constants. Now we apply the same method to the upper major circle.

Fig. 9: Detail view of the upper-bout construction.

The same tangency relationship applies in the upper-bout. We can therefore derive:

Calculating the Height of the Body

We now have two expressions for R₁ and R₂ in terms of the body constants and rₗ. As I mentioned, we need these two radii to calculate the height of the violin. We want to know: what height will result from a given choice of k-values?

Fig. 10: The height of the body.

As shown in Fig. 10, we need both R₁ and R₂ and another height, hc, to fully define the height (H) of the body.

hc is defined by the tangency relations of the four bout circles, independent of R₁ and R₂ (recall that for Eqs. 5 and 6 we isolated the lower and upper bouts). To calculate hc, we therefore need to consider the construction of the waist (Fig. 2).

Fig. 11: Detail of the waist construction.

From the dimensions in Fig. 11, we can construct two right angled triangles.

Fig. 12: Right-angled triangle construction in the waist.

From Fig. 12, we can derive the following expression for hc:

Substituting Eq. 8 into Eq. 7 gives us:

We now have an expressions for R₁, R₂, and hc, which allows us to calculate H. Recall, however, that H is typically one particular value for a given size of violin. We therefore already know what H should be. What we really want to know is how to make the body that height for our choice of k-values. If I want my body to be 356mm tall, how big do I draw the body circles?

Recall also that all of the k-values are scalars of rₗ. We therefore want to find an equation to relate H to rₗ only — all of the other circles will scale. We can find this relation by substituting Eqs. 5, 6, and 9 into Eq. 7:

This result means I can pick whatever k-values I want, and for a given height, I can calculate rₗ, and hence all other rₓ.

Using the Generalised Equations

We have now generalised the shape of a violin to five equations. The procedure for finding the dimensions to draw a body would look something like this:

1. Pick an H and some k-values (in Part 4, I discuss this in more detail).
2. Use Eq. 10 to determine rₗ.
3. Use Eqs. 1–4 determine all the remaining body-circle radii.
4. Get out your compass and straight-edge.

These five equations all me to draw any body without having to adhere to “nice” ratios. This gives me much more flexibility in my design choices, and allows me to create a much greater variety of shapes than would be possible from a specific equation.

In the next section, I generalise the equations for the corners to complete the construction of the violin.

Sections

Part 1: Generalising the Four-Circle System

Part 3: Corner Equations

Part 4: Analysing Historic 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