The “Four Circles” of the Violin: The Corner Equations
In Part 2, I derived five generalised equations to describe the body of violin, based off Kevin Kelly’s Four Circles system. In this section, I will complete the generalisation of the violin outline by deriving expressions for the corners of the violin.
Terminology
Adhering to my terminology in Part 2, “body” refers to the area enclosed by the lower, centre, and upper bouts, while “violin” denotes the entire soundbox and includes the corners and the body. I will use “corner” to refer to the aesthetic projections on the sides of the violin.
Defining the Corner Structure
We start by drawing a circle, radius A₁, tangent to the waist circle of the body. Let this be the “major” corner circle (not to be confused with the major body circles).
As the body geometry is defined in terms of rₗ, let A₁ be expresed as:
From the origin of the major corner circle, we add four radial lines, two of which terminate in the upper bout and two in the lower bout. Let these lines be known as u₁, u₂, l₁, and l₂ respectively. Further, let u₁ and l₁ be referred to as the major corner lines, and u₂ and l₂ be the minor corner lines. The purpose of this distinction will be explained shortly. The distances of u₁ and l₁ to the centres of their associated bouts are xᵤ₁ and xₗ₁ respectively, while for u₂ and l₂ they are xᵤ₂ and xₗ₂ respectively.
The lengths xᵤ₁, xᵤ₂, xₗ₁, and xₗ₂ are defined in terms of a scaling constant (cₓ) and the radius of their corresponding bout.
As an aside, Kelly uses some of the intersecting arcs in the body construction to define the above distances. While convenient, this limits the flexibility of the corner geometry by constraining the dimensions in the x-axis to predefined body geometry. To maximise the possible corner configurations, I have chosen to create independent variables to define the major and minor corner line end points.
We then add a minor circle concentric within the major circle, radius A₂, which is defined as:
Where a major corner line and the minor circle intersect, we add a horizontal line. I will refer to these as dᵤ₁ and dₗ₁ for the upper and lower bouts respectively. Where a minor corner line intersects the lower/upper-bout circle, we add another horizontal line. These will be dᵤ₂ and dₗ₂.
For the avoidance of confusion, I am using major and minr to refer to the “functional hierarchy” of these lines in the construction, and not to their relative positions. More on this later.
Determining the Corner-Circle Radii
Using the vertical distance between dᵤ₂ and dᵤ₁, we can determine the radius (yᵤ) of the upper upper-bout corner circle, and in the same manner, use the vertical distance between dₗ₂ and dₗ₁ to determine the radius (yₗ) of the lower lower-bout corner circle. Let these circles be the primary corner circles.
We now add the secondary corner circles to each bout. The lower upper-bout corner circle, radius aᵤ, is added tangent to the upper-bout circle, and coincident to the intersection between A₂ and xᵤ₁. The upper lower-bout corner circle, radius aⱼ, is added tangent to the lower-bout circle, and coincident to the intersection between A₁ and xₗ₁.These radii are a function of their related primary corner circles:
The reason for using different radii for the “inside” and “outside” of the corners is to allow for the “hooking” of the corners seen in many violins. The centre bout often exhibits a more pronounced curvature in the corners than the upper and lower bouts. Further, the addition of these radii does little to complicate the geometry, as they simply have a linear relation to the primary corner circles, and do not require additional construction lines.
Relating the Corner Variables to Primary Corner Radius
This derivation is very long-winded, and (unlike the body circles) completely unnecessary to determine anything “useful” with regard to drawing the corners. Eqs. 11–18 are sufficient to define and draw the corner circles. I derived these relations because I wanted to implement this model in Excel, but they are not necessary fro drawing the construction by hand. For those who do not wish to read reams of algebra, feel free to skip to the next section.
I will start by considering the lower-bout corner construction to determine the vertical height yₗ, which is the lower-bout primary corner radius yₗ.
From Fig. 6, we can see that yₗ is dependent on the vertical components of the lower major and minor corner lines. Let the total length of the major corner line be denoted by Lₗₘₐⱼₒᵣ and the total length of the minor corner line by Lₗₘᵢₙₒᵣ.
*Note that A refers to the body constant, and is unrelated to Aₓ.
Substituting the relevant body/corner constants we get:
From Fig. 7, we can see that Dₗ₁ can be easily calculated by subtracting the radius A₂ from Lₗₘₐⱼₒᵣ:
Calculating Dₗ₂ is less straightforward, however. We will have to switch to a cartesian coordinate system and derive an expression for the intersection coordinate of a line (the minor corner line) and a circle (the lower bout).
In cartesian coordinates, any line may be expressed as y-y₁=M(x-x₁), and any circle by (x-A)² +(y-B)² =R². Note that these variables have no relation to any of the body or corner variables. To avoid (even more) tedious algebraic manipulations, we will solve the coordinates of the intersection of a circle and a line generally, and then substitute in the relevant variables.
We rearrange both equations for x as only the y-coordinate is of interest.
Solving for the y-intersection:
For convenience let y₁+Mx₁ -AM =C:
Using the quadratic formula:
Substituting C back in, we get:
Now that we have a general expression, we just need to substitute in the variables specific to the lower minor corner line and the lower-bout circle.
Starting with coordinates, let the origin of the lower-bout circle, (A,B) be the cartesian origin (0,0), R=rₗ, and the terminus of the minor corner line (x₁, y₁) be (xₗ₂,0). Substituting these values into Eq. 26:
We now need the gradient, M. From Fig. 7:
For space, I will not show the substitution of Eq. 28 into Eq. 27. Suffice that we have expressed the y-coordinate of the intersection purely in terms of the body and corner constants — i.e. in terms of known variables.
We can now calculate yₗ by subtracting the vertical distance Dₗ₂sin(θ₂) from Dₗ₁sin(θ₁). From Fig. 7, θ₁ may be expressed as:
We have already calculated Dₗ₂sin(θ₂) in Eqs.27–28. Therefore:
The same approach may be applied to the upper bout to calculate yᵤ, substituting the relevant body/corner variables. I will do this in an abbreviated manner.
The lengths of the upper major and minor corner lines:
The length Dᵤ₁:
Converting to a cartesian coordinate system, taking the origin of the upper-bout circle as (0,0):
Where:
Expressing θ₃:
And finally deriving yᵤ:
Blunting the Corners
The final step in the construction of the corners is to “blunt” or cut-off their tips. The most common method I have observed is to use a diagonal line. I think, however, a more geometrically elegant method (though perhaps not aesthetically) is to use the termination of the major corner lines as the origin for a circular arc to trim the corners.
I prefer the circle-method because it is simpler to draw. The terminus of the major corner line provides a reference point for the compass, which can then be adjusted to the desired distance without additional construction lines. Geometrically speaking, each major corner line runs directly through the tip of its corner, and therefore the “blunting circle” will be close to normal to the arcs of the primary and secondary corner circles. At the extreme (where the arc is coincident to the intersection), the blunting arc is normal to the two arcs. This, I think, is geometrically more elegant than using what is essentially an arbitrary line through the corner circles. Furthermore, I think an arc is more coherent with the overall visual language of the violin, which is defined by arcs.
This approach does have limitations when the corner is “short”, as the curvature of the blunting arc becomes quite pronounced. Additionally, the fact luthiers appear to employ alternative methods for blunting the corners means my model does not always map neatly onto different corner shapes. I think however that the simplicity of my method outweighs its limitations in certain fringe cases.
The blunting circles are defined by the relations:
Where:
Using the Generalised Corner Equations
The procedure for using the generalised corner equations is somewhat more complex than the body equations as the function of each variable in determining the shape is less obvious.
I propose the following as a reasonable procedure:
- Choose a value for b₁. This controls the corner “splay”. The larger the value, the more “open” the corner. Typically, b₁ is less than k, which results in a more “closed” shape, with the corners pointing more steeply towards each other.
- Choose a value for b₂. The smaller the value relative to b₁, the more slender the corner.
- Choose a value for cᵤ₁ and cₗ₁. These determine how “high”/”low” the corner sits on the bout. The smaller the value, the closer to the centre bout. Smaller values also increase the amount of “hooking”.
- Choose a value for cᵤ₂ and cₗ₂. The larger the value, the smoother the transition between the body and the corner.
- Choose a value for bᵤ and bⱼ. The smaller these values, the more pronounced the “hook” of the corner.
- Choose a value for bₗ and bₖ. The larger the value, the pointier the corner.
In the next section, I will discuss, using this generalised framework, the works of historic luthiers and their choice of body and corner constants.
Sections
Part 1: Generalising the Four-Circle System
Part 2: Body 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
