Recently, I checked out a copy of Geometry, Proportion and the Art of Lutherie (review here) and was inspired to analyze some historical lyras to look for design patterns.
Tracing old instruments is useful for drawing inspiration when setting out to make a new instrument. However, in each step there is a risk of introducing errors (when tracing the lyra, using the tracing the create a template, projecting the template onto a piece of wood). In addition, old instruments may be slightly asymmetric, either because they were built that way or because of dimensional change in the wood over time.
To begin, we normalize the images so that all have a 40mm height difference in string lengths. Although this changes the absolute dimensions, I am most concerned in this analysis with relative proportions. These photos are not distortion-free, but I selected the ones that are most straight relative to the top of each instrument. Each string is ~15mm apart from the next.
First, we create a bounding box around each pegbox. We immediately see integer and root proportions emerge: 4/3, sqrt(2), and sqrt(φ), from left to right. These rectangular proportions are commonly found in art and architecture.
Next, let’s draw some circles. Today we’ll focus on 2003.378, and we draw a circle with a 92mm diameter (shown in blue):
Generate another circle by multiplying the diameter by sqrt(φ): 92 x 1.272 = 117mm. This ratio nicely mirrors the proportion of the bounding box. The larger circle is tangent to the smaller one, and is shown in red:
A nice result is that the (right) red circle’s center is at the left edge of the pegbox, and vice versa. In other words, the two circles’ centers are one radius apart. The resulting shape is called vesica piscis.
Note that you can scale this analysis as you like, by beginning with the desired width of the instrument and choosing this as the radius of the red circle, computing the others according to the proportions here.