The other weekend I was introduced to a young lady who had made a Waisted Vielle. It’s not a particularly peculiar instrument, being a precursor to a modern Violin or Viola, but the construction for such a thing provides a great opportunity to look at how Boethian Proportioning is used.
Following Boethius’ musical theory, a Parisian named Jerome of Moldavia wrote a treatise for the tuning of the Vielle in the 13th century. He instructed that a Vielle should be tuned to d, G (gamma ut), g, d’, g’ for secular music. That is, a non-ascending arrangement of Fourths (4:3), Fifths (3:2), Consonances (1:1), and Octaves (2:1). The total range of this tuning is an Octave and a Sixth. So we’re going to use these proportional intervals to layout the body of our vielle.
Ignoring the remains of previous layout attempts on this sheet of paper, we have the length of the body laid out as the center line for our instrument body. This line will be divided into parts of seven so that we can divide it into a Perfect Fourth. That is, a segment that is 4 parts and a segment that is 3 parts (4:3). We have also marked the centerpoint of the line, which divides the length equally. This is a consonance (1:1).
We will then establish two circles (which it isn’t important to actually draw all of them). The “Major” element is drawn at the bottom of the body, and it is an arc with a radius of 3 segments. The second hemi-circle creates the top of the instrument and is the “Minor” element. This has a radius of 2 segments. Between the top and bottom, we have generated arcs with an interval of a Perfect Fifth (3:2). We’ll truncate the width of the body by drawing two more arcs from the “2” position at the bottom, with a radius equal to the midpoint to the first segment. This arc relates to the others.
With the radius of the dividers set to 2 of 7, we can lay out the vertices for where the waist will meet the existing arcs of the upper and lower portions of the soundbox. This is done first by crossing the existing arc at two points, and then with the same radius finding where an arc from those points will intersect.
Centered on that vertex outside the body, we can now draw in the concave curve that creates the “Waist” of this Waisted Vielle. Repeating this process on the other side will finish the outside form. Notice that this creates a compound cyma curve with an interval of 2:1:3, or an Octave and a Sixth.
Now we’ll have a go at placing the soundholes. For this we’ll divide the lower half of the body into parts of Five. The radius is two parts of five, or again that Perfect Fifth (3:2). We’ll center at the Midpoint of the body, 2 marks from the bottom of the body, then find our corners from radius at 2 marks below the midpoint, and 3 marks from the toe. Striking symmetrical straight lines between each of these vertices will generate the spring line for our final two arcs, and provide the decorative “back” at the same time.
We can find the midpoint of our symmetrical spring lines by marking the intersection of each corner to it’s opposite, then striking a line across the center. The intersection of the mid-line we’ve just struck and the spine of the instrument body gives us the vertex, and radius of the arc that will be used for finishing the sound holes. This will just happen to create a chord that is a Sixth (of a circle) on each half. It also happens to figure pretty darned close to a musical Major Sixth, which has that seemingly magical interval of 10:6.
With the extraneous lines and marks removed, a little faring of the arcs, and some decorative shadowing, we now have the final layout of our Vielle. And, this matches the tuning used for the strings. Now, those aren’t imperical tunings, but rather tunings based on the lowest tone that can be generated by the G (gamma ut) string. Everything else derives from it. What about the neck? Well, It seems most necks have a length that is a Fourth. If the whole body is 4 of a module, then the neck has a length of 3 modules. These are then overlapped by one module. The neck extends from the body with a ratio of 3:2, considering the overlap of one. Both a Fourth, a Fifth, and a Consonance.
Yeah. Greek Music Theory, who knew?