Golden rules

Violin maker Wolfgang Schiele argues that the influence of the ‘golden ratio’ on painting, architecture and lutherie supports the idea of a perfect form for instruments

Points of interest to violin and bow makers

FIGURE 4 The overall instrument circle, containing all constructed measurements and the inside shape of the violin

Both music lovers and art connoisseurs will agree that the shape of the Baroque violin is perfect. It survived all efforts at improvement during the 19th and 20th centuries, and even today’s luthiers are using the form to create not just musical instruments with exceptional qualities but also highly prized works of art. However, the question remains: how could this ‘perfect’ shape have been developed so successfully by the artisans of the early Baroque era?

In my efforts to (re-)construct the development of the violin’s shape and proportions, I have worked out what I consider to be an ideal design from instruments by Nicolò Amati, Andrea Guarneri and Antonio Stradivari, all dating from around 1650 to 1670. All measurements, starting with the overall length of the instrument, are in specific proportional relation to one another, and all relate to the so-called ‘golden ratio’ of classical architecture.

During the Renaissance, there were many studies of and theories concerning the proportions of the human anatomy. At that time, people considered the human being to be ‘in God’s image’ and therefore anatomically perfect. Consequently, architects aimed to develop a clear and harmonious relationship between all parts of a building, and to base them both on human dimensions (as divine proportions) and on the ideal proportions derived from the golden ratio (1.618 : 1, figure 1).

Because humans appeared to be ‘designed’ based on this ratio, and also because it can be found over and over again in the natural world, it seems logical that people find this ratio of proportions harmonious, even simply from an optical standpoint. For instance, all pentagonal five-petalled flowers, as well as hexagonal honeycombs and snowflakes, are directly related to the golden ratio. Consequently, buildings, pictures and sculptures based on these laws, whether consciously or unconsciously, are similarly balanced.

In 1509 the mathematician and theologian Luca Pacioli introduced the term ‘divina proportione’ for the first time as the title of a printed work, which he used to define the golden ratio. ‘Divina proportione’ for him stood symbolically with its three parts for the Holy Trinity. Leonardo da Vinci is said to have coined the term sectio aurea (golden section); however, others claim that this term only originated in the 19th century.

FIGURE 1 Basic construction of the golden ratio, which divides a given distance A–B into two sections – major : minor (1.618 : 1)

FIGURE 2 Equilateral triangle in a circle. Aline drawn halfway up the triangle divides the circle into minor (above) and major (below) proportions

FIGURE 3 In this figure, the diameter of the circle is divided into the Major (the side of the square) and two corresponding Minors

ALL DIAGRAMS WOLFGANG SCHIELE

THE CONSTRUCTION

Just four geometric elements are needed to construct the perfect violin model. These are the circle, the equilateral triangle (figure 2), the hexagon and a ‘square in a semicircle’ (figure 3), demonstrating the golden ratio on the diameter of a circle in a very special way. These elements have been known for 5,000 years, and in fact became the basis of a Greek philosophy on harmony and the cosmos. In the Middle Ages these geometric elements had symbolic meanings attributed to them. For instance, the circle stood for the heavenly or divine unity, while the equilateral triangle represented the Trinity on different levels (father, mother and son; the human being as body, mind and spirit; the universe as heaven, earth and humanity). The symbolic meaning of the square is the Earth (as opposed to the circle for the heavens). The hexagon – the double triangle or Star of David – indicates that every true analogy has to be applied inverted: as above, so beneath. It also represents the unification of opposites (male and female, positive and negative). In Christian symbolism it means perfection, completion, and the six days of the Creation.

THESE GEOMETRIC ELEMENTS BECAME THE BASIS OF A GREEK PHILOSOPHY ON HARMONY AND THE COSMOS

Needless to say, our current metric system did not exist at the time of the Amati family. Hence, for the construction shown here, the basic unit of measurement is the Cremonese ell (484mm), which had been in use in this northern Italian region since the eleventh century. It can still be seen carved in stone on the clock tower of Cremona. Quite clearly, there is no part or dimension of a violin that can be related even as a partial length of the Cremonese ell. But although it may now seem completely arbitrary, by extending the Cremonese ell by 20 per cent of its length, this new length (580.8mm) gives the overall length of the Baroque violin (and it should be said that this overall length varied quite a bit during that period).

Looking at figure 4, we can see that applying the square-in-a-semicircle construction for the first time gives us both the maximum width of the upper bouts and narrowest width of the C-bouts.

Then the maximum width of the lower bouts is obtained simply by doubling the width of the C-bouts. Here it is important to mention that it is the inner outline of the body that is being constructed – the crucial part for the sound.

This works as follows. The square within a semicircle can either be constructed or calculated mathematically, by dividing the diameter of the circle by the square root of 5, as shown here: Diameter of the overall circle: 580.8 : √5 = 259.7 : 2 = 129.8

Major (of the overall instrument circle) = |MP1 H | = 259.7mm

Minor (of the overall instrument circle) = |MP1 K | = 160.5mm = width of upper bouts

(the minor of the overall circle corresponds to the major of the triangle MP2-V-V1)

Minor2 = | K H | = 99.2mm = width of C-bouts

Minor2 x 2 = 198.4mm = width of lower bouts

Since this construction is for the internal dimensions, you will need to add around 7–8mm. This would give measurements of around 168 / 106 / 206mm, with a body length of 351mm (as measured with callipers).

Scroll length = 580.8 - 129.8 - 343.6 = 107.4mm

This construction results in a neck length of 106mm, according to today’s method of measurement: from the top edge of the top to the lower line of the upper saddle. This corresponds to the length of a Baroque neck, which is slightly shorter than the modern one. It should again be noted, though, that the length of the Baroque violin varied quite considerably.

Now we turn to the second part of the construction, in which the inside length of the body is constructed over the largest lower width using an equilateral triangle, which serves as the diameter for the body circle. The determination of the height at which the greatest upper and lower widths will be situated is determined using the basic construction of the golden ratio on the right-angled triangle depending on these widths.

4 x m2 = base length for equilateral triangle for the body

4 x 99.2mm =396.8mm x 86.6% =

343.6mm = inner length of the body = L on | A B |

Again, the height of the equilateral triangle can either be constructed and measured, or it can be calculated mathematically by multiplying the length of one side of the triangle by 86.6%.

The intersection of these two goldenratio triangles results in the position of the narrowest mean width. Violins by Nicolò Amati as well as most of his students show a deep hole in the back at this point, drilled with a funnel-shaped tool. This was sometimes even visible on the outside. Later, after the graduations of the back were finished, the hole was filled up with a small dowel. This has therefore always been assumed to be an important construction point. It would be nice to collect data on whether the dowel is actually situated at the narrowest mean width of the backs – so if colleagues have information about this, I would be pleased if they would share it with me. It is certain that it appears at the strongest point of the back. In Stradivari models, the narrowest mean width is situated slightly higher towards the neck.

Several subsequent steps are required to construct the linear expansion of the body’s central area, as well as the reversal points between the concave corner areas and the convex outline areas. These also require the use of second square-in-asemicircle construction.

Once these have all been constructed, the outline can be added using relatively simple means. Each of the three sub-areas of the upper, middle and lower sections consists of three circle segments that can be easily constructed with a compass.

FIGURE 5 Showing the first part of the construction (figure 4) in more detail without the scroll measurements

FIGURE 6 The construction to find the position and size of the f-holes

THE HEIGHT OF THE SIDES, BACK, TOP AND BRIDGE CAN ALL BE FOUND BY USING SIMPLE GEOMETRIC MEANS

The third construction section is used to calculate the position and the size of the f-holes (figure 6). It is essentially achieved through another square-in-a-semicircle construction, in direct continuation of the body circle. This f-hole construction has come down to us in several versions for various instruments by Antonio Stradivari, as we know them from the Museo del Violino in Cremona.

It will probably never be possible to prove that the following part of the construction (figure 7) was used as early as the 17th century. But it shows in an astonishing way that the height of the sides, back, top and bridge can be found by using simple geometric means. Even the position of the bass-bar and the soundpost can be determined. The constructed heights of the back, top and sides correspond to the common dimensions of historical instruments, just as the constructed positions of the soundpost and bass-bar correspond to today’s common and acoustically sensible positions.

To construct the cross-section of the violin body at the location of the body stop, we use the width of the body at this point. This represents the diameter of a cross-sectional circle. With the help of the radius, two equilateral triangles overlapping in opposite directions are obtained. The sides of the equilateral triangles that taper to points A1 and A2 are halved in length. By connecting the points found in this way and extending these distances up to the circle (golden ratio on the equilateral triangle in a circle), the height of the violin’s sides is determined. The intersection points of the horizontal sides of the equilateral triangles ( | C1 C2 | and | D1 D2 | ) with the central axis result in the points C3 and D3, and thus mark the highest points of the top and back respectively.

FIGURE 7 The construction to find the heights of the sides, back, top and bridge is based on a hexagon shape

ALL DIAGRAMS WOLFGANG SCHIELE

The arched shapes of the back and the top itself have no connection to the outline construction. For the sake of completeness, however, I would like to mention that the curvatures drawn here are so-called ordinary cycloids. They are created when a point is rolled along the radius of a circle. This principle was presented in detail by Quentin Playfair in the November 1999 issue of The Strad. I also use it in the construction of my instruments, as I am convinced that the bridge pressure can be dissipated most evenly in this way.

The time when I discovered or rediscovered that the shape and even the third dimension of the violin could be constructed by these described geometrical means, more than twenty years ago now, was a highly exciting one. Space restrictions meant that I had to cut short the descriptions of the construction steps, but I hope this article has allowed me to communicate my fascination for this dimension of violin making.