are Based on the Golden Mean Spiral

This is a myth, and one that appears to have survived for a long time. I've never seen it written down but have heard it repeated any number of times. My suspicion is that in most cases this myth is based on a misunderstanding of just what the golden mean spiral is. When the spiral of a typical violin scroll is compared side by side with a golden mean spiral it is quite apparent that the two are vastly different.

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The golden mean or golden section is an irrational number usually represented by the Greek letter Φ (Phi, pronounced fie, rhymes with pie) and is approximately 1.6180339887. Historically this number was considered quite significant insofar as it described well typical growth patterns of both plants and animals. As such it was and still is considered by some to be the basis of a natural and “hard wired” aesthetic, that is, aesthetics that are not culturally formed, but that are universal in that they are derived from a basic ratio of nature. In fact the reason that we use Φ to represent this value is that it is the first letter of the name of the Greek sculptor Phidias, who was supposed to have made use of this ratio in sculpting the proportions of the various parts of the human body. Pretty heady stuff, no?

The golden mean spiral is based on a rectangle the sides of which are in the proportion of 1: Φ. If you divide such a rectangle such that one of the parts is a square, the other part will be another golden mean rectangle. And if you divide that rectangle in the same way so one of its parts is a square the other part will be another golden mean rectangle. For all practical purposes you can keep doing this division until things are so small you can't see them. From a mathematical perspective you can do this forever. Doing it 5 times will result in something like this:

And drawing a 90° circular arc in each square such that the central most corner of the square is the center of the arc approximates a golden mean spiral and looks like this:

Now we can compare this spiral to that of a violin scroll. I made an approximate copy of the spiral of a violin scroll done by Guy Rabut which appeared in a nice article he did on scroll carving which appeared in American Lutherie #52. First the golden mean spiral, sans construction lines:

And then the violin scroll spiral:

Even though the golden mean spiral above has fewer turns (2) than that of the spiral taken from Rabut's scroll, and even though each violin maker's scroll is a little different, it should be pretty obvious that the typical violin scroll is not based on the golden mean spiral. If you take the perspective that the spirals are spiraling inward, it is clear that the golden mean spiral closes in much more quickly than does the typical violin scroll spiral.

So how did the
myth that violin scrolls are based on the golden mean spiral start?
I can’t answer that, but my suspicion is that it was simply a
matter of incorrectly extending an explanation of aesthetics to a
different domain without actually checking things out, i.e. if the
golden mean can explain why certain aesthetic proportions in sculpture
are pleasing to us, and the shape of the violin scroll is aesthetically
pleasing to us, and there is such a thing as the golden mean spiral,
then violin scrolls must be based on golden mean spirals. But
even if the golden ratio is the basis of some aesthetically
pleasing natural and human made structures, the facts that humans
find the aesthetics of the violin scroll appealing and that it is *not*
based on the golden mean spiral are strong indicators that not all
appealing spirals are based on the golden mean. Here is a good
place to point out that the whole natural aesthetics thing is based
largely on myth, too. Naturally occurring spirals such as that
of the nautilus shell are said to be golden mean spirals. This
is decidedly false (the spiral of the nautilus describes a non golden
mean logarithmic spiral, see below), but a simple Internet search on
the terms “golden mean spiral” and “nautilus”
turns up an amazing number of websites, mostly of a New Age-y
orientation but some unfortunately from academic sources, that assert
this to be true.

Another possible explanation for the myth is that the construction of the violin scroll spiral can make use of the golden mean, and this got misconstrued to mean that the scroll follows a golden mean spiral. There is a superb treatise on the drafting of the violin scroll by Robert J. Spear which appeared in the third part of an exquisite three part article on drawing the violin in American Lutherie #95. Spear makes a compelling case that the golden mean can be used extensively in generating drawings of many parts of the violin.

But if a typical violin scroll does not describe a golden mean spiral, just what kind of spiral does it describe? For purposes of description spirals are generally divided into two major classes. Arithmetic (also called Archimedean) spirals increase in size by adding (or subtracting) a fixed distance from the pole per unit of turn, for example, 1” per each 90° of rotation. The result is a spiral that has bands that do not increase in width as it spirals out:

The other major class of spiral is the logarithmic spiral. Spirals of this class increase in size by multiplying (or dividing) the distance from the center pole by a constant amount per unit of turn. The bands of such a spiral increase in width by the multiplication factor as the spiral spirals out. The golden mean spiral is just one case of such a spiral. By changing the multiplication factor it is possible to create a spiral that expands quickly or slowly as it spirals out. But unfortunately for those that would use a simple algorithm for drawing a violin scroll, there is no multiplication factor that will result in a spiral that will even roughly approximate that of a typical violin scroll. When I copied Rabut’s scroll for the diagram above, I made use of a nice computer graphics algorithm for drawing approximations of scrolls by Taponecco and Alexa. Very simply, the algorithm approximates spirals by displaying them as tangent circular arcs, with their size relationships described by an arbitrary function. When I copied the violin scroll I did it using tangent 90° circular arcs of whatever size was needed to fit the section of the spiral I was working on. That the relationship of the lengths of the radii of these arcs could not be described by simple multiplication is a clear indication that the spiral of a typical violin scroll cannot be described by any logarithmic spiral. It is highly likely that a more complex function could describe the increase as the spiral spirals out but as my initial goal for this exercise was to find a practical formulaic approach that could be used by even non-technical luthiers I did not pursue this further.

Coates, in his book __Geometry Proportion and the Art of Lutherie__
posits that violin scrolls are based on a logarithmic spiral from
classic architectural theory known as the Ionic volute. The first known
mention of the construction of this spiral is in writings of Vitruvius,
thought to have been produced around 20 BC. Coates provides drawings of
many scrolls taken from real instruments and overlays those with a
suitably sized Ionic volute for comparison. None match, but some do
come close.

All of this of course begs the question of how one should go about drawing the spiral for a violin scroll. Perhaps the best answer I have heard comes from master violin maker Joseph Curtin. His suggestion is to draw a scroll that looks good and respects, to the degree desired, classical models.

Spirals and the Golden Section by John Sharp for the Nexus Network Journal
vol.4 no.1 (Winter 2002)

A very complete and wholly accurate
discussion on the math and history of the golden mean spiral.
Quite accessible to the non-technical audience.

Golden Ratio
- from MathWorld

A good discussion of the golden mean from MathWorld.

Piecewise Circular Approximation of Spirals and Polar Polynomials

The
paper by Taponecco and Alexa describing the computer graphics
algorithm I used as a model when copying Rabut’s violin scroll.

Ake Ekwall, "Volutes and Violins", Nexus Network Journal, vol. 3, no. 4 (Autumn 2001)

A very nice general discussion of the classic spirals that may have been used as the basis for instruments scroll.

Kevin Coates, __Geometry Proportion and the Art of Lutherie__, Oxford University Press (out of print).