My model, arches, and graduations are all based on "Golden Proportion" (.618) geometry, for centuries designated as a phi (F). Its presence can be found in the sacred art of Egypt, India, China, Islam, and other civilizations. Also many aspects of nature like organic life, the human body, lightning, and sound evolve through the laws of Golden proportion. The Cremonese used this knowledge in the construction of their instruments. incorporating the same principles in all my instrument's archings and graduations gives them a unison of voice that when played together is rarely heard.
"There must be no decoration, only proportion." Quote from St. Bernard of Clairavaux, who inspired the architecture for some of the most incredibly resonant acoustic twelfth-century churches.
The thicknesses or graduations of the top and back plates are very influential on the sound. Violinmakers start out using widely accepted standard graduations. With experimentation and experience they proceed to change their graduations by to thinning or thickening certain areas of the plate for desired acoustic effects. Most times separate schemes are developed for graduating the plates of different instruments.
I took a different approach. I graduate my top and back plates using consistent patterens based on Golden Proportion geometry. This allows for uniformity of plate flexing and optimum velocity of vibrations throughout the plate. Of course the density and tuning of the plate is always taken into account, and the whole pattern is made geometrically thinner or thicker accordingly.
There are many books, some of which I have listed at the bottom of this page, that have been written on the Golden Proportion and how it was used to construct the instrument form. But until now there was nothing written about how to geometrically construct arches with the Golden Proportion. I am sure there are many geometric and mathematical paths to achieve the same result. This is just a condensed description of the way I design my arches.
6/11 is the classical proportion that dictates where the bridge placement or Menzure of a violin will be. 16mm is the classical violin arch height
The line where the arch begins is set up by F ^5 the distance from Menzure to plate edges and from center line to plate edges.
These lines will determine the center of the arcs that comprise the upper and lower bout center line. For the top plate mark the F line from Menzure to arch line, and F^2 for the back. The height of the F line is determined by subtracting the Menzure height * F^5 from the menzure height.
The top and back plate centerline arcs are constructed as 3point arcs, mirrored from the base of the top Fposition to the arch line.
The centerline is finished when the midpoints of the upper and lower arcs connected through the Menzure.
Mirrored 3 point arcs are then connected through the centerline with the arc tangent to the arch line.
The middle section of the arcs (which will be used in the final instrument arch) are determined for the top plate by using the golden division of 1/2 the arc (centerline to edge), hile the back arcs use the golden division ^2of the arc.
This middle section of the arc (that goes through the centerline) is saved while the rest is cut out to the desired recurve.
Notes of interest: If looking strictly at the centerline, the lower bout seems fuller. But the upper bout is narrower crossways than the lower bout. So the arcs formed crossways on the upper bout are fuller than the arcs on the lower bout. Therefore the fullness of both bouts look the same when looking at a finished arch from the top.
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Books of interest to read on this topic:
The Divine Proportion
Geometry and the Visual Arts
Geometry, Proportion, and the Art of Lutherie
Art Method of the Violin Maker