Violin Design Method of Stradivari
by Eugene Boyd Bidwell
It is a tribute to Antonio Stradivari that after more than 250 years since his death in 1737 his method of violin design can still be discovered. He executed the construction of his violins so precisely that the method can be determined with confidence. Thus, Stradivari may also prove to be the key to understanding the mysteries of the other great masters of his time.
Apparently the technique of designing violins used by the great masters was lost shortly after Stradivari's death. Just fifty years later, Antonio Bagatella in his Treatise on the Construction of the Violin written in 1786 used circular arcs, from many unrelated origins, to approximate the outline of a Stradivari violin mold. Using a compass, very many arcs would be required to accurately reproduce just the outside edge of a violin. Today, violin makers choose an instrument of one of the great masters and trace it. W. Henry Hill, Arthur F. Hill and Alfred E. Hill are probably the most respected authors on the subject of Stradivari's work. They have written that there is no known method by which the master created his design. Nor is it known how he designed molds to hold the instruments during assembly.
This article will show that a three bar mechanical linkage can be used as a drafting tool to exactly reproduce the outline of the Stradivari violin known as the "Betts". The placement of the fixed points of the linkage is found by a simple construction method using a compass and straight edge. The placement of the fixed points of the linkage and the linkage dimensions have simple relationships to each other for the individual curves, upper bout, center bout, lower bout, corners and the f holes. The geometrical shape, proportion and relative placement of the parts are based on the golden section ratio and the ratio of the sides of simple triangles such as the 30° ,60° ,90° and the 45° ,45° ,90° triangle. Just as important, each part is drawn by a single curve with a single setup of the linkage. Both sides of the upper bout is a single curve. The same is true of the lower bout. The corner on each side of the bouts is one continuous curve. Both C bouts are drawn with the same setup of the same linkage! That is, the exact shape and position of both C bouts is drawn as two curves by one linkage in one position. These eight curve segments of the outside edge are shown in Figure 1.
The linkage makes drawing the curves quick and precise in much the same way as compass facilitates drawing a circle. With the materials at hand, the complete figure of the violin can be drawn within minutes. A methodical, repeatable method such as this allows the maker to improve on the design, both acoustically and visually. Each aspect of the composition is determined by the relative dimensions and placement of the linkages which can be determined by various geometric schemes, giving a more or less pleasing result. Finding a simple, self-consistant geometry as the basis of a design was the goal and distinguishing ability of Stradivari. Tracing does not afford this opportunity.
The units of measure used by Stradivari has been the subject of much investigation. Many analysts speculate that the various widths and lengths of the instruments would have certain simple whole number values if measured with the proper ruler. A knowledge of the units of measure is not required for the method being described. All lengths and positions are determined as proportions of the length of the violin body.
Stradivari probably drew the outline directly upon the flat underside of the slab of wood. If a design is drawn on paper and then used to trace the outline on the blank wood, the surface must be perfectly smooth to keep from distorting the result. On the other hand, a linkage sweeps out the outline without touching the surface of the wood, the same as a compass would do.
Having found the geometric steps required to give the dimensions and placement for the linkages, it can be seen that Stradivari was combining geometry and the relationship of notes on the musical scale. In particular he used the relative lengths which for a string would give the musical interval of the half-step. Figure 2 shows the simple construction technique for the length of a string which will produce a note a semitone below that of a string with a given length. This is the same geometric technique used to construct the golden-section ratio. As will be seen, the relationships of the major musical intervals, 1/2, 2/3, 3/4, 3/5, 4/5, 4/9 are also used.
There are many more considerations to building a violin than just the construction of its contour. However this is the only aspect of the violin which I have studied. The shape of the plates may very well be as important to the sound of a violin as it is to its appearance. The fact that the outer edge of the bouts can be drawn by a single linkage shows that there is a simple mathematical relationship between the points on these curves. This simple association probably has a profound effect on the acoustic properties of the instrument.
The following topics will be presented:
· How the linkage works · How to make a linkage · How the linkage is used in the design of the "Betts" violin · Observations and conclusions · Future projects planned to answer some of the many questions concerning the history of the violin and of its makers
The Drafting Tools
The only instruments needed to design the shape of the violin are a straight edge, a compass and a linkage of rods. A linkage is a connected set of rods. The particular linkage being described consists of only three rods, the simplest type of linkage. It has two rods hinged to fixed points and to a third rod called the free rod.
Figure 3 shows a linkage arrangement and the curve it draws (a figure eight) when the pencil is located in the middle of the free rod. Where the point density is thinnest is where the pencil point is moving the fastest.
Two positions of a linkage are shown in Figure 4. Two patterns are generated with this linkage arrangement; a circle and a figure eight. The figures are connected at the extreme left and right sides. In reality the two are one continuous pattern. When the linkage gets to a point where the two figures meet, it can take either path. Usually it requires a slight amount of pressure to take one path or the other due to inaccuracies in the construction of the hinges of the linkage. To draw both shapes with the linkage shown, two revolutions of one fixed-hinge rod is required while the other rod is rotated first in one direction and then in the other. Some linkage arrangements will produce two separate curves as in Figure 10 below.
The linkage rods can be made from various materials and the hinges made in many ways. A simple method is to use strips of hardwood for the rods and straight pins for the hinges. For a fixed hinge, push the pin through one end of the rod and into the drawing surface. For a free hinge, pin the two rods together and then clip off the protruding end. A small hole can be drilled in the free rod for the pencil point. It is important to place the pencil hole precisely on the line between the two pin holes on the free rod, otherwise the generated curves will be asymmetric. The positions along the rod for the pins and pencil must also be located accurately.
For a description of other linkage arrangements, interesting applications and history of their usage see the book Mathematical Models by Cundy and Rollett. They report "James Watt is said to have been more proud of his link-motion, which he discovered in 1784, than of his steam-engine."(Cundy) They are speaking of a particular arrangement of a three-bar linkage which produced near straight-line motion as part of its path.
In the figures and definitions to follow, symbols representing lengths will be underlined while the symbols specifying points will not be underlined. A constant method will be used herein to label the parts of the linkage and to specify the drawing point along the freely hinged rod (see Figure 6 below). The symbols and their definitions are:r - The length of the shortest rod with a fixed hinge. That is, the smallest distance from a fixed hinge location to the hinge location on the free rod.
R - The length of the longest rod with a fixed hinge. That is, the greatest distance from a fixed hinge location to the hinge location on the free rod. If both fixed-hinge rods have the same length, then this bar will be identified by its position relative to the design being drawn.
D - The distance between the hinge positions along the free rod.
P - The point on the free rod where the pencil is located to draw the figure.
L - The distance along the free rod from the hinge point of the r rod to the drawing point P.
H - The distance between the two fixed hinges.
Therefore, point P represents a place (position) while the other variables represents lengths.
Design of the "Betts" Violin Figure 5.
Following is a step by step description of how the linkage is used in the design of the edges of the "Betts" Violin. The procedure to determine the fixed points for the linkage and the dimensions of the linkage rods is given for the outside edge of the violin. The construction is shown against measurements made from a full sized picture of the violin.
The dotted outline of a violin shown in the figures of this section are measurements taken of the back rather than the belly of the violin. The back was chosen because the top of the upper bout is not obstructed by the finger board as it is for the belly, and the back is better preserved than the belly. The "Betts" was chosen because it is one of the best preserved and authenticated instruments still in existence.
Start by positioning the image on the surface:
(1) Draw the center line of the instrument longer than the desired length of the final body. Mark the approximate mid-point of the line C.
(2) Using a compass (one of the linkage rods makes a good compass), set it so that the diameter gives the desired length of the violin. For the "Betts" violin the length to choose is 356 millimeters. Place the compass point at C and mark the points A and B where it crosses the line at the bottom and top. This length is likely to have been derived from the length of the strings to be used, as has been proposed by several analysts. Unfortunately the original neck of the instrument has been replaced with a modern style neck. Thus, the original string length cannot be determined exactly.
(3) Divide the length AB into thirds. Setting the radius of a compass to 1/3 AB, mark the points E and F at that distance from C on the centerline. Points E and F are key points in the determination of all of the linkage positions for the violin.
Lower Bout - outside edge
Figure 6 shows two positions of the linkage being described. First the locations for the two fixed-hinge positions of the linkage which describes the lower bout will be determined.
(4) The fixed-hinge position for the smaller linkage bar r is point E. The other fixed-hinge position is G, which is on the center line at a distance EW below point A (see Figure 5). EW is Ö 2 times the length CE. The triangle _CEW is a 45° , 45° , 90° triangle.
This completes the determination of the positions for the fixed hinges of the linkage for both sides of the lower bout. Now to assemble the linkage. The terminology defined above will be used to specify the rods. In these terms, the value of H is already found. H is the distance between points E and G. R has also been determined, it is the length GA = CW.
(5) Bisect the length GA to get r. Thus, r is R/2.
(6) Generate the length D by constructing the circle centered on A with a radius of CA (see Figure 7 ). Label the point X where this circle intersects the original circle used to set the points A and B. The distance from B to X is the length D for the linkage. D is in the ratio of Ö 3 : 2 to the length of the violin AB. This is the length of the longer of the two remaining sides of a 30° , 60° , 90° triangle with a hypotenuse of AB.
(7) L is the distance along the free bar from its hinge with the bar r circling point E to the drawing point P. As can be seen from Figure 6 the free bar is longer than D by the distance L. The length of L is equal to the distance AZ. This is the amount of overlap between the circles generated by r and R.
The linkage can now be assembled. Place the fixed hinges of the linkage on the points E and G. Draw the lower bout from one side to the other past the corners. When the corners are constructed, the extent of the lower bout curve will be set.
Summarizing the procedure above; the linkage and its placement for the lower bout is obtained from the length of the instrument AB. Simple divisions of the instrument length and their projections on a 30° , 60° , 90° triangle and a 45° , 45° , 90° triangle is all that is required.
Upper Bout - outside edge Figure 8.
Construction of the upper bout is very similar to that of the lower bout. The fixed-hinge position of the r bar has already been found, point F. In order to find the other fixed hinge, first find the length of R:
(8) Construct the perpendicular to the center line at E (see Figure 9 ) with length of CE. Mark the end point Y. Draw the line FY. The triangle _YFE has sides in the proportion 1 : 2 : Ö 5.
(9) With a circle of radius EY intersect the line FY. With another circle of radius from F to this intersection, divide the line segment EF at V. The length FV is the length of the linkage rod R. FV divides EF into golden section.
(10) The other fixed-hinge length r is half the length of R.
(11) The position of the other fixed hinge (J) is on the center line above point F at a distance of R + r. Thus H for the linkage is FJ which divides AB into golden section.
(12) The free bar of the linkage has a length of 2 R.
(13) L is of length r - FB. This is the amount by which the circle of radius r overlaps the end of the instrument at B.
The linkage can now be assembled and positioned to draw both sides of the upper bout.
Summarizing the procedure above; the linkage and its placement for the upper bout is obtained from the length of the instrument AB. Simple divisions of the length and a golden section ratio (0.6180) of a division is all that is necessary.
Center Bout - outside edge Figure 10.
This is the part that really convinced me that Stradivari used a linkage to design his violins. The exact shape and position of both C bouts is drawn as two curves by one linkage in one position. Some of the same points and proportions of the violin length as used to create the upper and lower bouts are also used to make the C bouts.
(14) Using the length EF previously determined, mark the point M on the centerline at this distance above the midpoint C. This is the location for the fixed hinge of the R bar.
(15) Construct the length equal to the golden section ratio (0.6180) of the line EA. This is the length of the r bar of the linkage. The procedure of dividing a line into golden section was done in step (9) above for the upper bout.
(16) The other fixed-hinge position (N) is located on the center line at a distance r below point E.
(17) The R bar has the length of twice the distance CN.
(18) The free bar is relatively long, with length D which is Ö 3 times the violin length AB. This span can be constructed by doubling the length BX constructed in step (6) above.
(19) Finally, L = ( 1 - Ö 2 / 2 ) * D. This is also constructed from BX using a 45° right triangle. In Figure 11, L is the line XP.
Both C bouts can now be drawn. This linkage is mechanically restricted to one side or the other. In order to draw both C bouts one hinge point of the linkage must be disconnected and the point P moved to the other side. Alternatively, one of the fixed hinge points can be lifted, the linkage adjusted, and then reset the fixed hinge point were it was.
An enlarged view of the right C bout from Figure 10 is shown in Figure 10A showing the close match of the generated and measured points.
Lower Bout Corners - outside edge
Once the linkage is assembled and positioned it will appear as in Figure 15 below. In order to determine the fixed hinge positions 1 and 2, construct the length Et (see Figure 12 ):
(20) This is done with the same procedure used in step (6) above but starting with 2/3rds of the violin length, EF. Label the point s where the circle centered on C with radius CF intersects the circle of the same radius centered on F. Using Cs as the radius of a circle with center at E intersect the center line at t. Et is in the ratio of Ö 3 : 2 to the length EF. Triangle _Est is a 30° , 60° , 90° triangle.
(21) Construct a 3:4:5 right triangle with Et as the hypotenuse as shown in Figure 13 . With a compass centered on t and set for a radius of 4/5ths of Et draw the circle.
With a compass centered on E and set for a radius of 3/5ths of Et draw the circle. Point 1 is at the intersection of these two circles. Complete the circles to both sides of the centerline so that the fixed-hinge position of the other corner is determined at the same time.
(22) The other fixed-hinge position 2 is located at the distance H from point 1 along the line to E. H is of length tC (see Figure 14 ).
Determining the linkage dimensions will be easier than it was to find the fixed-hinge positions:
(23) The r bar, the R bar, and the D bar all have the same length. This length is wt as shown in Figure 13 above, Et - E1. There are other ways of determining this length; it is also half the length t1.
(24) Finally, L is 4/9ths of Et. This length could also be determined with the method of generating 17/18ths since 4/9 = 17/18 - 1/2. L is also 5/9ths of t1.
As can be seen from Figure 15 , the curve generated for the lower bout corner also conforms to the body of the lower bout for a considerable distance. Thus the linkages for the corner and for the lower bout have a certain compatibility. I believe that Stradivari decided upon the linkage parameters such as to maximize the bond between the contours they produce. This seamless overlap of the two curves gives the appearance of a single piece.
Only a small portion of the linkage figure needs to be drawn. The C bout determines the extent required on one side, while the lower bout curve already done limits it on the other side. In Figure 15 the linkage is shown for a point beyond the corner in order that the view of the corner itself is not obscured by the linkage.
Upper Bout Corners - outside edge
With the creation of the upper bout corner to be done now the complete outside edge of the "Betts" violin will be accomplished. The upper bout corner is determined in a manner similar to that of the lower bout corner. The position of the fixed hinge points is found by using a 3:4:5 triangle as was done for the lower bout corners. Figure 18 below shows the linkage, its position, and the resulting curve. Just as the fixed-hinge positions for the lower bout corner fall on a line to the 1/3 length of the violin (E), the fixed-hinge positions of the linkage for the upper bout corner fall on a line to F.
(25) Construct a 3:4:5 right triangle with CB as the hypotenuse as shown in Figure 16 . With a compass centered on B and set for a radius of 4/5ths of CB draw the circle.
With a compass centered on C and set for a radius of 3/5ths of CB draw the circle. Point 3 is at the intersection of these two circles. Complete the circles to both sides of the centerline so that the fixed-hinge position of the other corner is determined at the same time.
(26) The other fixed-hinge position 4 is located at the distance H from point 3 along the line to F. H is the same length as the r bar of the upper bout found in step (10). Figure 17 shows this step.
(27) r, R and D of this linkage all have a length of 1/3 of Et as constructed in step (20) above for the lower bout corner and shown in Figure 13 above. r is also 3/4ths of L as constructed in step (24) for the lower bout corner.
(28) L is constructed using H from step (11) for the upper bout. L is H× Ö 2/4. Thus H must be bisected twice and used as one side of a 45° right triangle giving L as the hypotenuse of that triangle. L is approximately the distance from point B to the fixed hinge position 4, differing by only 1/5th of a millimeter. Either method of constructing L gives about the same result.
This completes the construction and positioning of the linkage. In Figure 18 the linkage is shown for a point beyond the corner in order that the view of the corner itself is not obscured by the linkage. The curve generated for the corner also conforms to the body of the upper bout almost to the top of the violin. Only the portion of the figure around the corner needs to be drawn. The C bout determines the extent required on one side, while the upper bout curve limits it on the other. The assembly of only five linkages can produce the complete outside edge of the violin. Figure 1 shows the results of the work to this point. Notice the smooth conformation of the corners to the bouts.
Table 1 below lists the dimensions of the parameters described above with which you can construct the linkages without going through the steps of their derivation. The values are in millimeters. In order to combine the bouts the positions for the linkages must still be determined as presented above.
Table 1 - Linkage parameters for the "Betts" violin outside edge.
|Violin outside edge||r||R||D||L||H|
|Lower bout||83.9||167.8||308.3||332.9 1||227.1|
|Lower bout corner||82.2||82.2||82.2||91.3||86.9|
|Upper bout||73.3||146.6||293.3||307.3 1||220.0|
|Upper bout corner||68.5||68.5||68.5||77.9||73.3|
|C - bout||36.7||310.6||616.5||180.6||392.6|
1 measured from the R bar rather than the r bar. (the value as described above is D - L).
Observations and Conclusions
As has often been observed, mathematics and music are closely related. Greek thinkers in the sixth century B.C. knew that the most harmonious intervals of musical notes were those with the simplest ratios of the lengths of the cords producing them. Stradivari applied this precept by using simple geometric relationships in his design. The resulting musical quality of his work is history.
Stradivari's method of sizing and placing the linkage is simple yet elegant. His geometric technique indicates that he was well educated in the system of design employed by the ancient Greeks and the masters of Renaissance architecture. All of his linkage dimensions and placements were derived from a single length, the length of symmetry. He employed the golden-section ratio and the proportion of the sides of simple triangles as well as the relationship of musical intervals to divide this length.
A linkage provides a repeatable, quick, and accurate method for the design of the violin. As such, the linkage is not just a construction tool but a means by which Stradivari could comprehend the relationship of one design to the next. For a new design Stradivari may have maintained some elements while changing the geometry of other elements. For example; one bout (or the whole instrument) can be made larger or smaller by reworking the dimensions and placement of the linkage. Since the complete form is determined by a small set of geometric relationships, he may have learned to evaluate the effect of certain arrangements on the final outcome.
From the extreme symmetry of the generated curves one can appreciate the accuracy that Stradivari was capable of. He obviously took great care to carry out his intended design.
The linkage curve is related to its fixed hinge positions much the same as an ellipse is related to its focal points. The fact that a single linkage can produce the complete outline of a bout shows that all of its points are related in a simple way. This relationship is appreciated by the eye even though the mind has been unable to explain it. This geometric relationship may enhance the acoustic properties of the plates by focusing the sound waves reflected from the edges.
Knowing how the violins were designed may prove useful for:
· restoration of damaged instruments
· validation, to help authenticate the maker
· determination of the period of a master's work
· tracing teacher-apprentice relationships
· designing new violins
It is obvious that Stradivari and the other great makers continuously varied their technique. They each had a distinctive style, but hardly any two violins by the same maker is the same. Many more instruments will need to be examined in order to find the progression of ideas embodied in them. However, the design presented here may have many features in common with Stradivari's works that followed as the "Betts" 1704 design was both unique and the model for Stradivari's future designs. "The year 1704 marks, as far as our experience permits us to affirm, the last of those violins with pronounced long corners: we know of no specimen of later date. ... the violins of the years following 1704 show, by various parts of their construction, more especially the model, that Stradivari had settled upon certain points from which he henceforth but rarely deviated". (Hill p. 52-53) Stradivari was sixty years old at the time, so there are many previous examples of his work which need to be analyzed.
If the exact design methods can be found for the early Amati family members and Jacobus Stainer, maybe we can determine if Stainer derived his methods from a member of that family. We may be able to trace the progression of Stradivari's methods from that of the Amatis' to his own unique design.
Detailed Measurements of the "Betts" Violin
The measurements used for the analysis presented in this document can be found in the files bet600bu.mea and bet600bl.mea. Lists of X,Y coordinates are given for the outside perimeter of the back of the instrument. The data was obtained from a digital image scan of an actual-size photograph published in The Strad magazine of May, 1989. The device used was the Hewlett-Packard ScanJet Plus scanner. The resolution is 600 dots per inch. X values increase from left to right, Y values from bottom to top.
Cundy, H. Martyn & A. P. Rollett, Mathematical Models, New York, Oxford University Press, 1961
Hill, W. Henry, Arthur F., & Alfred E., Antonio Stradivari His Life & Work (1644-1737), New York, Dover Publications, Inc., 1963.