

Construction idea When first making an instrument 30 years ago, the most common explanation on how an instrument functions was in conflict with the observations I had made. I have done research in order to find out what is false and also to figure out what the correct function of the instrument might be. The two sketches below illustrate different conditions that are simple to verify by practicing yourself.
On the left sketch we see the Conventionally Described Function. We see a person representing the shape of the sound post and the bridge, as they become the function of a supporting column in the instrument. When the person pulls the string that is fitted at the base of the trees, a downward load is produced. The trees represent part of a large solid structure. The soil underneath the feet, depicting the belly and back plate, becomes compressed and forced downward and causes the shape of the belly and back to deform. This is only possible when the trees (the end blocks on the violin) hold a stationary location. The misconception is that the trees representing the end blocks are not a stationary structure on the violin. On the right sketch, we imagine the end blocks depicting the trees as not being part of a solid stationary structure. They are part of the violin structure itself and may move their location. In this state, the center column is compressed by other circumstances and remains the stationary location. What we see is that the person still representing the bridge/sound post stands on a curvedshaped beam as it represents the back plate of the violin. The string is fitted on the end points of the beam. When the person pulls the string, the end points on the beam start moving. The end points move differently depending on the stiffness of the beam structure on the left and right side of the person. A reaction is occurring under the feet where no movement is produced. The only structure above the feet that does not move at all is the person (the sound post). That structure becomes compressed. The complete structure on the left and right side of the column can be deflected, with the exception of the area under the column. If we now imagine not only the structure of the beam but also the complete instrument structure, it becomes very difficult to have a comprehension of how the complete shape becomes stressed and disfigured. Understanding the complex state of disfiguring is the main importance for the violin maker to understand and handle. We must understand that we have to deal with two structures, basically two instruments, one on each side of the supporting column. These two structures are intimately joined to one functioning unit. Each shape may have different qualities since they are of a different width and length. We must accept that stresses that arise on the arching produce different Operating Deflection Shapes and their function thus becomes different. Observe that the animation below shows conditions where the left and right side hold equal shape thus equal deflection. Equal structural deflection and dynamic behavior The main issue to detect is the function of each structure (upper and lower bouts) on the real violin and how they can be brought to function as one unit. In order to come to some understanding of the differences in shape, I investigated the outside curve shape on the arching of each instrument by holding and moving a ruler (straight object) on the surface. With back light illumination, a shadow is projected and a curved shadow line on the arching surface becomes visible. This shadow shape can be compared with others by changing position of the ruler. I did this in order to find similar curvature shapes. Looking at different angles of equal shadow shapes, their location and shape could not be verified with exactness. The most important observations I made were four very thin straight tangent lines on the surface. When the ruler closed the gap, there was no visible backlight underneath the ruler. Pairs of such conditions are found in diagonal directions with a length of about 23 inches on the upper and lower bout shapes. These flat straight shapes are the only two pairs that have an accurate equal cross sectional shape. While moving the ruler from this location, one side of the line has an increasing convex shape and the other side an increasing concave shape. It occurred to me to adopt a function for these straight tangent lines by regarding them as extremely narrow structural elements. Because of the location between convex and concave shapes, the straight tangent lines act as hinges. On both of the concave and convex sides, there may be an impact on the reaction and each of the shapes may become increased. This might happen when the arching structure becomes stressed and begins to deflect while the straight tangent line structural elements inside the arching do not. This extremely simplified idea attracted me to search for a 3 Dimensional (3D) geometric layout that produces this structural quality with straight tangent lines on the arching surface.
In my own analysis, two expressions require special definition. An isoline, in accordance with normal topographic usage, represents the altitude of points having equal distance above a reference level, in this case the upper side of the instrument's rib. Many isoline levels create a topographic map and enable us to examine different structural sections systematically. A baseline is defined by connecting all points where the extended cross arching shapes (vaults) intersect at the rib level. All shapes used to produce the isolines are simple geometric shapes  straight lines, circular arcs, etc. No calculations are needed.
A computer program was written which could exactly calculate X and Y coordinates on the isoline, together with the Z (vertical) coordinate in order to produce a topographical map. It was easy to adapt the program for different baseline layouts and the longitudinal arch shape in order to obtain a quick solution to the various properties of the arching shape. The shape of the baseline and the longitudinal arc in the centerline were manipulated in order to find the dual element property (cross arc shape) covering half the arching. All geometric shapes should fit together in a simple geometric process without the need of making calculations. For a comparison, I examined the drawings by Sacconi and found incomplete "fansectors" with dually arranged curveshaped cross sections. I assumed that Sacconi must have had some notion of a possible 3D geometric construction since he partly applied the equal "cross section" layout in his isoline layout. He achieved a limited state by 2D modeling in the Cbout area. However, his arching layout seemed to be a good start in the search for a complete 3D geometric description. In my initial experiments, I constructed the baseline using six different circular arc sizes, following the outline near the deepest point in the scoop and close to the Sacconi arcshaped baseline. To find the Sacconi baseline, I had to calculate a number of points which occur where the extended circular arc of a transverse cross section intersects the zero level. Many such points produces the shape of the baseline. On the Sacconi baseline, I found no relation of circular arc size that fit together on a natural basis. However, I used the layout in the starting process. Only half of the instrument was needed to find the fanshaped arching structure containing dual structural cross arc shapes between the upper and lower bout areas. The computer program produced isolines like a topographical map. With the typographical map, it become possible to check where equal cross arc sections arise in the upper and lower bout structures. This final geometrical concept produced the state I was looking for which consisted of a baseline with three identical circular arcs connected at angles of 60 degrees. The chord lines thus have equal length. This baseline is rotated in order to produce different upper and lower bout widths. It does not matter how much the rotation is the result always produces the quality I searched for. Later on, I found the key to a complete understanding of the complex measured relationships of the instrument when we look at the 2D underlying construction of the 3D arching construction. The Otto Möckel book contains isoline drawings and topographical maps of Stradivari and Guarneri del Gesù instruments. These isoline patterns of Möckel’s were in surprisingly close congruence with the geometric model when I checked the quality I searched for. There were only very slight deviations which can be explained by agerelated, long time deformations of the arching shape or by less accurate craftsmanship when the isolines where mapped. However, the isolines measured by Möckel on the real instruments are in closer agreement with the geometric model than the average values assumed by Sacconi.
When the planegeometric model (2D) was completed with its mirrorimage layout on the longitudinal centre line, it became clear to me that there also exists an obvious planegeometric congruence with the instrument – the relationships of width, string length, Cbout corner location, Fhole location etc.
The outline of the instrument is not defined by simple geometry. In order to find an outline, solid geometry is required in order to study possible scoop shapes. This is shown in the following pages. 1. Dual congruent crosssectional arc shapes in the upper and lower bout shape are found in a fanshaped layout. 2. Straight tangentline quality engendering a structural framework like a pyramid is found in the arching structure. 3. A 2D geometric layout is found that connects the location and size we normally find in an instrument.
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