David Beard

Hi Francois, I believe you and I view the over hangs very differently. Don't you normally employ a fixed overhang? ****** I think the overhang is largely consequence rather than cause of anything. I my view: Design a mold, involves anticipating the 'inset' from outer edge to the 'purf line' as 3 purfling widths in. In older style work, the purfling sits outside this 3p line. In the new style work, the purf sits inside the 3p line. In all cases, the 3p line in concept corresponds to the mold line. And when calculations are from the purf instead of the outer edge, the 3p line is generally used. The insets can be view as a fraction of Body. Wider (lower ratio) insets were more common earlier, and higher ratios narrow insets became more common later. 1/80 is a middle ground. Extreme examples and special cases land between 1/60 and 1/120. So the concept of overhang is not present here, but certainly implied. The common choices end up implying overhangs that are far from 2p. Ok. Now you make a mold. And build some sides. In these things, the overhang is a none issue. But then, you situate your actual sides on a back plate in preparation to design and actual plate/instrument outline. Here you will estimate overhang as 2p. You will push and pull your sides, and twist them around their pinning for alignment. In these things, you are seeking a disposition of the actual sides that will allow you a desirable outline design for the instrument. These means considering many aspects, including the bout and body ratios, the corner levels and disposition that you have only very limited control of at this stage, and you want a good flowing '4 circles' guide behind your design. Once you have clamped and pinned and pushed, twisted, and pulled the sides into a disposition you can live with; only then do you etch the outline of these sides into the plate. Now you remove the actual sides and are free at last to work your actual instrument outline. In working your plate outline, you do not need to follow etched sides outline for an even overhang. You can take a gentle slight liberty in relating your outline work to etch line at simply near to a 2p overhang, drifting and changing some to accommodate your desires for the outline. **** So, in that light, the over hang is just consequence.

Ok. Since it popped into the thread, let's look at the 1705 P mold and its design. As is typical in all the mold examples, mostly same principles as in mold design -- but not quite identical. Note in the upper bout, that the corner line is located (3/4)UBm from the top of the inner outline. This is a common way to look the corner lines in both molds and plate, by ratio with the bout width. Then note that the notch line is place simply 1/2 from bout line to corner line. This is the older style of locating the notch line. This is a very curious practice in a way. It places the notch line in consideration of the mold only. But in all the molds, this distance from corner line to notch line gives the a nominal radius for the outer corner circle in the mold design. As noted before, the corners in actual plates don't have to follow this. But still, to me, it seems a curious aspect to Cremona practice. The lower part of this mold doesn't follow that earlier practice. It uses a common convention in working the plates to give the outer corner circle radius and the distance from the corner line to notch line. This circle is sized by the units of the Vesica division of LB. The placement of the corner line is also uncommon, differing from most or all earlier practice. Locating the lower corner level was often down by ratio from the bout width, similar to what we see in the upper part of this mold. It also was common to locate the lower corner level either as 2/5 BODYo or 2/5 BODY(m=p). Neither of these common methods is used here. In these case, the corner level line seems to ride on top the vesici circles. Since these are vesica picis, that is also the same as saying the line is 1/3 LB above the bout line. In this mold, the main center bout circle has a diameter equal to LB width. As is common, the outer corner circles are based on a key circle that relates to vesici division of the bout. While the while the concentric circles of the inner corner approaches are based on a key circle that relates to the radius/diameter of the main cBout circle. Notice the lower bout risers use the very common Cremona practice of a radius one unit greater than the vesici circles. The upper bout instead simply rises with the vesici circles themselves. This is more common in molds than in plates. And more common in the upper bouts than in lower bouts. Also note. Corner treatments, and the connection from the initial riser arcs to the corner circles are often more elaborate and sophisticated in the plates compared to the molds. The connections from risers to to circle can involve additional circles of double radius, and segments of straight line, as well as other less common complexities. But molds generally use only single outer corner circles, connected very directly to the initial standard riser arcs.

I had no intent to avoid irrationals. I took no steps to avoid them. I just kept asking; 1) is there observable structure in these examples that I can understand?? 2) are there larger more unifying ways to understand the structure I've found so far?? 3) Is there more detailed structure observable that I've missed so far? 4) is there a more self-consistent way to see the structure I've found so far? I keep repeating these questions, and asking more similar questions. I went in expect to find phi. After a while, I admitted to myself that I just was not finding irrational ratios in use. I was finding very straight forward simple integer patterns. ***** After I while, I also recognized that a great many modern people are phi blind. They want to see it everywhere. So, now I'm standing in a rather anti-phi posture as far as classical work is concerned. And I recognize now that certain historical figures like Da Vinci and Vetruvius that often mentioned in conversations about phi were in fact very focused on simple integer relationships and little or not at all on irrationals. Durer I haven't investigated enough. I know enough about he to know he also was very interested in integer ratio relationships, but not enough to know if perhaps also had interest in irrationals. *****Once again. Would have been completely happy to find irrationals used in the examples I've observed. I went in expecting to find uses of irrationals. I DID NOT FIND THEM IN THE PATTERNS OF DESIGN WORK I OBSERVED. DID NOT FIND. Would have been happy to. Just didn't.

What are we explaining here? I don't see anything remotely meaningful to explain. Once again, it's all very pretty. But is it supposed to mean something?????? The top diagram you superimpose a spiral drawing on the P mold. Your drawing presents main different shapes of rectangles. You show us two that almost fit the upper and lower notches. So what. Does this follow a pattern you understand? Did you know those rectangle would nearly/almost match the notches? Or did you just find that fitting as an accident, some thing incidental? Incidental fits in single examples don't meaning anything. These design are in fact highly structure. As a result, they will present many many incidental fits to many other structured things. Such accidents don't mean anything. Does this represent a pattern? Is there some logical variation of this fit that you also find in all the other existing example molds from classical Cremona? Is this fit somehow a key to more fully and completely understand the mold designs? And what are the next drawings showing? Again, you have a mysterious creative spiral drawing overlaid on the P mold. Meaning what? You drawing shows the eyes of two spirals. The eyes have centers, so you can draw a line through them. You can draw a line through any two points. Compared to the orientation of the mold, this line appears as a diagonal. Is that supposed to mean something? What does that mean? You then show an addition drawn curve of unknown or no relation to the two spirals. This extra line travels very roughly along the bout path, but certainly not accurately. The diagonal random cuts into the block notches. What is supposed to need explaining here?

Could have. But then the geoemtry would drift out of focus. The fact that a very specific usage of geometry and ratios is there to observe in all the examples from all the generations of a couple centuries of Cremona making suggests that even though they didn't have to, even though they could have done otherthings instead, nevertheless it appears the actually did work through these methods of ratios and geometry.

Here is some (probably too much) detail about the ratios I did find: The ratios actually observed in Cremona classical making seem to be based on nothing grander in concept that the notion of ‘a part’ and differences based on a part. These appears consistently in the ratios for almost every ( or actually all?) feature. Let’s look at some examples: The span across the upper eyes (between for celli) is executed as ‘the Stop Unit’ less a part’. So this can mean anything from 7/8, 6/7, 5/6, 4/5, 3/4. Notice also that if the ‘part’ is ‘none’, then this pattern includes the StopUnit itself. The upper eye diameter then is in turn simply ‘a part’ of the EyeSpan. In practice this is usually 1/8, 1/9, or 1/10. But in ‘theory’, we can see that this series of options natural expands as 1/7 or 1/6 for a bigger eye, or 1/11 or 1/12 for a smaller one. The patterns couldn’t be simpler or more practical in use. The lower eyes in turn are ‘a part’ bigger than the upper eyes. In practice, 2 to 3 is most typical, but we see many other choices like 3:4, 5:6, etc. --Always on the same consistent principle. We also see the same sorts of ratios used in the ‘guide rules’ that help decide execution choices, but are not actually used to carry out the work. We see this example of this in the EyeSpan choice. In earlier generations of Cremona, the eyes usually don’t stray far from 1/4ths across the CenterBout width. And, in a good number of instruments from later Bros Amati through early Strad, the upper eyes general don’t stray far 1/3rds of the UpperBout width. These relationships to bouts have a nice appealing logic in considering the horizontal spacing and sizing of the upper eyes. Yet, we observe in the actual example instruments that true fits to these ‘guide rules’ are only sporadic. Nor are the eyes consistent placed symmetrically to the sides. While still a bit loose in accuracy, the eyes are more aligned to the center line of the instrument then to the naturally asymmetric bout and corner lines. And, in a far from immediately obvious relationship, the EyeSpan consistently derives from ‘a part less than the StopUnit’. This is interesting, because it builds in a relationship between the string and stop lengths and the soundhole system. However, in most cases, the ‘part’ chosen, i.e. 7/8th versus 3/4th is chosen to create an EyeSpan close to the 1/4th cBout guide, the 1/3rd UB guide, or both. Let’s consider relationships in the head and scroll. The sizings here key from the scroll height. The scroll height derives again from the StopUnit. As it turns out, the StopUnit is a very important key in the Cremona instruments. Once again, we see the ratio for this relationship is based on the difference of ‘a part’. Most typically, the Volute or head height is ¾ the Stop Unit. However, particularly in violas, we also see the choice of 2/3 ratio. Again, the pattern could be logically extended if a making a special instrument that somehow presented motive for a size significantly larger or smaller than the standard choice of ¾ verus 2/3 allowed. In turn, the volute frame, length:height, is typically in Classical Cremona work almost always simple 3 to 4, or 4 to 5. Notice also, that the turns and eye of the Cremona volute can be looked as presenting two more nested frames around the mid and inner volute turns. These frames also present the 3:4 or 4:5 ratios. And, in Cremona work, the reduction of diameters from one turn to the next tend very strongly to be based on 1/3rd reductions. In contrast, in volutes outside of Cremona, ¼ reduction are a reasonably common alternative. So still, we are seeing a pervasive use of ratios with a ‘difference of a part’ concept running through all. Now lets look at the stop unit, since this is such an important key. Also, there are interesting complications here that illustrate how Cremona work tends to stretch rather than break a habitual principle. So, in most violin instruments, the basic body stop to neck stop ratio is 2 to 3. And, as with many of our modern numeric standards, the old ratio still lurks within. The often quoted modern standards for neck and body stop of 130 to 195 are in fact in a 2 to 3 ratio. Of course, the body stops and neck stops varied more in actual measure in the old Cremona work, but the used the 2 to 3 ratio. So what do I mean by the ratio unit? The 1 in for example 2 to 3. In the modern numeric standard, the unit is 65mm. x2 65= 130. and x3 65 = 195. But what happened when the old makers wanted a shorter or longer neck. For example in a piccolo, you likely want a longer neck than the normal ratio to fit the hand better. And in a big tenor, you likely want a shortened neck. Give what we looked at so far, it shouldn’t be too surprising that the practice we can observing them using was to adjust the neck length by ‘a part’. They used ‘a part of the Stop Unit’ for this purpose. So we see things like (2 + ¼)::3, or (2 – 1/3)::3. Notice that the fairly common modern neck stop ratio of 7 to 10 equates in terms of the old system to: (2 – 1/10)::3 The there is more we could look at, all again illustrating the same basic ideas, lets leave this by looking at some of the ratios arising in the body outline work. First, lets notice that the typical Cremona ratios for Body Length to Width can be viewed in the same ‘part less’ light. These ratios amount to: ‘Take the body as twice the width, less a part’. So, if the width is 4, twice would be 8, and less a part gives the traditional/observable 7, so the ratio is 4::7. We can see that all the Cremona width to body ratios follow this: 2::3, 3::5, 4::7, 5::9. Now, as with the neck stop, we again can see that in special cases that can’t fit, the preferred to derive a solution from the standard ratios, rather than just choose a special ratio for the special case. So, Strad’s Pochette can be observed to be based on the standard 3::5, used commonly in stout shaped celli, violas, and viols, etc. But to get the skinny Pochette, it is used as (3/2)::5. Now let’s consider the Vesici divisions of the bout lines. These have more significance than just the vesici, because commonly the initial arcs radii rising from the bout lines toward the corner areas are based on the units and logical locations created on the bout line by the vesici. Similarly, the circle diameters for the outer corner circles tend to be based on the Vesici division of the bout line. The basic conceptual or ‘guide’ idea of the lower bout Vesici is 1::1::1, the vesica piscis where the overlapping circles divide the line equally in three parts. However, the most common variants observed in classical work for this are 4::3::4, and 5::4::5. But how are this continuous in a simple ‘difference of a part’ pattern? While we have no way of knowing, we can consider that possibly they work or thought of 1::1::1 in terms of the equivalent 4::4::4. This idea certainly did not occur to me in the early stages of my research. But, as I moved along and accumulated more and more examples and details of their choices, the possible suggested itself. For one thing, the ‘bodega’ book, which is clearly not about the methods observable in the generations of Cremona work, but is about a system not greatly unrelated; that book often mentions ratios in unreduced form. So, not inconsistent with 4::4::4 instead of 1::1::1. Perhaps that book opened my thoughts to this notion. But then there are other points. 5::4::5, 4::4::4, 4::3::4 do form a pattern based on differences of ‘a part’. So that is consistent. Further, initial riser arcs in Cremona work, from both upper and lower bouts general are centered in one of three ways: on the bout center, coinciding with the vesici circle itself, or centered and with radius ‘one vesici unit larger than the vesici circle’. However, viewed as 1::1::1, a number of the Vesica Piscis cases appeared to present the risers anomalously as centered mid way from bout center to vesici circle center. But, if we understand these cases as 4::4::4, then the risers present the normal ‘one vesici unit larger’ situation seen most commonly in all the Cremona work. Adoption of the notion also help in understand the corner circles in a consistent way. The last circles giving the corner circles appear to be based on overlapping sets of concentric circles. The circles on the outer edge and circles on encompassing the purf are the main circles in this. And they meet and overlap cleanly in forming the corner. The corner designs involve two pairs of these circles overlapping, one pair outside the cBout area, one inside the cBout area. In each pair, one circle will use a ‘key’ radius. The key radii in the out circles generally derive from vesici divisions of the bout line. The key radii for the cBout corner circles generally derive as ‘a part’ of the radius of the main cBout circle. Once again, understand the 1::1::1 vesici examples as 4::4::4 helps us see the consistency of the observable practices. Otherwise, a common sizing for the lower outer corner circles appears to be ¼ the lower bout, and not obviously related to division of the bout as 1::1::1. But, using 4::4::4 as the idea of the division puts these 1/4LB key corner circles in relation to the vesici divisions, just like all the other examples of key outer corner circle radii. Ok. Enough uber nerdy ratio detail for now.

Yes. But then you're making a table. The concept of success for a table is somewhat limited, and significantly aesthetic. Have looking in great detail at the recipe choices across so many violin examples from every generation of classical Cremona making, my opinion is that the ratios used were simply about repeating the recipes that had worked before, and about allowing a structure to recipe variations that could then be repeated or abandoned based on how variations succeeded. I don't see them picking their working ratios based on any conceptually or theoretical ideas about the ratio choices used to execute work. On the other hand, ratios in guide ideas might perhaps at times have some theory or concept to them. So, for example, in the lower bout a 1:1:1, vesici pesci, is certainly the concept. And this concept of equal divisions and three parts might have had many fancy notions attached. But in practice, and especially with time and evolution of the practice/tradition, they were more likely to actual execute the work using a 5:4:5 vesici division of the bout.

Those things may be valid for your scheme. But when copying old instruments, or other more general contexts, rhat is not a safe assumption.

These attractive and logical notions are not at all what a found actually used by the old Cremona makers. By all appearance, the ratios used were just a mechanism for sizing and placing things, used habitually and consistently -- but not with special meaning. One can not.know their thoughts, but be appearance from the patterns of use, the simply used what worked before. Or made a selection that would nudge things in a direction they wanted to try. I believe the various elements of their tradition effectively yielded a design developement by evolution. Repeating what worked. Trying minor recipe variations. Slowly migrating to what worked better. No special numerology, nor overt imitation of music ratios.

Each cross template is only valid within a particular context. Apply a template precisely, and in its natively correct context, you will insure a precise result that is good as the template source. That is of course the theory. But how often are template actually in contexts the fully and truly the source? I would say not usually. Use a template out of context, you get a pricise error.

Yeah. It fact, by adding complexity almost can be made to trace almost anything. In itself tracing means very little about the nature of the shape tracing. It's only when the tracing yields special simple patterns that it means something. As when lines are used to trace a square. The square is genuinely within the family of straight line shapes.

I'm claiming that the existing examples of their instruments and molds allow us to observe many detailed things about their actual practices. And that examples given entirely sufficient demonstration that they carried out the design workers with specific choices of circle geometry and ratios. These are things we can examine and test today. Now, I asking you what you claim about the historical relevance of the methods you are proposing. ????

Which are you saying? Do you have a system to make violin like shapes that DON'T reflect historical practice and shape? Or are those who question any historical relevance to your work wrong, like flat earthers. You don't get to say 'my work isn't historical, it's the true history.' Which way? They are mutually exclusive.

It might be helpful to talk about some of things that make various families of curve making different. The subject can easily be confusing. Partly this is because depending on how we use a ‘curve family’, many different results are possible. When we use the curve families in certain complex ways, they start to overlap and lose their distinct identities. But used in certain simple ways, the differences reveal themselves. It is in these limited and more simple usages that we get to say that a certain shape belongs to one curve family, and not to others. To start, let’s just consider what we can draw with bits of straight line. For starters, we can draw polygon shapes like triangles and squares. And we might be inclined to claim that simple polygon shapes are natively ‘straight line shapes’. On the other hand, if we use a large number short bits of straight line, we can follow along the shape of any curve. And, just by increasing the number of short bits of line we use, we can trace along curve at all, and we can get as close as we like. But, just because we can trace along a curved shape as closely as we like by using enough small bits, that doesn’t make curved shapes native to the ‘family of straight line shapes’. So, we distinguishes are tracing of a square with straight lines from our tracing a circle with many bits of straight lines? For one, our trace of the circle is always imperfect. If we have infinite resolution available, then when we zoom in on the detail we will always find our bits of line don’t ultimately match the curved shape we’re tracing. We can also reduce the error, and can always get more accurate, to any degree we seek, but this increase of accuracy will require increasing the number of line bits we use. In contrast, when we use our lines to trace a square, it turns out we only need 4 lines, and we reach a perfect trace. So we can see how these traces differ. For the square, there is what we might call a ‘simple’, ‘unique’, and ‘finite’ ideal trace with our line bits. This is the basic reason we ‘claim’ or ‘recognize’ that the square is a ‘native’ ‘straight line shape’, or that it ‘belongs’ to the ‘family of straight line shapes’. In contrast, the curve shape can also be traced with our bits of line, but it isn’t native, it doesn’t belong to the straight line family. The manifestation of this is more and more accurate traces require more straight line bits in an ever more complex construction. With the square, let us note a further important distinction. Our trace revealed a simple four line structure, but also the lines joined at the ends in right angles. So the trace reveals a clear meaningful geometry pattern. Just reusing this pattern allows us to generate all sorts of rectangles. In fact, we could define the idea of a rectangle using this generate pattern. And then, even further, we can observe in our trace of the square that all 4 lines were equal in length. So the trace also reveals a simple ratio pattern governing the square shape. All the lines are in 1::1 ratio! Once again, we can generate all squares from this pattern information. We even could define the idea of a square in this way if we wished. And if generate a square using the same length as in our trace, we will generate a shape identical to the original. So, we can further say that the shape of the square we traced ‘simply’ and ‘rationally’ ‘generates’ in the ‘family of straight line shapes’. Now this is exact what I’m claiming about Old Cremona violin making shapes and ‘Circle geometry’, or if you prefer, ‘the family of shapes from Circles and Straight Lines’. Cremona Violin making shapes ‘simply’, and ‘rationally’ ‘generate’ from such ‘circle geometry’. This is why I say merely ‘tracing’ is meaningless. Some curve families that can trace ‘classical Cremona violin making shapes’: Sine Waves Straight Lines Bezel Curves Linkages French Curves Freehand Spirals Volutes Circle Geometry Bent sticks or splines Zig Zag lines But only the ‘family of circle and lines’ shapes can ‘simply’ and ‘rationally’ ‘generate’ the ‘classical Cremona violin making shapes’!

Yes. You are demonstrating the spirals can trace the shapes. Not news. Not meaningful. What is the logic of the size and placement of the spirals? Tracing and generating mean very different things!