Tuesday, August 2, 2016

Breaking the Stradivari Code.

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A System of Geometry and Simple Ratios.

As it turns out, there is a system of geometry and simple ratios behind the shapes of classical violin making.

This geometry can be cracked and deciphered -- once we find a few starting keys into the code.  And, like the DNA codes that shape life, classical violin making's codes of geometry and proportion not only determine the form of single instruments, but also play the central role governing changes and development of instruments across the generations of classical making.


In its way, this geometry forms a code or DNA behind the creation of the great old instruments.


In living things, evolution involves an interplay of variation and selection across generations of reproduction.  Genetic code plays a governing role.  Any individual's grow is controlled by the genetic code they inherited, but is also influence by various factors during development.  Thus 'identical' twins can't be very different.  Their growth is limited by identical coding.  Yet they always are somewhat distinct and individual in their actual growth.   The same turns out to be true with classical instruments.   Two instruments made with the same 'design code' (the same choices of geometry and proportions), can't be very different, nor can they be identical.  Subtle but important variations will arise in the progress of their making.

Perhaps 'Breaking the Stradivari Code' is a bit of an overblown title, but I couldn't resist. A recent MIT research article on the evolution of sound holes in violins helped inspire the title.  At the time I read the article, I was considering how best to make my own research public, and what might be a good title.  The MIT article made me realize that by working only within particular geometric constructions and proportions, the old Italian makers had in fact tethered the variations in their making to something akin to a DNA code.  

DNA code (or our classical making code) limits and governs the development of an individual.  The vagaries of an individual's development add additional elements, but can only stray in a very limited way from the coded design.  Such code then allows repetition across generations.   Build again with unchanged code and the results will substantially be a repeat.  So the coding enables repetition.  It also limits the extant of small changes.   Make an instrument with almost the same code, with just a few slight variations, and you can expect the new instrument to be substantially a repeat -- with just a few slight variations.  In this way, such 'design code' helps act as a ballast against random unhelpful change, and also acts as a mechanism for repeating variations and limited changes when beneficial.

The MIT study identified an evolution like pattern to the way the shape of sound holes in bowed instruments developed from medieval times through to the longer f shaped sound holes of a Del Gesu violin.



But the MIT study didn't identify any violin making parallel to DNA genetic code.  In discussing their bowed instrument making parallel to evolution, they pointed to elements of variation and selection, but not to a parallel for genetic code.  When I read the study, I realized the geometry and proportion choices I was seeing across generations of classical Italian making were like a code, and would serve a similar role to DNA in the MIT 'evolution' scenario.  Hence, I had the idea to write up my research results under the title "Breaking the Stradivari Code."

My last blog post used the story of a fictitious classical Italian maker building and designing a violin from scratch.  This was done to hypothesize and sketch a system of designing and building violins that can 'make what we see in the classical examples' using only 'historically consistent methods'.  In this post, I'd like to flesh out that sketched system, and discuss some of it's implications.

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A hypothesized system behind classical violin making and design. 

The system sketched shows the design process united to the building.

The building process seen is somewhat loose, and the little variations arising during the build are allowed to push the design around some.  Prior to building, there isn't a blueprint or similar complete drawing of the design in an idealized form.  The design exists as more a recipe.  And the recipe presents choices, some of which can be decided on the fly as building progresses.

Just from these first few aspects of the system, we can see that this approach will never produce identical instruments, even if we repeat a build with exactly the same design recipe in every detail.  The variations arising from the loose build will naturally be slightly different each time.  

Nature too is this way.   Flowers are wonderful examples.   The flowers on a single plant all share the same genetic code, yet they are each varied subtly.  Small variations during development are allowed to push the design around slightly.  As with the treble and bass sides of a classical violin, no flower will even be exactly symmetrical in itself.

And so with classical violin family instruments.  The design recipe insures a certain character and structure to the final results, but on close examination small irregularities and asymmetries arising uniquely during the build will be found incorporated thoroughly into the instrument, pushing the idealized design around some.

We might summarized the principles behind the sketched system in three main principles:

  1.  All the shapes are based on simple workshop friendly geometric constructions and simple ratios.
  2.  The design geometry is applied during the course of the work, embracing the vagaries of the particular build as it progresses, and the natural movement of wood.
  3.  Choice of geometric constructions, guides, and ratios is generally traditional.


And expanded more completely:
  1. All the shapes are based on simple workshop friendly geometric constructions and simple ratios.
    • Simple integer ratios govern the constructions
    • The constructions are based on circle arcs, straight lines, and simple ratio proportions
  2. The design geometry is applied during the course of the work, embracing the vagaries of the particular build as it progresses, and the natural movement of wood.
    • Variations arising in a build are mostly incorporated as the process moves forward
    • Positioning and sizing of next elements follows off of actual prior elements
    • Freehand smoothing plays a limited role in shaping, but always following off guiding geometry and proportions.  Such smoothing parallels the tradition of 'fair' curves in boat building.
    • Elements of shape can get adjusted slightly to reconcile design to the actual build, and for visual balance overall.
  3. Choice of geometric constructions, guides, and ratios is generally traditional.
    • Every shape and size in a build is guided by choices of geometry and proportions.
    • Any feature has only one or a very few geometries traditional to it, and with only a limited range of traditional proportions to choose from.  
    • Variations between instruments, makers, generations, and even types of instruments will be found to be simply variations of choice within these traditional geometries and proportions.
    • Finagling the application of guides and constructions is a basic aspect of the tradition, allowing a limit measure of freedom to the maker.  For example, a maker can stick to the traditional proportion but make a feature slightly bigger or smaller by choosing to include or exclude a margin in the calculation.  By measuring perhaps from the purfling line versus the outer edge of the outline.
    • Some elements are more freely varied instrument to instrument, but for other features the guides are highly consistent across many families and generations of making
    • As a basic part of the tradition, we see a constant experimentation and pushing within the system, and mutation and evolution of standards across generations and regions.  So a new variant approach to a feature usually produces nearly identical results to the main method when first introduced.  But with time the new method may become the standard tradition. Variations from that new standard may produce results that the older standard never could.  Thus, sometimes features change substantially over the generations.  .  
    • Variation within the traditional geometry and proportions is common, as is finagling the application. But breaking or abandoning the traditions is very limited and rare. Step by step, the changes almost always are incremental and evolutionary rather than dramatic breaks of tradition.

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Why hypothesize this particular system?

It seems the simplest, most workshop practical, and historically consistent way to generate the range and kind of results evidenced in the existing classical instruments and artifacts.

As example, consider the basic outline shape composed from constructions of lines and circle arcs.  A number of recent and historical researchers and sources suggest this basic idea: Kevin Coates, Francois Denis, Sacconi, Bagatella, etc.  

However, more difficult than identifying the basic construction is understanding the choices the old makers made in arranging the elements of such a construction to get the specific results seen in their work.

This 'decoding' of makers' specific choices is made more complicate in that the classical instruments don't display a single answer to this question, but rather a range of answers.   A good understanding then must account for the range of instruments we see, not just a few cases.   And the issue is further obscured because of the looseness and build variation characteristic of classical work.  This variation obscures the idealized design recipes hidden behind the final results.  Beyond this, understanding the existing evidence is further obscured by the alterations of time.  Various repair workers have thinned, straightened, polished, mended, and altered the instruments we can now see.

But some makers worked cleaner than others.  And some instruments have survived less altered than others.  Also, some makers represent more of a 'starting point', and others more of a 'late evolution'.  We can start work by examine clean examples from the 'starting point' of the tradition.  A handful of instruments from the Amati family and earlier generations of the Guarneri family help us find our way into an initial understanding.   From their we can expand our study.  When we can see the practices across all the generations of Classical Cremona making as variations on a small set of traditional practices, then we start to have a good understanding of the methods.  And indeed, we end up finding the earliest Amati work doesn't spring from a vacuum, but in itself is variation within traditions previously existing.

We are actually talking about a long span of practice.  Even if we focus entirely on 'Classical Cremona' making, that runs approximately from 1540 to 1790.  But the fundamental approach to instrument design seen in Cremona seems to be shared more broadly across Italian, and begins before the violin family.   

In a way, the system we've found in studying the old instruments isn't all that surprising.   It relies on simple convenient tools of the time: straight edge, compass, dividers.   We see these tools frequently depicted in the art of the times.  These were basics of the craftsman's world, but also basics and important symbols for the architect, military man, navigator, mathematician, scientist, and artist.  Special compasses with number scales etched into their arms were even the 'calculators' of the day.  This was the high tech gear of the times. 

And these tools were very much part of the broader woodworker's kit.   We see this evidenced in early paintings depicting woodworkers, and in the decorations on their existing work, largely consisting of carved patterns laid out with elaborate compass work.

The focus on proportions based in simple integer ratios is also unsurprising.   The geometry and mathematics of these things was the intellectual fashion of the day.  Part of the revival of knowledge from the ancient world.  And they understood that harmonies, musical intervals, and resonances all relate to simple integer ratios.

Now, we might be tempted to understand the classical making based only on the principle of geometric constructions and simple ratios.   But we soon discover that the old instruments rarely fit the kind of perfectly symmetric design we can easily layout on paper.  The shapes of classical making deviate from ideal geometric design to a significant degree.

To understand these deviations, we need the second principle of our system.  This gets at the characteristic way the classical makers applied their geometry and proportions.  Rather than working out a complete design perfectly ahead of time, the design geometry was instead applied bit by bit as the work progressed, and based off any irregularities that had already developed.

In this approach, the design is more like a recipe for the work.  We don't see them using the more modern approach of completing and perfecting a design ahead of the work, and then following the whole of that ideal design as closely as possible.  Rather, the classical design recipes says how a next feature will relate off of previously established features.  If those previously established features deviate from an idealized design, the new feature will still follow off the actual size and position of the older features.

The third principle of our system is needed to understand the constant experimentation with design seen in classical making, the way classic practice varies and retains a continuity at the same time.  They experiment and change details of design constantly, but in a characteristic way that greatly respects and honors traditions.

The simplest way to understand the consistency behind the many facets and variations of classical Italian instrument making is expressed in the three principles we've elaborated and hypothesized.


  1. All the shapes are based on simple workshop friendly geometric constructions and simple ratios.
  2. The design geometry is applied during the course of the work, embracing the vagaries of the particular build as it progresses, and the natural movement of wood.
  3. Choice of geometric constructions, guides, and ratios is generally traditional.


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So how can we begin developing a confident understanding of the geometry choices behind the old instruments?


Let's begin by looking at the outline geometry in the violin family. 

Given the prior work of others, and the clear visual evidence of vesici geometry in the instruments, it isn't difficult to conclude that vesici are a basic element in violin design.   But when we start trying to apply Vesici to real examples, we quickly run into complications.   As we've seen, the classical Italian instrument makers used their geometry as a recipe, rather than as a blueprint.

So let's look at a real example.  Stradivarius is a cultural paragon for ideal craftsmanship.  So let's look at one of his instruments.   Among the best examples, is the Lady Blunt violin from 1721.  This a wonderfully made instrument that comes down to us in very good condition.

How then exactly are the vesici applied in the design of the Lady Blunt?  

To help us see the fit precisely, we use concentric circles and try to fit both the purfling shape and the outline shape at the same time. 

Here using the Lady Blunt Strad, it's easy to find a good clean fit for concentric vesici circles:


The Lady Blunt outline is indeed based on Vesici, and more than that on Vesica Piscis measured from the outer edge of the purfling.  The Vesica Piscis is shown in blue below, dividing the width of the Lower Bout equally in a 1::1::1 proportion.

Exceptionally clean examples like this might inspire us to continue looking for perfectly executed ideal designs in the classical instruments.  And our modern bias surely would prefer that.  Something in our culture would prefer to find the greatest symbol of ideal craftsmanship, Stradivari, working perfectly to ideal design blueprints.  Our modern sensibility is more inclined to this ideal than to the messier 'recipe' approach to design that we've hypothesized.
But even such a cleanly executed classical instrument as the Lady Blunt illustrates the 'recipe' approach adjustments away from perfectly idealized design that characterize classical Italian violin making. 

The Lady Blunt provides a nice example of this typically pushed around asymmetry in its Upper Bout vesici. The Lady Blunt is about as cleanly precise as classical making ever gets.  Yet the upper curves of its outline are distinctly asymmetric.

The actual vesici circles found in the Lady Blunt's Upper Bout are shown traced below in black.  For comparison, the vesici circles for the idealized design are also shown, marked in blue. The center of the actual treble side upper vesici is shown marked in red, significantly off center from the idealized position marked in blue.


We can see that the actual Vesici are asymmetric, adjusted from the ideal design when Stradivari reconciled the back plate to fit the actual sides he'd made.  The blue circles show the idealized 2::1::2 vesici common for Cremona violin upper bouts.  We see the bass side follows this, but the radius and center of the treble vesici circle had to be altered to reconcile and fit the back to the sides.

This asymmetry should not be viewed as carelessness or error. Rather, this illustrates classical Italian violin making's characteristic approach to design as a 'recipe' carried out bit by bit as the building proceeds, rather than as an ideal worked out ahead of time completely and perfectly to be followed like a blueprint.  That would be the modern paradigm.  But this asymmetry illustrates a different ancient paradigm, essential to understanding classical making.

The ribs for the Lady Blunt were made first, bent onto a wooden mold.  That wooden mold was shaped by similar design processes.  The wooden mold helps determine the final shape of the ribs.  But characteristic of classical making, it leaves some play for wood movement.  And in the case of the Lady Blunt, that wood movement lead to a deviation in the upper rib shape.  Rather than ignoring that deviation, or pushing it out of existence, Stradivari did the characteristic thing and adjusted the vesici of the outline design to accommodate the actual rib shape.  I suggest that for Stradivari and other makers in the tradition, the error would have been to not adjust the outline shape.

Tracing the geometry behind many classical instruments, from many makers and different families and generations reveals both the characteristic constructions and proportions used, but also the patterns of adjustment that we've sketched in our hypothesized system.

To account for all the various things seen in classical work, we can't just look for idealized design.   We instead see the old masters making design adjustments in a particular curious way that isn't really freehand.  As we see in the Lady Blunt's upper bout, the curve is still a circle arc, just as the design recipe calls for.  Rather than being abandoned to freehand adjustment, that element of the design has been pushed around some to create the adjustment needed to follow another feature.  In this case, the center of the bass side upper bout vesici was pushed a little off its idealized location in order to achieve a better fit to the actual sides Stradivari made in building the Lady Blunt violin.

Such adjustment and interaction of elements is characteristic of classical work. The hypothesized system does exactly the things we see evidenced in the actual classical instruments.


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Even with identical design, every classical build will vary, giving unique results.

We sketched a picture of classical violin making, with the design and building processes not separated or insulated from each other.

It is inherently NOT a copyist system.  It doesn't provide the level of control a copyist needs.  It's a system that insures good characteristically classical corners, but not the exact placement of those corners, or the exact shape of the outline approaching those corners.  So a curious aspect of this approach, is that a copyist can't use these same classical methods to exactly replicate the details of a particular classical instrument.

The classical methods we've seen insure a character and category of result, but the exact results emerge not just from the design recipe, but also from the building process, or the 'cooking' so to say.  And the results therefore are unique every time.

But a copyist needs to predict the exact outcome of details.  So there is an inherent conflict or divergence with classical making.  In order to closely replicate details of a specific classical instrument, a copyist is forced away from using the integrated design and build processes of classical making.

To draw a parallel, the master artisan baker prepares dough, and shapes the dough in a particular way for the oven.  But this is by hand every time, and subtly different each time.   The recipe insures a characteristic continuity from batch to batch.  But if the baker hand prepares the dough, then even the dough will vary.  Each batch will be subtly different.  Using naturally prepared raw materials will add further elements of variation from batch to batch.  And then the hand shaping of the dough to go into the oven will also be subtly unique each time.  The baker is aiming for approximate uniformity, but by artisan hand methods uniformity will never really be achieved.  And there's nothing wrong with that.  Characteristic results are more important than equal results.  There is a tradition for his bakery of how the dough is prepared and shaped for the oven.  That tradition does not scrub away all variation, only enough to insure a characteristic shaping.  Then in the oven, each loaf swells, and the crust browns and cracks.  Each loaf's final exact shape is reached in a partially uncontrolled way.  Exact uniformity is not the aim.  Exact predictability of any one loaf is not the aim.   Only consistently characteristic results are important.

When exact copying is a priority, a greater degree of control must be imposed.  Thus classical results can't be explicitly copied by classical methods.  Classical methods only insure characteristic results, not exact results.  This divergence between classical methods and copyist methods can easily become very significant.

Conspicuously, our sketch exposes large differences between most modern making and classical methods.   First is that the classical method does not share a modern concern for absolutely controlling final results.  As with the artisan baker, the methods of classical Italian making only insure characteristic results, not exactly predictable and totally controlled results.

This manifests both in not avoiding or controlling away wood movement, and in using a 'recipe' approach to design that interacts with the building as it proceeds.  As for example, in classical making's use of a thin inside mold to shape the ribs.  This kind of mold only partially controls the wood for the ribs.  It leaves some wiggle room for wood movement, especial the vertical squareness of the rib structure is not highly controlled in this method.

Similarly, modern violin making has moved away from the use of water processes that release wood movement and tensions within the wood.  Such movement works against the copyist's need to control results.   But classical making doesn't need this same level of control in order to produce its characteristic results.  In classical making, there would be no need to avoid water processes.   We must assume that hot water sizing, which was ubiquitous in woodworking and artisan crafts of the time was normal to violin making, not avoided.

And in the classical design processes, we see that the various geometry constructions and guides were applied during the course of build, and were calculated off the actual size and placement of elements at that stage of work, continuing off of and embracing whatever deviations had already emerged.  This aspect of classical design runs contrary to the modern assumption of a completely resolved and determined design worked out before building -- 'a priori', and contrary to a copyists need to completely control results.  Rather, this characteristic interaction of the 'design recipe' with the building process lends variation and life to classical making.

Classical methods insure characteristic and beautiful results, but not exact uniformity or repeatability.  In fact, natural variability is an essential characteristic of classical work, contributing to the humanity and glory or the results.  It's easy for us to not realize how much the modern focus on uniformity and repeatability is not an aesthetically driven choice, but rooted instead in commerce.  Repeatability, precision, uniformity: these are essentials to mass production.

So the classical approach we sketched is neither the completely freehand work of the inspired lone genius individual of a 19th century artistic ideal.  Nor is it the control and uniformity of a modern or engineering ideal.  Rather, it is a very characteristic artisan craftsman's ideal of centuries past, of the time before industrialization and Napoleonic political, social, and economic change.

From our modern viewpoint, the way a classical build constantly interacts with design as a 'recipe' is alien.  But this is the key thing that allows classical making to embody both a structured order and a living natural character. 

This aspect of looseness also lets the maker's hand and eye into the process.  A cooking recipe intended for unskilled amateurs is likely to relieve the need for judgement by saying very specific things like 'add 1 tablespoon pepper' --and by being fairly complete, where a master's recipe is more likely to allow liberty and discretion by being less complete and saying looser things like 'season to taste'.  Part of the modern drive toward uniformity of materials, and measurements, and temperatures, and control of all variables is to allow an inexperienced hand to predictably create a result based on instructions from a more skilled person.  Many of the older art or varnish recipes aren't so helpful in this modern way.  They are more like the 'season to taste' instruction.  But that very looseness gives room for an experienced master to let their judgement and hand enter into the shaping of the work.

Consistent with this modern way of externalizing the deeper levels of skill and understanding, the copyist approach similarly defers design judgement to the master maker of the instrument being copied.  The background idea motivating copy work is to capture some of the virtue of the original.  Of course, the historical emergence of copy based violin making is more complicated than that, with social and economic forces also very important.  But behind all the other elements is the idea that classical instruments are more valued and better than contemporary instruments, and that copying can help the modern maker borrow some virtue from the classical masters.  This motive is rooted in a concession that the contemporary maker's knowledge and understanding are insufficient to make as good an instrument as the classical examples.

But when we turn to copy work, we deepen this lack of understanding.  We turn away from the need to understand even the 'what' of the old recipes of design, let alone the 'how' or 'why' understanding that might emerge if we first understood the 'what'.  When we turn to copy work, we no longer need or have use for the original design recipes, the example we copy becomes the design.  And like a blueprint or other modern style design, this is a complete 'a priori' design.  The example shows all details, before we start.  This runs completely against the way of interactive building and design recipes that is central to the classical character of making. And by going down this copyist road, it becomes more and more difficult to return to the classical methods, as they appear more and more alien to our modernized eyes.

For those today who want to revive old methods, we can start by avoiding by methods understood to not be historically consistent. But then we have the very large task of finding historically consistent methods.  If we take the additional very important step of rejecting the copyist approach, then we have in front of us the even larger task of finding historically consistent recipes for every detail of the instrument designs.

The past work of people like Roger Hargrave, Sacconi, and others has helped open our understand of classical building methods.   While work by Coates, Denis, and others has helped our understanding of aspects of the design.   The further work presented in these blog posts fills in enough of the gaps to enable a 'revival' approach to making.

Let's start looking at more of the design recipes needed in making an instrument.

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Are there relationships relating the length and width of the violin body?

Yes, classical violins often show a 4 to 7 ratio between the overall body length and width.

Using a 1744 violin by Stradivari's apprentice Bergonzi as example, we see the 4 to 7 ratio in width to length in the body.

Here the ratio appears in a very straight forward way, measuring both the Lower Bout width and the Body length from the outer edge of the outline.

In Stradivari's 1700 violin 'Ward', we again find the 4 to 7 ratio of Lower Bout width to Body length cleanly expressed, but this time measuring from the purfling.



In this case, the 4 to 7 relationship appears based on the line of the outside edge of the purling.  Why would this line be important in the design?   We saw before that in Stradivari work, and much Guarneri family work, this line falls 3 purfling widths in from the outline edge.  This is the same distance that marks the inside edge of purfling in Amati family style work.  And throughout classical work, this 3 purfling inset line is important to the design relationships, as well as the outline edge itself.  Even though overhangs are not absolutely even, this distance basically corresponds to the inside of the ribs.  Even though the ribs are pushed and pulled around when oriented to the back plate, this inside rib line also corresponds to the line of the mold -- at least in concept.

Here we see the same 4 to 7 ratio of width to length in Ruggiero's Milanolo violin.  But here the relationship is based on the outline edge.


And we see this also in patriarch of classical making.  Here in a violin by Andrea Amati, we see the 4 to 7 width to length ratio based on the outline.


Even in the last years of Del Gesu's work, we can still see this ratio in action.  Here in Del Gesu's circa 1744 Ford violin:





But not all the instruments use the 4 to 7 ratio for width versus length of the body.  We also see instruments with a 5 to 9 ratio instead.

Here for example is the piccolo violin by the Brothers Amati.


Other, stouter shaped instruments, and particularly some more archaic instruments show a 3 to 5 ratio instead.

Here we see a 3 to 5 ratio of width to length in the body of this Lira by Giovanni Maria, 1525:





And here in a Lira by the brothers Amati, we see a 2 to 3 ratio:



Is there something unifying these choices of ratios?

Indeed, all these ratios make the instrument width a little more than half its length.  2::3, 3::5, 4::7, and 5::9 do in fact share a characteristic.  Each is one united short of 1::2.   So for example  5 = (3x2)-1, 7 = (4x2)-1, etc.   So the concept behind these ratios is a very simple relation, 1 to 2, altered by a single unit of division.  We will find similar choices in other aspects of classical making.

If we put these ratios together for comparison we can see a little more about them.  2::3 and 1::2 appear as the conceptual bounds of this series of ratios, and 4::7 is the average of these extremes.


Most classical instruments appear to fit this scheme cleanly.  Because so many examples fit this so well, it seems unlikely to be chance.  This seems to be an idea running through the old classical making.  And the idea is all the more confirmed when see that the older and odder forms also fit.

Here for another example is a Zanetto viola:




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What relationships can we find running along the length of the whole violin? 

This question is made more difficult to probe since most of the old instruments have had their necks altered, and many of the larger instruments -- cellos and violas -- have even had their body shapes cut down.

Luckily, one of the very few measures in modern making that still retains a ratio version is the neck stop to body stop.  The neck stop runs from the upper edge of the top plate to the nut, viewing the instrument straight on.  The body stop runs from the top edge of the plate down to the bridge line, which traditionally relates to the nicks in the f holes.

https://en.wikipedia.org/wiki/Violin_construction_and_mechanics


In modern terms, the violin neck stop and body stop are often given as 130mm versus 195mm.  But there is still a remembrance that these numbers represent a 2 to 3 ratio.

Because this same ratio governed in both modern setups and when the instruments were in original condition, many of the violin length relations survived modernization.   Still, in the process the original scrolls and pegboxes were removed from the original necks and spliced onto the new necks.  This gave both opportunity for them to slightly misplaced length wise, and opportunity for them to be tilted backwards excessively.  So sometimes the length relationships were obscured or lost, but not always.

Here are the length wise ratios still found in a 1749 Montagnana violin:

2::3 Neck Stop to Body Stop:



1::2 ratio between Box Length versus Neck Stop


1::2 ratio between Volute Length versus Box Length



Combined, these ratios yield a pleasing large scale 1::1 relationship between the body stop and the placement of the scroll:



Overall, these ratios give this Venetian violin a pleasing harmony of length wise ratios:



But let's look at how some special variations on these proportions were managed.

Only a handful of original Cremona and old Italian setups have come down to us in unaltered condition.  And often these are on instruments of unusual sizes that fell out of fashion, and were therefore spared the on going pressure to adapt them to current playing standards.

Here in the 1613 Brothers Amati piccolo violin we can observe an unaltered original setup, but in an oddly proportioned special sized instrument.

We see the same ratios between neck and head, and between pegbox and volute:



But this instrument has a specially short string length.  To accommodate this, the normal ratio of neck stop to body stop is reduced.  As we've seen before, the old maker's preferred to create variations on a standard ratio by reducing it by one unit, or a reasonable division of that unit.  Here the ratio is reduced by 1/4 of the ratio unit, or 1/12 the normally expected Body Stop.


By making these alterations to the normal ratios, the Brothers Amati made a piccolo with a very short string length, but a less abnormal neck.  This way a normal hand can better navigate the small instrument.

With tenor violas, we see classical makers negotiating solutions to the opposite problem.  The string length of a tenor must be exceptionally long.  But a normal hand must play the instrument.  So they reduced the neck proportions from what would normally go with that giant body stop.  And they used the same sort of methods we saw in the piccolo violin.

Here the neck stop is reduced a 1/4 unit from the normal expectations, but Andrea Guarneri also reduces the features of the head.  In this way he gets the pleasing 1::1 from the bridge stop to correspond with the top of the volute, instead of hitting at the chin as with saw with the Montagnana violin:




And we see similar things even in Zanetto:






In Ruggiero's Milanolo violin, we see the same ratios as were in the 1749 Montagnana:



We should note that the geometry behind the head and scroll follows the same kinds of principles seen throughout Italian making, but gets complicated. And as with the soundholes, Italian traditions encompass quite a few variations in the scroll and head geometry.  A future post will explore these details.  

We should also mention some of the vertical to horizontal relationships in the instruments.  Again, time has somewhat obscured these.  But looking across many instruments we can make an attempt to understand the relationships that were at least nominally there in the past.

It seems that maximum height of sides as ribs plus edges versus ribs alone normally has a ratio of 10 to 8 in violins.   And that this maximum height including edges, versus the body length normally runs 1 to 9 in violins and violas.  Though the Brothers Amati piccolo for example seems to run 1 to 8.  Also, though there is much variation and finagling on this point, nominal plate height might be viewed as 1/2 rib height.  And bridge height also might be view as related to side height, at least as a background idea.

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What additional relationships can we find in the instruments? 

Beside the overall ratio of width to length, there are other relationships seen between bout widths and body length. 



For example, very commonly, we see a 4 to 5 ratio present between the upper and lower bout widths.

Here is an absolutely clean example of this 4::5 ratio between the width of the upper and lower bouts.  This is a 1624 violin by Nicolo Amati.



And here in the 1693 Harrison Strad, we see the same ratio based on the outer edge of the outline:



We also very commonly see the center bout related either to the lower bout or to the
upper bout.  In the case of the 1624 Nicolo Amati, the Center Bout appears as 2/3 the Upper Bout.



And we see this 3::2 relation between the Upper and Center bouts again in the Harrison Strad:


When the Center Bout is related to the Upper Bout, it is usually in a 3 to 2 ratio.  However, particularly with later generations of makers it is very common to see the Center Bout related instead to the Lower Bout, on a ratio of 1 to 2.



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Surveying the range of bout ratio choices in classical making.


Across the entire period of classical making, we see quite a lot of variation in the choice of ratios and the application in setting the bout width of instruments.  The choices fall within rather limited ranges, but within this playground we see the makers continuously experimenting.

The overall proportion of an outline is given by ratio between the length of the body and the width of lower bout.  For the entire 250 years plus of classical making, the instruments all exhibit just a few ratios between body length and the widest part of the lower bout.  For normal violin family instruments the LB* to BODY ratios seen are only: 5 to 9, 4 to 7, and 3 to 5.    

(*LB for Lower Bout, UB for Upper Bout, CB for Center Bout)

What do these ratios have in common?  They are simple reductions from 1 to 2.  Each of the traditional LB to BODY ratios is a slightly squat reduction from the outline being twice as long as wide.   So in each case, a 1 to 2 ratio is reduced by 1 unit.  So instead of 5 to 10, we see 5 to 9.  And instead of 4 to 8, we see 4 to 7.   In other words, these are 1 to 2 reduced by a fifth and a fourth respectively.   Similar if we reduce 1 to 2 by a third, 3 to 6 becomes 3 to 5.

So there is an idea behind the traditional length to width ratios in classical making, in each case the outline is almost twice as long as wide.

Even the exceptions demonstrate the rule.  If we look at very squat Lira's made by the Brothers Amati, we find the ratio 2 to 3 -- which is simply the most squat version of the idea possible.   And when we look at a very skinny Pochette made by Stradivari, we will find all the width to length ratios are simple 1/2 widths from the standard range of ratios.

Here these ratios are arranged for comparison, from narrowest to widest: 5/9, 4/7, 3/5, 2/3.


In classical making, we also see the Center Bout and Upper Bout widths sometimes related to the Body Length by ratios.

The Upper Bout is various related to the Body Length with a ratio of UB to BODY of 4 to 9, or 3 to 7, or sometimes even 1 to 2.

Similarly, we see the Center Bout to Body occurring in classical work variously as 2 to7, or 1 to 3.

But we also sometimes see the Center Bout and Upper Bout related to other bouts instead of to the Body Length.  So the Upper Bout is often in a ratio with the Lower Bout.  And the Center Bout is often in a ratio with either the Upper Bout, or the Lower Bout.

All together, there are basically 16 ratios variously seen with the bout widths in classical work.  These are:

  • Body Length to
    • Lower Bout: 2 to 3, 3 to 5, 4 to 7, 5 to 9
    • Upper Bout: 1 to 2, 4 to 9, 3 to 7
    • Center Bout: 1 to 3, 2 to 7
  • Lower Bout to
    • Upper Bout: 3 to 4, 4 to 5, 5 to 6, 6 to 7, 7 to 8
    • Center Bout: 1 to 2
  • Center Bout to
    • Upper Bout: 2 to 3

In an analysis of a broad collection of 88 instruments, including 13 cellos, 8 violas, 60 violins, and 5 small or special shaped violins, and 2 Lira, some of these ratios occur frequently, and some only rarely.

Also, many instrument show only 3 ratios, the minimum to establish the bout proportions, but in many cases the makers managed to show additional redundant ratios.  So for example, if the upper bout and lower bouts both relate to the body in the ratios 4 to 9 and 5 to 9, then they also relate to each other in the ratio of 4 to 5.   In a very few instruments, the makers managed to show as many as 8 relationships with bouts.

So, here is a break down of how frequently different ratios appeared in this sample of 88 instruments:

More than 50 occurrences:
       4 to 7 :: LB to Body,     4 to 5 :: UB to LB

Between 30 to 45 occurrences:
       5 to 9 :: LB to Body,     4 to 9 :: UB to Body,     2 to 3 ::  CB to UB,     1 to 2 ::  CB to LB,
       2 to 7 ::  CB to Body

10 to 15 occurrences:
         3 to 5 :: LB to Body,    5 to 6 :: UB to LB,    6 to 7  ::  UB to LB,   1 to 3 ::  CB to Body

Rare, under 5 occurrences:
         2 to 3 :: LB to Body,     1 to 2 ::  UB to Body,   3 to 7 ::  UB to Body,     3 to 4 ::  UB to LB,
         7 to 8 :: UB to LB


The application of these ratios is complicated a step further by the choice of calculating from the outside edge, or from the purfling line (3 purfling widths in from edge).

Here is a typical example of these ratios in applied in a 1669 Nicolo Amati violin:


In these case, the maker has managed to redundantly calculate the Lower Bout width as 4/7 BODY measuring from outer edges, and as 5/9 BODY measuring from the inside purfling lines.   The width of the Upper Bout from outer edges is then found as 6/7 the Lower Bout measured from purfling lines.  And lastly, the Center Bout is found as 2/3 the Upper Bout, measuring from outer edges.

As we noted before, the sequence of ratios seen between LB and BODY are all reductions of 1 to 2.  In other words, the Lower Bout is always somewhat wider than 1/2 the length of the BODY.

Similarly, all the ratios between UB and LB are reductions from 1 to 1.  3::4, 4::5, 5::6, 6::7, and 7::8 are all one unit short of a 1::1 relationship.   So in other words, the Upper Bout is always almost but not quite as wide as the Lower Bout.  And the Upper Bouts relations with the Body Length are consistent with this. They are all a little less or equal to 1/2 BODY.

Looking at the Center Bout, all the various ways of finding the Center Bout width ends up placing it at more than 1/4 BODY and up to 1/3 BODY.

Here we have a very straightforward approach in a 1672 violin by Francesco Ruggieri:

Of course, by basing LB and UB as 4/9 and 5/9 of BODY, there is automatically also a redundant 4 to 5 ratio between UB and LB.

And in his 'Milanolo', we see Ruggieri going a step further with redundant relations.  In this circa 1680 violin, Ruggieri reconciles and shows the historically most favored ratios 4 to 7 :: LB to BODY,  4 to 5 :: UB to LB, and 2 to 3 :: CB to UB.


In' Stradivari's 1683 small violin, the decorated 'Cipriano', we again see a very simple set of ratios:


And in the Tuscan-Medici Strad viola of 1690 we see the most favored ratios present:


We see the same principles used in Guarneri family work also.  Here we see a 1711 filius Guarneri violin:


This 1735 Bergonzi uses redundant relations to show 7 of the traditional ratios present:


In Del Gesu's 1735 small violin the Chardon, we see a 3 to 5 ratio of LB to BODY used to give the instrument extra width for its short body length.  Otherwise, its ratios are traditional favorites for violins.


And again, we see the same pallet of ideas used throughout Italian Classical, not just in the main Cremona families.   Here is a 1740c Santo Seraphin violin made in Venice.  It shows a simple plan, using the same traditional ratios.


And even in Del Gesu's wildest instruments from his last years, we still find these principles operating.  Here in the 1744 'Ole Bull', Del Gesu uses redundant relationships to show 6 of the most favored traditional ratios in one instrument.


We should also note that the choice of ratios seems somewhat to have gone through fashions.   The 4 to 5 :: UB to LB,  4 to 7 :: LB to BODY, and 2 to 3 :: CB to UB ratios were frequent from the beginning, and even seem to predate Amati family work.   But other ratios like the 5 to 9 :: LB to BODY,  4 to 9 :: UB to BODY,  and  2 to 7 :: CB to BODY gained frequency later on, ultimately becoming very common.

The use of redundant relations also seems to have emerged later.  The first experiments with this seem to start late in the Brothers Amati work, and then continue through Nicolo Amati's career.  This is also when the 1 to 2 :: CB to LB ratio appears.  After Nicolo Amati, there seems to be a lull in these trends until the early 1690s when such experiment resume in full force.

Along these lines, the 5 to 6 and 6 to 7 ratios for UB to LB are never very common, but in the years from 1690 to 1735 they become at least semi-frequent.

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What principles apply to the violin molds?

Among the various artifacts left from old Cremona, we have 18 instrument molds that are believed to be from Stradivari's workshop.

Here is the PB mold:


I've highlighted some of the characteristic features on this mold:


Besides the center line, there are four horizontal lines.  These are found on all the Cremona molds.  These lines intersect with the outline, defining and documenting the end points of the slots for the corner blocks.

There also are ten circular holes drilled through the mold.  These are shown in orange.  These holes are used to insert round sticks that are part of a system using binding chords and outside clamping blocks to pull the rib wood tightly onto the mold for shaping and gluing.

The two arcs and center point shown in yellow are found on all the molds.  These document the main rib height, and the reduce rib height used in the upper bouts.

Stradivari's markings identifying this mold are shown circled in blue.   Several 'P' molds exist, and this one has been further distinguished with an additional 'B' marking.

All these features are typical for the known old Cremona molds.

The outline and sizing of these molds needs to relate to the intended outline of the final instrument, but doesn't necessarily need to be a simple inset from the planned outline.

The risers on the mold for example can sometimes come in tighter.  And there can be logical inconsistencies between the vesici choices for the mold and the outline.  Also, the corner circles can be of larger radius than might be expected, allowing a somewhat less dramatic bending of the ribs.  As long as the mold supports the final outline appropriately, then all is well.

Lets start looking at the some of the geometry and proportions in these molds.

Consider the arcs marking the primary and reduced rib heights.   We've noted in earlier, that the violin ribs plus the edge heights seems to general give a combined height corresponding to about 1/9 the Body Length.  Though, because of wear and other factors, this ratio is often obscured in existing instrument.  Nevertheless, there are many clean examples.

Both the primary rib plus edges height of 1/9 BODY and the reduced rib height near the neck are evidenced in this 1710 Hieronymus II Amati violin.


So how would this same principle manifest in the rib height markings on Cremona molds?

We've also noted before, that for the most part the actual ribs themselves have a height 8/10ths that of the combined ribs plus edges.  Combining these bits of info, we might expect a Cremona maker to derive the primary rib height by first dividing the Body length in nine parts, then taking 8/10 of this to mark onto the mold.

We see exactly this evidenced for example in the P B mold:



In almost all the molds, the same Vesici proportions of 2::1::2 for upper bouts and 1::1::1 for lower bouts are used to shape the mold outline, just as shown in the P B example.    However, there are some exceptions where a rounder shape to the lower bouts is achieved by using a 4::3::4 proportion.

The bout width ratios are variously calculated from the mold edge, or from the final instrument's outline edge.  It should be noted that in concept, the mold edge corresponds to the 3 purfling width inset line that we've already seen is generally important in classical violin proportions.  At least nominally, this width also corresponds with edge width and with 1/10 of the 1/9 divisions of Body that gave the rib height markings for the molds.

The bout ratios seen in the molds are a conservative selection from the same one's seen in the instrument outlines.   Mostly, the mold Lower Bouts are based on 4/7 BODY.  Some use 5/9 BODY instead.   Most of the mold Center Bouts are based on 2/7 BODY.  Though 1/2 LB is also seen.  The Upper Bout ratios are more diverse.  2/3 CB, 4/5 LB, 6/7 LB, and 4/9 BODY are all seen.

Very importantly, the molds are the thing that actually sets the corner levels in the final instruments.  We see that the notches for the corner blocks are treated as one of the most important features of the classical molds.  The horizontal lines document and record these notches.  The end points for the corner blocks are precisely placed at the intersections of the these horizontal lines with the mold outline.

Given everything else we've seen, we might expect these horizontal lines to also be placed with method and tradition.

What we find is a mix of several methods.  Sometimes the corner levels are set by a ratio with the bout width, measured in from end block.   But we also find corner levels set by ratio with the Body Length.

Again, using P B for example, we see the upper corners based on 3 to 4 ratio with the upper bout width, measuring down from the top block.   And we find the lower corners placed at 2/5 Body Length.


Calculating variously from the mold line or the instrument outline (projected by adding 3 purfling widths), most of the lower corner levels are set based on a 2/5 BODY ratio.  And most of the upper corner levels are set on a 3 by 4 ratio with the upper bout.  But we also see examples using a 4 by 5 ratio with the bout.  And we see examples placing the lower corner levels by methods similar to the upper bout, in 3 by 4 or 4 by 5 ratios with the lower bout.

The secondary horizontal lines determine the length of the block notches.  For the lower blocks, these are usual set at 1/3 the distance from the lower bout line to the lower corner line.  In some of the upper blocks, this is similarly set as 1/2 the corresponding distance.  But sometimes the upper blocks merely copy the lower block length.

In the P B mold, we see the corner block lengths set as fractions of the distance between the bout and corner lines.



But, particularly with later molds, we see Stradivari experimenting with an additional method for setting block lengths.  Here in the G mold we see him using a different application of a 1 to 5 ratio to set the length of the upper blocks.   Though everything else about the corner blocks and corner levels is more traditional.


We see a consistent geometry behind the mold outline's shape in the center bout area.   This is consistently constructed using a circle of diameter based on the Lower Bout width.  Approaching the corners, this is joined tangentially by corner circles with 1/4 radius for the lower corners, and 1/5 for the upper corners.


Here this construction is shown for the G mold:

Since the CB widths are set by planned ratios, the large circles for the center bout curves are already placed horizontally, but have a degree of freedom vertically that needs to be set.  In many of the Classical molds this appears to be set by having the circle just touch the lower corner notch, at the intersection of the outline and the secondary horizontal line -- as seen in the G mold above.   But just as often, the large circle is aligned to the upper corner notch instead.

Though not exhaustive, we've covered most of the principles seen operating in the existing classical molds.  In addition to what we've noted, there are two apparently redundant relationships that are consistently seen in the molds that are believed to have been made before 1700.  First namely, in all these molds the line of the lower bout vesici occurs 1/5 along the length of the Body, variously calculating from the outline or the mold line.   And second, the upper corner line corresponds with 1/4 the distance from the upper bout line down to the lower bout line.  Sometimes a 2/5 distance is seen.


*******

We can uncover the system of geometry and ratios behind classical making, but how did they go about using and applying this system?

Even after we understand the geometry behind a scroll, or an f hole, we still don't necessarily know how they made use of that geometry.   Did they use a compass directly on the instrument as they carved?   Or did they prepare the shape on paper and then use a template to apply the shape as they worked the wood?  Did all the workmen understand the geometry, or just the master?

And which shapes did they construct first?   We know they took the sizing of the next part as a ratio of a previous part, so the sequence of working can matter.

These kinds of questions can actually be harder to work out than it was to uncover the geometry.  To some extent, we may never be able to unravel all of these kinds of details.  For one, the sequence of work doesn't necessarily leave any evidence, particularly if the work is clean and symmetrical.

But in many cases, there are clues and hints that tell us that one shape or size is dependent on another.  When this dependency is clear, it strongly suggests that the dependent element was created after the element it depends on.

And sometimes the classical variations in method gives some clue about the sequence of work.  For example on the molds, the corner block lengths are normally set as a proportion of the distance from the bout line to the corner line.  In itself, this suggests that both those lines are established before the block length is set.   But in some cases, the makers set the length of the top corner blocks by just following the length of the lower corner blocks.  In all cases, the lower corner block lengths are found by some kind of ratio, but in these few cases the upper blocks merely follows the lower block example.  This implies that the lower blocks were worked out first, then the upper blocks were worked out after.

There are cases of concrete sequence evidence in the working of the molds, and some less concrete cases.   All these bits of evidence seem to suggest the molds were mostly worked from the bottom up.  Also, the bout ratios tend to depend on the body length choice.  And, the circle geometry for the outline curve of the mold in the center bout depends on the exact placement of the corner blocks.  When we take all these bits of evidence together, there is a suggestive but not conclusive case for the sequence of work in these classical molds.

First, a center line is scribed and a body length is set.  Everything else depends on this choice.  Then a plan for the bout ratios is made.  The lower bout line is drawn at the 1/5 Body mark (calculated either by outline or mold line).  Then the lower vesici are constructed on this line, following the planned bout ratios. The length of the long arc joining the vesici is chosen to reconcile the bout line to the planned body length.   After the lower vesici are scribed, the lower corner line is scribed on a 2/5 Body mark.  And the secondary block line (setting the lower corner block lengths) is set by proportion, generally on the 1/3 mark from from bout line to corner line.   The corners must also be set horizontally.  We haven't mentioned this before, but these are usually set on an arc drawn up from the bout line.

Continuing the work from the bottom up, we can set the width of the center bout by planned ratio. Now we set the upper corner line using a 3 by 4 ratio from the top of the Body length and using the planned upper bout ratios.   We can now place the upper bout line in relation to the upper corner line.  Following the relationship seen in most all the earlier molds (pre 1700), we take 1/3 the distance from the upper corner line down to the lower bout line, and we place the upper bout line that distance above the upper corner line.  The upper vesici are now scribed on this line following the bout plan.  As with the lower bout, the radius of the long joining arc is selected to reconcile the upper vesici to the planned body length.   Again, the corners are set horizontally by drawing a cut off arc based on the upper bout line.

To finish the mold geometry, we need only work the curve for the center bout.   For this, we take 1/2 the lower bout width as radius for the main center bout circles.  The main circle is drawn to give the planned center bout ratios, and touching the planned corner notch (either upper or lower).  We then use tangential corner circles of 1/4 and 1/5 radius for the lower and upper corners to complete the curve shape into the corners.  Using the same 1/4 radius for the outer corner circles, we then draw up appropriate risers from the bout lines into the corners.

Lastly, we calculate the main rib height as 8/10 of 1/9 the Body length and mark a corresponding guide arc onto the mold.   One way to find the secondary rib height is to reduce the main rib height by 2 purfling widths.  Nominally, this can be taken as 2/3 of 1/10 of 1/9 the Body length.

Now the mold geometry is finished.

Was this work done on paper first, or straight onto a board for a mold?   Perhaps an effective method, particularly for a new original model, would be to work an outline geometry on the back board for an instrument.  Then use paper to transfer the body length and bout width choices to a board for a mold?


We can't know if this really was the actual work sequence for the molds, but its a plausible and effective sequence that fits the evidence well.   

*******

Conclusions:

We've found choices of geometry and ratios behind every feature of classical violin making.   In its way, this is like a DNA code behind classical making.

It seems that the classical makers were comfortable freely changing their choice of ratio, geometry, or the way they applied the geometry in order to modify any specific feature.  But from all we've seen, it also appears they would never even imagine working a feature without deciding its geometry and proportions.  Freehand was not their way.  This approach through geometry and proportion characterizes classical Italian violin making.

One consequence of their approach is variations are limited to choices within a system.  This lends a finite aspect to the range of choices, and makes the choices 'discrete' or 'notched' rather than continuous and infinite.   The complete collection of these discrete choices in an instrument is essentially its DNA code.   This provides an new avenue for studying classical instruments.  And it provides a clear avenue to contrast differences in the choices made in two instruments, or by two makers.

Their approach also ensures a certain structured nature to all the shapes of classical making.  It also means that the different components of instruments are put into relationship with each other.

Why would any of this matter or contribute to the success of classical making??   The ratios we see behind classical making are the same kind of simple integer ratios we see behind the pure intervals of musical harmony.  Perhaps that matters?   Also, the characteristic looseness and small asymmetries seen in classical work will translate physically into small dampenings and deflections of vibrations in the instruments.  For the physics of resonances and driven vibrational systems, that means softened and broadened responses.  Perhaps that matters?   The acceptance and even encouragement of natural wood movement during the build means fewer tensions locked into the fibbers of the wood.  Perhaps that matters?

But most of all, the system of geometry and proportions we've seen in classical Italian making provides mechanism both for repeating successes, and for exploring variations.  Again, this is like DNA.  It seems that across many generations, and many instruments, this mechanism helped classic makers create great instruments.  This system provides a very fine grained encoding of choices behind their making.  This classical approach ensured good and characteristic results even in the hands of the less famous Italian makers, while facilitating both the propagation of effective making, and the development of better making.






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