The question that forms the title to this page might seem a strange one.
After all, a violin doesn't seem to be a complicated contraption with moving parts, except for the tuning pegs, and it's fairly obvious what they are there for.
However, when they are bowed, the strings of a violin obviously do move, at least a very short distance; not only is this visible, but in the absence of vibration, a violin would make no sound.
And the body of a violin is a hollow box which allows the violin to produce a louder sound than the strings would produce by themselves. This already needs some explanation. After all, a regular acoustic violin doesn't need batteries, so without a source of power, how can it amplify a sound?
Of course, it doesn't. In electrical terms, what the body of a violin behaves like is a transformer: it performs impedance matching between the strings and the air, so that more of the energy of the strings' motion can impart motion to the air. The strings are small; they move the much larger surfaces of the violin, so now they're grabbing hold of more air.
When the strings are bowed, they move from side to side, in a pattern closely resembling a sawtooth waveform. And thus, on a simple synthesizer - or the SID chip of a Commodore 64 computer, as I can vouch for from personal experience - one chooses a sawtooth waveform when one wants to make a violin-like sound.
The strings move from side to side, with the violin beneath them. How, then, does the bridge of a violin do anything more useful than slide along the belly of the violin from side to side?
The rather crude diagram above attempts to answer that first question.
The tension of the strings holds the bridge firmly against the body, so the feet can't easily slide from side to side. But the shape of the bridge is chosen so that the side-to-side motion of the strings tends to cause the bridge to tip over from side to side. Thus, its two feet move vertically, in a direction perpendicular to the surface of the belly of the violin, thereby being effective in making its whole area push on a large amount of air.
There's just one problem. While one foot pushes down, the other one is going back up. So it looks as if a lot of this motion, although it's in the right direction, is going to cancel out.
To see how this problem is dealt with, we need to take a look inside the violin.
Although a violin looks nearly symmetrical from the outside, on the inside the two feet of the bridge are treated differently.
One foot has the "bass bar" beneath it.
The belly, or front surface, of a violin is made of a soft wood like spruce or pine (Stradivarius always used spruce). The bass bar is made of a hard wood, like maple, and it runs across a large portion of the height of the belly of the violin.
This helps the foot of the bridge that lies above the bass bar to move a large portion of the belly of the violin.
The back of a violin is made of a hard wood like maple, sycamore, or pearwood (Stradivarius always used maple). The other foot of the bridge sits almost on top of the "sound post", a wooden peg made of a hard wood, like maple, which connects the belly of the violin to the back of the violin.
Since the back of the violin is made of a harder wood than the front of the violin, that foot of the bridge now won't move as far as the other foot, which pushes on only the front and not the back. So the two feet don't cancel each other out.
The back of the violin does move, though, and in the opposite direction to the front of the violin. So together they either expand or contract the interior of the violin at the same time, with a bellows action.
The only problem with this otherwise ideal arrangement is that the sound directly radiated from the top of the violin will, therefore, be opposite in phase from the sound coming out of the f-holes in the top of the violin. Given the usual manner in which a violin is held by a performer, however, this is more of a problem for the cello and the double bass than for the violin or the viola.
This analysis is an oversimplification; the speed of sound is finite, and given the frequencies of notes played on the violin, this is significant for its acoustic behavior; so instead of being opposite in phase to sound from the belly plate, the phase relation of sound from the f-holes to sound directly radiated from the belly plate will depend on frequency.
However, based on this understanding, I wondered if the violin could be improved by replacing the f-holes on the belly with an aperture of the same size elsewhere; for example, on the upper bout, near the neck of the violin, on the side opposite the chin rest.
I see, however, that a similar idea has been had elsewhere, although in a case where the circumstances are different: the German luthier Thomas Ochs produces his own version of the Kasha guitar design by Michael Kasha with the sound hole placed on the side of the instrument.
Also, in the immediate postwar period, Julius Zoller made violins which had a set of small sound holes in the side instead of f-holes; however, these violins differed from conventional violins in other respects as well; their shape was different, and they had an extra string for sympathetic vibration.
The diagram below illustrates the notes played on the open strings of the violin, and also for two other members of the violin family, the viola and the cello.
The strings of the violin are tuned a fifth apart, as is true for those of the other instruments described in this diagram; those of the double bass are tuned a fourth apart, so it and related instruments are described in another diagram.
In addition to the violin, viola, and cello, the tunings for six of the eight instruments in the Violin Octet of Carleen Hutchins are shown in this diagram.
An earlier proposal for additions to the violin family, by Dr. Alfred Stelzner, is also noted in this diagram. He had also proposed a change in the shape of the instrument to improve its acoustic properties; among other things, the difference in size between the upper and lower halves of the body was increased slightly.
The violin family as we have it today does present some difficulties.
This image which I produced from an old photograph of a violin and a viola side by side shows that the viola is quite a bit larger than the violin. However, the viola is tuned a fifth lower than the violin; it may be larger, but it is not one and a half times larger in every direction.
To improve the viola, Jean-Baptiste Vuillaume invented an instrument called the contralto. This instrument basically looked like a viola with its body split in half, with an additional rectangular piece inserted, thus making it wider without changing the shape of its sides. While the result was found to have a satisfactory tone, and the principle of its design was intended to make it possible to play it in a similar way to which the viola is played, the instrument was found to be unwieldy.
More recently, the pellegrina was invented by David Rivinus in 1993 as a way to reduce the large size of the viola, thus helping to protect viola players from injuries such as carpal tunnel syndrome. The instrument has a somewhat unattractive appearance, looking like two left halves of a violin joined together, one upside down.
At first, I wondered if it might be possible to achieve the same thing with a shape like that of the contralto, but with the bridge, strings, and neck in an off-center position. On further consideration, in order to enlarge the instrument, but have everything reachable, while retaining a symmetric appearance, perhaps a solution would be to turn both sides of the instrument upside down, so that the smaller round part is nearer the player, which would then allow for the instrument to be widened as the contralto. Another option would be to thicken the instrument, as we will see below was done for the violoncello, except with a part at the bottom cut off so that a chin rest could still be used.
On the left, there are images of the front and the side of a violin, from an old photograph; on the right, images of the front and side of a cello, from another old photograph.
The images are not to the same scale; for comparison, the two instruments have been made the same height, whereas a cello is in fact much larger than a violin.
As can be seen from this image, a cello is considerably thicker, in proportion to the size of its front and back, than a violin.
This was done to increase the internal volume of the cello, thus lowering some of its resonant frequencies, while limiting its size.
So what is illustrated here is that the wide range of musical pitch, of audio frequency, covered by the violin family cannot be matched by a proportionate increase in size, because the awkwardness of having larger instruments is not felt to be a reasonable disadvantage to accept for the improvement in tone that would be obtained. Of course, since the cello exists, a larger viola that would have to be played like a cello could still be played; and since the double bass exists, a cello that has a larger front and back, but more violin-like proportions, again, could still be played; but in both cases, making the instrument larger would limit what players could do with it in performance, and it would make it awkward to seat as many players of each instrument in an orchestra.
So a balance was found between the notes each type of instrument would produce and how much of the agility of a violin player performers in that range of sound would need to enjoy for the purpose their parts would serve in music.
To allow a clearer comparison of the sizes of the violin, the viola, and the cello, the diagram on the right shows several dimensions of such instruments, following the notation used in the book Antonio Stradivari: His Life and Work by W. Henry Hill, Arthur F. Hill, and Alfred E. Hill. This book included, as an appendix, tables of the dimensions of a number of instruments by Stradivari and others.
Here are just a very few of the measurements given there, to give an idea of the relative sizes of these three instruments:
A B C D E Stop Violin 14 1/16 8 1/4 6 5/8 1 1/4 1 3/16 --- Stradivarius, 1720 Violas 18 7/8 10 3/4 8 5/8 1 11/16 1 9/16 10 1/4 Stradivarius, 1690 (Large size) 16 3/16 9 9/16 7 5/16 1 1/2 1 7/16 8 5/8 Stradivarius, 1701 (Small size) Cellos 31 3/8 18 1/2 14 1/2 4 3/4 4 1/2 16 3/4 Stradivarius, 1690 (Tuscan) 29 7/8 17 3/8 13 5/8 4 5/8 4 1/8 15 3/4 Stradivarius, 1711 (Duport)
The Duport cello, being made after 1710, is an example of one made using the "Forma Buono" cello mold, and these proportions are now considered the ideal for a cello.
As for the viola, current violas are all in the "small size" class. If the size of a viola were in proportion to its pitch, it would have a body size of 21 3/32 inches, and a thickness at its base of 1 7/8 inches, which would be completely impractical, unless it were to be played as a cello is played.
Using sizes from the table above, which don't necessarily correspond to the actual instruments the old pictures of which were used in preparing this image, here is an image providing a graphical comparison of the size of a violin, a viola, and a cello:
If a cello were made as a scaled-up violin, its body height would be 42 3/16 inches, but the thickness at its base would only be 3 3/4 inches, less than that of today's cellos, which indeed, as noted, use a larger depth as a substitute for a full height.
Looking at an adult playing a cello, one might think that there would be no problem in making a cello in that larger size resulting from maintaining the same proportions as a violin, since the metal rod reaching down to the floor from the bottom of the cello, called the pin, is certainly much more than one foot in length. However, this would also enlarge the distance between the points at which the fingers would have to press the strings to the fingerboard to play different notes. This is the reason why the strings on a double bass are tuned in thirds rather than in fifths, so that fewer notes need to be made with any one string.
Of course, one could imagine a cello of this form being made with its strings tuned an octave higher than the strings of a double bass. The lowest note that could be played would be E instead of C, but it would cover much of the same range.
A viola that was a scaled-down cello instead of a scaled-up violin would be 14 15/16 inches in height, but 2 5/16 inches thick at the base; perhaps such an instrument is too thick to hold under the chin, but as there appears to be room to put a shoulder rest under a viola, perhaps not.
Note that all three instruments are slightly wedge shaped, being thicker at the bottom than at the top (in the sense of towards the neck, not towards the belly).
The diagram below shows the tunings for the double bass and related instruments.
Shown there in addition are the two remaining members of the Violin Octet, and the Octobass, proposed by the noted luthier Jean-Baptiste Vuillaume.
To assist with the discussion that is to follow, I am adding two diagrams which illustrate the different parts of a violin. Incidentally, rather than attempt to draw a picture of a diagram myself in a paint program, I have used illustrations from old books in the public domain as my starting point. It may be of interest to note that the photographs of a violin used in the diagram immediately below are of the Earl Stradivarius, named after the Earl of Westmoreland, and they came from a book with photographs of several of the violins in the collection of Royal De Forest Hawley.
In the first diagram above, one thing to be noted is that two completely different parts of the violin are called the "saddle" of the violin by different sources.
According to a diagram of the parts of the violin I saw on one site, the curved portion of the neck of the violin, near where it joins the body of the violin, is the saddle.
Other diagrams use that term to refer to a small rectangular piece of wood which protects the belly of the violin, and its purfling, at the bottom from the pressure of the tail gut. This part is sometimes also called the "rest", not to be confused with the chin rest, not shown in these diagrams.
As we've already noted, the body of a violin has a back carved from a piece of maple, or sometimes another hard wood, and a front, called the belly, carved from a piece of spruce, or sometimes another softer wood.
Around the sides of the violin, maple is also used. In this case, thin strips of maple are bent using heat, to form the upper bout, the two center bouts, one on each side, and the lower bout. These are held in place by carved solid blocks of maple, the corner blocks.
As well, the upper end block and the lower end block brace the upper and lower bouts.
The top nut is a piece of metal against which the strings rest as they come from the tuning pegs and go to the bridge. This minor part serves two important functions: it defines the lengths of the open strings (that is, it defines how long the vibrating part of the string is when the fingers aren't pushing the strings against the fingerboard to shorten them for playing a higher note), and it ensures the strings are a distance above the fingerboard right from the start on their way to the bridge.
The tailpiece, which holds the far end of the strings, isn't permanently attached to the body of the violin. Instead, it has a metal rod, bent into a U-shape, and also bent towards the back of the violin, called the tailgut, which hooks on the tail button of the violin. The tension of the strings holds it in place, against the saddle at the bottom of the violin.
The fingerboard is usually made of ebony. It does not have frets, unlike the similar part of a guitar; one of the features of a violin is that, like a trombone, it can play notes that are continuously varying in pitch, with no striking change in sound quality between pitches that correspond to the piano keys and those that are between them.
Incidentally, note that the corners of the violin, due to the fact that the corner blocks are solid, are not part of the shape of the interior of the violin. This has been used in the shape of some modern violins made of carbon fiber, which omit the corners in their external shape - and, as well, in the nineteenth century experimental violins made by François Chanot had this form, as well as having f-holes of a simplified shape.
I was able to find an old image of a Chanot violin, but it had a heavy Moiré effect, so I had to process it somewhat heavily:
The earliest known bowed instrument is the Ravanastron of India. It is very similar to the Erhu of China, or the Haegeum of Korea. It has two strings with tuning pegs, and a small round sound body with a membrane as its upper surface. The strings are made of silk.
Other old bowed string instruments are the Rebec, and the Viol, which was the immediate predecessor of the violin.
The oldest surviving violins are those made by Andrea Amati of Cremona; at one time, violins made by Gasparo Bertolotti, usually known as Gasparo da Salo, in Brescia were thought to have been older, but recent historical research has overturned this belief.
Another person once mistakenly believed to be the earliest known violin maker was Gasparo Duiffoprugcar; it is now believed that while he made the bowed instruments which preceded the violin, he did not make any actual violins. His last name is a variant spelling of Tiefenbrucker; a different German surname, Tiefenbacher, found its way to Canada in a different variant spelling: Diefenbaker; so he does not belong to the same house as a former Canadian Prime Minister.
In Cremona, Andrea Amati was succeded by his son Gerolamo Amati, and then he in turn was succeded by his son Nicolò Amati. Each of them produced violins that were considered to be even better than the violins of their fathers.
It has been generally accepted that Antonio Stradivari was one of the apprentices of Nicolò Amati. However, I am inclined to accept the conjecture offered in the book Stradivari's Genius: Five Violins, One Cello and Three Centuries of Enduring Perfection by Tony Faber, and also echoed by the book Cremona Violins: A Physicist's Quest for the Secrets of Stradivari, by Kameshmar C. Wali, which is about the researches of William F. "Jack" Fry, that Stradivari only worked for Nicolò Amati in the capacity of a woodworker, formerly apprenticed to the woodworker Francesco Pescaroli, to produce decorated violins, and then he later became an apprentice of another violin maker, likely Francesco Ruggeri, based on similarities in their technique of carving the back and belly plates of the violin in order to tune them properly.
Francesco Ruggeri and Andrea Guarneri were undoubted apprentices of Nicolò Amati. Andrea Guarneri was succeeded by his sons Pietro Guarneri and Giussepe Guarneri; the latter had a son also named Giuseppe, famed as Guarnerius del Gesu. Although today a Guarnerius is recognized as a reasonable alternative to a Stradivarius, while Stradivari was successful and wealthy, Giuseppe Guarneri the younger struggled to make a success of his business.
Antonio Stradivari was succeeded by his sons Francesco Stradivari and Omobono Stradivari, who hired Carlo Bergonzi, already an established violin maker, to assist them.
Also, Lorenzo Guadagnini made violins in Stradivarius' workshop, and his son Giovanni Battista Guadagnini is often considered to be the third greatest maker of violins after Stradivarius and Guarnerius.
The old violins of Cremona are held in very high repute, particularly those of Antonio Stradivari; as they were considered the finest violins available at the time they were made as well, about a thousand were made, and a commonly quoted figure is that about 650 of them survive. Those of Guarnerius del Gesu, of Giovanni Guadagnini, and some of those by Carlo Bergonzi, and a select few other early luthiers, are also considered to be nearly as good, but there are far fewer of those instruments extant, so these violins, and other fine violins from this era of violin making, do not greatly reduce the scarcity of violins of the first rank from what it would be if no other violins had even approached the excellence of a Stradivarius - although the scarcity is such that we must be thankful for each fine violin that exists.
Fritz Kreisler is associated with a Bergonzi, and Vanessa-Mae with a Guadagnini.
One name that comes up is that of Domenico Montagnana; he was a luthier in Venice, rather than Cremona or Brescia; while the violins of Stradivarius are praised, among other things, for making it easier for a performer to obtain the best tone from them, those of Montagnana are notoriously difficult to play; yet, they are still highly valued because of the beauty of the tone that can be obtained from them by a sufficiently able performer. Yo-Yo Ma is a performer associated with one.
The earliest surviving Stradivarius was made in 1666, and already bears the stamp of his genius, but his earliest violins were not necessarily superior to those of Nicolò Amati. By 1683, this had changed, and his violins were already superior to any others, but they had not yet reached the peak of their quality, as displayed by the violins of his "golden period" from 1700 to about 1725.
While the violins of Stradivarius were recognized as the best violins available during his lifetime, as the years passed after his death, his reputation passed into obscurity. It was revived by the efforts of Luigi Tarisio in rescuing forgotten Stradivarius violins, and incidentally making a fortune through buying them cheap and selling them dear, and by a concert by Giovanni Battista Viotti in 1782 where he played a Stradivarius violin made in 1709.
Before 1782, it was widely considered that violins such as those made by Jacob Stainer represented the summit of the art of violin making; as Stainer died in 1683, he was a contemporary of the Cremonese violin makers, but he worked in the Tyrol, part of Germany.
It is generally believed today that not only are the violins of Stradivarius and the other great Cremonese masters are the best ever made, but that the violins of today, even those made by the best luthiers of today, do not even come close.
There is clearly only one possible explanation for this: space aliens!
In fact, many explanations have been offered. One of the earliest is that Stradivari used a special secret varnish; there is even a legend that his rival Guarneri once arranged a break-in at the Stradivari shop to steal its formula. Another old theory is that violin production led to the extinction of a tree known as the Balsam Fir; but, apparently, this had never actually happened.
Three recent theories that attempt to account for the superior sound of the old Cremonese violins are based on observations that the variation in the density of the wood associated with its grain is less pronounced than in modern wood.
This is variously attributed to the trees growing during a period of low solar activity, resulting in cooler weather, to a bacterial infection that the trees suffered (with a proposal that a particular fungus can be used to more safely produce a similar effect), and, by Dr. Joseph Nagyvary, to a preservative bath given to the wood.
In the case of Dr. Nagyvary, he found that the techniques he tried of applying a preservative bath to wood did not result in the level of mineral content he found in old violins, so instead he uses wood salvaged from the waters of Lake Superior which does have the desired characteristics.
Looking at information about wood preservatives considered safe to use today, most of them were chemicals with long names that doubtless were not available in the time of Antonio Stradivari. The only exception to that I noticed was boric acid, a substance also used in dilute solution as an antiseptic eyewash. It is suitable for use for items that will be used indoors, but water can remove it from wood - so it shouldn't be added prior to "stewing", to be explained below.
In 1938, the physicist Frederick A. Saunders reported that a violin by Franz Josef Koch had a sound confirmed by measurement as closely resembling that of the fine old violins. Franz Josef Koch made violins in Dresden during the 1920s, and had designed them based on his own scientific studies. In particular, he used "a resin that imparted uniform qualities to the wood"; thus, reducing the effect of wood grain has a long history as being considered as an important step in attempting to attain the heights reached by the Cremonese masters.
Another old technique to modify the wood used in a violin was called "stewing": the wood was gently heated in a salt solution. The purpose of this was to accelerate the degradation of hemicellulose in the wood that would take place as the wood ages normally. Hemicellulose absorbs moisture to a greater extent than other parts of wood, so reducing the amount of hemicellulose in wood helps to prevent changes in sound quality, or even cracking of the wood, resulting from changes in humidity. This technique was known and used in old Cremona; there is reason to believe that it was used by Guarneri del Jesu, but there is also reason to believe that it was not used by Antonio Stradivari.
A brochure from Yamaha notes that a technique called A. R. E. (Acoustic Resonance Enhancement), used with their premium Artida series YVN500G and YVN500S violins, accelerates the natural maturing process of the wood, and so other approaches have been attempted to obtain the improvement that time brings to a violin's sound.
The book by Simone F. Sacconi, I "segreti" di Stradivari (The "Secrets" of Stradivari; the English translation has the same title, but without the quotation marks) denies that Stradivarius himself had any such secret; his superiority to other Cremonese makers of his day was solely due to his own superior craftsmanship.
Since there is a narrow gap between Stradivarius and Guarnerius violins, but a wide gulf between them and other fine early ones (Bergonzi, Guadagnini, and numerous others) and later ones, that could well be entirely true, while yet leaving a lost Cremonese secret that needs to be discovered.
While no "gimmick" would enable an indifferent luthier to make violins the equal of a Stradivarius, if the most experienced and accomplished luthiers of today are unable to approach the excellence of the violins of the Cremonese masters, it is not at all unreasonable to consider the possibility that some unknown factor, such as the lack of the right wood or the right varnish, is standing in their way.
But there is also the null hypothesis: perhaps all the fuss about Stradivarius is just due to hype, and we've all been fooled by Luigi Tarisio and his successors, who have been laughing all the way to the bank.
While some blind listening tests of recent date have lent some support to this notion, I am inclined to reject out of hand an idea that basically requires that every single one of the world's greatest violinists is either a liar or a fool.
Given that the excellence of a Stradivarius lies more in its playing qualities than its sound, and that to get the best out of even a Stradivarius requires time to become fully familiar with its individual qualities, blind testing under controlled conditions could easily be missing what is important.
But, in any case, objective scientific measurements carried out by Heinrich Dünnwald show that there are very real differences between the classic Cremonese violins and the violins made by the master luthiers of the present time.
A famous graph, reproduced in a number of books and papers, showing the responses of a large number of different violins divided into three groups shows the following:
The frequency response of the old Cremonese violins does not look, from a casual visual inspection, to be too much different from that of inexpensive modern factory-made violins. But there are definite differences.
From about 900 to 1800 Hz, the response of the old Cremonese violins is significantly lower. Excessively prominent resonances in this region tend to make a violin sound "shrill", and so reducing them is an improvement.
From about 2000 Hz upwards, all the way to 7000 Hz, as far as the measurements were taken, the response of the old Cremonese violins is significantly higher. Stronger response in this area is associated with a "silky" tone, hence a better sound, and with a violin that is able to project its sound better in a concert hall.
The frequency response of the violins by modern luthiers, on the other hand, looks strikingly different from that of cheap factory-made violins.
After about 2500 Hz, the frequency response of a cheap factory-made violin goes downhill, and so does that of an old Cremonese violin, but more slowly. In the case of the ones by fine modern luthiers, though, the frequency response hardly declines at all from 2500 Hz up to at least 6500 Hz.
So the violins by modern luthiers seem to, at least in this respect, have an even better sound, and definitely their sound should project better.
But in the area from 900 Hz to 1800 Hz, where the frequency response of the old Cremonese violins is reduced, that of the violins from fine modern luthiers is not, but is instead even higher than that of cheap factory-made violins.
Incidentally, at least one paper has been published criticizing Dünnwald's method on the basis that it obscures many real differences between violins that can be made evident by bowing their strings rather than hammering their bridges. This, however, does not reduce what I find significant about his findings; it would make questionable any conclusion based on his measurements that some distinction does not exist, but it doesn't diminish the validity of a conclusion that a difference does exist, if the test that found it is limited in its sensitivity.
Here are the conclusions I draw from this:
The differences between the old Cremonese violins and those made today are real, and not due to a placebo effect.
However, since the fine luthiers of today are able to make the sound characteristics of their violins differ from those of cheap ones to an even greater extent than the old Cremonese masters did, the suspicion naturally arises that it is not that they aren't making violins that sound like those of Stradivarius because they are unable to, but because they aren't trying to.
Given the high prices that a Stradivarius violin can fetch, that is something that requires an explanation. And the most obvious possible explanation would lead to the conclusion that there is a grain of truth to the very hypothesis that these graphs have disproven - that the Stradivarius phenomenon is due to hype generating a false mystique.
If they aren't trying to make violins as good as those of the Cremonese masters, it could be that they feel that were they to do so, the virtues of their violins would not be recognized, and so they're better off concentrating on improving sonic aspects of their violins that are more obvious during a short demonstration in a violin showroom.
That may not be the only explanation, though; reducing the response in the 900 Hz to 1800 Hz area may be a particularly difficult task, or, at least, it may be particularly difficult to do so without affecting the sound of the violin at other frequencies in such a way as to make it sound "dead". So there could still be a lost secret, even if the secret is not a simple one, but which instead relates to how a violin is tuned by carefully shaving away wood from its top and bottom plates.
Two researchers, who have both been praised by their supporters and advocates as having approached the classic Stradivarius sound, stand out above the rest.
The luthier Carleen Mayley Hutchins, who worked with the physicist Dr. Frederick A. Saunders until his passing in 1963, although she is most famous for proposing a set of eight proportionally-sized instruments to replace the four instruments of the present-day violin family, made an extensive study of plate tuning, showing how and where the thickness of violin plates should be varied to cause their modes of vibration to resemble those of the fine old violins.
The physicist Dr. William F. "Jack" Fry also studied the tuning of violin plates, paying particular attention to how the fibers in the wood affected the behavior of the plates. One of his early basic findings was that the back of the violin should be made thinner on the side opposite the sound post, so that the part of the back around the sound post could move as a unit. In dealing with the portion of the front of the violins between the f-holes, which he found to be relevant to the violin's high-frequency response, he did find it useful to increase the stiffness of the wood across the grain. However, he achieved this without using exotic ingredients; he used normal casein-based wood glue, diluted half-and-half with water to improve penetration.
(This is not necessarily an ideal solution; it works by adding stiffness across the grain, whereas wood preservatives and the other items suggested tend to reduce stiffness with the grain. Since varnishing a violin adds stiffness, and is detrimental to its tone, a question is raised. However, if stiffness were all bad, we would be making the fronts of violins out of rubber, or at least balsa wood; if the wood has more stiffness, then reducing the thickness of the wood will balance that. Since Franz Joseph Koch used a resin to equalize the wood, it is likely that his method also added stiffness rather than reducing it.)
The frequency range that leads to shrillness in a violin, was, according to him, associated with modes in which the bass bar tilts in response to impulses from the bridge. One way to reduce this would be to make the bass bar effectively symmetrical around the foot of the bridge (that is, since it extends further above that point than below it, make it thinner above the bridge so that the product of the amount of wood and its distance from the foot of the bridge, considered as a fulcrum, is balanced), but instead of doing that, he thinned the wood in selected regions of the front of the violin which he referred to as "Stradivari holes", as they were areas which were observed to be thinner on some Stradivarius violins.
In addition, he devised tools which allowed him to continue removing wood from the inside of the top and bottom plates of a violin after it was assembled, by reaching with the tools through the f-holes.
Subsequently, on a visit to the Stradivari Museum in Italy, he saw tools used by Stradivari which he believed were intended to perform a similar function.
I am particularly impressed by that particular finding, as it is indeed obvious that assembling a violin will allow its actual sound qualities to be heard. Of course an experienced luthier can anticipate what a violin will sound like from the sound of the plates, but there might be gaps in that knowledge, while having the actual sound available to hear obviously diminishes the room for uncertainty and error.
If contemporary luthiers have, essentially, all been making violins with one hand tied behind their backs, it's hardly surprising that they have been unable to match the achievements of Stradivarius!
However, Dr. Fry's explanation of the workings of the violin has inspired me to propose a radical redesign of the violin.
Not nearly as radical as some suggestions as have been proposed, but none the less more radical than anything that he advanced.
The back of the violin would continue to be made of maple, of two pieces side by side with the grain running vertically.
But since wood is stronger along the grain, and in the front of the violin, still made of spruce, sound is transmitted from the bass bar, I propose that most of the front of the violin should be made from wood with the grain running from side to side. One result is that this would remove the need to stiffen the wood between the f-holes with diluted glue, as the natural grain of the wood would now favor the direction in which sound needs to be transmitted.
The red lines in the diagram distinguish between pieces of wood with their grain running in different directions; most of the front has vertical lines, but some has horizontal lines.
The blue lines are the boundaries between the pieces of wood from which the front would need to be constructed. No glue would be used to join the pieces of wood along the two blue lines that are entirely within the material to be removed for the f-holes. The slight slope of the top and bottom of the pieces with vertical grain is so that while a vertical pressure keeps the three pieces with horizontal grain pressed against each other as the glue dries, a horizontal pressure can keep the pieces with vertical grain also tightly pressed against the pieces to which they are being glued without conflict or interference.
Instead of the grain running strictly in a vertical direction, the green line in each piece shown with vertical grain show a possible actual direction for the grain so that these pieces perform their intended function of providing a strong, effective connection between the top and bottom of the violin.
Note also that the large top and bottom pieces with horizontal grain may need to be divided into two pieces because of limits to the size of available pieces of wood with grain running in that direction. In that case, the split would be along a horizontal line in the middle of each piece.
In addition to the four traditional instruments of the violin family, and various intermediate instruments proposed from time to time, such as Carleen Hutchins' Violin Octet, mentioned above, violin family instruments in a number of other sizes exist.
These, however, although they differ in size, have their strings tuned to the same pitches as the regular instruments to which they correspond.
Thus, you may see advertisements for violins which, instead of having the standard size, called 4/4 size, are in sizes ranging from 7/8 down to 1/32. The fractions are conventional names for the sizes; a 1/2 violin is not actually one-half the length of a regular violin. These are intended to allow young children to have instruments suited to them as they learn to play the violin.
A Sears, Roebuck catalog from 1912 notes that one of the violins offered is also available in 3/4 size, "especially adapted for children from 9 to 14 years of age", and another is also available in "ladies' or 7/8 size"; the 1/2 size and 3/4 size for violins are also referenced in a 1917 catalog of John Friedrich and Brother, so this fractional notation for violin sizes predates the introduction of the Suzuki method of violin teaching in the postwar era.
A French book, "L'Art du Luthier", from 1903, gives an extensive table of recommended dimensions for different instruments of the violin family, in millimetres, from which I will give the value for the length of the body:
Violin, 1/4 size 297 mm Violin, 1/2 size 302 mm Violin, 3/4 size 332 mm Violin 358 mm Viola (large model) 473 mm Cello, 3/4 size 653 mm Cello 770 mm Double Bass 1m 130 mm
An even older source, from 1895, gives 14 1/8" as the body height of a full-size violin, 13 7/8" as that of a ladies' violin (not explicitly referred to as 7/8 size), and 13 1/8" as that of a three-quarter size violin, and I have now seen references in English to the three-quarter violin as early as 1866.
A German book, from 1892, gives these dimensions for a number of instruments from the violin family:
Violin, 1/2 size 320 mm Violin, 3/4 size 335 mm Violin 360 mm Viola 400 mm Cello 760 mm Double Bass, 1/2 size 1m 105 mm Double Bass, 3/4 size 1m 108 mm Double Bass 1m 110 mm
Here, the viola is definitely one based on a small-size model, Stradivarius' small model being 411 mm long.
Another source gives the body size for a 7/8 violin as 343 mm, but it gives that for a full-size violin as 355 mm rather than 358 mm.
A web site giving modern values for different fractional sizes of the violin, however, gives considerably different values for the sizes, and so does a second modern web site, and then I found slightly different values in a PDF document on yet a third site, and significantly different ones on a fourth site in French, and I give the sizes from all four below:
Violin 356 mm 355 mm 356 mm 355 mm Violin, 7/8 size 345 mm 345 mm Violin, 3/4 size 335 mm 335 mm 335 mm 330 mm Violin, 1/2 size 310 mm 320 mm 320 mm 317 mm Violin, 1/4 size 280 mm 285 mm 285 mm 292 mm Violin, 1/8 size 255 mm 255 mm 260 mm Violin, 1/10 size 235 mm Violin, 1/16 size 230 mm 230 mm 216 mm Violin, 1/32 size 215 mm
and the first of these sites also notes that Suzuki sizes for 1/10 and 1/16 violins are significantly smaller than the traditional sizes, being similar to 1/16 and 1/32 sizes on that scale respectively; the sizes given on the fourth of those sites seem to correspond to that.
The dimensions from the third source at least seem to reflect the best dimensions available from the first two, but I'm not decided as to whether I should regard it as authoritative, or possibly derived from an inconsistent mixture of sizes from other sources. Perhaps plotting these sizes, and the old French ones, on a graph might help me see if there is some system to this.
Of course, neither 302 mm nor 310 mm nor 320 mm is anywhere near half of 358 mm; if we were to divide 358 mm by the cube root of 2, the result is 284 mm, so the fractions do not even represent volume.
Since 1/4 size is 285 mm in some cases, I suppose one could try the sixth root of each fraction. That would give:
4/4, full size 356 mm 7/8 348 mm 346 mm 3/4 339 mm 336 mm 1/2 317 mm 3/8 302 mm 299 mm 1/4 283 mm 3/16 269 mm 267 mm 1/8 252 mm 1/10 242 mm 238 mm 1/16 224 mm 1/20 216 mm 212 mm 1/32 200 mm
These sizes don't correspond too badly to the modern values given in the previous table, except that the "1/32" violin would have to be called a "1/20" violin instead, which perhaps splits the difference between "traditional" and "Suzuki" sizes.
On further reflection, it seems likely to me that the fractions should be considered to be merely conventional sizes, for steps in a uniform logarithmic scale of sizes, with the additional 7/8 fraction indicating an intermediate step. So either 3/16 or 1/5 would indicate the same size, halfway between the 1/4 and 1/8 sizes. Sizes on that basis are shown in the second column; only those for the intermediate values are changed, generally becoming slightly smaller.
The values for 1/2 size and 1/4 size in the French source seem to be so close to each other (and to the size given as that of a 3/8 violin above) that it doesn't seem that those sizes could be the basis for a systematic scale.
In proportion to the ideal size for a viola, the actual size of the instruments commonly in use is equivalent to that of a violin 275 mm in length. Thus, a viola is really a "1/4 size" viola, or thereabouts, which makes it quite understandable that its sound quality is compromised. While the cello and double bass are also available in smaller sizes described in fractional terms, smaller size violas instead have their sizes given directly in inches of body size.
Incidentally, the usual size for a double bass is considered not to be the full size for a double bass, but instead its 3/4 size. Full size is considered to be 1 metre, 160 mm, and 3/4 size 1 metre, 110 mm, so the size for the double bass given in the old French book quoted above is intermediate between these two values. On the other hand, the old German book quoted above gives exactly the current 3/4 size usually used as the full size for a double bass.
The 1912 Sears, Roebuck catalog noted above advertises both 3/4 and 1/2 double basses, but no full size ones, so this would seem to corroborate that the usual size is designated the 3/4 size.
Also, Antonio Stradivarius made at least a few small violins. Two of them, the 'Cipriani Potter' from 1683, and the 'Belle Skinner' from 1736, have a body height of 339 mm, which would seem to make them 3/4 size violins (although I have seen the Cipriani Potter referred to as a 7/8), and another, the 'King Maximillian Joseph' from 1702, has a body height of 347 mm, which would seem to make it a 7/8 violin. The 'Gillott' from 1720, having a 267 mm body size, doesn't correspond to one of the more usual values, yet it is close to being midway between the size of a 1/4 violin and a 1/8 violin, which suggests that a standardized system of sizes for violins has long been in existence.
A paper discussing violin sizes which I encountered showed examples made by three different members of the Amati family, along with another example by Lorenzo Storioni. Another source notes that Andrea Amati, in addition to 355 mm full-sized violins, also made some of 342 mm, or 7/8 size.
I was surprised that the sizes Antonio Stradivari used for small violins were, almost to the millimetre, the same as those I had listed in my final attempt at laying out standardized values for the fractional sizes of violins.
Of course, the fact that the ratio between the sizes of a full-sized violin and a 1/2 size violin was the sixth root of two, and thus if one adds in the 3/4 size violin in a uniform logarithmic scale, the ratio between successive sizes is the twelfth root of two - which is, of course, a value very familiar from earlier pages in this section, as it is the ratio between frequencies, and also wavelengths, and therefore string lengths, for successive notes one semitone apart in the equal-tempered scale.
But at first this seemed to be an amusing coincidence, or due to perhaps some sentimental attachment on the part of violin makers. But perhaps it might simplify memorizing the appropriate placement of one's fingers when playing a violin and moving across different sizes of instruments.
Stradivarius also made a few guitars. And what would be more reasonable than that if there is a need for reduced-size violins for younger players, the same might apply to guitars? And why not use the same set of sizes for both instruments?
And then the obvious dawned on me. If the ratio between instrument sizes is the same as the ratio between string lengths for successive notes, then a single template or jig would suffice either directly, or for the making of jigs for the individual instruments, to control the placement of frets on the fretboard of a fretted instrument such as a guitar.
Of course, on a violin, it isn't the size of the body that determines the spacing between fingers on the fingerboard, it is the distance between the nut and the bridge. And an analogous rule applies to the guitar. In the case of the violin, one can see from the image above comparing the violin and the cello that not only is the cello proportionately thicker, but its neck is proportionately shorter. And some small-sized violins have necks that are larger in proportion than on a regular violin, to make fingering more convenient: the instrument may be intended as an easily portable one for an adult player instead of one suited to a child.
However, it may be that the fractional sizes I have worked out, while they were at one time the traditional sizes for smaller violins, are not the sizes used most often at present. After all, violins don't have frets, so the issue I've identified that would encourage the particular sizes I've proposed for guitars doesn't really exist for the violin.
I have seen the figure of 92.5% as the ratio between successive sizes quoted in a few places.
And if the fractions are merely conventional sizes, could it be that the steps between 4/4, 3/4, and 1/2 are the same as those between 1/2, 1/4, and 1/8?
Given that the sizes normally seen today are 4/4, 3/4, 1/2, 1/4, 1/8, 1/16, 1/20, and 1/32 - and also providing a nice explanation for the fact that when 1/20 was inserted for instruments used under the Suzuki system, it had the same size as 1/32 as used by others... could it be that 3/4 is no longer being treated as at a half step, and instead the size of a full step has been shrunk, so that things average out, making the sizes of at least some of the violins in the sequence still similar?
The image above compares violin body sizes, along a logarithmic scale, between, on the lower scale, the traditional system I have inferred, and on the upper scale, the possible modern system that I am now suspecting may exist as an alternative.
Fractional size instruments suffer in sound quality. But players of the viola often end their careers with injuries, and this is not even uncommon to violin players.
Given that the cello has a smaller body size than its pitch would call for, but its sound is still satisfactory, but with a sweeter quality, due to its additional thickness, it seems worthwhile to examine in detail how greater thickness can compensate for smaller body size.
Here, then, is a chart of how this could work out:
Violin Viola Cello A: 382 mm A: 764 mm 34 mm E: 68 mm 4/4 A: 357 mm A: 535.5 mm A: 714 mm A: 1071 mm E: 35 mm E: 53 mm E: 70 mm E: 90 mm 7/8 A: 346 mm A: 692 mm E: 35.5 mm E: 71 mm 3/4 A: 336 mm A: 672 mm E: 36 mm E: 72 mm 1/2 A: 317 mm E: 37 mm 1/4 A: 274 mm A: 411 mm E: 39 mm E: 60 mm 1/8 A: 253 mm A: 380 mm A: 506 mm A: 759 mm E: 41 mm E: 62 mm E: 82 mm E: 105 mm 1/10 A: 238 mm A: 357 mm A: 714 mm E: 42 mm E: 64 mm E: 128 mm 1/12 A: 231 mm A: 346 mm A: 692 mm E: 42.5 mm E: 64.5 mm E: 129 mm 1/16 A: 224 mm A: 336 mm A: 672 mm E: 43 mm E: 65 mm E: 130 mm
The existing size of the violin is shown in the 4/4 row, but that of the cello is shown in the 1/8 row. The existing body size of the viola is shown in the 1/4 row, but with the thickness increased to 60 mm from 37 mm.
Instead of the increase in thickness of the cello being proportional to the square of the departure of its body size from the ideal, it is less than the square root of the change in body size (the 2.33 root), so the demand for increased thickness as we reduce the body size of an instrument is modest.
Not all possibilities have been shown in the table above; just the top row, with the proportions of all the instruments matched to those of the violin, those with the proportions of all the instruments matched to those of the cello, and those others that seemed likely to be useful.
Thus, a 7/8 violin made slightly thicker is shown, and the violas in proportion to a thickened 1/10 and 1/8 violin are shown, as those appear likely to be ergonomically useful; as well, the cello sizes in the 1/10 and 1/16 rows are shown, as they would correspond to a 3/4 cello and a 1/2 cello, and, of course, younger players would play those.
For the intermediate instrument between the viola and the cello, tuned an octave lower than the violin, body sizes corresponding to some reduced sizes of the cello are shown, as they would be relevant to younger players as well.
A 1/12 size, halfway intermediate between 1/10 and 1/16, is shown, so that a viola with a body size equivalent to that of a 7/8 violin could be shown.
Above the full size or 4/4 violin, the body size of the Mezzo Violin from the Violin Octet is shown, as the intermediate instrument could be made larger in proportion and still be playable like a cello. However, the Alto Violin in the Violin Octet, which corresponds to the viola, has a body length of only 508 mm, not 573 mm, so I may not be following the correct rule in scaling up the instruments.
The Baritone Violin, which corresponds to the cello, has a body length of 866 mm but a rib height of 138 mm, so both the body length and the rib height are greater than that of the conventional cello, so I cannot use it as the basis for deriving a different scaling rule by treating it as the scaled-up violin to be compared with the normal cello.
In a description of the Baritone Violin, however, it is noted that while it has a powerful tone with respect to its lowest string, unlike the cello it does not have a more powerful tone with respect to the A string, which is the highest string on both instruments. Thus, perhaps the cello, like the viola, is undersized, rather than being, as I had assumed, like the violin, ideal for the pitches it plays. Therefore, in order to derive a scaling rule from the Violin Octet, perhaps the thing to do would be to scale up the Mezzo Violin, and compare the result to the Baritone Violin, treating the Baritone Violin, like the conventional cello, as the example of how to exchange thickness for increased body height.
However, it is not necessary to do that, as the Violin Octet was explicitly designed in accordance with the scaling rule given in J. C. Schelleng's paper The Violin as a Circuit.
Of course, this might just be exchanging neck injuries for arm and wrist injuries, at least in the case of the viola, but if a thicker instrument could be played without problems, this shows a way to extend the violin family while respecting the health of players in addition to sound quality.
And, of course, thickness is not an issue for the cello or for the intermediate instrument intended to be played in a similar manner.
The simplest thing to do would be to ignore everything but the main air resonance, using the formula for the Hemholtz resonance of a sphere with one hole. In this case, the wavelength is proportional to the square root of the volume, so doubling the wavelength only requires dimensions to be increased by a factor of 1.5874.., or two to the power of two-thirds. As for the front and back plates of the instrument and their area, their frequencies can be changed by changing their thickness; however, making them thicker raises the frequency, and it is not practical to make them thinner for larger instruments, as this would lead to them becoming fragile.
This implies that the basis for an enlarged instrument could either be one that has the body height increased by the basic scale factor, but the rib height unchanged, or one that has the body height increased by the square root of the basic scale factor, and the rib height increased by the basic scale factor. But then when the body height is reduced, the rib height would be increased so as to keep volume constant.
Using the latter alternative, the chart above would be modified to the following:
Violin Viola Cello 4/4 A: 357 mm A: 535.5 mm A: 714 mm A: 1071 mm E: 35 mm E: 35 mm E: 35 mm E: 35 mm 7/8 A: 346 mm A: 692 mm E: 37 mm E: 37 mm 3/4 A: 336 mm A: 672 mm E: 39.5 mm E: 39.5 mm 1/2 A: 317 mm E: 44 mm 1/4 A: 274 mm A: 411 mm E: 59.5 mm E: 59.5 mm 1/8 A: 253 mm A: 380 mm A: 506 mm A: 759 mm E: 70 mm E: 70 mm E: 70 mm E: 70 mm 1/10 A: 238 mm A: 357 mm A: 714 mm E: 79 mm E: 79 mm E: 79 mm 1/12 A: 231 mm A: 346 mm A: 692 mm E: 84 mm E: 84 mm E: 84 mm 1/16 A: 224 mm A: 336 mm A: 672 mm E: 89 mm E: 89 mm E: 89 mm
However, while this exacts a heavier penalty in increased thickness for reduced body size, it still leads to the cello turning out to have an ideal size which is actually thinner than its existing size. That is unlikely to be the case.
If we take the claim that the A string on a cello is the one that has its sound enhanced by the instument to indicate that its resonance is in fact a factor of 27/8 too high, that would make this quite impossible indeed. Note, though, that the A string on a cello is actually tuned a whole note higher than the lowest string on a violin, its G string.
At this point, one might wonder if one might as well give the cello three higher strings, and one lower string, than it presently has, thus with seven strings tuned in fifths to F, C, G, D, A, E, B, and thus with no need to have any other instruments in the violin family.
If we multiply 105 mm by the square root of 27/8, we get an ideal thickness for the conventional cello, with a body height of 759 mm, of 193 mm.
Violin Viola Cello 4/4 A: 357 mm A: 535.5 mm A: 714 mm A: 1071 mm E: 35 mm E: 51 mm E: 66.5 mm E: 96.5 mm 7/8 A: 346 mm A: 692 mm E: 37 mm E: 70 mm 3/4 A: 336 mm A: 672 mm E: 39.5 mm E: 75 mm 1/2 A: 317 mm E: 44 mm 1/3 A: 289 mm A: 866 mm E: 53.5 mm E: 147.5 mm 1/4 A: 274 mm A: 411 mm E: 59.5 mm E: 87 mm 1/8 A: 253 mm A: 380 mm A: 506 mm A: 759 mm E: 70 mm E: 102 mm E: 133 mm E: 193 mm 1/10 A: 238 mm A: 357 mm A: 714 mm E: 79 mm E: 115 mm E: 218 mm 1/12 A: 231 mm A: 346 mm A: 692 mm E: 84 mm E: 122.5 mm E: 231.5 mm 1/16 A: 224 mm A: 336 mm A: 672 mm E: 89 mm E: 130 mm E: 245 mm
Using that empirical scaling factor as the basis, the above shows revised recommended rib heights for an ergonomic extended violin family.
As a check on whether this makes sense, a 1/3 size violin has been added, sized so as to correspond to the Baritone Violin in the Violin Octet. Calculating by this method leads to a rib height of 147.5 mm instead of 138 mm, which is at least roughly in the same ball park - and, of course, my calculations made some rather drastic simplifying assumptions.
So, applying the appropriate correction factors:
Violin Viola Cello 4/4 A: 357 mm A: 535.5 mm A: 714 mm A: 1071 mm E: 35 mm E: 49.5 mm E: 63.5 mm E: 90 mm 7/8 A: 346 mm A: 692 mm E: 37 mm E: 67 mm 3/4 A: 336 mm A: 672 mm E: 39.5 mm E: 72 mm 1/2 A: 317 mm E: 44 mm 1/3 A: 289 mm A: 866 mm E: 53.5 mm E: 138 mm 1/4 A: 274 mm A: 411 mm E: 59.5 mm E: 84.5 mm 1/8 A: 253 mm A: 380 mm A: 506 mm A: 759 mm E: 70 mm E: 99 mm E: 127 mm E: 180.5 mm 1/10 A: 238 mm A: 357 mm A: 714 mm E: 79 mm E: 112 mm E: 204 mm 1/12 A: 231 mm A: 346 mm A: 692 mm E: 84 mm E: 119 mm E: 216.5 mm 1/16 A: 224 mm A: 336 mm A: 672 mm E: 89 mm E: 126 mm E: 229.5 mm
Thus, in this case, using the Baritone Violin as a guide, the thickness of a cello with its existing body height would need to be almost doubled, from 105 mm to 180.5 mm, to re-position its resonances to correspond to its lowest string.
And, initially, I thought that increasing the thickness of a viola, to allow it to have a good sound while being no larger than a violin, to 64 mm, would already lead to an awkward instrument; now, an even larger thickness, almost twice as great, of 112 mm (even greater than the 105 mm thickness of a conventional cello!) seems to be required.
However, the figures in these last two tables are based on the premise that the conventional viola, with body height of 759 mm and thickness of 105 mm is an instrument whose resonances are actually higher than those of the much smaller violin, which seems hard to accept.
The sizes given on this page for violins and other related instruments are quoted in either inches or millimetres. But neither English units nor metric units were in use in the Cremona of Stradivari's day, so possibly expressing dimensions in terms of the units which he had used might avoid rounding errors. One place where this might be significant is in the case of the ratio between dimensions D and E, since this would affect the precise angle between the belly and back of the violin.
From the book Stradivari by Stewart Pollen, it is noted that a harp made by Stradivari has markings indicating that he used a Cremonese foot of 483 millimetres. Other sources give such a unit as well.
An official document, approved by a decree of May 20, 1877, Tavole di Raggulaglio de Pesi e Delle Misure Già in Uso Nella Varie Provincie del Regno col Peso Metrico Decimale gives the length of the "Trabucco di Cremona" as 2.901233 metres.
This leads to the following system of units:
Trabucco (6 Piede) 2.901 233 metres Piede (12 Once) 483.538 833 333 mm Oncia (12 Punti) 40.294 902 777 mm Punto (12 Atomi) 3.357 908 564 814 814 mm Atomo .279 825 713 734 567 901 234 567 901 234 567 mm
where the Piede is another name for the Braccio da Fabbrica, the unit in question, and the figures are prolonged sufficiently so the repeating decimals are visible without any claim to significance.
Of course, this was long after Stradivari's day, but it shows that the unit in question existed for long enough that a precise value was eventually established for it, to seven significant digits.
In Milan, the Atomo was further divided into twelve Momenti; in Piacenza, it was instead divided into twelve Minuti, each of which was divided into twelve Momenti, with each of those further divided into twelve Scrupoli, so this constitutes a duodecimal system of measurement. If we subject the Cremonese units to the Piacenzan mode of division (the Braccio in Piacenza was longer, 675 mm), we get:
Trabucco (6 Piede) 2.901 233 metres Piede (12 Once) 483.538 833 333 mm Oncia (12 Punti) 40.294 902 777 mm Punto (12 Atomi) 3.357 908 564 814 814 mm Atomo (12 Minuti) .279 825 713 734 567 901 234 567 901 234 567 mm Minuto (12 Momenti) .023 318 809 477 880 658 mm Momento (12 Scrupoli) .001 943 234 123 156 721 mm Scrupolo .000 161 936 176 929 726 794 695 930 498 mm
and so we would only have to repeat the division by 12 three more times to get a unit about the size of a nanometre, but we don't have to go any further to get into the realm of modern microelectronics; it wasn't that long ago that integrated circuits had feature sizes on the order of a Scrupola. A Scrupola is about 1.62 microns, and the original 80386 was made with a feature size of 1.5 microns.
In terms of the marks one might make on a ruler, to have a smallest unit comparable to 1 mm or 1/16 inch in size, instead of stopping at the Punto, presumably each Punto should be divided into half, with marks representing six Atomi.
The effect of this on the shape of a violin can be significant. Take the example of Stradivari's 1720 violin given above, with a body 14 1/16 inches long/high, with the neck side 1 3/16 inches high/thick and the button side 1 1/4 inches high/thick.
In units of 1/16 of an inch, one has two heights of 19 units and 20 units, separated by a distance of 225 units; if the figure of the violin's ribs is a symmetrical wedge, then the two plates are tilted, in opposite directions, from the axis by the arctangent of 1/450, or about 0.1273 degrees.
Convert those measurements to millimetres, and round them to the nearest millimetre. Then the body is 357 mm long, the thickness at the neck is 30 mm, and the thickness at the button is 32 mm. Now, the plates would be tilted by an angle of the arctangent of 1/357, or about 0.1605 degrees.
If the actual dimensions of the violin were in units of 6 Atomi, about 1.68 mm, then we would have heights of 18 units and 19 units, separated by a distance of 213 units, so now the angle in question is the arctangent of 1/426, or about 0.1345 degrees.
What this really means, of course, is that measurements need to be made to a greater precision than one millimetre, whatever unit may be used, for the angles of the plates to be more than guesswork.