The “Four Circles” of the Violin: Analysing Historic Violins
In Parts 2 and 3, I derived a generalised series of equations for constructing the body and the corners of a violin soundboard, based off the works of Kevin Kelly. In this section, I will superimpose this generalised framework onto the works of Nicolò Amati, Antonio Stradivari, and Guiseppe Guarneri, and analyse the trends visible in their designs.
I add here that the purpose of this is not to reduce the works of these luthiers to mathematical equations, nor am I suggesting that this is the construction method used by any of them. I am simply trying to observe, over a cross-section of the “Golden Age” of Cremonese luthiery, what patterns, if any, are visible by superimposing my generalised model.
Method
I implemented the generalised expressions for the body and corners in Excel to create a graphic which I could superimpose onto photos. With hindsight, Excel is a program I would strongly advise against using for this — it is much easier to do it in something like Matlab or Python where you don’t need to calculate every single coordinate.
I then took photos from Tarisio Auctions and resized them to fit on the graphic, which has a standardised height of 1 unit. Excel does have the advantage of making it easy to position and resize images by the highly advanced click-and-drag method.
Adjusting the variables to fit the violins was done by eye. For lack of time, I didn’t attempt to compensate for the perspective of the images, though for the most part Tarisio’s images appear to be very close to plan view. With this in mind, the exact numbers in my data should be taken with a handful of salt. Moreover, it is possible to generate very similar shapes of violin using different combinations of body and corner constants. You may find, if you attempt to use my equations, that the ratios you generate are very different.
Choice of Luthiers
My choice of Amati, Stradivari, and Guarneri was due to their fame. I wanted to see (in terms of my generalised model) what trends might exist that “make an Amati an Amati”. Though it is overly simplistic and inaccurate to attribute differences in the sounds of historical instruments to shape alone, I thought it would be interesting to see if any correlations in shape existed that might partially account for the generally accepted tonal variances between these famous luthiers.
Amati, Stradivari, and Guarneri also represent a cross-section of Cremonese luthiery during its “Golden Age”, when considerable technical and quality advancements in the production of violins were made. Amati was teacher to Guarneri’s grandfather, Andrea Guarneri, and possibly to Stradivari — who was at least significantly influenced by him. There is, therefore, something of a historical evolution to be seen by investigating their works.
A Note on Dates
All of the dates in the graphs below are taken from Tarisio’s website and are circa a given year. This is partly because Tarisio includes the entire violin in their dating, and parts such as the scroll may have been reworked or made by a completely different luthier at a later date. Many of these instruments have also received alterations to accommodate the changing demands of music since their original construction. Another reason is that the chronological provenance of these instruments may be unclear due to historical records (or lack thereof).
I am using dates in my graphs as a convenient way of denoting a particular violin. It would be inappropriate to view the dates I useas representative of a given luthier’s work for that year. Stradivari, for example, likely made around a thousand violins — my sample covers only around 6% of his work. I generally only have data from a single violin from a given year, so my sample is less than comprehensive.
Nicolò Amati (1596–1684)
After the Italian Plague of 1629–31, Nicolò Amati was left one of the surviving fine luthiers in Cremona. To cope with demand for instruments, he became one of the first luthiers to take apprentices from outside his family, resulting him becoming a significant influence on names such as Guarneri, Rogeri, Stainer, and Stradivari. Amati is known for his “Grand Pattern” violins, developed around 1630. The Grand Pattern is characterised by a slightly wider soundboard than typical Cremonese design, with long, hooked corners.
My data covers 20 violins made between 1628–1683.
Starting with the body variables, there are two notable features. First is the use of kw values greater than 1, which makes the waist of the violin wider, and “squashes” the lower bout, making it more prominent. Second, the use of a median kc value of 0.9, which produces straighter sides to the waist (or centre bouts) of the body. The overall result is a more trapezoidal shape: the upper bout is less pronounced while the lower bout is accentuated, and the transition from both to the waist is smoother.
It is interesting to note that the ratio of lower-bout:waist:upper-bout (L:W:U) is relatively consistent throughout my sample of Amati’s works at 1:1.05:0.8. The ratio of lower-bout:centre-bout:upper-bout (L:C:U), however, varies over time. There is an upward trend in kc, until around 1670, after which Amati seems to use values covering the whole range he used from 1630–1670.
Amati uses a modest median b₁ value of 0.7. Recall that the median kc=0.9, which means that the corners are more steeply pointed inwards in relation to where the would “naturally” occur on the centre-bout circle. The use of a b₂ of 0.5 results in the corner transitioning to a point over the relatively long distance of 0.2 of rₗ. Post-1650, the data shows the slenderness of Amati’s corners increasing, evidenced by the increasing difference between b₁ and b₂.
Corner hooking is evident in his use of small bᵤ and bₗ values, which means the “inside” of the corners on the centre-bout circles are much tighter than on the outside. It is worth noting that the upper corners are more aggressively hooked than the lower ones. bᵤ takes a median value of 0.53, while bₗ is 0.65.
Amati’s style of long corners can be seen in his choice of bₖ and bⱼ values. bₖ in particular, median 1.21, is larger than the median values of Stradivari and Guarneri.
Amati also appears to favour corners that are located very low on the upper bout, and at a moderate height on the lower bout. cᵤ₁ in particular, takes values less than 1 (median cᵤ₁=-0.04), which means, interestingly, the major corner line crosses over onto the opposite side of the bout. The effect of this is to produce a lower “shoulder” on the upper bout, which, I think, is more in keeping with Amati’s tendency towards a straighter upper-bout-to-waist transition.
There are some interesting trends visible in the choice of cᵤ₂ and cₗ₂ values. Recall that these variables control the smoothness of the transition to the corners, with larger values producing smoother transitions. A decreasing trend is visible in Amati’s work over time, with the result that the corners, particular the upper ones, become more prominent. In Fig. 6, the length of Amati’s corners remained relatively consistent over his career, but the reduction in cᵤ₂ and cₗ₂ would have the effect of increasing the aspect ratio of the corner, making the appear longer and thinner, as they are less blended into the upper bout.
As the data shows evidence of increasing/decreasing trends across Amati’s career (and a distinct change in b₁ and b₂ values), I think it is reasonable to construct two median figures as typical of Amati’s work pre-1650 and post-1650.
As discussed above, there is little change in the overall body variables over Amati’s career, however it does appear that rc increased in the latter half. Changes to corner geometry are more obvious, with the upper corner becoming more slender, and the lower corner becoming longer and slightly wider. Note in general the slightly squashed lower bout and generally wide soundboard characteristic of Amati’s Grand Pattern.
Antonio Stradivari
Antonio Stradivari is arguably the most famous historical luthier, and is often considered to be the greatest violin maker in history. He was an incredibly skilled crafts-person, creating highly intricate purflings and ornamentations for his instruments, and generally showing great precision and attention to detail. Whether or not he was an apprentice of Amati is debated, but the latter at least had a significant influence on his work. Stradivari is widely credited for creating the basis for the violin as we know it today —narrower and less highly-arched than earlier Cremonese designs. He is also known for his “Long Pattern”, characterised by an overall slender body shape with highly curved upper and lower bouts.
My sample covers 62 violins made between 1666–1737.
Two things are noteworthy in the graph of Stradivari’s body variables. First, the consistency of their use throughout his career. kₘ, kᵤ, and kw show very little variation over the time period sampled, and even kc remains relatively constant with some spread in the later years of his life. The second is the emergence of a different variant of shape between 1690–1700. This is clearly visible in both kc and kₘ as a “step” in the data trend, with both values increasing and clustering together over the ~10 year period. The effect of this change is to make the instrument much narrower and with much more rounded upper bouts than the “standard” pattern used by Stradivari. I think this distinct change corresponds with what is known as the “Long Pattern”.
It is interesting to note that while the L:W:U of the “long” Stradivari is very similar to that of an Amati (1:1.05:0.8), the use of a large kₘ radius of 0.75 makes the soundboard narrower (particularly in the upper bout) by increasing the curvature of the upper and lower bouts. The effect of a larger kₘ is effectively to “cut off” a portion of the outer edge of the soundboard by rounding it off.
Stradivari constructed corners with more splay than Amati, initially using b₁ values between 0.5–0.7, before eventually settling on an intermediate value of around 0.6. The median splay of Amati’s corners was 0.7. The value of b₂ fluctuates between periods of consistency, though overall produces a corner slenderness similar to Amati’s. Though the clustering is not particularly strong, it does appear that, corresponding with the “long” body data between 1690–1700, b₁ and b₂ decrease slightly. This would correspond to the corners on the “long” Stradivari pointing more inwards, with greater tapering.
Stradivari appears to favour more strongly hooked corners than Amati, with an overall median bᵤ=0.43 versus 0.53, and bₗ=0.56 versus 0.6. The visual differences, however, are subtle, as shown in Fig. 15 for the “long” data.
The upper corners are almost always more strongly hooked than the lower ones, which is consistent with the general shape of Stradivari’s instruments being more compact in the upper bout region. It is notable, however, that, coinciding with the “long” shapes from 1690–1700, the difference in hooking between the upper and lower bouts was very slight, with bᵤ and bₗ taking very similar values during that period. The value of bₗ decreased significantly to converge with the value of bᵤ. Interestingly, in the final years of his career, Stradivari appears to have moved back towards this similar degree of hooking between the upper and lower corners.
There appears to be some clustering of significantly reduced bᵤ values in the late 1720s, corresponding with an increase in upper corner hooking, but this does not seem to be reflected in the lower corners, or correlated with a change to the geometry of the body.
The value of bⱼ remained relatively consistent throughout Stradivari’s career while bₖ varies considerably more. In his later years, however, Stradivari appears to settle on a value of bₖ around 1.15. Both bⱼ and bₖ are lower than those used by Amati, which corresponds to shorter corners overall, especially in the upper corners where overall median bₖ=1.15 compared to 1.21. Interestingly there is no significant change in corner length to corresponding to the “long” body data. As shown in Fig. 21, the corners are “pulled” inwards by the decreasing body width, while maintaining proportionally the same corner length. Recall that all radii are relative to rₗ, which decreases in the “long” data.
My data suggests that Stradivari preferred to have his corners situated low on the upper bout (overall median cᵤ₁=0.05) and very high on the lower bout (cₗ₁=0.5). Distinct clustering of datapoints is visible in the 1690–1700 period, where Stradivari positioned his corners higher on the lower bout (cₗ₁=0.6 to “standard ” 0.5)and upper bout (cᵤ₁=0.15 to 0.05). The effect of this is to make the upper-bout corner slightly more prominent, and to de-emphasise the lower-bout corner by placing it in the centre-bout-waist transition.
Stradivari appears to have progressively decreased the smoothness of his upper and lower corners (increasing their prominence) until about 1710, after which he moved in the opposite direction to increase their smoothness. There is clustering visible in the 1690–1700 period, with corner smoothness of both the upper and lower corners increasing significantly. I speculate that this is to compensate for the “natural” decrease in corner smoothness that would have occurred had these variables been left fixed while the body width was reduced.
There is also some clustering of higher cᵤ₂ values in the late 1720s, which correlates with the changes to upper corner hooking observed in Figs. 18–19. By increasing the corner smoothness, Stradivari would have been able to emphasise the hook of the corner as the inner radius bᵤ decreased.
Given the discrete regions in the data observed above, I think it is reasonable to divide my Stradivari data into three variants: the “standard” and “long” regions corresponding to the significant changes to body and corner variables between 1690–1700, and a “late-1720” region corresponding to the clustering of some corner variables. The “late-1720” region should be regarded as a subset or “corner variant” of the “standard” pattern, as there was no substantial change to the body variable during that period. Note that the “standard” median excludes the period form 1725–1730, which comprises the “late-1720” median.
Fig. 26 shows the close relationship between the “late-1720” and the “standard” patterns, which are both have significant overlap in the body region and are clearly distinct from the “long” pattern. It is interesting however that the “late-1720” figure is something of an intermediate between the “long” and “standard” patterns in terms of rc.
In terms of the corners, however, the “long” variant shares more similarities with the “standard” variant than does the “late-1720”. As discussed above, the “long” variant corners are effectively translated inwards from the “standard” version. The “late-1720” variant on the other hand has significantly more tapered lower corners and slightly less tapered upper corners, with both being shorter than the “standard” pattern. Stradivari appears, during this period, to have increased the prominence of the lower corner hooking (by decreasing bₗ) while reducing the prominence of the upper corners (as reflected in the cᵤ₂ value). Interestingly, despite the noticeable reduction in bᵤ discussed above, the increase in hooking in the “late-1720” pattern is only slight.
Note in general the more curved upper and lower bouts compared with Amati’s designs, and the reduced visual impact of the upper-bout.
Bartolomeo Giuseppe “del Gesù” Guarneri
Giuseppe Guarneri was a contemporary of Antonio Stradivari, and was, at the time, overshadowed by the latter’s fame. Today however, his instruments are often more coveted than those of his rival; prized for their unique tonal qualities and aesthetic daring.
While Stradivari is known for his consistency and attention to detail, Guarneri is reputed to have created individual, dissimilar instruments, and showed a general lack of interest in creating a beautiful finish. His scrolls and f-holes in particular, have been described as “crude” and “hasty”. Given the unique design of each of Guarneri’s instruments, it is perhaps unsuitable to attribute a characteristic to his work. Generally however, Guarneri violins are considered to have longer centre-bouts than Stradivari and Amati.
My sample covers 25 violins made between 1715 and 1744.
Surprisingly, for a maker reputed to have eschewed templates for artistic flair, Guarneri is fairly consistent in his choice of body variables. In the early years of his career, Guarneri made violins that were very similar to the “standard” Stradivari body variables. By the end of his career however, Guarneri’s violins were noticeably wider with a kw=1.05 to Stradivari’s 1.03, though the other variables remained the same. Though the trend is not particularly noticeable, I think there is reason to argue that Guarneri’s variation in kw divides his violins into three distinct types: “pre-1730”, “1730–1735”, and “post 1735”. The corner variables provide stronger evidence for this, which will be discussed shortly. Fig. 30 illustrates the clustering of kw visible during these time periods.
Regarding the often-cited trait of elongated centre bouts, my data suggests that this is not because of changes to the body variables, as these are very similar to those of Amati and Stradivari.
Guarneri’s overall median b₁=0.68 and b₂=0.5 are very similar in value to Amati’s, corresponding to a more open centre bout and more slender corners than Stradivari’s. From around 1635 onwards, Guarneri’s b₁ increases slightly as the centre-bouts become more open, while corner slenderness decreases as the difference between b₁ and b₂ decreases. Both of these changes correspond with making the centre bouts more elongated in appearance, both by widening the distance between the upper and lower corners, and decreasing their prominence.
Both bᵤ and bₗ increased over Guarneri’s career, starting at 0.36 and 0.57 respectively (making them even more strongly hooked than Stradivari’s at 0.45 and 0.6), increasing to 0.52 and 0.63 around 1730, and increasing again to 0.55 and 0.66 after 1735. bᵤ in particular, shows clustering in three distinct periods: pre-1730, 1730–1735, and post-1735. I think this reduced hooking, resulting in relatively open corners compared to Stradivari and Amati, is the reason why the centre bouts on Guarneri’s instruments appear elongated, especially in his later years.
Guarneri generally favoured corners shorter than Amati and Stradivari, particularly in the upper corners. bₖ decreased from 1.22 to 1.08 over his career, while bⱼ remained relatively constant at 0.89. The reduction in bₖ seems to correlate with the choice to reduce corner hooking, as shorter corners further “open up” the centre bout, making them appear longer.
Continuing the tendency for more open centre-bouts, cᵤ₁=0.1, positioning the corner relatively high on the upper bout. cᵤ₁ increases significantly post-1730, which would correspond with the general trend for reducing the presence of the upper corner on the centre bout. Interesting, cₗ₁ only takes a moderate value of 0.47, and is relatively consistent throughout Guarneri’s career. I think the reason cₗ₁ does not take a lower value (to move the lower corner down and further elongate the centre-bout) is because the lower bout is already somewhat squashed by the use of a large kc, and hence placing the corner lower would cause visual “imbalance” by further emphasising this squashing.
Both cᵤ₂ and cₗ₂ decrease over Guarneri’s career, corresponding with a reduction in corner smoothness. cᵤ₂ decreases from 0.75 to 0.58, while cₗ₂ decreases from 0.68 to 0.6. I think the reason for this is that the decreasing length and hooking of Guarneri’s corners necessitates increasing their prominence from the body to prevent them from being “lost”. Furthermore, increasing the prominence of the corners emphasises the length of the centre bout by defining more clearly where the bout “ends”.
As mentioned above, the data, especially with regards to corner hooking and length, suggests three distinct patterns: “pre-1730”, “1730–1735”, and “post-1735”. The overall trend is for an elongation of the centre bout, caused by a reduction in corner hooking and length. As clearly shown in Fig. 42, the corners move apart and “rotate” into the body, opening up the centre bout. While some changes do occur in body shape with regards to width, the body shape overall remains very similar.
Overall, the trend in Guarneri’s design is for an Amati-esque squashed lower bout and wide soundboard, but with a uniquely open centre-bout with de-emphasised corners.
Direct Comparisons
Comparing the most “mature” designs of Amati, Stradivari, and Guarneri, reveals subtle, but distinct differences. For reference, I am comparing the patterns my data suggests each luthier was using at the end of their careers: the “post-1650” Amati, the “standard” Stradivari, and the “post-1735” Guarneri.
The differences in body shape are noticeable but much more subtle than the numbers themselves suggest. The most obvious differences occur in the centre-bout/waist region, where Stradivari (magenta) is the narrowest and the Guarneri (blue) the widest. The lower bout shape is also narrower on the Stradivari than the Amati and Guarneri, owing to the former’s use of a smaller kc. Looking very closely, the Stradivari has a slightly narrower and curvier upper bout due to the larger kₘ value. Overall, the Guarneri body shape is much more similar to Amati’s than Stradivari’s — who appears to have “evolved” towards a noticeably narrower instrument.
The difference in corner design are much more distinct. Starting with the lower corners, Amati’s are the most slender, with the narrowest tips, while Stradivari has the least slender, with the widest tips (Item 1, Fig. 47). Amati’s lower corners show the strongest degree of hooking, while Guarneri has the least (Item 2, Fig. 47). Stradivari has corners positioned higher on the lower bout than both Amati and Guarneri. Interestingly, despite the greater hooking of Amati’s lower corners, the corner terminates at a similar location to Stradivari’s (Item 3, Fig. 47). Guarneri’s lower corners are the lowest, both in positioning on the bout and in terms of the outer termination of the corner (Item 4, Fig. 47).
In the upper corners, Stradivari’s are the most slender with the narrowest tips. Amati’s are relatively similar, while Guarneri’s are the most divergent with the least tapering and the widest tips (Item 1, Fig. 48). Stradivari also has the most strongly hooked upper corners, and Guarneri the most open (Item 2, Fig. 48). The lower corner termination point (Item 3, Fig. 48), is quite similar between all three luthiers compared to the lower corner. Notably, despite this similar lower termination point, the outer edge of Guarneri’s corners are significantly higher than both Amati’s and Stradivari’s (Item 4, Fig. 48).
The overall trend is for Stradivari to have the most closed centre bout, with the corners closest together, while Guarneri’s is the most open.
While the above observations pertain only to the later works of Amati, Stradivari, and Guarneri, I think these comparisons are reasonably representative of the generally accepted “characteristic styles” of the three. In the case of Stradivari, the overall consistency of his designs means the “standard” pattern in my data has traits characteristic of most of his work. With Amati and Guarneri, while both showed evolution throughout their careers, the instruments produced later in their lives evidence many of the characteristics for which they are known. Hence, I think it is reasonable to conclude there are quantifiable differences between the accepted visual characteristics of these three luthiers.
Limitations of the Generalised Equations
In general, the generalised equations are a good approximation of the shape of the instruments measured. Because, however, the equations are “perfect” they cannot account for non-geometric curvature that has occurred as a result of artistic choices or historical wear, and I did not account for asymmetry (which was present in some instruments).
Furthermore, the actual geometric construction methods used by these luthiers are likely different from my modified version of Kelly’s. Some of Amati and Stradivari’s instruments appear to rely on an additional circle to complete the construction of the upper corners, which was not present in my equations (Fig. 51). Furthermore, my use of “blunting circles” in the lower corner does not seem to fit several of Stradivari’s instruments very well, as he seems to have cut those corners with a line running non-perpendicular to the lower major corner line (Fig. 52).
Concluding Remarks
Mapping a generalisation of Kevin Kelly’s construction method onto historical violins reveals some interesting trends in the works of Amati, Stradivari, and Guarneri, which correlate reasonably well with the general consensus on the visual characteristics of their designs.
Ideally, with more time, I would be able to enlarge the data set beyond one instrument per year for each luthier to draw more comprehensive conclusions. It would also be interesting to expand the study to include more luthiers, historical and contemporary, to see what, if any, evolutionary changes have taken place since the Cremonese “Golden Age”.
Sections
Part 1: Generalising the Four-Circle System
Part 2: Body Equations
Part 3: Corner Equations
References
Kelly, K. (2019, January 14). The ‘Four Circles’ System of Violin Making. Retrieved from The Strad: https://www.thestrad.com/lutherie/the-four-circles-system-of-violin-making/8545.article
Tarisio Auctions. (n.d.). Nicolò Amati. https://tarisio.com/cozio-archive/browse-the-archive/makers/maker/?Maker_ID=12
Tarisio Auctions. (n.d.). Antonio Stradivari. https://tarisio.com/cozio-archive/browse-the-archive/makers/maker/?Maker_ID=722
Tarisio Auction. (n.d.). Bartolomeo Giuseppe Guarneri ‘del Gesù’. https://tarisio.com/cozio-archive/browse-the-archive/makers/maker/?Maker_ID=240
Appendix
