THE WELL HARMONISED MOULD

The logic governing the structure of Stradivari’s violins remains a mystery. André Theunis and Alexandre Wajnberg take a fresh look at his moulds to find an intriguing system of proportions, utilising the tools and measuring systems of his day

FIGURE 2 Stradivari’s ‘PG’ mould. Note the four horizontal thin lines perpendicular to the main axis; these connect the corners and the ends of the C-bouts

COURTESY MUSEO DEL VIOLINIO

What was it that guided the designs of Antonio Stradivari’s instruments? Existing studies on the structure of the violin give complex answers with regard to the knowledge and practice of luthiers in the 17th century. This may be the time to apply the principle of Occam’s razor, namely that ‘preference should be given to the simplest assumptions’.

We re-examined the dimensions of the Stradivari moulds from the point of view of the violin maker, taking into account the available tools, woodworking techniques and measurement systems. By reasoning in a spirit of simplicity, we can highlight sub-structures in his moulds with rational proportions that are easy to draw and possess important musical and symbolic meanings.

The standard unit of measurement in Cremona was the braccio (‘arm’), equivalent to 483.5mm according to the official table of weights and measures published by royal decree on 20 May 1877. First of all, it should be noted that the distance between the upper saddle and bottom saddle of the Baroque violin is very close to 483.5mm, to the braccio cremonese, a round number (figure 1). Note also that the width of the upper part of the PG (MS21) mould – 161.5 mm – is half the length allowed for the Baroque vibrating string. This prompted us to look for other proportions of the Cremonese violins expressed in simple fractions of local units. This research was carried out on ten original Stradivari moulds, now housed in Cremona’s Museo del Violino. Our measurements were taken from the 1:1 scale photos printed in François Denis’s 2006 Traité de Lutherie, and Stewart Pollens’s book The Violin Forms of Antonio Stradivari, published in 1992.

STRADIVARI’S MOULDS

In violin making, moulds are used as templates for the construction of the instrument. They give material form to its interior structure, and define its proportions (figure2). First, we measured the length of each mould’s axis of symmetry between the top-and bottom-block (which were not included). From this, we found an average length equal to 321.3mm, or 2/3 of a braccio –a simple fraction (within 0.3 per cent).

FIGURE 1 The dimensions of an Amati: almost equal to 1 braccio, from saddle to saddle; with the vibrating string very close to 2/3 of the braccio

This value is also very close to the length allowed for the Baroque vibrating string: 322mm (i.e. 6mm shorter than the current string). What a coincidence! We hypothesise that Stradivari chose this length as the working unit of reference (a ‘modulus’) for the design of his instruments. However, owing to the uncertainties over the length of the Baroque vibrating string, our own modulus was the axes of the moulds (without their blocks), which we can measure today.

THE RATIONAL MOULD

Lines (possibly drawn by Stradivari) perpendicular to the central axis are still visible on both sides of all the moulds examined. They could well have been used as reference marks to place the ends of the two C-bouts and the four corners with the same gesture. Therefore, what we are studying (on the front of each mould) are the segments of the central AA’ axis determined by the points of intersection C, D, E of these lines with the AA’ axis, for each of the moulds (figure 3, page 50).

Table 1 shows the lengths of all ten moulds, along with the measurements to their upper corners, lower corners, and the right angles of the upper corners. Strong trends emerge once again: each of the measured segments corresponds to simple fractions of their own AA’ modulus. For example, the line HH’ joins the two upper corners, and intersects AA’ at point D. On average, the length DA’ (upper part of the violin) is equal to 1/3 of AA’, another simple ratio (within 1.2%). Likewise for the lower corners and the lower edge: AC = 2/5 of its axis (within 0.5%); and for the right angles of the upper corners with the upper edge of the mould: EA’ = 1/4 of its axis (within 0.4%).

NATIONAL MUSEUM OF AMERICAN HISTORY

Table 1 Measurements of the sub-structures of the ten moulds (front) and average of the ratios to their central axis. The measurements along the ten central axes are taken on the ridge of the moulds, blocks excluded, with a margin of error of 0.1 mm

ANTONIO STRADIVARI SPENT HIS ENTIRE LIFE IMMERSED IN A MUSICAL UNIVERSE OF RATIONAL PROPORTIONS

FIGURE 3 The length (AA’) of the axis of symmetry, between the blocks (blocks excluded) is the unit from which all our other measurements were taken (the ‘modulus’). It is in relation to said modulus that the mould’s various substructures are compared: A’D, A’E and AC

FIGURE 3 ANDRÉ THEUNIS

Moreover, by comparing the same lines drawn on the fronts and backs of each of the ten moulds, we noted significant differences, to the order of -2.5mm to +1.3mm, i.e., ±0.6% (from -0.8% to +0.4%): this gives us an indication of the luthier’s margin of precision!

THE COMPASS MOULD

Although the ten moulds vary in length, their internal proportions seem to be little affected. For the upper corners, the average of the deviations from the simple rational ratios is 1.2 per cent; for the lower corners, 0.5 per cent; and for the right angles of the upper corners, 0.4 per cent. This is a remarkable result considering the degree of precision of the woodworking, the wear of the moulds (depending on their use) and the potential warping of the wood after more than 300 years.

While continuing his experiments, the Cremonese master seems to have kept the different sections of the moulds in simple rational fractions for the positioning of the corners. These intervals are easy to trace with the tools of the time: a compass; a proportion compass, also called a Galileo compass, commonly used by craftsmen and architects; and a ruler graduated in oncie.

In this regard, it should be mentioned that Stradivari knew the architect and surveyor Alessandro Capra very well, as the latter was the ex-father-in-law of his first wife, Francesca Ferraboschi. Capra was known for his ingenuity, his capacity for synthesis, the simplicity of his constructions and the multiplicity of his interests (which included military architecture, solar clocks and theatre machines). He had just published one of his treatises, Geometria famigliare, et instruttione practica, in the year 1671, that is to say at the very beginning of the ‘first period’ of Antonio Stradivari who was just 27 years old. They drank prosecco to celebrate the release of the book! On the page shown in figure 4 Capra details the units in use in Cremona: braccio, oncia, punto, linea. The oncia equated to a twelfth of the braccio.

There are twelve 40.29mm oncie in the 483.5 mm braccio. The mean value of the central axis AA’ (321.3mm) is 8 oncie, to the nearest 0.3 per cent. With the ruler graduated in oncie and the proportional compass, the luthier’s gesture appears immediate to us, and the construction of the moulds becomes obvious

To give a final example in support of our hypothesis of simplicity: the Museo del Violino collection includes the paper pattern for a harp made by Stradivari. On the pattern for the pillar of this harp (figure5, page 52) Stradivari has written ‘Bracia duo – 2 Lungo’ – again, whole units. So, the Cremonese unit of measurement used by Stradivari and his peers is indeed a key that renews our understanding of the structure of his moulds.

Table 2 Some sub-structures of the moulds expressed in millimetres and in simple fractions of the different units: Cremonese units (braccia and oncie) and ‘luthier units’ (our AA’ modulus). Average values for the ten moulds

FIGURE 4 Traditional Cremonese units of length, as published in Geometria famigliare, et instruttione practica by Alessandro Capra, ex-father-in-law of Stradivari’s first wife Francesca Ferraboschi (Cremona 1671; reissued in 1717).

IT IS NO COINCIDENCE THAT THE SOUNDBOARD IS CALLED THE ‘HARMONY BOARD’ IN FRENCH AND ITALIAN

We have summarised our results, expressed in millimetres, and in simple fractions of the various old measurement units: braccia, oncie and AA’ modulus (Table2).

Our results show firstly that the moulds have remarkably stable rational proportions, with uncertainties compatible with the margin of error observed between the front and back of the moulds. Secondly, when using the units of the time – braccio, oncia and the modulus (the AA’ axis) – these rational fractions are expressed in small whole numbers. These units are related to each other, but the most significant is the AA’ modulus because it produces the simple fractions with the smallest numbers. This also has a deep musical meaning, which will support our hypothesis.

FIGURE 5 Inscription by Stradivari for the height of the column of his harp: ‘Bracia duo–2 Lungo’

MARION POLLARD, MUSEO DEL VIOLINO

IT IS LIKELY THAT STRADIVARI TURNED TO THE MUSICAL PARADIGM AND ITS MUSICAL RATIONAL RELATIONS RATHER THAN TO THAT OF IRRATIONAL MAGNITUDES

THE HARMONISED MOULD

Ever since the experiments by Pythagoras on vibrating strings, we have known that numbers and musical notes are intimately linked. The shorter the string, the higher the note (the frequency): vibratory frequency is inversely proportional to length. In the 16th century these concepts were mathematised by Gioseffo Zarlino (1517–90). For example, a string twice as short (1/2) vibrates twice as fast to give the second harmonic (the octave); if it is three times shorter (1/3), it vibrates three times faster to produce the third harmonic (the fifth at the octave) etc. This is the ‘harmonic series’, the basis of the Zarlino scale, which would spread throughout Europe at the beginning of the Baroque period, during Stradivari’s lifetime. In this natural scale, the intervals between two consecutive notes are not equal.

The tempered scale, in which the musical intervals are equal but mathematically irrational, would not become popular until much later, with Bach’s The Well-Tempered Clavier, published in 1722. Antonio Stradivari was 78 years old at the time and had spent his entire life immersed in a musical universe of rational proportions: a veritable paradigm.

The laws of ‘harmonious’ proportions were also applied to the visual arts, architecture and nature, etc. They gave meaning to the world. For example, one of the theories of the ‘Harmony of the Spheres’ made it possible to ‘explain’ the circulation of the planets, whose different orbits were harmonically in tune: a semitone between the Earth and the Moon, a minor third between Venus and the Sun, etc., the whole forming the full scale of six tones (five tones and two semitones).

Note that in classical architecture, to satisfy these laws of rational proportions, it was common to dispense with the usual units and to choose a specific ‘modulus’ taken as a unit – for example the diameter of a classical column – to serve as a guide when constructing by means of fractions or multiples of this modulus. This principle is seen in the works of Vitruve and Alberti among others. Our idea to choose a ‘violinistic’ modulus to understand the construction of the violin is therefore perfectly consistent with the thinking and practices of the time.

With this context in mind, the simple proportions we found in Stradivari’s moulds acquire highly significant musical value. Let’s start with the vibrating part of the Baroque string, equal to 2/3 of a braccio. This means its vibratory frequency, compared to the frequency of a similar string measuring 1 braccio, is the inverse of 2/3, or 3/2: musically, this violin string vibrates at the perfect 5th of the braccio – or, one might say, the perfect 5th of the length between the two saddles.

This line of reasoning may sound brazen, but there are several examples showing this is what architects, craftsmen and luthiers of the 17th century thought. And this is the other reason for choosing this precise length of the Baroque vibrating string (322mm) as a possible modulus unit for the moulds’ construction. The musical consequences in relation to the architectural balance of the moulds are astonishing:

• The longitudinal axis of the moulds AA’ is the root note – or in other words, the tonic

• The upper bout width is equal to half the modulus – in other words, its octave, the octave of the longitudinal axis

• The distance between the upper corners and the upper edge of the mould (DA’ on figure 3) is equal to 1/3 of the modulus; converted into vibratory frequency, this gives 3/1, the triple, that is to say the third harmonic of the root, therefore the perfect 5th to the octave of the axis

• The same reasoning applied to the distance between the lower corners and the lower edge of the mould (2/5 of the modulus) gives the major 3rd to the octave

• Finally, for the distance between the right angles of the upper corners and the upper edge of the mould (1/4 of the modulus), we find its double octave

It is amazing to note that these last four values – octave, 3rd to the octave, 5th to the octave and double octave – are precisely those of the perfect major chord to the octave of the central axis of the mould. It is here that the harmonic genius of the conception of the violin family is expressed.

We have not yet included in our reasoning the distance AB (between the right angle of the lower corner and the lower end of the mould) because the relation with the modulus AA’ is less clear: we are close to the 5th but the precision is much lower than for our other data. We do, however, find other interesting proportions: with the width at the lower bout and the width at the C-bout. This compels us to look for the same kind of proportions for the upper right angles.

The musical significance that, in a very 17th-century spirit, we attribute to these proportions of the moulds does not imply that, for Stradivari, the front and back plates vibrated like strings. We now know that the laws of acoustic vibration in two dimensions (for the plates) are much more complex than in one dimension (for strings). However, this was not known at the time. The luthier could therefore think that these laws were of the same order. And the violin board nevertheless vibrates in order for the sound to project. The plate vibrates between the blocks, which play a musical role equivalent to that of the saddles for the strings. It is no coincidence that the front of the instrument – the soundboard – is called the harmony board in French (table d’harmonie) and Italian (tavola harmonica). The ‘harmonious’ proportions of the design and sound frequencies come together perfectly.

Research on the use of the ‘golden ratio’ (1.618) or of the square root of 2 (1.414) to understand the architecture of the violin remain relevant. Indeed, our musical ratios in simple fractions are only valid for the moulds, which is to say for the interior violin space. The total external length of the body of an instrument by Stradivari varies, depending on th thickness of the upper and lower blocks (the thicker they are, the greater the difference between the central axis AA’ and the overall length). A number of previous studies (such as those by François Denis, Simone Sacconi and Stewart Pollens) attest to this, and these allow us to understand the external architecture of the violin as being governed by the architectural parameters of the time. However, it is very likely that, to design the moulds that would give the soundboard its structure, the luthier turned to the musical paradigm and its musical rational relations rather than to that of irrational magnitudes frequently used in architecture.

FIGURE 6 The pegs are not equidistant from each other, but arranged in simple fractions of the unitary modulus, the vibrating string. The peg of the G string is located at a distance from the saddle equal to 1/20th of the modulus; the peg of the D string is at 1/6; that of A at 1/5, and the peg of E at 1/10

Our research on the proportions of the moulds therefore does not invalidate the existing works on the external proportions of the instrument, but rather it opens up this new perspective: the architectural vision for the exterior; the musical ‘harmonic’ paradigm for the interior. Indeed, Stradivari’s moulds appear to be well harmonised.

CONCLUSION

We have shown that the corners are positioned ‘in 3rds and 5ths’ with respect to the central axis, in a ‘major perfect chord’ mould structure. With these simple proportionality ratios, the mould is constructed through a few gestures with the compass, and there is a triple justification for its dimensions: artisanal (for the simplicity of construction with the tools and measuring systems of the time), vibratory (for its musical relationships) and symbolic (the harmonic paradigm).

This logic is similar to that observed in the pegbox of Cremonese violins, copied by later luthiers (Making Matters, The Strad, June 2019): the respective positions of the pegs are also in musical harmony, presenting 3rds, 5ths and octaves (figure 6). It is possible to think that this logic was not only limited to the pegbox but also governed the conception, the drawing and the construction of the instrument as a whole, thus contributing to its sound qualities.

The authors would like to thank Gunnar Gidion of Albert-Ludwigs-Universität Freiburg, Germany