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The Golden Proportions of Cremona

(The vibrating string as the basic module)

The story starts in 1972. I was a young luthier at Mittenwald School and Simone F. Sacconi had in that year published I Segreti Di Stradivari . In this publication, he used as a basic module the total length of the Stradivarius mould.  Given the complexity of his demonstration, I thought I would try to simplify it. I had noticed that if, instead of taking the total length of the mould, one took the distance measured between the blocks, it resulted in a simple ratio close to ½, between the bottom bout width and the distance between the blocks.

# This idea needed time to ripen. Today after 48 years of research, and following my study on the tuning of sections of strings positioned in the peg box see: The well-tuned Universe, in The Strad, June 2019 vol.130 No.1550 p.66), I am ready to suggest a new hypothesis.

Let us consider the architecture of the violin by taking the length of the vibrating string as the new basic module     already connected to the Braccio Cremonese     and examining the consequences that ensue.

Braccio Cremonese

During the Amati era, craftsmen used measuring standards which sometimes varied from one town to another. They based some of their systems on the length of the forearm (from the elbow to the tip of the major finger): braccio means forearm.

In Cremona, the Braccio, converted into the metric system, equated to 483.5mm, as seen in the original document below (figure 1 : Tavole di ragguaglio dei pesi e delle misure già in uso nelle varie provincie del Regno col sistema metrico decimale / approvate con decreto reale 20 maggio 1877, n. 3836, p. 267).

## Fig. 1: Braccio da fabbrica cremonese 0.483539 m.

This standard gauge can still be seen in Cremona, engraved in the marble at the base of the 14th century Cremona Bell Tower, ref. S. SACCONI who gives 484mm in his book I Segreti Di Stradivari

Fig.2 : At the museum of Cremona the original plan of a Stradivarius harp bears the inscription : "Brasa duo – 2 lungo".

Fig.2: Original plan of a Stradivarius harp  bearing the inscription  : â€œ  Brasa duo â€“ 2 lungos  Â»

During the Baroque period the length of the vibrating string was approximately 6mm shorter than nowadays. It measured 322mm, or exactly 2/3 of the Braccio Cremonese which is Musically a quinte. It is this length of 322mm which we suggest considering as the basic module in the construction of the violin.

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So what about Stradivari’s moulds? Despite their varying dimensions, they present noticeable consistencies.

Some moulds are still visible in the Museum of Cremona. (cf. article by Andrea ZANRE & Philip IHLE, in The Strad, june 2019, vol.130 No.1550 p.30; François DENIS, Traité de Lutherie, 2006).

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In contrast to the usual methods, we measure the length of the mould between the blocks; therefore, without including them in the measurement. This gives us an interestingly stable measurement of 320mm.

This is 2mm shorter than the length of the vibrating string (322mm).

However, it is not merely a coincidence. We understand it easily if we take the elevation into account. The string is not parallel to the sound board and, connected by the bridge, forms an angle with it.

Viewed in profile/ in sideview, the string and the sound board form a right-angled triangle (see figure 3 : triangle ABC ). The vibrating string is the hypotenuse AC (322mm); slightly longer than the long side AB measured along the sound board (320mm); while the bridge forms the short side of the right angle BC

(usually around 32mm).

#### fig. 3

It is logical to think that Stradivari conceived his moulds (2mm shorter than the vibrating length) with the precise purpose of obtaining, through the elevation of the vibrating string, the fifth of the Braccio Cremonese. By doing so, he would achieve a coherence between the length of the mould, the vibrating string length and the Braccio Cremonese.

It is interesting to note that a straight line between the top nut A and lower saddle D corresponds exactly to the “Braccio Cremonese”. fig. 3

The violin’s other measurements naturally follow on from this: the Master worked in harmonic fractions (fifths, octaves, thirds, etc); in other words, simple fractions of the Braccio Cremonese which can be transferred to illustrate the different parts of the violin through its construction.

Fig. 4 :  average proportions of Stradivarius mould

• Distance between the blocks = AA’ Length of the basic module

(Distance between top and bottom blocks)

• Upper width DD’ = AA’ / 2

• Lower width BB’ = AA’ / 1,618 (golden ratio)

• Width of the centre CC’ = BB’ / 2

If we consider the average measurements of Stradivari’s violins (fig.4), the stop length, in other words the distance from the top of the sound board to the bridge, is in golden proportion to the distance between the blocks: (320 / 1,618 = 197,7).

These ratios can also be checked on violins from the Amati family. The lower width of the mould equates to the stop length. Fig. 5 :

Fig : 5  Violin by Nicolo Amati 1682

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This begs the question:  Why use as the basic module the distance between the upper and lower blocks since this measurement is no longer visible once the instrument is closed?

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The sound board could be considered as a vibrating string delimited by the upper and lower blocks acting as a saddle.

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Likewise the ribs, demarcating the internal widths of the sound board, are tuned in relation to the vibrating string: an octave for the upper width and almost a fifth for the lower width.

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In conclusion, the subtle combination of harmonic relationships and golden ratios merges all the instrument’s properties and reveals just how sublime the concept the violin is, acoustically as well as aesthetically.

A special thank you to Marion Pollart and Alexandre Wajnberg for their invaluable help.

Brussels 21-03-2020

André Theunis

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