Some Notes in the Margin about Mimmo Peruffo’s Paper in FoMHRI n°143, sept. 2018 : Theory and Practice of Twisting Strings

Charles Besnainou charles.besnainou@upmc.fr

Scientia non est humor in exitium iecur (Rabelais)

The title [1] of Mimmo Péruffo’s paper would like to convince us that his arguments demonstrate without possible reply the rightness of his thesis on the use of the loaded gut in the manufacture of the old strings for bass string instruments such as the lute. This peremptory assertion suffers from several weaknesses that these marginal notes will attempt to enlighten.

This answer is motivated by the fact that I am the only person with whom Mimmo Peruffo is polemical in his article by attributing to me totally grotesque and improbable facts without the slightest reference to what I have been able to write here or there, which means that he probably does not know the physical mechanisms of rope construction. I would take advantage of this response to criticize his approach and to publicize my results.

Introduction

To begin, never a scientist would dare to say that his hypohesis is the ONLY possible ! Hubble when he actually measured the precession of galaxies, thereby demonstrating the theory of the expansion of the Universe of Lemaître, had used throughout his historical article only the conditional … leaving open the debate [2]. We know today that this theory has found many applications in astrophysics that proves its solidity.

Mimmo Peruffo is probably unaware that a single counter-example is enough to destroy his hypothesis as unfounded. It is that we will strive to show with a multitude of examples. But before continuing, we must clearly define some essential words for the continuation.

HYPOTHESIS: Provisional explanation of an observation, phenomenon, or scientific problem that can (must) be tested.

CLUE : Trace or apparent and probable sign that a thing exists or has occurred and that the interpreter associates with the possibility of a reconstitution.

EVIDENCE: That is which imposes itself on the mind as a truth, or a reality, without the need for any proof or justification.

PROOF: Fact, testimony, reasoning capable of irrefutably establishing the truth or the reality of something.

FACT: Knowledge, information or any objective element of the reality of what happened, accomplished

The hypothesis of Mimmo Peruffo (that is to say, by densifying gut one can manufacture strings of smaller diameter, thus more flexible, such bass strings could be better than those of natural gut) [3] is a very interesting proposition based on the laws of physics. And this could be an alternative to so-called historical re-enactments. As a modern process, this idea is a valuable contribution to the renewal of the lute and modern instruments ; as a rediscovery of a forgotten ancient process, it must be argued with clues, evidences to become proof.

As soon as it was introduced, Mimmo Péruffo claims, for the bass strings, there are « only two hypotheses … » one resorts « to the construction of ropes ... » and the other « that the density of gut was increased by a treatment with metal compounds … ». If we agree that the second is a hypothesis, to be tested by historical documents. On the other hand, his first « hypothesis » is not a hypothesis since a widely known iconography attests to the minimum that the bass strings possessed well a texture evoking that of the marine cables. This iconography is in itself facts suggesting several clues that we will present.

Part 1

1-Iconography

This iconography extends from antiquity to the present day as well as from Europe, Africa and Asia. There are four types of iconography : engravings, paintings, sculptures and photographs. Each of them must be analyzed with different criteria. First of all, we must be aware that all representations whatever are depending on the artist’s desire to bring out a meaning ; when he signs a detail, he does so with the intention of being understood. On the other hand, the absence of a detail does not mean the proof of its absence in general ; the artist chose, perhaps, a simplification to emphasize something else. Painters are not photographers. In most part of the cases, their representations can only suggest some thing. Some times, once a while, we found some details that are too relevant to have not been expressly wanted by the artist. I have now many of them to present.

1.1- Engravings

The engraving is an art of the conciseness, it aims to make representation relevant by eliminating redundancies and without omitting of important information, here is some examples.

1.1.1– In a Latin manuscript of the 11th century depicting « King David and his musicians » the engraver drew the strings of the king’s lyra and those of the musician’s harp with a texture that obviously evokes curls, such as shriveled hair. This clue means, in my opinion, a desire of the engraver to represent what he saw. That is to say strings with twisted clearly visible and probably none polish (bumped, corrugated).

Figure 1 King David and his musicians, Bibliothèque Nationale, MS latin n° 11550, Paris
Saint Germain-des-Prés, vers 1070

1.1.2 in Syntagma Muscike (1620,) Michael Praetorius shows the bass strings of the viol with a structure of ropes looking like marine cables (?). In addition, this view gives us detail how the strings was tied to the tailpiece, it seems that the strings have been splitted in two strands to attach them to the tailpiece.

Figure 2 : Michael Praetorius, Syngtagma Musicae (1620), Viola di Gamba, planche XX

1.1.3- In Harmonie Universelle (1636), Marin Mersenne represents the bass strings of the cello and the viol with a lay up string structure (stranding). It may also be noted that the direction of the twists for the viol are opposite to that of the cello, it’s probably not as if by chance… Would it have a link with the different holding of the bow : for the cello, the bow is held over and therefore the accent is marked by drawn, while for the viol the bow is held from below and so the accent is marked by push ?

Figure 3a : Marin Mersenne, Harmonie Universelle (1636), p. 184
Figure 3b : Marin Mersenne, Harmonie Universelle (1636), p. 192

1.2 Paintings

The paintings require a more demanding reading because we must not confuse a detail with brushstrokes that are the mark of the artist’s own touch.

1.2.1 The rest during the flight into Egypt, by Caravagio ; a close inspection reveals the artist’s peculiar fascination with intricate and eccentric details : an end of twined string hanging from the pegs of the violin.

Figure 4a&b : Caravagio (1571-1610), The rest during the flight into Egypt, galerie Doria-Pamphili, Rome, Italy.

1.2.2 Young man playing lute, by Caravagio. The detail here presented requires special attention : the bass of the sixth choir does not show structure itself, but the author’s experience has been attracted by the size abruptly larger than the other strings, as if it’s bumped structure it has less density and requires larger diameter to reach its pitch.

Figure 5a&b : Le Caravage (1571-1610), Jeune homme jouant du luth, Musée de l’Hermitage, Saint-Pertersbourg

1.2.3 Triumphant Cupidon among Emblems of Art and War, by Paul de Vos & Willeboits Bossxhaert. This painting shows a remarkable number of objects, all very finely executed. It takes a very close view to discover that the artist has taken the trouble to paint the two bass strings of the cello with a finesse of neatly stunning details revealing an obvious rope structure.

Figure 6a : Paul de Vos (1591-1678) & Thomas Willeboits Booschaert (1613-1654) : Triumphant Cupidon. (private collection)
Figure 6b : Detail amplifyed (linéar transversale anamorphosis). Notice the tailpiece hook.

1.2.4 Still life with many musical instruments, Evaristo Baschenis .

Here again, it takes a trained eye to discover the precision of the artist’s touch, revealing parts of the cello strings, also like ropes.

Figure 7a&b : Evaristo Baschenis (1617-1677), Still Life with Musical Intruments, exhibition catalog, édited by SKIRA (private collection)

1.2.5  Bass viol, music notebook and sword, Michel Boyer de Rebeval

This painting has this remarkable that at least the four bass strings of the viol have a tight twined appearance that we can not fail to notice.

Figure 8 a&b : Michel Boyer (1668-1724),Bass, musick notebook and sword (1693), musée du Louvres (réserves), Paris. [pictures, Ch. Besnainou]

1.2.6 El Majo de la Guitarra, Lorenzo Tiépolo

This painting is interesting because it concerns the guitar and it shows that in the middle of the 18th century the practice of twisted cords was still alive, and not only for the bass strings confirming the following quote : more than a century after the invention strings spun [4] we find this comment in « L’Encyclopédie Méthodique » [5] : « … the spun bourdons have the disadvantage of dominating too much on the other strings, and to make lose the final sound by the duration of theirs, mainly in guitar batteries ». Which means that the only acceptable strings for the guitar of that time were all gut.

Figure 9 a&b : Lorenzo Tiepolo (1736-1776), El Majo de la Guitarra, Palacio Real, Madrid
 
Figure 10 a&b : Anonimous, Haydn playing quartet, in « La légende du violon », Yehudi Menuhin, Flammarion editor, Paris 1997.

1.2.7 Haydn jouant en quatuor, Anonymous, in « La légende du violon », Yehudi Menuhin.

The safety of the hand of this painter is attested by the whole composition of the painting. So if the C of the cello is marked with punctuations, it is certainly not a hand trembling, but a detail that the artist wants to suggest.

  1. Photographs

The photographs, despite their reputation as direct testimony, must also be carefully analyzed.

  1. Pluri-arc, Pierre Sallé, Musique du Gabon

The practice of twisted two-stranded vegetable strings is very common in the African musical instrumentarium. This multi-bow is a typical example.

Figure 11 : Pierre Sallé, Deux études sur la musique du Gabon, Travaux et documents de l’ORSTOM n° 85, Paris 1978.

We still meet today many African instruments like the cora with such twisted ropes as examplified by the next slide.

1.3.2 On this photo we immediately notice the structure in helix. The wide pitch of the helix suggests that to be functional it must be re-twisted again on the instrument to obtain a tight spirals related to a sufficient tension on the instrument.

Figure 12 a&b : Cora strings,snapshot by the author at the First Arts museum, quay Branly, Paris.

1.3.3 The well known koto silk strings made as ropes

Figure 13 a&b :  Corean koto, musée de la Musique, Paris.

1.3.4 This picture of Nepalese sarengy strings is to be compared with Paul de Vos’s painting (fig.6) of the cello strings, Triumphant Cupidon, the resemblance is astonishing. Which suggests the persistence of the techniques of twisting musical strings in time and space !

Figure 14 : Nepalese sarengy, photo Jean Galodé (1978) in La légende du violon, Yéhudi Ménuhin.
  1. Sculptures

And last but not the least sculptures from Antiquity. As for engraving, the sculptor’s gesture goes to the essential. The two following sculptures, two thousand years old, are as precise as a pen drawing.

1.4.1 A bas-relief of Etruscan antiquity in marble with a musician and his lyra, the structure of the strings is obviously that of ropes.

Figure 15 : Etruscan unrn II-I siècle before JC, Strings détail of a cithare, Musée Etrusque « Garnacci », Volterra, Italy.

1.4.2 We notice here the alternately opposite twisting of the strings.

Figure 16 : Engraved silver cup, Roman Empire, 1er siècle after JC, trésor de Berhouville, Normandie, Cabinet des médailles, Bibliothèque Nationale de France.

All these documents are not irrefutable proof in themselves, but clues that should not be overlooked. Nevertheless, the preceding figures show that the manufacture of bass strings (at least) in the 16th and 17th centuries had features in common with the texture or structure of marine ropes. This is a fact that needs to be deepened further. Does this mean that they were made so ? We will show later that it is not, but beforehand we will study all the techniques of making ropes and (re)discover a completely forgotten technique that goes back to ancient times …

Part 2

2 -Mimmo Peruffo’s his theory

I use the word theory here in the sense that it is more a hypothesis based on information or limited knowledge, in one word a conjecture, not as a set of statements or principles designed to explain a set of facts or phenomena, in particular that have been repeatedly tested or are widely accepted. So, from the diameter of the holes, he infers = > that the diameter of the strings are necessarily at more than 85% (why not 95%, mystery) of that of the holes, then he infers => that to get the right pitch and the right tension with such diameters, is needed a denser material, otherwise the strings are slack, unplayable and false (inharmonic), finally he infers = > that it is necessary to densify the gut, without giving any recipe, to complete its demonstration. By the way, he tries to scratch the supporters of the structured strings that look like ropes.

2.1 -The lute bridge holes question

In such an important article that purports to definitively prove the correctness of his theory (« the only possible hypothesis… »), it is surprising that Mimmo Peruffo provides us with so little details about his measurements. Nowhere is the least histogram of its measurements in relation to the number of choirs, the date of manufacture of the instruments or the length of the diapason. The only elements he provides tell us that about 50% (!) of its measures were eliminated, without giving any criteria. It would have been very useful to describe what are the significant elements which knocked-out 50% of the measures. Is there not a confirmation bias here that focuses solely on some data to approve its initial hypothesis ?

Why would it not be possible to attach a string with a diameter larger than the hole? It suffices, for example, to split in two strands the twisted cord (polished or bumpy), then to tie a strand passing through the hole – whose diameter corresponds to the hole – with the other strand, as evidenced by many iconographic sources, for lutes and viols. However, once again iconography brings us elements that can not be neglected under penalty of completely missing the essentials.

Figure 17a : Rutilio Manetti (1571-1639), Triomphe del amore (Siena,1624), National Gallerie, Dublin.

In a painting, richly composed with several musical instruments, Rutilio Manetti (Triumph del amore, Sienna, 1624) offers us a close-up view of the bridge of a lute with strings fixation. We can immediately observe the diameter of the strings from the bass to the treble which confirms the observations of Mersenne, namely a big bass that compared with the chanterelle can well measure between 2 or 3 mm. But the most interesting is to note that the strand coming out of the bridge hole is much thinner than the string itself. If the painter actually represented what he saw, then the conclusion is that the string was divided (splitted) into two pieces to allow one strand to penetrate the hole while the other passes by the outside to tie the knot. Thus, the mystery of the diameter of the holes in the bridge of a lute is lifted.

Capture d’écran 2018-11-25 à 14
Figure 17b : Detail, strand coming out the bridge hole is finer than the diameter of the big bass string.

This simple observation and the fact that we can easily reproduce even today this kind of fasten with gut strings is likely to completely invalidate the theory of Mimmo Peruffo. The fact that in his article Mimmo Peruffo published this view without realizing that he had in front him the answer to his holes’ hypothesis of the diameter constitutes a blatant cognitive bias that speaks volumes about his blindness to prove his theories [6] [48].

2.2 The question of the maximum diameter of the bass strings

Indeed, why say that the bass strings of the lute were necessarily less than 1.3 mm diameter. While one of the most reliable sources of the 17th century says the opposite ! Marin Mersenne, the secretary of the Savant Europe of his time and whose scientific results still force admiration today [4], gives in Harmonie Universelle (1636) in the second book (pages 7 to 72), the relationship between diameter, height and string length. On page 51, he even gives the diameters of the gut strings he measured on the lute engraved ; the 11th bass C has a diameter of one line, which corresponds to about 2.5 millimeters (yes, gut…), which confirms the evaluation that we made with the painting of Manetti. By following its calculation of proportion, we obtain a chanterelle of 0.37 mm, which is very easy to obtain. Unfortunately, Mersenne, generally so precise in his observations, gives no indication of the string length. So I made a calculation with a length interval between 72 and 67 cm and I found tension around 4.5 kg with a tuning diapason of 415 Hz, with 398 Hz, I found 3.5 kg. Which allows a good playbility. I rather trust the Mersenne measures [7] ; all his treatise is based on measures, rejecting speculations.

2.3 The question of the traces left by the strigs inside the holes

It is now necessary to examine the hypothesis of the densification of gut by metallic charges to complete the reasoning of Mimmo Peruffo. Indeed, this hypothesis can be seriously considered to go in search of historical documents that would confirm it. But, faced with the reasonable question: « Can holes contain traces of metals or oxides added to the string ? » Mimmo Peruffo refuses to jump the Rubicon by deciding « … not to take samples », probably fearing « … to make mistakes », which would contradict his hypothesis. What a shame !

So far, having written a quantity of articles on the subject, Mimmo Péruffo has not found any historical documents. The absence of such historical documents does not mean that it does not exist and that its hypothesis is definitively closed. The fact that he himself managed to densify gut with which he made strings probably proves that this is plausible ; why would it not have been possible for the elders ? For scientific research lack of response to a question does not remove the interest of this question, we simply have to find another point of view and above all do not hesitate to take samples when they are needed

2.4 The suggestion of the strings color as a proof of gut densification

Mimmo Peruffo suggests that the color of the strings could indicate various treatments, among which the densification of gut. He even thinks that the strings in brown or black hues would be the most convincing sign of densification results. Guts as well as animal skins consisting essentially of collagen are likely to be tanned. This tanning has the effect of increasing the mechanical strength. The tanned modern intestines which meet as surgical sutures have a more or less brown color. This treatment allows slower resorption than untanned sutures, but there is no significant change in their densities.

In the published article (FoMRHI 143, p24), Mimmo Péruffo honestly acknowledges: « it is true that at the moment there is no direct evidence related to a process of loading of gut (a string maker’s recipe ; a document that mentioned that the basses was treated in some way to make it denser, for example) ». So, in the current state of our knowledge, the practice of the densification of guts remains a hypothesis

2.5 About the surface condition of large lute basses

‪Peruffo Mimmo says: « It is only today that we find bumpy strings described as historical » ignoring all the iconographic examples (cf. §1) that attest to structures resembling marine ropes. He probably kindly wants to make fun of me by attributing to me the making of strings directly (!) on the bridge of the lute by the musician himself … [8]

Unfortunately his belief that the bass strings were all smooth (polish) comes from a restrictive definition of the word smooth which has many different meanings depending on the context. The common sense in the 17th century is that of a supple and soft fabric under the finger

In Thomas Mace’s Musick Monument [9] , we find dozens of occurrences of this word as adjective, adverb and verb. ; with the sense of a flat surface [MM p.59]; the action of cleanly cleaning a collage [MM p.59] ; as well as the action of marbling the end of a string to soften it to fasten easeier the knot of the frets [MM p.69] ; to lead a bow without shaking [MM p. 248] ; to sound his lute with tenderness [MM p.130]; to drag a fret onto the neck [MM p.50] ; to describe a properly cut rose [MM p. 49] ; the lute play technique that Mace calls « close play » allows flawless speed [MM p. 85] ; to find a flexible string without « hull » (without knots) [MM p.67].

To answer the question whether there were large, smooth (polished)  or bumped bass strings, we must first examine the various meanings of the word « knots » that we often meet in the description of these Venice-Catlins. In the context of a visual description it is certainly not real knot (yaw knot) but a spiral shape that winds along the axis of the string. We are helping in this by distinguishing two kinds of spiral loops: long and short : « Venice-Catlins are made up, in short double Knots » and « The Lyon String, is made up in a double Knot; but as Long as the Minikin » [MM p. 66]

This means that visually two spirals (double Knots) wrap around each other with a wide (long) step for the Lyons or with a tight (short) step for the Venice-Catlins

It is also understandable that Minikinds are not made up of two elements but with a skein of gut simply twisted: « Minikins are made up always, in long-thin-small Knots », which is the usual way to make gut strings.

Lyons would consist of two elements such as wet and still ductile Minikinds that are smoothed during drying to give a smooth final surface (see fig. 42).

The Venice-Catlins, they would consist of two elements already strongly twisted in the final phase of drying which give a dented surface (see fig.42). We will see in § 4.1.3 why these ropes are radically different from marine ropes. Indeed when T. Mace characterizes mechanically them by « the Smoothness, and Stiffness to the Finger« , he speaks of soupleness (flexibility) and elasticity (stiffness).

Which are the main mechanical properties for musical strings (see § 5.1.3)

2.6 Mimmo Peruffo invents a new scientific paradigme

He states : « There are rather indirect clues and that can give a clear vision of how things probably were done at the time. » Probably the word clear is exaggerated… He pursue «… indirect evidence that became direct evidence through calculations, like some of the arguments here, I believe ». For him, a clue or indirect evidence is transformed into direct evidence simply because the calculations that support it are correct. It is not enough for a hypothesis to be logical for whatever to be true. Mimmo Péruffo in his approach confuses the internal logic of his demonstration with the proof of the demonstration. He calls on LeVerrier and Tombaugh who have perfectly conjectured the existence of the planets Uranus and Pluto before their actual observations (material fact) ; as long as they had not been observed by their fellow astronomers, their existence was hypothetical. The internal logic of their demonstrations was based on very complicated mathematical calculations which could not be disputed by their peers. And it is thanks to their calculation that the telescopes could be pointed in the good direction. This does not mean that is always the case. For example today, the « Strings Theory of Univers » which has already been gratified by hundreds of papers in peer-reviewed scientific journals –i.e. verified by intransigent referees on the rigor of calculations– received no confirmation by cosmological facts that astrophysicists could have observed. These theories stay hypothetical, the quest continues. [10]

What Mimmo Peruffo offers us is not the result of a calculation or any « thought experiment » [11] but an amalgam of two unrelated elements. He writes : « I would like to point out the very interesting presence of barrels of hide glue in some stringmakers workshops of the 17th century » [and] « Containers with red-dye are [also] able to [be] mentionned » and he add « we cannot know whether that was employed for dye, or loading the gut. », nervertheless he argues that « I can say that glue was never used in the traditionnal or even in the modern gut strigmakers’s art ; instead, it is absolutely necessary, for many reasons, when one is making loaded gut strings today ». QED !? What is the artisan workshop that does not have hide glue  pot ?

2.7 Mimmo’s demonstration can summarized so :

Premiss : bass gut strings was made of loaded gut, because they allows smaller hole diameter in the bridge.

Fact : the holes shown in the historic bridges are too small to accept a larger diameter –to a no-loaded gut– means that only a smaller rope diameter is suitable for these holes.

Conclusion : bass gut strings was made of loaded gut., which has the effect of reducing its diameter.

What is a typical circular reasoning.

Circular reasoning is often of the form : « A is true because B is true ; B is true because A is true. ». Circularity can be difficult to detect if it involves a longer chain of propositions. see more in : https://en.wikipedia.org/wiki/Circular_reasoning

It is exactly the same process Mimmo use, he accumulate a huge number of documents, quoting, iconographies, personal thoughts [12] without logical links to arrive to what he desires. Like the Sophists of ancient Greece, he prefers to ask questions that are formulated in such a way that an affirmative answer to his hypothesis is given. He looks for the consequences that one would observe if their hypothesis were true, rather than what would happen if it were false. Fortunately for us now, Aristotle created the « LOGIC » against Sophists who used circular reasoning for political purposes to muddle unsuspecting minds.

Finally, having found no historical sources to his densified gut theory, Mimmo Peruffo for good measure adds : « Actually, no direct information [13] exists at all from the 16 and 17th century string makers concerning their art in general ; this is not just true for the bass gut string technology. For example, we do not even have a clear, direct description concerning how roped strings were made by string makers ; we just have just proof of the presence of the ‘orditori’ or ropewalk machines in some 16th and 17th string maker’s workshops. » . Thus, he returns back to back his two initial assumptions. After lengthy developments where he says one thing and its opposite, he decrees the results of the « World Strings Championship : Loaded Gut vs Roped String = 0-0 ».

Part 3

Results of our historical & technical researches

It’s time to turn to serious matters, we are going to show that these sources exist, as «direct information » and that they are perfectly understandable for who is willing to give the effort to analyze them. In what follows, I have endeavored to be as explicit and concise as possible, with examples that all readers can reproduce very easily with a minimum of tools.

Now, can we ask ourselves the following question : is it possible to make strings with pure gut that meet the criteria of a good musical string, from the point of view of tension, acoustic performance and playability ?

3.1 A short story of an inventor who discovers that his invention was known since ancient times !

I play the lute for almost 60 years and like all the pioneers I suffered a lot of disappointments in my self-taught learning. From the beginning, the thing that was the most painful to me was the decay of its bass that buzzed like airplanes in my ears, especially because of the big spun strings guitar very tense on an-historic instruments. I did not know of course that during the 17th century lutes were exclusively set up with gut strings. Like many of my budding lutenists friends, I tried to dampen intensity and duration of free bourdons with tricks such as rubber to nut, with more or less happiness…

How can bass gut strings meet the requirements of sound balance with top strings ? That is, the duration of which does not cover the duration of the treble strings, while being harmonic.

3.1.1 From twisting to strings layed as ropes

In the mid 1970’s, Djilla Abbott and Ephraim Segermanen [14] provided some answers, noting that the sound qualities of the gut strings could be greatly improved by twisting them (moisten) sufficiently, unfortunately this process had its limits and as soon as the diameter of the string became important the profits were not there. And so they proposed for the big strings a marine rope structure that would improve the flexibility of the large bourdons and thus the sound qualities of the bass strings.

With this information, I embarked on the systematic experimentation of the laying of ropes. Helped in this by a book of mechanics [15] unearthed in the library of my laboratory, in which I discovered the mysteries of the structural stability of ropes. After having squandered, in sheer loss,  a fortune in gut strings without being really satisfied with the results, I turned to a less expensive experiment using nylon fishing threads. This choice, sacrilege, will prove to be extremely happy thereafter.

For several years, I have made thousands of 2, 3, 4, 5, 6 strand ropes by systematically recording the parameters of the stranding, the number of twisting turns, the force applied to the swivel, the applied resistance force on the top… I was always very dissatisfied. [video n°1]

Video n°1 : Regular laying machine for ropes

3.1.2 From transgression as source to disruptive invention

Then, on a feverish night in February 1982, disillusioned by my mediocre results, I made, by dare, a thing strictly forbidden by the theory and practice of laying. Instead of twisting the rope in the opposite direction to that of the strands, in accordance with the principle of structural stability, I forced the strands in the same direction, which is an unstable configuration ! Of course the rope resisted this unnatural operation, the more I twisted in the wrong direction the more it took force, until what was to happen happened : the destruction of the test machine and an hank of threads and counterweights mixed together.

The miracle in this indescribable hank was that I discovered on a portion of a few centimeters the three strands perfectly entwined helicoïdally and moreover in a stable configuration. In a single glance I understood what I saw in front of me ! By the transgression of the established rules of the laying opened thus another perspective in the quest of the flexibility of the musical strings.

Having been, in my youth, a fan of models of motorized planes with twisted rubber strips, I knew that beyond a certain number of twisting turns, loops are formed which are called knots that progress regularly along the elastics thus constrained [16]. By releasing this constraint, the rubber strips unravel by driving the rotating propeller that propels the aircraft model in the air. This transition is called the solenoid phase or super winding.[17]

Was it possible to get the same organization in loops (knots) with a simple yarn of nylon twined ad hoc ? Well yes, it’s even pretty easy to do, just the right fingering to acquire … (video n ° 2). But the most unexpected thing is that after releasing the stress of necking this helicoidal structure persists I had invented the spiral single-strand string that combines both elasticity and flexibility. Immediately installed on my lute, its sound qualities immediately agreed to me right away.

Video n° 2 : Principle of making a nylon coil spring by rotary necking

The spectrographic analysis of the sounds produced by a modern spun string and the « spiral » string of the same height shows : 1) the quenching time of the spiral string is much shorter than that of the spun string, 2) the harmonic spectrum of the coiled cord is richer and is higher in the spectrum than that of the spun string, much of whose energy is concentrated in the fundamental. For the same note pitch, the spiral string perception is denser and clearer while being shorter than the spun string which has a shrill sound due to the excessive presence of the harmonic 2. It immediately appeared that the sound qualities of this string were due essentially to its elasticity, premise of a remarkable flexibility even tense.

Figure 18 : Spectrographic analysis [42] of a spun string and a spiral string of the same height (february 1982)

Very quickly, the two-stranded spiral string followed, by screwing into each other, two hélical mono-strands made separately (video n ° 3, two strands spiranyl). Then a moderate cooking makes it possible to fix the maximum constriction of necking, which means that one can adjust simultaneously the stiffness and the elasticity of the string by adjusting the good torsion according to that the string is designed to the lute (low tension) or for the viol (high tension).


Video n° 3 : Spiral string with two strands

3.1.2.1 Transverse and Longitudinal Wave Frequency Optimization

There are essentially three types of vibrations that run through the string : transverse vibration, longitudinal vibration involving compressional waves in the string material, and torsional vibration under the action of a bow or a finger. Each of these vibrations has a value that depends on the length, the density, the flexural and tensile moduli of the material, the tension, as well as the moment of inertia. There exists a set of values of these parameters for which the frequencies are in harmonic relations : then the resulting sound is the most « full » the most « plain », in a word the most satisfactory for a musical ear…

This helicoidal winding can be obtained with any more or less ductile material. Polymers (nylon, polyester, PVDF, etc.) are particularly suitable for necking. Especially since the starting point of the loop is the seat of a significant rise in temperature followed by a sudden cooling keeps the helical form naturally. It is thus possible to form gut strings moistened in an ad hoc alcohol bath at 72 degrees of concentration with a subtle fingering ; the osmotic pressure ensures the right amount of humidity in the material that gives the good ductility.  Then, you just have to slowly dry the spiral rope so that it keeps its shape. In the same way, it is possible to obtain the winding with a correctly moistened cotton cord. It is also possible to apply the same procedure to a metallic wire provided that one has an induction oven to soften the metal, but this is not within the reach of everyone… in his kitchen.

Very proud of my discover I filed a patent [18] of my lucky find. Two of my students took the study of this process (Figure 19) as subject of memory for their graduation. Thus we have developed a machine that can continuously produce wriggle and build a high performance tennis racket. At that time Internet did not exist and I did not worry anymore if this discovery had a history.

I was happy to have finally found satisfaction in the sounds produced by my lute [19]. Some of my friends enthusiastically embraced these new strings, while the learned purists of the lute dismissed them as non-historical. Then my research activities dragged me elsewhere.

Figure 19: Fonda & Jaoui, Study and manufacture of tensile springs with tights loops made of polymers, PFE, ENSAM, Paris (1984)

3.1.3 And the light sprung !

Twenty years later, the lutherie school of Puurs in Belgium asked me to give a lecture on the mechanical and acoustical properties of musical strings [20]. It was then that I connected to the Internet, as it should be since the Wide World Web exists, to complete my bibliography. What was my surprise to discover dozens of references to the questions posted about the strings of instruments of the 16th and 17th century. One of them left me completely stunned ! John Downing had discovered a text by Agostino Ramelli, dating from 1588, which enigmatically mentioned the big strings of big violins ! [21] Thanks to the Internet, a digitization of Ramelli’s book makes it possible to consult it [22]. Ramelli thus decreed this string: « … crossing through the middle of iceux : a big rope & double, made in the way of the big strings of bass-contre of the big violin which is well retorted and tense« . We understand that we are here in the presence of some clues that if they are intelligible, would give the key to building of musical strings during the 16th century.

The engraving of the plate CXC shows, in the jumble of a battlefield, a trebuchet whose ropes are finely drawn. At first glance, thanks to my experience in rope laying, I realized that the fiber organization of the rope designated by Ramelli had an unusual orientation for those who are familiar with marine ropes (fig.20a&b).

Figure 20 a&b : In Ramelli’s, a) regular laying ; b) rope in the manner of big string violin

The resolution of digitization does not allow to be absolutely affirmative. I had the chance to consult directly this 16th century book preserved in Paris in the library of the Museum of Arts and Crafts.

Figure 21 :  Trebuchet by Ramelli (photo Ch. Besnainou)

Figure 21 shows a very small surface of a few centimeters embedded in an In Quarto of more than 50 cm large ; the engraver very carefully drew what he saw (or what Ramelli asked him to represent). The fineness of the lines, without any useless draw, attests to the invaluable importance of this document. Ramelli’s rope anticipated my so-called invention for several centuries.

Comparing the different directions of the components of Ramelli’s rope with my cotton replica, they coincide in all points. What matters is its structure, like that of a helical spring, which gives this rope its hyper elasticity. The nature of the material only adds a little more to its astonishing properties, in the increasing order of tenacity combined with elasticity: hemp, silk, gut, ox nerve, sinew, etc.

Figure 22 compares the engraving with the string I had known how to make for twenty years !

Figure 22 : Details and comparison of fibers and strands direction with the author’s reply

Ramelli explains, in a convoluted way, that this rope is used to brake and dampen the fall of the balance of the machine which otherwise would be destroyed by the violence of the shock. Indeed, Figure 23 shows the extreme elasticity of such a rope manufactured by me ; it is easy to obtain more than 40% elongation. [23]

Figure 23 : Ramelli’s cordage hyper-elasticity

N.B. 1 : The replica of fig. 22/23 is not of course a big string of big violin, but a replica of Ramelli cordage ; to obtain such a structure we must understand and master techniques behind SUPER WINDING which are those used to make big musical strings.

The fact that I had rediscovered, by chance but also by perceptual and musical necessity, an old process which had a link with the musical strings and the war machines like the catapults, the ballists and the trebuchets meant that this way of twisting the threads had a very old history and pushed me to look for historical traces.

Part 4

4 Seeking for historical documents

In Vitruvius (1st century BC)  we find a particularly interesting description that combines music and the art of war machines. He recommends that both arms of a balistas be stretched out equally so that the projectile is pushed straight ; the shooter must make sure that the bandaged gut (tense) make the same sound –the same height– when they are struck, before shooting [24]. It is the oldest quotation that indicates that the twisted ropes of ballists were likely to sound like strings of musical instruments. Neither Vitruvius nor Ramelli give any explanation as to how to obtain such ropes, which can probably mean that it was obvious to their readers and that it was perfectly well known [25] [26].

Thus, this unexpectable sculpture in a castle in the south of France, which I could not show in the paragraph on iconography, before having established that a hidden link existed between stringed musical instruments and instruments of war (24). It is an archer bending his bow. The hidden link lies in the hyper elasticity (see fig.23) of Ramelli’s (assumed) rope whose elastic force comes in addition to the elastic force of the two branches of the bow (same for the ballista) which increases its jet power [27]. For a musical string, elasticity is the determining factor that guarantees harmonic sounds.

Figure 24 : Archer, Archery Museum, Saint Izaire Castle, France

4.1  The first scientific monograph on marine ropes

The most important source on the production of ropes, which embraces all the known techniques of making ropes in the 18th century and which gives an answer to the making of musical strings, is the absolutely essential work of Henri Louis Duhamel du Monceau, member since 1738 of the Academie Royale des Sciences, of which he was elected three times président. When he was appointed Inspector General of the Navy in 1739, he worked on the qualification and standardization of the manufacture of marine rope (maneuvers) as an area of strategic importance for France [28]. Its goal, homogenize, and also improve the production of quality ropes manufactured in the arsenals of Brest, Rochefort and Toulon. For this purpose he will put in place scientific procedures in competition between several teams each including: a supervising officer, an officer in charge of systematically collecting the data and a group of servants who practically perform the instructions given to make quantified experiments on the mechanical behavior of the ropes.

In his 544-page book, with many engraved plates, he describes all the steps that lead to the realization of high-quality ropes : from hemp crops, from stages of processing to obtaining all the ropes thinkable, each for a specific destination. It would be too long and tedious to describe them all here. I will stick to the basic principles.

4.1.1  Practice and theory of laying ropes

The making of ropes (big or not) with several strands of smaller sizes is attested since prehistory. Figure 25 shows the steps involved in making a rope as hunter-gatherers practiced it thousands of years ago. We observe that when two elements wind around each other in the Z direction, they must be twisted separately in the S direction to maintain the cohesion of the set (conversely Z transforms into S and S transforms in Z). With these data we have said almost everything about the structural stability of the ropes whether they are built with any number of strands ; a rope becoming a strand (torons) for a rope of greater size, and so on.

Figure 25 : Principle of laying ropes

Under these conditions there is structural stability of the whole, that is to say that the cord will not unwind. That’s why, the needelwoman’s drawstring (fig26-4) is the archetype of all ropes [video n ° 4].

Video n°4 : Needelwoman’s drawstring

4.1.2  About twisting

All the techniques are based on twisting threads. Figure 26 shows the various shapes that can appear when applying a torque on a wire and at the same time we apply a more or less well controlled axial force, obviously when we release the torque , the thread unfolds. Figure 26-2 shows a hull-loop (kink) -–which is a defect– that sailors know well when ropes are manhandle. Unfortunately, Birbent [29] did not continue the twisting moment to reach the super-coiling stage (figure 26-5, author’s addition) which would have allowed him to reconnect with a forgotten rope [22]

Figure 26: Different configuration taken by a wire in necking under axial stress ; Figure 25-5 has been added by the author, Birebent had not spotted it …

Thus, when several strands previously twisted, are associated (co-laid together => stranding) they unravel by wrapping around each other in the opposite direction. Figure 27 gives principles of a machine : the torsion hooks, the topper whose resistance, more or less great, determines the rope’s pitch, more or less tight, and the swivel, hooked with a weight, which serves to accompany the winding motion of the strands on one another (see video n°1).

Figure 27 : Diagram of a machine to lay REGULAR ropes

4.1.3 : Did you say : « garochoir »?

Then at the turn of a page of his treatise, Duhamel du Monceau mentions a rope which does not obey the principle of use, since it is a rope whose lay of the strands is carried out in the same direction of torsion as that of the separate elements, although he has insisted a few pages previously on the instability of this process [§. XIV, p. 196]! He named it: Rope en garochoir or torso hand [p.197]. The book contains dozens of experiments that use the rope en garochoir [30]. In fact, there are two types of ropes that do not comply with the principles described in 4.1.1 & fig.27.

The first is known as the current packaging twine or as the metal cables according to the Lang process [31] (Figures n° 28a & b). The schematic diagram (Fig. 29) and the video n ° 5 show that each twist of cable layed, the strands not layed get a twist which must be released by the swivels to maintain the stability of the cable, otherwise they would accumulate a twist that will tend to unroll the cable or brakes it.

Figure 28 : a) Simple packaging twine;        b) Cableway cable with rolled compacted strands
Figure 29 : Diagram of the LANG laying

Under these conditions structural stability of the string is obtained. Lang laying is mainly used in the manufacture of metal cables. This procedure was known to the ancients. We find the proof in Mersenne [HU, second book, p.99]: « the biggest chord of the 3 & 4 rank [of the cistern] is twisted, & made of a doubled and folded chorde, so to make sounds more filled, and fed. »(fig.30a). Just like Manetti who gives a striking image of it on the cistre (fig.30b) which composes the painting Triomphe del amor (Fig. 17a)

Figure 30 : a) String cistre Mersenne ; b) String cistre Manetti Triomphe del amor
Vidéo n° 5 : Machine Lang

The second rope en garochoir is much less obvious to understand. Duhamel writes on page 197 : « The ropes which are called hand torso, and in Rochefort-City [32] garochoir, … have their strands twist in the same direction as the threads ». He goes on to say: « The threads, by rolling one over the other, acquire a certain degree of tension which binds their fibers like springs, which by their reactions tend to straighten up … » and page 199 « we must look the torso hand as being made with  extremely twisted threads». Video n°6 shows how to proceed it from a seamstress’s cord. When one forces the setting in the same direction of torsion as the thread, it is the topper, by the resistance which it opposes, which straightens the loops, it is the principle of the garochoir [etymology : garochoir originates from the verb « garroter » which means to strongly twist a bond with a tourniquet to increase its tightening] ; note that halfway the thread still has enough twist to lay itself regularly. Note how loops are straightening up as being exactly described by Duhamel. In the conditions of the experiment, we obtains the structural stability: it is the change of direction at mid-point, between the garochoir and the needlewoman drawstring which blocks the garochoir segment (fig. 31), otherwise it unrolls.

Video n°6 : How to switch from en garochoir to needlewoman drawstring

From needlewoman strawsting to garochoir

Figure 31 : a) the thread that will be used to make b) a needlewoman strawstring, that is to say a direct lay on, or c) a « en garochoir » that is to say reversed lay on, extended by a needlewoman strawstring.

4.2 Ramelli’s rope is made from two elements en garochoir twisted in opposite, so in a stable state

It’s time to return to Ramelli’s rope. There are many ways to build this rope. The rope en garochoir is never used alone. Because it is extremely twisted it tends to untwist, so we will use it as the element « strand » of a rope normally lay on (that is to say in the opposite direction of its torsion), then the whole is stable. The hauser thus obtained has the remarkable property of elasticity (see fig. 23) which is needed to slow down the fall of the catapult balance or to moor a boat. What could have been a use that has been forgotten (lost) with the disaffection of the sailing navy (?), but whose need is still current for pleasure boating [33].

Video n ° 7 shows a method that I adopted to come to it ; there is no reason to think that it was not known to the ancients when one imagines the millions wriggles that have been tortured for centuries ! When comparing the various directions of the components of this rope, they correspond exactly with those of the engraving figure 22 ! QED ?

Video 7 : Ramelli’s rope in practice
Figure 32: Summarizes the different types of lay on which shows the direction of the fibers to identify them, as well as their mechanical properties.

4.3 And how to interpret the descriptions that Skippon reports in his travel diary ?

In the travel diary of Skippon [34], a young English aristicrate traveling through Europe in the 17th century, found one of the few descriptions of a viol strings maker workshop in Padua, Italy.

Figure 33: Schematic of the twisting of gut reported by Skippon

A skein of gut strips is hung in i in its middle, while its ends are hung on the hooks V and V, hooks that are driven by the gear S which implies that the two elements are rigorously twisted with the same number of turns. Some have concluded that two strings were simultaneously made with identical characteristics. We can ask ourselves the question : why not design a gear system with more hooks (2, 4, 6…) that would increase productivity ? This was common practice at SOFRACOB [35], until a few years ago.

4.3.1 Attempt to extend what Skippon did not see… or was not allowed to see

It is likely that Skippon did not attend all the big bass strings operations. By not revealing the entire manufacturing process, the craftsmen often protect themselves thus to keep their secrets.

Remembering the video n ° 6, one can imagine the following process which takes exactly the same elements reported by Skippon and completes them.. The following video n° 8 is perfectly explicit, especially, by comparing the return in « i » of the skein of gut of figure 33 with the « a » of figure 34 !

Video 8: Proposal that completes Skippon’s description for making large strings

To be mechanically perfect, it must be taken into account that when, in a first step, the two strands previously prestressed by necking (I) are laid together, in a second step (II), the strands not yet laid take on an additional torsion each turn, it is therefore necessary to release this supplement by an opposite movement of the hooks b and c (fig.34).

Figure 34 : Alternative process from Skippon…. in two steps : (I) prestess ; (II) stranding

This example makes it possible to understand that between Lang’s process and the garochoir there is simply an additional twisting pre-stress that is used to « straighten up the fibers like springs » (Duhamel du Monceau).

Part 5

5 Sound qualities of musical strings « en garochoir » : acoustic study

Having demonstrated that one is able to reconstruct Ramelli’s cordage does not mean that the big lute or violin bass strings were so made (video n°7). It must now be proved that the strings that I think to have rediscovered (part 2) by judiciously using twisting can answer to the sound and playability qualities that musicians need.

5.1 Prerequisites for the qualification of sound properties of musical strings

The European cultural area has privileged harmonic sounds to compose its music, unlike other cultural areas that have created musical wonders with inharmonic sounds [36], such as percussion (musical bow, balafon, gamelan, gong, bell, concret music…) [37]. The need for harmonic sounds comes from the choice of musical scales that are built to make chords ; each sound being in correspondence with the intimate components of associated sounds. The seven notes which constitute the range of Western musics are all the transpositions of the harmonic components of a fundamental note, that is to say whose frequencies are multiple integers of the fundamental vibration frequency of 1. It will not have escaped to anyone that the components 4, 5, 6 and 7 engender the major perfect chord (i.e. C, E, G and with the seventh B flat). So all Western music works with chords without beats [we will not go into the charm… of temperaments here]. The essential reason is that the sustained sounds –wind, bow– always produce harmonic sounds out of physical laws necessity. But, none sustained sounds, plucked or hammered, are by physical the same laws necessity none harmonic [38].

5.2 The key-role of bending stiffness on the eigenmodes of a string

To be able to marry with harmonic sounds, non-sustained sounds must have an almost harmonic spectral composition, as close as possible to the harmonic series. The physical property that determines the quasi harmonicity of a plucked string (or harmmered) is its stiffness, more precisely the radius of curvature (fig. 34) that it can take when vibrating waves propagate throughout. Each component is characterized by nodes and bellies –eigen modes– that divide the string into 1, 2, 3, 4, 5 … Thus, the minimum radius of curvature determines the smallest possible deformation of the string, so the maximum possible frequency of the sound, beyond there is no more components.

Figure 35 : The real string is rounded at the pinch point, thus limiting the size of the maximum permissible deformation, so the maximum frequency of the nth eigenmode

But at the same time « the stiffness constitutes for each partial a restoring force, like any restoring force, it increases the frequency of oscillation [38, p.71] » ; in other words, the stiffness has the effect of shortening the distance between the nodes (Fig. 36), which has the effect of shifting (sharpened) the frequencies of the eigen modes of the string which are moving away from the harmonic series: the note sounds false. Even if the ear is tolerant, from a certain degree of inharmonicity the consonant chords are no longer acceptable.

Figure 36 : Shift of the frequencies of the eigenmodes with stiffness: ideal vs real string

When the ratio length to diameter of a string is very large, the radius of curvature becomes almost negligible. For example, an Italian harpsichord string of 40/100 mm diameter and 2 meters long produces harmonics up to 18 000 Hz without difficulty, the radius of curvature at the pinch point is less than 0.2 mm. The same diamètre on a cistern of 52 cm diapason, is out of intonation, unless an ad hoc texturing improves flexibility (see video 5) as Mersenne recommends it [39].

5.3 How to increase the flexibility of a string ?

Let’s take the time to see what physics can teach us. For a homogeneous material wire, the flexural modulus is deduced from the tensile modulus (Young’s modulus) [40]. The challenge is to increase the flexibility of a large diameter rope regardless of the Young’s modulus of the material constituting it…

A first solution is given in Figure 37 showing a 10 mm diameter gut string that was found in the case of a 17th century violin double bass (Musée de la musique de Paris). It is observed that the gut strips are not glued to the core which allows the helical loops to slide one part relative to each other and thus increase the flexibility and reduce the radius of curvature, so a better harmonicity of the string.

Figure 37: The helical shaping of the strips thus becomes a deformable structure that gives elasticity and flexibility to the string. The marine rope is a particular helical shaping.  Musée de la musique, Paris. (photo Ch. Besnainou)

A second solution can be conceived by noting that a spring, maintained in extension, simultaneously allows longitudinal and transverse deformations (fig. n° 38) with the smaller radius of curvature possible.

Figure 38 : A spring combines at the same time elasticity and flexibility
 

5.3,1 Mechanical properties of the « garochoir » musical string

Thus, the garochoir rope combines these two properties of transverse flexibility and longitudinal elasticity [41]. To demonstrate this, we made the following experimental set up (fig. 39).

1- We spun fibres, of different colours, to make an initial strand,

2- with which a rope was laid in the regular direction,

3- and then with this same strand a en garochoir rope.

Figure 39 : The same initial strand was used to make two ropes with different properties, the regular lay and en garochoir lay.

It is immediately noticed that the constituents of the initial strand in the two ropes have perpendicular directions recognizable at first glance (fig. 32).

Incidentally, if one compares the clenched fingers due to Dupuytren’s disease (Fig. 40), one immediately grasps why the garochoir is also called, in French, torso hand, that is to say twisted hand.

Figure 40 : The French word « main torse » means by metonymy the similarity of form between a rope en garochoir and the clenched fingers due to Dupuytren’s contracture desease.

The following videos 9(a-b-c-d) show the initial strand stretched between two fastening points, as well as  regular rope and en garochoir ; with the pliers, we try to deform them axially. Neither the strand nor the rope has longitudinal elasticity, while en garochoir can deform easily by 1 cm, like a spring.

Vidéo 9a initial strand 
Vidéo 9b regular lay
Vidéo 9c en garochoir 
Vidéo 9d two strands coton garochoir

It is this property of longitudinal elasticity which makes that the laying en garochoir allows a radius of curvature of the order of magnitude of diameter (fig 41) guaranteeing a perfect harmonicity to the sound. In addition, by adjusting the twist of the rope before passing in the peg, we can adjust its stiffness/elasticity on the instrument and therefore its sound quality (see 3.1.2.1).

Figure 41: The radius of curvature of this lute bass en garochoir is the size of the diameter of the string (2.9 mm)

5.4 Sound qualities of the musical string « en garochoir » : acoustical study

To be complete this study ends with a comparative acoustic analysis of the sound qualities of several gut strings :

1-a current commercial string made of gut with collagen strips glued to the heart, without any flexibility

2- a gut string designed by us according to the Lang. From my point of view (§ 2.5) it is these strings that T. Mace calls Lyons. (fig 42)

 3-and a two-stranded garochoir gut string. From my point of view (§ 2.5) these are the strings that T. Mace & J. Dowland calls Venice-Catlins. (fig 42)

Tied up on the same bench (fig 42), these strings have the same mass per length unit, the same vibrating length (60 cm), the same pitch (70 Hz), the same tension (2.3 Kg) and are pluged in the same way

Figure 42: Above, a gut string according to Lang process (Lyons ?) ; bottom, a garochoir string,

The spectrograms [42] of figure 43 allow to analyze finely the sounds associated with each of the strings of experience

Figure 43: Comparison between sounds emitted by three different gut strings by sonagraphic analysis, in abscissa time; on the ordinate frequency. The red line has its frequency = p1 x 6
Figure 43 soundtrack

Pi are the partial components of sounds. The three sounds have the same fundamental tuning on 70 Hz (Csharp1). We can notice that the frequency of the sixth component of the sound3 is exactly 6 times that of the fundamental, which means that the partial components of the sound3 are indeed harmonics of the note. While for the sounds 1 and 2 the frequencies of the partial components move away from the harmonic series, which is confirmed by the listening (audio 1). These sounds are poor, sound wrong and have no sustain. While the listening of the sound3 gives to hear a comfortable sound in duration and harmony.

These examples show to what extent stiffness is decisive both in terms of spectrum –harmonic richness– and duration –sustain– on the musical qualities of the strings. They make it possible to understand how stiffness directly affects the duration of vibration extinction. The quality of the sustain is directly related to the fusion of harmonics between them.

Because of the stiffness, the radius of curvature, reducing the effective vibration length of the string’s natural modes, results in a higher propagation rate of the high frequency components than those of the low frequencies (dispersion) [38].

Thus, the n mode wave propagates faster than the n-1 mode and eventually catches up with it, so they interfere, which causes them to cancel faster … And so for n-1 with n- 2, then n-3 … Here is the explanation of the shortened decay time of inharmonic strings. This dynamic damping is therefore added to the thermal structural damping due to the friction of the molecules between them during the propagation of the deformation waves. The following examples show how the inharmonicity inherent in stiffness, even if very weak, affects strings of small diameters and different densities (fig.44). Listening to the sounds produced, even a non-musician ear immediately perceives the difference in timbre due to inharmonicity (audio 2).

Figure 44 : Comparison between two strings of the same height = 220 hz which differ only in their stiffnes. (Red lines frequencies are multiple by 5 to the fundamental)
1- Nylon, length = 60 cm; Φ = 1.1 mm, density = 1
2- FluoroCarbon, length = 60 cm; Φ = 0.8 mm, density = 1.8
Figure 44 soundtrack

The two strings, which have the same mass per unit length, as well as almost identical tensile modules (~2500 N/mm2) differ only in their diameters. The thinnest string therefore has a smaller radius of curvature at the pinch point. The resulting sound is more harmonic and lasts longer by about one second, which is considerable for the ear.

5.5 Discussion on the respective merits of density and stiffness

The preceding paragraphs show that a good musical string must be above all harmonic whatever the material with which it is made. If we compare the 0.8 mm diameter PVDF string of fig. 44 with a steel string, all else being equal (linear density, length, frequency and tension) of 0.38 mm in diameter, then a simple qualitative calculation [43] shows that the radius of curvature at the pinch point is much smaller for steel than for PVDF because the Young’s modulus of steel is about 1000 times greater than that of PVDF which means that the PVDF string is more souple and then more harmonic than the steel one.

Therefore, the increase of the density of the material and correlatively the decrease of the diameter of the string are not necessarily beneficial to the harmonicity of the string. We always have an interest in favoring flexibility.

Beyond a certain diameter, whatever of the material, the bending curvature radius affects the harmonicity and the string loses its musical qualities. We have seen that one way to increase flexibility is to create an ad hoc geometric texture. What is Mersenne’s proposal to wriggle the cittern strings. Compare Young’s module of a 50/100 mm iron monofilament string with a a double iron twisted strands of the same mass per unit length. Such a textured string is obtained by laying two 31/100 mm threads together using the LANG process (fig 29).

Under a strain of 10 kg (100 N), the 50/100 string (1m length) extends by 0.6 mm ; while the string of equivalent linen mass extends by 2.5 mm (video 9). This means that Young’s module of the twisted string is 150 times weaker than that of the monofilament [44]. This is how the twisted strings « make sounds more filled, and fed. » Mersenne dixit.

Vidéo n°10. Allongement remarquable d’une corde tortillée de 2 x 0,31 mm en laiton.

Nota bene 2 : In the most common cases, Lang process is required to form multi-stranded cables (>> 2) that wind together in a helix with an extremely wide pitch, which ensures their stability but their longitudinal elasticity is reduced to that of the strands. But in the case of 2-wire rope twisted as « Mersenne », the metal wires form extremely tight loops and the pitch of the helix is of the order of magnitude of the diameter of the wire, hence a structural amazing elasticity (two springs nested one inside the other) of these strings. These cords are conveniently placed in the string of cistre, bandora, orpharion, chitarrone, clavichord, spruce and even fortepiano.

Nota bene 3 : When a cable regular laid (§ 4.1.1 Fig. 25) is subjected to a tensile force, the loops of the strands tend to tighten on the center, which has the effect of increasing the friction, therefore the depreciations that ruin the musical qualities of such strings. But the strings in garochoir exhibit a completely different behavior, a pull on a single-strand or double-strand string has the effect of widening the pitch of the turns (like a spring, fig.35) without tightening them on the center ; this is why it is advantageous to adjust the twisting preload of the string on the instrument before tying on the bridge according to tastes, game and the possibilities of the instrument.

5.6 The part of the damping in sound qualities

There is no relationship between density and Young’s modulus. There are materials that have close densities without having close young’s modules. For example : copper and nickel density = 8900 kg/m3 and Ycu = 128,109 N/m2 ≠ Yni = 207,109 N/m2; It can be inferred that copper having a smaller young module, allowing a smaller radius of curvature compared to the same nickel string, will give a more harmonic chord.

But in the sound qualities of a string comes another very important factor : the dekay of the vibration after the attack transient is internal damping. This damping of the  wave propagation at the heart of the material has several origins.

The viscous friction of the vibrations of the string with the air, these losses are preponderant for the low frequency components of the spectrum ; the material’s own viscoelasticity acts mainly on the high-frequency components ; thermal losses due to the compressed and stretched areas affect the intermediate range between low and high frequencies of the components of the vibration spectrum of the chord [45].

Figure 45 : Comparison of the average Q values obtained on threads of different composition and metallurgy [37].

That is why there are such differences between a copper tread and a steel one. Figure 45 shows the average of the quality facteurs (inverse of the damping) of each of the vibrations of a string vs frequency for three materials, copper, iron and steel. These curves give the relative importance of the components in the perception of the timbre. A steel string is much brighter , even more aggressive, than an iron one while the tone of a copper string is perceived as less clear, rounder, softer, with more fondamental.

5.7 Inharmonicity caused by damping

In all good physics books we find oscillator equations which show that the natural frequency of a mode depends not only on mass and stiffness but also on damping (Fig. 46). Unfortunately none gives concrete examples, one is reduced to « believe » in the power of mathematical calculation.

Figure 46 : The peak frequency of the resonance of an oscillator decreases with damping

But the loaded gut strings produced by Peruffo Mimo offers a remarkable example that deserves to be included in all physics books. The sonograms in Figure 46 show that all the partials of the string have frequencies that are lower than the harmonic series (red equidistant line). Damping is an added inertial force (i.e. mass) that lowers the frequency of the mode.

Figure 47 : Two loaded strings by Peruffo M. a) density=2290 kg/m3 ; L=60 cm, d=1.25 mm, f=69Hz, T=2.8 Kg and b) density=2650 kg/m3, L=60 cm , d=1,65 mm, f=54 Hz, T=2.9 Kg
(Red lines are the harmonic serie to the fundamental)
Figure 47 soundtrack

This inertial force, which confers a virtual increase in mass lowering the mode frequency is different from the restoring force due to stiffness (§ 5.1), which increases the spring term of the mode, which has the effect of increasing its frequency.

By analyzing sonogram fig 47b) (the lowest sound), a trained eye will notice that the first eleven partials are lower than the harmonic series, while the 14th, 15th 16th partials are higher than the harmonic series [46]. This means that the inertial force due to damping no longer acts and that the upward shift is due to stiffness. The laws of physics stays awake… even for extremely short durations.

These strings certainly do not correspond to John Dowland’s recommendation that the basses of his lute should not be doubled by octave [47] because these sounds (audio 3) are short, very deaf and without sustain.

CONCLUSIONS

From iconography to craftmen’s know-how, mechanics and acoustics, the common thread of this research was the twisting. By replicating Ramelli’s enigmatic rope in the organization of its intimate fibers and strands as well as in its elasticity qualities, we have shown that under certain conditions, instead of ruining the qualities of the rope, the solenoid phase (super-coiled) –correctly used– allowed to obtain ropes of very large diameters which remained remarkably supple (elastic), and therefore conducive to harmonic accuracy essential for musical use.

Without being absolutely affirmative, it is conceivable that craftsmen of the 16th and 17th centuries who were familiar with the art of twisting in the making of gut string had one day the experience of the super-coiled phase and that they had it then applied to « big strings of big violins » and lutes ; that they have kept the secret is in the spirit of their time [Skippon].

What is remarkable about the solenoid phase is that it can be applied to a wide variety of materials (gut, cotton, synthetic polymers, DNA… etc.). Today, we prefer to set up on our lutes and viols with these polyester strings, indistinguishable by the ear from gut strings and much less sensitive to humidity.

We were also able to remove an ambiguity of language. Both, Thomas Mace [39] and John Dowland describe their « catlines are double knots joyned together », which, in our opinion, would mean « two strands entangled together, inside each other, in super-coilling« . Indeed, even today, in the model-making community, speaking of twisted rubber motors, Anglo-Saxons use the same formula: « double knots » ; so « knot » must be taken not in the sense of « tie » but « loop« .[16]

We have rediscovered that there had existed since time immemorial a technique, today completely forgotten, of laying ropes « en garochoir », whose main property was to confer a phenomenal and adjustable elasticity to the strings so laid ; that the energy accumulated in this kind of springs could be used for jet weapons ; mooring ships ; but also as a suspension for the carriages instead of the commonly used wood or metal blades [23] ; that the musical strings of very large diameters retained a correct intonation (harmonics) thanks to elasticity

We showed that we could hang very big strings in the small holes of the lute bridge, it is true with a little imagination… We also showed that one could be « a university graduate » [Peruffo dixit …] to know how to enlarge the holes in the bridge of a lute [48] to make it pass big strings and to check that the big nodes thus obtained were a big source of damping of the vibrations and inharmonicity. To close this file, we showed that one could make the strings of his lute himself in his kitchen (private joke with Mimmo) [49].

Last but not the least

It is not excluded that one day, perhaps a scholar will discover in an old grimoire the track that will lead to the densification of gut as a historical process, like the others ? This does not mean that the Mimmo Peruffo’s research was useless.

The determination he showed in pursuing his idea of loading materials finally paid off ! His work has had some collateral damage… i.e. significant and useful benefits : he invented the loaded polymer threads, which in my opinion are the best musical strings ever seen, better than nylon, polyester, PVDF and even the pure gut currently available for lutes, guitars (romantic and modern) and even the bow instruments. There is no doubt that its charged polymers will become the 21st century’s strings, congratulations.

C’est fini.

* Charles Besnainou

Retired, researcher-engineer, Laboratoire d’Acoustique Musicale du Centre National de la Recherche Scientifique (CNRS) ; Université Pierre&Marie Curie, UPMC-Paris 6

Former professor of the musical acoustics class at the Conservatoire National Supérieur de Musique et de la Danse de Paris (CNSMDP).

Aknowlegments

I would like to thank my former students of the musical acoustic class at the CNSM in Paris for making the videos that formed part of the practical work of teaching.

APPENDIX 1

The point of view of the concert performer Christophe Coin, viola da gamba teacher at the CNSM in Paris

Figure 48 : Christophe Coin’s Henry Jay viola da gamba 1626

« It is an instrument of 1626 made in London by Henry JAY, by a famous maker of viola da gamba. This instrument is what we call  » the Consort Bass « , it is the biggest instrument of the family of viols before the « violone », the double bass.

It had been transformed into cello in 19th century (regenerated as said its label) and we put it back in the most plausible original state thanks to an instrument of the same maker coming from the collection Kastler, slightly smaller by four centimeters, which is a « DivisionViol » of 1621; and we were so able to reconstitute the original drawing and it gives a length of vibrating string of practically 80 cm, what is enormous ; it is between the extensions of the viol and those of the double bass, what obviously raises problems of tension.

Until now the instrument was set up with open-spun strings, that are wounded with a thread of brass, which was not certainly the case at that time, because we know that they did not still exist and thus the challenge today is to put strings non-spun to the grave.

What I find interesting in this type of strings, it is that we have a great deal of fundamental sound, and, I would say of  » under fundamental  » like a 16 feet in the sound. Furthermore we have an equal resonance which does not decrease too fast, which remains stable till the end, and when we pass to the not laying ropes, we have an excellent homogeneity.

I think this is a progress and we can start to find something plausible, while remaining historical, as close as possible to something original.

The sound emmision is pleasant, that is to say that the bow hangs very quickly with a minimum of rosin on the strings and white hair. The speed of emmision is perhaps not yet optimal. I think we still work on the tension and the curvature of the bridge and that we can improve it.

And so I think we are on a right trail and that we can abandon the strings half spun, for this type of instrument (before the late 17th century, because we know they did not exist), while having something musically plausible that works and gives both a rich harmonic burst to the treble and at the same time largeheavy fondamental. »

APPENDIX 2

Here, me  too, I would like to issue some questionable thought… about the so-called « half-spun strings ».

During my iconographic research, I discovered two paintings with extraordinarily precise details that clearly suggest spun and half-spun strings. These two paintings represent viols of seven-stringed gambes from the 18th century.

Figure 49 : a) détails « Nature morte, gibier, fruits et viole de gamble » by François Desportes (1661-1743), Château de Giens, Musée International de la Chasse, France..
b)Violiste « Jean-Baptiste Forqueray » by Jean-Martial Frédou, 1745 (photo Pierre « Mathias » Jaquier, private collection)

These representations (fig. 49 a&b) are remarkable : the 7th and 6th strings of the two instruments are obviously strings spun with a fine silver thread, moreover the 5th of a) and especially the 5th and 4th of b) have much wider punctuations suggesting a large diameter silver wire. Could this be the clue of wide-angle spinning as it is commonly practiced on the strings of clavichords, spinets and fortepianos ?

In the opinion of the current musicians, the half-spun chords in imitation of the strings of clavichord are extremely fragile, the rubbing of the bow ruins them in a very short time because of the metallic line in over-thickness on the gut ; they are practically not used today.

A comparison between J-B Forqueray’s viol strings and those in Boyer’s painting (§ 1.2.5) indicates very different manufacturing techniques (fig. 50 a&b). For my part, Boyer’s are made with the « garochoir » twisting while Frédou’s is asking how such a large metallic line – about the size of the diameter of the core gut on which it is wound – can-it holds and above all do not hamper the bow playing and much more do not wear out quickly ?

Figure 50 : Comparison of the textures of the strings painted by Boyer (17th century) and Frédou (18th century) ; while all the strings at Boyer are in twisted gut, Frédou’s is based on spun and half-strung strings.

It was on this occasion that I made the connection between the torso columns of the altar of St. Peter of Rome and the designation of the torso hand (french) to describe the garochoir. Indeed, when one applies oneself to stretch well a gut garochoir during the drying process, one obtains a stable structure in all points identical to that of a torso column (fig. n°51 a&b).

Figure 51 a&b :a) Altar of Saint Pierre of Rome ; b) garochoir stretched

Remembering that during some services the torso columns are decorated with garlands of flowers wrapped in the furrow, everything became clear, including the size of the spinning line.

The string made of half-threaded garochoir (fig. n°52) became perfectly obvious to built !

Figure 52 : The half-wired string with a varnished copper wire wrapped on the garochoir (silver is too expensive)  

For bowed instruments (viola da gamba, baroque cello, cello da spalla) these strings are of remarkable sound quality, excellent underhair pickup and as the line is embedded in the groove of the garochoir the wire does not tend to slip and wear.

There would still be a lot to write, but we really need to stop.

I dedicate this work to the memory of my comrades at the bench :

Pierre « Mathias » Jaquier, luthier

Ho Xich Tué, computer scientist, viol and lute player

John Wright, musician and organologist

NOTES

[1] Mimmo Peruffo , « Why the load of gut for bass strings is the only hypothesis that fulfils the requirement of seven criteria arising from consideration of historical evidence »published in FoMRHI Quaterly n ° 143 pp 4-31, september 2018. Article that can also be found in the magazine of the Dutch Luth Society: « The Lutezine n°126, july 2018 », pp 37-84.

[2] All quotation in square brackets and in italics are taken from the article by Mimmo Peruffo (MP)

[3] Edwin Hubble, « A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae », Proceedings of the National Academy of Sciences, vol. 15, 1929, p. 168-173

[4] Introduction to the skill of music, John PLAYFORD. London 1664.


[5] L’Encyclopédie Méthodique : l’Art du faiseur d’instruments, Paris 1785.

[6] During the round table organized by the Dutch Luth Society (August 31, 2018) in Utrecht on the question of ancient lute strings. Mimmo Peruffo having learned my invitation at the last minute and having probably understood his cognitive dissonance, was quick to show us a photo of Manetti’s painting grossly make up to make us believe that the strand coming out of the bridge had the same diameter as the string and that he pretended to have himself taken during one of his travels to Dublin National Museum. Since then I challenged him to publish this « piece of evidence ». What he does not know is that with the Photoshop software we are perfectly able to go back to all touching up of an image.

[7] Marin Mersenne, l’Harmonie Universelle, Paris 1636, livre second p. 99.

[8] May be even in his kitchen after making sausages out of sheep’s guts ? On reflection, it is a simple way to weigh down gut especially if these sausages are heavy to digest on the stomach…

[9] Thomas Mace, « Musick Monument, the Lute Made Easy », 1676, London.

[10] « The proof of the pudding is by eating it … »! What are the indirect proofs of pudding ? Stale bread, brandy, sugar, raisins as indirect proof, do not make a pudding.

[11] The general approach that presides over experiences of thought is formulated by the question: what would happen if …? https://fr.wikipedia.org/wiki/Expérience_de_pensée

[12] That it would be too long and tedious to dissect here !

[13] Once again here M.P. replaces the notion of evidence with that of information when it suits him.

[14] Djilla Abott & Ephraïm Segermann, On twisting gut strings, Early Music, Vol. 4, No. 4 (October 1976).

[15] Henri Bouasse, « Cordes et membranes« , Albert Blanchard éditeur, Paris (1927). Particularly prolific author who left to posterity more than 45 volumes, dealing with mathematics, physics, mechanics with hundreds of examples drawn from the operation of musical instruments, and in particular on musical strings.

[16] Robert Morris « Twist and Writhe near Max Turns in Rubber Motor », Free Flight Quarterly April 2011 ; https://www.hippocketaeronautics.com/downloads/Twist_and_Writhe-Morris_v2-2.pdf . Quote : « I do remember noticing the rows upon rows of knots that formed during winding and dissolved during the power run ». Which discribes perfectly the process.

[17]A. Ghatak & L. Mahadevan, Solenoids and Plectonemes in Stretched and Twisted Elastomeric Filaments, Physical Review Letters, vol. 95 ; 2005 (figure above)

[18] Patents, FR 8303921 (10/3/1983) ; FR 8320763 (26/12/1983 ; EU 84400493(12/3/1984).

[19] Charles Besnainou, « Les cordes et leurs mystères », in Tablature, revue de la Société Française de Luth, Juillet 1987.

[20] Charles Besnainou, « La fabrication des cordes et en particulier comment répondre aux questions posées par les cordes anciennes », lecture, Corde Factum, Puurs, May 2008).

[21] John Downing, « Roped Gut Bass Strings », FoMRHI Quaterly, C-1318 Jan. 1995#77 and « More on Roped Strings and other Knotty Problems ». FoMRHI Quaterly, C-1318 April 1995#79

[22] Agostino Ramelli, « Le diverse et artificiose machine », Paris, 1588.

http://cnum.cnam.fr/SYN/fDY3.html

[23] This property of phenomenal elasticity will be resumed in the 18th century for « substituted these strings for the springs [metal] of post chaises & other carriages, and they have done very well » which « M. le comte d’Herouville used It particularly, either to collect what the ancient Greek and Latin tacticians had on catapults, ballists, and other machines of war to which they used the cords of nerve » in Encyclopedia Diderot & d’Alembert, see volumes III & IV, pp. 18 & 207-208, Paris (1752)

[24] Vitruve, « Les dix livres d’architecture de Vitruve », dans la traduction de Claude Perrault en 1673, éditeur Bibliothèque de l’image, préface de Antoine Picon, Paris 1999

Quote from Vitruvius: « As for Music, it must be consumed so that it knows the Canonical & Mathematical Proportion for the proper bending of war machines like Ballists, Catapults & Scorpion (trebuchet), whose the structure is such that, having passed through two holes through which the arms of the catapult (a ballist) are also stretched, and one of which is to the right and the other to the left of the capitals of these machines, cables are made gut strings that are banded with vindas or reels & levers; these cables must not be stopped to put the machine in a state of unchecking, that the master [of shooting] does not hear them make the same sound when we touch them, because the arms that we stop after being bandaged, They must strike with equal force, which they will not do if they are not stretched equally, and it would be impossible for them to push straight up what they must throw. »

[25] There is a rich iconography that supports this thesis, but has no place here.

[26] The quotation from Vitruvius close to Ramelli’s text fully justifies John Downing’s hypothesis (Catapult Cordage – PART 2, More Speculation, FoMRHI Quaterly # 81, Oct. 1995, Comm 1395, p.21) that the word « catline » refers to catapult cordage as a contraction of the word « cat(apulte)line » and of course the word « catgut » has retained the prefix « cat » (perhaps for the twisted structure) and to indicate the material, the gut. It is a shame that some opponents have tried to debunk this great idea, which today finds consistency elements ! See : Ephraim Segerman, On Downing’s speculations on catgut in Comm 1751, FoMRHI Quaterly #106, January 2002, Comm 1791, p30.

[27] Saint-Izaire Castle and Archery Museum, 12480, Saint Izaire, France.

https://www.saint-izaire.com/archerie/

If one observes the rope well, it is formed of two twisted segments in opposite direction When hanging a rope with a load, it tends to untwist. Having the two segments of the bow string twisted in the opposite direction means that when the archer strikes his bow the two segments go to untwist in the opposite direction, that is to say, to prevent the rope from rolling. under the fingers.

[28] Henri-Louis Duhamel du Monceau, L’art de la corderie perfectionné, seconde édition dans laquelle on a ajouté ce qui regarde les cordages goudronnés, Reproduction fac-similé de l’édition de 1769 : https://ia800304.us.archive.org/17/items/bub_gb_gROUgnp92V4C.pdf

[29] Birbent, Résistance des fibres végétales filées ou commises, Annales de la faculté des sciences de Toulouse, 3ème série, tome 21 (1929), p. 43-137. Birebent was professor Henri Bouasse’s assistant [9]. http://www.numdam.org/article/AFST_1929_3_21__43_0.pdf

Here we have the very example of a phase transition. When we inject energy into a system, it tends to return to its equilibrium position ; but depending on the boundary conditions it can switch to another more or less stable equilibrium. Figure 26 shows the different bifurcations that appear when a wire stretched according to a certain force is subjected to a necking. The example shows four almost stable states..

[30] I did not find in the English specialized literature the description of such a rope, so I keep its word in French en garochoir.

[31] From Dutch langslag (same direction) contrary to kruisslag (oppossite direction), formerly Albert’s lay, Wilhelm Albert (1787-1846) german engineer who adapted this concept to mine cables

[32] I live a few kilometers from the Corderie Royale de Rochefort, and when I asked the curator of this museum to show me a garochoir, he was completely unaware of its existence. Like what, oblivion is the worst of purgatories. https://www.corderie-royale.com/

[33] The need for elastic moorings is still current, we find on the market this type of spring system for pleasure boating. (photo, www.bateaux.com)

[34] Philip Skipon (1641-1691), An Account of a Journey Made Thro’ Part of the Low-Countries, Germany, Italy, and France, London 1732.

[35] SOFRACOB, Société française de cordes en boyaux, ZI 38121Reventin-Vaugris, France

[36] In music, inharmonicity is the degree to which the frequencies of overtones (also known as partials or partial tones) depart from whole multiples of the fundamental frequency (harmonic series). Acoustically, a note perceived to have a single distinct pitch in fact contains a variety of additional overtones which contributed to pitch and timbre.

[37] Pierre Schaeffer, Traité des objets musicaux, édition du  Seuil. 1966, Paris

[38] C. Valette & C. Cuesta, Mécanique de la corde musicale, Hermes éditeur, Paris 1993.

https://exofessubhe.firebaseapp.com/2866013638.pdf

[39] Marin Mersenne, l’Harmonie Universelle, Paris 1636, livre second p. 99.

[40] https://en.wikipedia.org/wiki/Young%27s_modulus

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

Ideally, flexural or bending modulus of elasticity is equivalent to the tensile modulus (Young’s modulus) or compressive modulus of elasticity for homogeous materials. In reality, these values may be different, especially for polymers (by non-linéarities) and textured ropes.

[41] A simple analogy makes it possible to understand this notion of elasticity under tension: consider a metallic chain, this one is very flexible not tensioned and becomes inelastic and non-flexible when it is under tension

[42] The spectrographic analysis (sonagram) of sounds is a time/frequency representation that can be read as a musical staff. Moreover, this representation shows the intimate spectral composition of the sounds simultaneously to the pitch. See free software :

https://wavesurfer-js.org/

[43] For a beam embedded, the deformation funder a load P is given by the formula f=PL3/3EI (E Young modulus ; I moment of inertia)

https://fr.wikipedia.org/wiki/Formulaire_des_poutres_simples

[44] For a simple calculation, see : http://www.formules-physique.com/categorie/306

[45] C. Valette & C. Cuesta, opus cité, pp 83-132

[46] The lack of the 12th and 13th partials indicates that the string was pinched exactly between 5 cm (60/11) and 5.5 cm (60/12) from the bridge. It is inferred that the string was pinched with a 5 mm wide plectrum ; with a inch thumb, the contact would have been much wider (about 20 mm) and the absence of components would have affected more partials…

[47] in Robert Dowland, A Varieties of Lute Lessons, London, 1610. John Dowland, « Other necessary …to set a small and a great string togetheris left, as irregular to the role of Musicke. » p.14

[48] To definitively close the question of the holes that haunted Peruffo Mimmo for years, the following figures should reassure him:

On the left, evidence that a « graduate of the universities » can make a hole 2.5 mm diameter in the bridge of his lute to tie a gut string 2.3 mm diameter ; in the middle, the same rope 2.3 mm unwrapped in two strands (time : 2.5 minutes) to achieve a knot as in the painting of R. Manetti ; on the right, still in gut, a 3 mm diameter en garochoir string tied to the bridge in a 1.3 mm diameter hole. QED

To conclude: this sectional picture of a lute bridge indicates to us –red arrow– that the overhang one meets very often on the historical instruments can be used to better block the other strand previously burned to make a small anti-slip ball .

[49] « Hello Peruffo… Scientia sine humor, quam odiosis »