8. Equal tension, equal feel and scaling tension

It will have surely not escaped the observant reader that the string diameters hitherto described do not at all lead to stringings with a system of equal tension but instead to one of the scaled type (for comparison, an arrangement in equal tension, starting for example from a chanterelle E of 0.70 mm, would give the following diameters: E = 0.70 mm, A = 1.05 mm, D = 1.60 mm).

Today it is commonly held that a correct stringing for the violin or another instrument must have all the strings at the same tension (in other words, with the same kg), but in fact this is not at all how things stand.

Before pursuing the analysis of the documentation we must therefore tackle this fundamental point, for it affects the way we reconstruct the stringings of all the plucked and bowed instruments of the Renaissance and Baroque – not only the violin.

Let us begin our discussion of this subject with the concept of ‘tactile sensation of stiffness’. For it needs to be stressed that when a musician applying the pressure of his fingers evaluates the tension of the strings of his instrument, he is actually not evaluating the kg of tension at all, but instead the sensation of tension, which is quite another matter.

It comes natural to ask what criteria were used to evaluate a stringing in the past. This, for example, is what certain seventeenth-century treatises write about the lute:

"Of setting the right sizes of strings upon the lute. [...] But to our purpose: these double bases likewise must neither be stretched too hard, nor too weake, but that they may according to your feeling in striking with your thombe and finger equally counterpoyse the trebles" (82).

“When you stroke all the stringes with your thumbe you must feel an even stiffnes which proceeds from the size of the stringes" (83).

"The very principal observation in the stringing of a lute. Another general observation must be this, which indeed is the chiefest; viz. that what siz'd lute soever, you are to string, you must so suit your strings, as (in the tuning you intend to set it at) the strings may all stand, at a proportionable, and even stiffness, otherwise there will arise two great inconveniences; the one to the performer, the other to the auditor. And here note, that when we say, a lute is not equally strung, it is, when some strings are stiff, and some slack" (84).

From the treatises of the time one deduces therefore that the criterion for choosing the strings in a given stringing responded above all to principles of empiricism: the strings were expected to be neither too tense nor too slack but to have a just degree of tension; and what is important, this tension was expected to be evenly distributed among all the strings. It goes without saying that any judgement of the degree of tension is merely subjective. A different matter, on the other hand, is the search for evenness of tension between the strings, which is the true, shared criterion of reference.

In conclusion, when the early documents use the words ‘equal tension’ (and we find them until at least the end of the eighteenth century) they consistently mean ‘equal feel’ and not equal kg, as instead is done today.

A pertinent example is the following passage from Galeazzi: "la tensione dev'esser per tutte quattro le corde la stessa, perchè se l’una fosse più dell'altra tesa, ciò produrrebbe sotto le dita, e sotto 1'arco una notabile diseguaglianza, che molto pregiudicherebbe all'eguaglianza della voce" (the tension must be the same for all four strings, because if one were more tense than another, that would create under the fingers, and under the bow, a considerable inequality very prejudicial to the equality of tone) (85). Here tension clearly means feel; as is equally plain in Bartoli's treatise: "Quanto una corda è piu vicina al principio della sua tensione, tanto ivi e piu tesa. [...] Consideriamo hora una qualunque corda d' un liuto: ella ha due principj di tensione ugualissimi nella potenza, e sono i bischieri dall’un capo, e '1 ponticello dal1'altro; adunque per lo sopradetto, ella è tanto piu tesa, quanto piu lor s'avvicina: e per conseguente, e men tesa nel mezzo" (The closer a string is to the beginning of its tension, the tenser it is. [...] Just consider any lute string. It has two beginnings of tension that are absolutely equal in power: the pegs at one end, the bridge at the other. As a result, it will be tenser the nearer it is to those points and less tense in the middle) (86).

To try and give some kind of scientific expression to the concepts of ‘even stiffness’, ‘equally strung’, etc. described in the treatises is in itself a somewhat complex matter, both because there is no conclusive proof that by ‘feel’ they all meant the same thing and also because that ‘feel’ can be also understood in a, so to speak, broader sense.

A preliminary distinction (when evalutuating the degree of ‘tension’) can be made, for example, by deciding whether in pressing down on the strings it is directly the fingers or the bow hairs, for in the latter case the thicker strings can oppose more resistance to the rubbing movement, thereby giving the musician the sensation of a certain unevenness. To resolve this specific problem the use of scaled tension was justified on the violin by Plessiard (87).

In the likely hypothesis that it is the fingers (and not the bow) that are required to assess the tension of the strings, we can again understand ‘feel’ in at least two ways. The first (that commonly accepted, by the present writer as well) considers the effort required to impart a certain meaure of lateral displacement to a string, which obviously opposes the pressure exerted. If we replace the finger with a weight acting at the same point, we can exactly measure the quantity of lateral displacement for every string examined. The second hypothesis, introduced by Segerman (88), considers that a thinner string, which digs more deeply into the finger tip pressing down on it, would produce a greater sensation of tension than a thicker string, which, having a wider surface, does not ‘dig into’ the finger to the same extent. According to this second interpretation, therefore, ‘equal feel’ involves more tension in kg in the thicker strings than in the thinner. As Segerman himself stresses, we have never yet had practical evidence that the bass strings have more tension than the higher ones.

Let us therefore examine the first hypothesis better: in other words, that which considers ‘feel’ to be the sensation of resistance given by a string pressed by the fingers and ‘equal feel’ to mean that this sensation is the same also for tuned strings of different diameters; in other words, that when the same weight acts at the same point, the lateral displacement encountered is the same. The vibrating length obviously has to remain constant.

According to the laws of physics such a conception of equal feel corresponds exactly to a stringing of equal tension (89). That is true, however, on condition that the initial diameters of the strings (as measured with the strings not yet mounted) remain unvaried even after they have been tuned, i.e. under tension. In pratice, however, and especially with gut, this never happens: once the strings have been tuned to the required note, their respective calibres have dimished in different ways. This happens because the material possesses a certain longitudinal strain which is related also to the diameter (which in gut is divided into recoverable strain and non recoverable strain: in practice once a new string has been placed under tension, it no longer reattains its initial diameter at rest). This reduction of calibre will therefore also imply a corresponding reduction in working tension. It is observed that the thinner strings lengthen more and hence diminish in calibre by a greater percentage than the thicker ones (it is generally known that the thinner strings require many more twists of the peg than the thick ones). And so it also follows that, after tuning, the respective working tensions (established as identical to start with) will no longer be equal but scaled: in other words, the thinner the string, the lesser the tension.

As a result, therefore, the ‘feel’ between the strings is no longer equal (because the tensions are now different) but instead unbalanced in favour of the thicker strings. In other words, on the thicker strings more pressure from the fingers is needed to obtain the same quantity of lateral displacement as on the thinner ones. Hence according to the laws of physics, if the tensions are not equal, nor is the lateral displacement; nor, therefore, is the feel even.

As an example, we tested two gut strings of medium twist calculated to have the same tension (8,3 kg at a pitch of 440 Hz) when brought to the required pitch (the violin ‘E’ and ‘D’ in this case). The vibrating length is obviously the same for both (33 cm). We obtained calibres of 0.65 mm for the ‘E’ and 1.45 mm for the ‘D’ when measured at rest, i.e. not under tension. Once they had been tuned and stabilized, we proceeded to measure their diameters: the calibre of the ‘E’ had reduced to 0.62 mm, whereas there was no noticeable drop in the ‘D’, even when measured by a micrometer. Hence while the thinner string had dimished in diameter by 5%, the thicker string be considered as unvaried for practical purposes. These values are of an experimental type: which means that strings made in different ways may provide different percentages of reduction. The constant factor, however, is that – manufacturing techniques being equal – it is always the thinner string that contracts more. In our case the tensions of the strings stretched on the instrument reduced to 7.6 kg on the ‘E’ and 8.3 kg on the ‘D’ compared to a calculation value of 8.3 kg in both cases.

In order to have ‘E’ and ‘G’ strings that retain the kg decided on initially when tuned to pitch, one must therefore increase the initial gauge of just the ‘E’ by 5%, i.e. 0.68 mm. When making the traditional calculation to obtain the tensions of this second pair of strings ‘at rest’ one detects a trend of the scaled type: namely 9.2 kg for the ‘E’ string and 8.3 kg for the ‘D’. To sum up: the experiment shows that calibers of 0.65 and 1.45 mm lead only to a theoretical state of equal tension; conversely, if one uses diameters of 0.68 and 1.45 mm, once the strings have been tuned (i.e. in traction) they will assume a new, and more reduced, diameter situation, such as would exactly produce equal tension, i.e. equal feel.

This situation was in fact verified – with the assistance of a micrometer – in a second experiment carried out on this second pair of tuned strings.

If one wants a stringing of equal feel, it is therefore necessary to use criteria of scaling when selecting the diameters of strings ‘at rest’ (i.e. not in tension). As mentioned earlier, one advantage of scaled tension is that the increasing attrition encountered when moving the bow from thin to thick strings (because of the larger contact surface) is much less noticeable.

If we respect the condition that there should be equal tension between the various strings at pitch, one concludes that scaled tension and equal tension (measured at pitch) express the same thing: equal feel.

Although the test reported in the first version of this article (in Recercare IX of 1997) produced substatially correct results, the interpretation of the data turned out to be wrong. The same consideration applies to another example cited there: that of an elastic band and steel string whose diameters were calculated to have the same tension values to start with. When tuned to the same pitch, only the elastic band will reduce considerably in section to assume a new, lower state of tension, in contrast with the unextendable steel string. At this point, therefore, the feel will be different.

Let us now turn to the cases of Serafino Di Colco and Leopold Mozart. (90)

Di Colco writes: "Siano da proporzionarsi ad un violino le corde […] distese, e distirate da pesi uguali […]. Se toccandole, ò suonandole con l’arco formeranno un violino benissimo accordato, saranno bene proporzionate, altrimenti converrà mutarle tante volte, sin tanto che l’accordatura riesca di quinta due, per due, che appunto tale è l’accordatura del violino" (The strings are to be proportioned to the violin [...] extended, and stretched by equal weights [...]. If by touching them or playing them with a bow they form an excellently tuned violin, they can be considered well proportioned, otherwise you will need to change them as many times as necessary to obtain fifths between pairs of strings, which is precisely the tuning of the violin).

Barbieri believes that in all likelihood these considerations are purely speculative. Mozart, on the other hand, drawing on the same concepts, suggests attaching equal weights to each pair of strings: if the diameters are well chosen, the open strings will give fifths; otherwise the diameters will need changing until that result is obtained.

-The cases of Mozart and Di Colco can lead to a certain interpretative confusion. Indeed it has been attempted to conclude hastily that they are stringings in equal tension: as if they had been worked out by ‘sitting at a desk’, so to speak, i.e. based on formulas.

Appearances, however, are misleading. The test recommended by Mozart takes place in conditions of equal weights (i.e. equal tension) that already work on the strings. This situation therefore does not at all replicate that of apparent ‘equal tension’ obtained by means of calculation by establishing the same kg in the formula for the strings with the purpose of obtaining all the diameters of the stringing (a tension that, as we saw, will be diversified because of the differences in the thinning of each string after tuning). In his case the pairs of strings are chosen in a state of actual traction, not of calculations done on paper. Seeing that this is a situation of true dynamic equal tension (because the weight always remains the same), we therefore find that the strings also display equal feeling.

In other words, the method suggested by Leopold Mozart achieves what we indicated above, though by a different route. It is evident that the strings chosen by Mozart as suitable for the purposes of tuning in fifths would present initial diameters ‘in the packet’ that theoretically display a profile of scaled tension, exactly as in the other cases described.

-We conclude by observing that the degree of scaling mentioned hitherto does not correspond to that found in most of the historical documentation. The tension slope is steeper. In other word, one cannot detect a situation of equal feel.

Unfortunately, at present there are no documents that can offer illuminating evidence for why this practical choice was made by the violinists of the time.

Huggins (91) advances two hypotheses: the first takes into consideration the pressure exerted by each string on the table of the instrument. He stresses that in a state of equal tension (but also of equal feeling, we add) the pressures in kg exerted by the first three strings on the underlying table are not at all equal; and this is because the angle of incidence of the string on the bridge is increasingly acute towards the thicker strings. Hence a greater thrust on the table. To obtain equal pressures on the table from every single string what is needed therefore is an ‘additional’ scaling to the condition hitherto considered. The second hypothesis considers the fact that as a rule the thicker strings are progressively more distant from the fingerboard: the result, therefore, is that in a condition of equal tension/equal feeling the fingers of the left hand must make a further effort when pressing down on the fingerboard. Hence the reduction of tension, with the aim of recovering evenness of feeling in the fingers of the left hand.

A third and final hypothesis that tends to suggest a (markedly) scaled tension concerns the search for the maximum possibile evenness of attrition vis-a-vis the bow hairs. This is propounded by Riccati already in the eighteenth century and later repeated by Pleissiard in the second half od the nineteenth century:

‘Egli è d’uopo premettere, che quantunque l’arco tocchi una maggior superficie nelle corde più grosse, nulladimeno la sua azione è costante, purchè si usi pari forza a premer l’arco sopra le corde. Questa forza si distribuisce ugualmente a tutte le parti toccate, e quindi due particelle uguali in corde differenti soffrono pressioni in ragione inversa delle totali superficie combacciate dall’arco.’ (Giordano Riccati ‘Delle Corde…’ op. cit, p. 129)

(‘First it is necessary to say that in spite of the fact that the bow touches a greater surface in the bigger strings, its action is nonetheless constant, provided that equal force is used in pressing the bow on the strings. This force is distributed equally to all the parts touched, and hence two equal particles on different strings undergo pressures inversely proportional to the total surfaces encountered by the bow.’ ).

Let us now resume our investigation of the documents.

The examination of the historical sources relating to violin stringings has prompted some scholars to assume that two systems of stringing coexisted in the eighteenth and nineteenth centuries: a scaled system and one based on equal tension (from theoretical calculation). In the light of what has just been discussed, the hypothesis of theoretical equal tension is no longer sustainable in relation to the practice of real stringing.

Here are a few pertinent examples:

- Fétis wrote that Tartini in 1734 found that the sum of the tensions of the four strings of his violin was 63 pounds (96). Quite apart from the matter of how Tartini arrived at this figure (and if it was then correctly converted into other units of measurement), it needs to be stressed that it does not lead to a confirmation of equal tension, but most likely to a scaled stringing, as is clearly indicated by the calculations given below.

Given that we are talking about a violin, we can assume as reliable a vibrating length of 0.32 m, while for the standard ‘A’ we can conjecture an eighteenth-century Venetian pitch of 460 Hz. If we also assume that 63 pounds is effectively equivalent to 31 Kg (Segerman, op. cit.) and we follow the hypothesis of equal tension, we therefore have about 7.7 Kg per string, which would give the following calibers:

E: 0.61 mm

A: 0.92 mm

D: 1.38 mm

G: 2.06 mm (expressed in equivalent gut)

As we can observe, however, the diameter of the chanterelle conspicuously exceeds the range of calibers obtainable with 3 lamb guts (which, as we know, is the basic historical fact).

Nor do things look better if, on the other hand, we start from a mean value for the ‘E’ of 0.70 mm (3 lamb guts…) with a stringing in equal tension, for then we would have an overall value of as much as 42 Kg. It is incidentally worth noting that the sum of the tensions of the three higher strings only (about 30 Kg) would alone be almost enough to reach the figure indicated by Tartini for all four strings).

The hypothesis of scaled tension – again departing from a mean value for the ‘E’ of 0.70 mm and adopting calibers for the ‘A’ and ‘D’ strings that are average (as found in the historical sources) – would lead to the following:

E: 0.70 mm (9.9 Kg)
A: 0.90 mm (7.3 Kg)
D: 1.16 mm (5.4 Kg)

Total 22.6 Kg

In order to achieve the 31 Kg indicated by Tartini one would necessarily need to have an overspun G generating about 6.5 Kg of tension. That would correspond to a theoretical all-gut string of at least 1.90 mm.

This would be perfectly feasible if the string was overspun according to Galeazzi’s indications. (See ch. 9: ‘The fourth string’)

-Filippo Foderà, in his manuscript violin method dated 1834, indicated string measurements in terms of the notches inscribed on a string-gauge: (92)

“Misura delle corde alla trafila delle grossezze” [String measurements at the slot]

“Violino di Guarnerio Grado della trafila delle grossezze” [grade of the the slot]

The terms "dritto" (front) and "rovescio" (back) would appear to refer to the notches marked on the front and back of the string-gauge (having them all on one side would have probably made the gauge very difficult to read); they probably refer to the maximum and minimum gauges available, or recommended, for the violin. Though we have no way of converting these figures into metres (the author lived in the Kingdom of the Two Sicilies and the unit of length used there has not yet been traced), if we assume the "front" value of the top string to be 0.70 mm (in accordance with historical data), the remaining values would run as follow. in mm:

Here our assumed factor of conversion is 3.3. A system of progressively increasing tension is evident, as is the fact that the fourth string must have been overspun. We also note a similarity between the string-gauge and Spohr's diameters; particularly in the degree of scaling and in the (external, we think) dimensions of the fourth string or "cordone". As the "back" measurement is expressed in hundredths, the unknown unit of measurement should then be 33-35 cm.

In the Maugin and Maigne book (93) there would already appear to be a profound contradiction between the data already given by the string maker Philippe Savaresse (on the number of guts to be used) and the tension in kilos indicated for each string at Paris Opéra pitch: 7.5 kg for E, 8 kg for A, 7.5 kg for D, and finally 7.25 kg for G. Assuming a nineteenth-century French pitch of 435 Hz and a violin-vibrating string length of 33 cm. (94)

Here are the diameters:

E = 0.63 mm

A = 0.96 mm

D = 1.40 mm

G = / (overspun)

First of all, we note that the working tension of the top string is strangely lower than that of the A. This might be just a printing error: perhaps 8.5 kg was meant, instead of 7.5 kg; if so, the diameter of the top string would be 0.68 mm, which is perfectly in line with the Italian and French traditions. But the most striking evidence of the unreliability of such working tensions is the breaking load of the gut strings: the first string (0.63 mm) would break at between 12 and 13 kg; the second (0.96 mm) at 15 kg, and the third (1.40 mm) at between 40 and 45 kg. Our findings show that the breaking load of current gut strings is between 31 and 38 kg/mm2 (mean value 34 kg/mm²) — values that we also found on the gut string samples dating to the early twentieth century. It is worth stressing that if this were not the case, no violin first string of the time could be tuned up to E with the typical vibrating length of 32-33 cm; it would immediately break once the breaking index for gut was exceeded (95). Now, according to the tensions indicated in the text, Savaresse's gut would have a breaking load of 38-41 kg/mm² for the E (33-36 kg/mm², with a diameter of 0.68 mm) — which is acceptable — but of only 21 kg/mm² for the second and as little as 17-19 kg/mm² for the third. As the breaking load of gut has been shown experimentally to be an element that is subject to scant variability — especially when the materials have the same provenance and are manufactured in the same way (as is always the case) — one might well ask what string diameters would break at the tensions indicated by Maugin and at the mean breaking load of 34 kg/mm2. The answer is 0.75 mm for the A and 0.98-1.04 mm for the D: calibers utterly different from those derived from the working tensions indicated by Maugin.

But that's not all. We also read that the second and third strings weigh respectively twice and three times as much as the first. Assuming that the diameter of the first string is correct and (obviously) that the density of the material is constant, we obtain diameters of 0.89 and 1.09 mm for the second and third strings respectively. Quite plainly these measurements correspond to a system of progressively increasing tension, and are perfectly in line with both the number of guts indicated in the same text and the information given by De Lalande.

Hence the system of equal tension advocated by Maugin and Maigne is heavily contradicted in the same text by Savaresse, who bases his conclusions on the manufacturing data of commercial strings.

Several english, german, francaise sources in the late nineteenth/early twentieth centuries quote the string making information reported by Maugin and Maigne. This includes, also, Angeloni’s manual, which presents these erroneous data. (65).

There are some other nineteenth-century authors who apparently supported a system of equal tension. Huggins, for example, after giving the theoretical gauges on the basis of the proportions between diameters and frequencies (implying a system of equal tension), goes on to write: "A violin strung with strings of the theoretical size was very unsatisfactory in tone"; immediately afterwards he mentions the diameters sold in Ruffini's sealed boxes, pointing out that these strings had a scaled tension and — an important point — that only by this system could one obtain perfect fifths (98). Like Huggins, many other English documents of the same period recommend stringings that follow a system of progressively increasing tension, with diameters again similar to those of Ruffini/Hart and, more generally, to the French and Italian traditions (99). We find exactly the same indications, above all in the English violin methods, right up to the onset of the Second World War.