Leopold Mozart suggests an original method for choosing a violin’s set of strings: to hang them in pairs with two identical weights. The right diameters will be the ones that, when played empty, will give an interval of open fifth. If the interval is wider, it means that one of the two strings is too thin or the other is too big, and vice versa if the interval is narrower than a fifth. Mozart does not specify either where this gauge selection test should be performed (on a violin? on a frame?) or how much should the loads weigh.

As a matter of fact, there’s an extremely similar example found in a book by Serafino di Colco dated 1690: one could assume (but not prove) that Mozart was indeed influenced by reading this text.

What Serafino di Colco describes, however, is apparently valid… in his mind only. In fact, the author of this text has verified what he suggested in the practice, finding that it does not lead at all to the stated result: for example, two strings, calculated at the same tension – expressed in Kg – of an interval of one octave (we used the diameter of 0.60 mm and 1.20 mm), when subjected to the same kilograms by means of two equal weights, did not lead to an interval of an exact octave, as expected by Mersenne’s string law, but gave an interval than was greater than one octave.

This is the experimental demonstration:

https://youtu.be/EjKXihgOM7o

All this happens because strings with different diameters will stretch differently when subjected to the same traction force: more for the thinner one. An elongation is nothing more than the manifestation of a reduction in diameter.

But according to Mersenne/Tyler’s string law, if a string becomes thinner (with the same density, vibrating length and tension in Kg) its frequency increases as a consequence. The 0.60 mm string, under the traction of the weight, will therefore lose a greater percentage of diameter than the 1.20 mm string: this explains why the theoretical interval of one octave is not observed in the practical verification.

As above said, strings of different diameters, put into traction by means of equal weights, lengthen differently: in Di Colco’s drawing, instead, we observe that the four equal weights lengthen the four strings of the Violin (which is strangely without neck!) of the same entity: this goes against the laws of Physics and the experimental test we performed.

It is concluded that Di Colco’s demonstration has no value, because it is physically, mathematically and experimentally wrong.

Let’s get back to Leopold Mozart. His method mentions equal weights; therefore, it seems to refer to the criterion of equal tension, but it’s not like that at all.

Let’s see why.

The first thing that can be noticed is that the choice of the diameter to be compared with the one that’s already under traction, changing the gauge until it gives the exact open fifth interval, is made when strings are already under the traction of equal weights; that is to say that both strings, due to the traction, have already lost a percentage of their original diameter, the one shown on their package.

This situation is completely different from the calculation that is made today, when speaking of equal tension settings, where all calculations are made considering the diameters of the strings still in the package, that therefore have not undergone traction yet.

As previously mentioned, putting in traction, either by means of weights or by turning the pegs, causes a certain gauge decrease which is maximum for the first string, lower for the second and almost nothing for the third and fourth (here fourth unwound string is meant). As a consequence, a setup calculated in equal tension in Kg, once installed will produce a slight inverse scaling of the tension values: the first string will have less tension than the second string; the second less tension than the third, just to mention an example.

Physics and Trigonometry laws state that when two strings have the same tension in Kg they also show the same (tactile) feel of tension (i.e. the same magnitude of lateral displacement with the same force applied acting and at a constant distance from one of the two constraints) as long as the tension in Kg is the same of the strings already in traction.

Experimental tests carried out by us have shown that the e-string by Aquila is reduced by about 5-6 %; a-string by about 2%; d-string by 0.1 %. We did not do the test with a plain gut C-string, but certainly the reduction is lower than that of d-string. The result changes little depending on how the strings are made.

So, what happens in Mozart’s case?

Let’s take the couple of e- and a-string as a reference and apply the principle of equal tension as we do today: let’s say that the first e-string is 0.620 mm; in a situation of equal Kg the a-string will be 0.93 mm and between them they will give a perfect interval of fifth but … will we still have the intervals of fifth (as calculated according to the law of the strings) even after having put them under two equal weights as Leopold Mozart suggests?

So far, we have done nothing else than apply the string formula introduced by Mersenne, and this is pure Mathematics, and Mathematics is not wrong.

But here is what happens instead: as we said before, we know that the first E-string put under weight (i.e. put in traction) reduces its diameter by about 6% while the second string reduces by about 2%.

So, once they have both been put under equal weights (let’s consider, for example, that both weights are 7.0 Kg; but nothing prevents them from being of another value) the E string will lose 6% of its diameter and thus become 0.58 mm; the A string will instead lose 2% of its diameter and become 0.91 mm.

Now let’s apply the law of the strings to this new set up of diameters (at 7.0 Kg of applied weight – which is a typical average tension of a Violin’s first string – and 0.33 meters of vibrating length) and let’s see what frequencies we get out of it:

They are 677.4 Hz for the E, and  431.7 Hz for A: but this is no longer a fifth interval!

In fact, if we start from the A string, that has 431.7 Hz, if we want a pure interval of fifth with the E string we must have 646.8 Hz and not the calculated 677.4 Hz: it is an interval greater than a fifth.

In practice, we are in the same situation as our experimental verification of Di Colco shown in our video but all in all also in the same situation mentioned by Mozart: if we obtain intervals greater than a fifth it means that the E string is still too thin or, vice versa, that the A string is too thick.

How to solve this?

First of all, in order to obtain the interval of fifth between E and A strings, when subjected to the same weight, it is necessary to start from diameters that are slightly scalar, not calculated in equal tension.

The procedure is then the following: does the E-string reduce its diameter by 6%? Then we’ll need to compensate that 6% reduction in our calculation by increasing the starting diameter accordingly. So, for example, an initial 0.62 mm diameter will become: 0.62 x 1.06 = 0.657 mm (that rounded to the nearest available diameter will be 0.66 mm).

The same logic can be applied to the A-string. Assuming that it reduces its diameter by 2%, the theoretical calculation will need to compensate that 2%. A 0.91 mm diameter will therefore become a 0.93 mm (as a result of 0.91 x 1.02 = 0.928).

When these two diameters (66 and 93) will be subjected to the same 7.0 kg tension, they will get reduced to the 0.62 and 0.91 initially expected, therefore the interval of perfect fifth will be achieved, exactly as suggested and described by Mozart (a good observer will have noticed that the diameters once in a state of traction must be exactly the ones obtained by theoretical calculation in order to have the perfect open fifth interval).

If instead, using Mersenne/Tyler’s formula, we calculate the purely theoretical tension of the ‘packed’ E string (of 0.66 mm) and the A string (of 0.93), whose diameter is measured in their ‘rest conditions’ with no strain, considering the frequency of 622.2 Hz for the E (at 415 Hz pitch) and 415 Hz for the A, we will see that the tension profile has a scalar nature: 7.58 Kg for the E of 0.66 mm and 6.74 Kg for the A of 0.93 mm.

Once again we’d like to underline that once the strings are subjected to a tensile strength, their diameters will decrease with different percentages (-6% on the E that becomes 0.62 mm thick, and -2% on the A that becomes 0.91 mm thick), and in these conditions, at the vibrating length of 0.33 m of a violin, they will both have a real tension of 6.74 Kg = equal tension = equal tactile feel of tension = open fifths; exactly what Mozart suggested.

As a conclusion, the true interpretation is that the method specified by Leopold Mozart in 1756 to determine the right gauges to be used on a violin does not concern applying a strict equal tension to the setting, as it is still believed by many today, but theoretically calculating a scalar tension setup that, once in tune, will provide an equal tension profile, expressed in Kg, that gives an exact condition of equal tactile feel of tension and will produces the open fifths: the theory we always supported.

Vivi felice

Mimmo Peruffo