I installed some new high-quality strings, the nut is perfectly polished and graphite has been correctly applied, but the first string keeps breaking when I get near to the final tuning: why?

“I bought a small harp for medieval repertoire, but unfortunately the strings on the first octave will break right after being installed, even if they are from a well-known famous brand: why does that happen?

Here is the fact: you might have followed the best way to install the strings, and you might have used the best strings on the market, but did you ever look into the Breaking Index (also called FL product) of your instrument?

The FL product
When a gut string is gradually stretched, it will eventually reach a specific frequency at which the string will break abruptly: such frequency is called “Breaking Frequency”.

Counter-intuitively, such frequency will remain the same even with different diameters of the string (the only variation will be in the tension of the string, expressed in Kg). The reason is this: if the diameter is increased of a certain percentage, the tension will also increase of the same percentage (and viceversa).  As a matter of fact, applying Mersenne/Tyler string formula, when changing the diameter and the resulting tension, it will be noticed that the frequency will remain unchanged.

As a consequence, both the following statements start from false premises, and therefore will result not true:

The string snapped: I decided to install a thinner one because it has a lighter tension and therefore it won’t break

The string snapped: I decided to install a thicker one because it is stronger

We have taken several measurements on gut strings of different brands, and the results show that the average frequency value at which a gut string with a length of 1 meter breaks is 260 Hz .

The Breaking Frequency (F) is inversely proportional to the vibrating length (L), so if the length of the string doubles and goes from one meter to two meters, the frequency will be divided in half, and vice-versa. In other words, the product between the two parameters F and L is a constant that, at the unit length of 1 meter, is defined as the Breaking Index, or FL.

When a luthier designs an instrument, he starts, instead, from the frequency of the 1st string: so, dividing the Breaking Index by the frequency in Hertz of the first (and therefore highest pitched) string, we can calculate the vibrating length at which the string will inevitably break.

Let’s make a practical example on a Violin:
The frequency of the first string, the E (considering the baroque pitch of 415 Hz), is 621.7 Hz
Therefore, the vibrating length at which the E string will instantly break is:
260 / 621.7 = 0.418 m = 41.8 cm (Breaking Vibrating Length)

Breaking Vibrating Length and Working Vibrating Length
As we have observed, we can obtain the vibrating length at which the string will instantly break simply dividing the Breaking Index by the frequency of the first string, regardless of the diameter that will be used: these are our Pillars of Hercules.

In order to have a string that does not break, it is therefore necessary to introduce a certain prudential shortening of this limit length, but by how much should we shorten it? If the shortening is excessive, the acoustic performance is compromised: the sound performance of a string, at the same tension and frequency, will be improved if its diameter can be reduced as much as possible by acting on the vibrating length (vibrating string length and diameter are inversely proportional). To find the right answer, let’s take a look at the following graph:

 

 

It can be noticed how the string initially stretches a lot (irrecoverable loss of elasticity, also called “false elasticity”) before following the linear stress/strain trend given by the Mersenne/Tyler law.
At some point that linear trend changes ratio, and the line gets steeper. This has only one explanation: the string has lost its ability to stretch; the breaking point is getting nearer, and it’s only 2 or 3 half-tones away.

Speaking of which, Daniello Bartoli wrote in 1692: “a string shall break when it can stretch no more”:

 

 

Praetorious (‘Syntagma Musicum’, 1612) provides us with both the length measurement unit and the various tunings of the instruments represented in his tables: after deducing the possible standard pitch, in most cases it was found that the various FL products reached the upper limit of the line (a clear reminder of the common rule of that times to tune the first string to the highest allowed), right before the final steep section; we are therefore at two / three half tones from the theoretical breakage of the first string.

This corresponds to a Working Index of 220-230 Hz·m on plucked instruments (2-3 half-tones less than breaking point); 210-220 Hz·m for bowed instruments (3-4 half-tones less than breaking point), excluding large sized bowed instruments, that work around 190-200 Hz·m (see the several works of Ephraim Segerman in FOMRHI bulletins).

 

 

This is also confirmed by our calculations that we have carried out in some historical Lutes and 5 course Guitars kept in museums, provided that the vibrating length has not been altered and that they can be traced back to a sufficiently identifiable standard pitch (see A. J. Hellis:’The History of Musical Pitch’, London 1880, and Bruce Haynes: ‘A History of Performing Pitch: The Story of “A” ‘ 2002).

This is the case of theorboes/ archlutes built in Rome in the 17th century (estimated standard pitch of 390 Hz), in Venice in late XVI century- beginning of the 17th century (Venetian standard pitch ‘mezzo punto’: 465-70 Hz), in France in the late 17th/first half of the 18th century (standard pitch around 390 Hz), and, finally, in Germany in the 18th century (Kammerton pitch, around 420 Hz), where the calculated Working Index range is 225 -235 Hz·m.

 

But how can this be applied, in practice, to our instrument?

Simple, by applying the traffic-light rule (this rule is only for plucked instruments; in the case of bowed instrument the orange light turns red, the green light turns orange): green light (little or no breakage risk), yellow light (possible medium risk of breakages, depending on intrinsic quality of the string, environmental conditions such as humidity and temperature, etc), red light (maximum risk, breakage is inevitable).

This is how to proceed.

Calculate the Working Index multiplying the vibrating length of the instrument, in meter,  by the frequency of the first string (vibrating string lenght x Hz):

For plucked instruments:

  • if the value is within 230: green light (the first string works in safety conditions)
  • if the value is between 240 and 250: yellow light (statistically the treble could break within a few hours/days, especially in high humidity conditions).
  • if the value is over 260: red light (the treble will break immediately, or within few minutes).

For bowed instruments:

  • if the value is within 200: green light (the first string works in safety conditions)
  • if the value is between 210 and 220: yellow light (statistically the treble could break within a few hours/days, especially in high humidity conditions).
  • if the value is over 240: red light (the treble will break immediately, or within few minutes).

For Doublebass and Violone: the Working Index should not exceed 190.

 

In our example of the Violin, the vibrating string length that generates the breakage must be reduced by three to four semitones.

The result is as follows: 34.5-33 cms (range of possible vibrating working length at the pitch standard of 415 Hz)

 

Of course, if the same instrument has to be tuned to different pitches, the calculation has to be repeated.

 

Essential fields of use

Harps in general (also modern harps)

This calculation is particularly useful on harps that, because of their great variety, might not respect this rule: one should mostly concentrate on the first octave, carefully verifying the FL product of either all strings or also in steps. This information should be taken into consideration by luthiers first of all, since they need to plan the harp according to known notes and pitch. Historically speaking, most harps work with the highest octave in conditions of yellow light

Medieval/reinassance instruments

Since no original instruments survived to this day (we make use of iconographic sources only), and the standard pitch of that time is unknown, it’s always worth verifying the FL product before buying any instrument. This information should be taken into consideration by luthiers when they are planning the instrument knowing the note of the first string and the standard pitch to use, as required by the customer.

Newly designed instruments of alleged historical reconstruction

The FL product should be considered in yellow light for Lutes, Baroque Guitars, Reinassance Bass Gambas; it should be considered in green light for bowed instruments, with or without frets (for Violone consider an FL product of 192-200 Hz·m)


 

Other fundamental applications of the FL product

How can a string maker understand when to change from a gut string to a wound one?

How to understand when a gut string will not have acceptable acoustic performances anymore?

I installed all gut strings on my bass Viola da Gamba, but the 6th string doesn’t perform at all

I’d like to install all gut strings on my Viola: can this be done?

I installed very good pure gut basses on my Lute, but they are too dull: why?

 

The FL product is the answer. If on the first string the FL product is also called Working Index, on the other strings this index itself can express the Inharmonicity degree of that particular string in the instrument, having the vibrating string length and the frequency.

Generally speaking, the Inharmonicity degree can be considered as an index of acoustical quality; it will be maximum on the first string, and it will gradually decrease on lower strings until the FL product, and consequently the acoustical performance of the strings, will be reduced to a point where human ear will not perceive it as acceptable anymore (it is widely known that strings of growing diameters, placed on the same vibrating length, will become more and more dampened, will be difficult to be brought into vibration, and will give bad acoustic performances).

At that point the only solution is to adopt a different type of string (wound, roped, KF, loaded, etc).

How can one predict when to adopt such different technological solution? Looking at the FL product, of course!

 

An example for the classical guitar (and all plucked instruments, in general):

The 3rd string – the ‘g’ – of a classical guitar is the last nylon string; its FL product is around 127 Hz·m (using a scale of 0.65 m and a frequency of 196 Hz for the ‘g’ note)

The 4th string – the ‘D’ – is instead a Nylon wound string; its FL product is around 95 Hz·m (on a scale of 0.65 m and a frequency for the ‘D’ of 146.8 Hz)

The principle behind this transition is that a nylon or gut string will not be able to give good acoustical performances when its FL product will be lower than 90-100 Hz·m (on the 5th course of the Lute, the FL product is around 70-80 Hz·m only, but the workaround is using two strings paired in octave).

The 6th course of a Lute will have an FL product of 59-60 Hz·m; the inharmonicity problem is here resolved only using two strings paired in octave (see Virdung 1511) , but this is the lowest limit: under 60 Hz·m the acoustic performance is so poor that even a paired octave will not do, therefore there’s the need to change to a type of strings that will work down to a limit of 39 Hz·m (wound strings, KF, loaded, Gimped, etc).

 

PLUCKED INSTRUMENTS (i.e. Lutes)

With the 1st string at 225-235 Hz mt of working Index (that is the best situation for the performance) , the 6th course of a renaissance Lute has an FL product of only 58-60 Hz·m: despite the fact that the inharmonicity problem here is limited by using the paired octaves, this can be considered, generally speaking, the lower acceptable limit for a gut string. In fact, under 58-60 Hz·m, the acoustic performance starts to degrade progressively until it becomes so poor that the paired octaves are no longer enough: it is therefore necessary to switch to a type of strings that will work under 58-60 Hz·m, till the lowest limit of 39 Hz·m, that is the FL product of the 10th/11th course (i.e. wound strings; KF, loaded; Gimped; etc).

 

BOWED INSTRUMENTS

On bowed instruments, thanks to the continuous action of the bow on the string, the situation is better: in this case, the transitional FL product can be considered around 90 Hz·m (that is the 4th Bass Gamba and the Violin 3rd). With that being said, it is still possible to have a good performance also when the FL product for the 6th string is of 57-58 Hz.m only, provided that the gut strings of the 5th and especially the 6th strings have a very high elasticity and/or density (roped structure/loaded gut/whole lamb gut).

However, care must be taken so that the FL product should never be under 55-56 Hz.m: under this value, wound strings must be used.

This is the situation that commonly happens with all those instruments whose FL product is less of 200-210 Hz·m.

For example, the 6th D string of a Bass Viol with a scale of 69 cm (at 415 Hz pitch standard) has an FL product of only 48 Hz·m:  if such instrument would have been designed to use only gut strings, using the correct FL product for the 1st string (i.e. 200-210 Hz·m), the FL product of the 6th would raise up to 57-58 Hz·m; as a consequence, the instrument would have a scale of 77 cm, and not only 69 cm.

A pure gut g-string for a Violin at 415 Hz shows an FL product of 61 Hz·m: this means that a pure gut string can be still used, provided that it’s of excellent quality (i.e. very high elasticity). On the other hand, it’s impossible to reach the low c on a Viola da Braccio that has a vibrating length of only 38 cm: its FL product is just 47 Hz·m.

To have a low c in pure gut – of the best quality –, the vibrating length of a Viola should ensure an FL product like the one for the low g of the Violin: 61 Hz·m (in any case, not less of 57-58 cm). Therefore, following the fore mentioned proportions, its vibrating string length should be 47-48 cm (or at least 43 cm as absolute minimum value, that ensures the same FL product of the 6th string on the Gamba family).

Vivi felice