I installed some new very high-quality strings, the nut is perfectly polished and graphite has been correctly applied into the grooves, 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 of the first octave 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 thin gut strings (.40 mm diameter) of different brands, and the results show that the average frequency value at which a gut string with a length of 1.0 meter breaks is 260 Hz (more or less 240-280 range).

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 half, and vice-versa. In other words, the product between the  parameters F and L is a constant that, at the unit length of 1 meter, is defined as the Breaking Index, or FL product.

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, that is 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 = .418 m = 41.8 cm (this is the Breaking Vibrating Length)

Breaking Vibrating Length versus 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.

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, at the beginning,  stretches a lot (this is the 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 in 1692 wrote: “a string shall break when it can stretch no more”:

 

 

Praetorious (‘Syntagma Musicum’, 1619) provides us with both the length measurement unit of Brunswich and the various tunings of the instruments represented in his tables: after deducing the possible standard pitch by the Praetorius’s organ pipes, 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, such as John Playford), right before the final steep section; we are therefore at two / three half tones far from the theoretical breakage of the 1st string.

This corresponds to a Working Index ( FL product) of 220-230 Hz/m on plucked instruments (i..e. 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 baroque 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 surviving theorboes/ archlutes (for example those of Grail, Buechenberg, Hartz etc ) built in Rome in the 17th century (estimated St. Peter standard pitch around 390 Hz), renaissance lutes made in Venice in late XVI century- beginning of the 17th century such as for example those of Venere, Sellas, Tieffenbruchker etc  (Venetian ‘mezzo punto’: 465-70 Hz or ‘tuon del cornetto’); 5 course guitars made in France in the late 17th/first half of the 18th century -such as the Voboam ones- (Royal standard pitch around 390 Hz), and, finally, 11,13 course  d minor german lutes in the 18th century ( Kammerton pitch, around 420 Hz), where the calculated Working Index ranged form 225 up to 235 Hz/m.

Is there a single Breaking Index, or more than one?

Until today, the average Breaking Index (i.e. the breaking point) of a gut string has been considered to be of 260 Hz/m (statistical average measurement taken by me on several modern gut strings with a diameter of .40 mm) while in the 1970-90 it was considered of 240 Hz/m (see Ephraim Segerman).

We have only recently come to understand that it is not possible to consider a single value for all sizes of plucked and bowed instrument: there are at least three Breaking Indexes that should be used (this finally solve the puzzle with the calculated Praetorius working index of the instruments of his tables)

Why?

Here is the deal: it is commonly thought that gut strings – whether they are thick or thin – are all manufactured following the same chemical procedures, applying the same twisting ratio; using the same type of gut and are finally produced following the same production phases.
Actually, this is not the case: in professional string making technology, at least THREE different types of manufacturing are (or should be) followed, which involve the use of different chemical baths, different twisting ratios, different types of raw gut and, finally, different manufacturing steps.
If we were not proceeding in this way, we could not effectively solve the two main underlying problems of a musical string: the breaking load (breaking point) and the Inharmonicity (i.e. the acoustic dampness, which is related to the degree of elasticity of the string, a parameter linked to the twisting ratio; type of raw material and chemical phases employed): these two parameters are in opposition to each other: increasing one will decrease the other (and vice versa).

Therefore, the different string diameters – especially those used as trebles – need their specific technology.
For example, it is not possible to produce a string of a certain thickness with the same technology employed for Lute trebles: the obtained string would be extremely stiff and hard, therefore totally dull.

On the other hand, it is not possible to adopt the technology used for the thickest strings (high twist) when manufacturing Lute trebles: the trebles would break long before they reach the required note.

 

The three types of gut strings

(these considerations are related to our own production. We cannot knows which is the situation of other stringmakers)

-The first manufacturing type concerns the super solicited and thinnest strings only – basically Lute and Baroque guitar trebles – where the one and only goal is to reach the maximum tensile strength and the maximum resistance to surface abrasion due to finger’s action. The ‘official’ Breaking Index of 260 Hz/m refers precisely to this type of strings. No one, here, take care of Inharmonicity (these strings are so thin that there is no problem)

-The second type includes strings that are still considerably under stress, but not at the extreme levels that are typical of Lute/Baroque guitar trebles.
Typical examples of this type are the Violin and Gamba family  1st (except Pardessus)
Here the goal of the stringmaker is still to look for a high tensile strength, but at the same time beginning to reduce a certain degree of the inharmonicity by modifying the chemical process used.

-The third type is represented by the first strings of bigger stringed instruments such as cello, bassetto, G and D Violone and double bass: with these types of instruments, it is no longer necessary to research for a maximum tensile strength, so the manufacturing techniques aim to reduce the inharmonicity of the string of a good percentage but still not the maximum.

There would actually be a fourth type: the thickest strings that are never used as trebles. In this case, everything is aimed at reducing the inharmonicity as much as possible, taking absolutely no care of the tensile strength.
This is the typical example of the plain gut 3rd string of the cello, the 2nd ,3rd and 4th of the G and D violone (and sometimes even the 5th and 6th) and the 2nd and 3rd of the double bass – sometimes even its 4th string.

As we have seen, today the commonly adopted value for the Breaking Index of gut is 260 Hz/m which in fact represents only of the first type of gut strings: the Lute trebles (in our company this concerns diameters from .36 mm up to .50 mm).
Always referring to our company, diameters between .50 and .90/1.00 mm instead are made according to the typical construction criteria of the second type. Experimental data of Breaking Indexes make us converge towards a value of 240 Hz/m.
The third type is represented in our company by strings of diameters larger than 1.10 mm and up to 2.50 mm. We have not calculated the Breaking Index yet, but by extrapolation we believe that it is further reduced to about 220 Hz/m.

To sum up:
First type of strings (.36-.50 mm diameter): Breaking Index equal to 260 Hz/m
Second type of strings (.50- 1.10 mm diameter): Breaking Index equal to 240 Hz/m
Third type of strings (1.10-1.40 mm diameter approx.): Breaking Index equal to 220 Hz/m

 

How to put into practice all the above on 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: the vibrating length of the instrument, expressed in meters, needs to be multiplied by the frequency of the first string and then:

Lutes, renaissance and baroque guitars (diameters between .36 and .50 mm)
-if the value is less than or equal to 220: Green light
-if the value is between 220-230 : Yellow light
-if the value exceeds 240 : Red light

Violin, Viola da braccio, Gamba family Treble, Tenor and Bass (diameters between .50 and 1.0 mm):
-if the value is less than or equal to 200: Green light
-if the value is between 210-220 : Yellow light
-if the value exceeds 220 : Red light

 

Cello, G and D Violone, Double Bass (string diameters thicker than 1.10 mm):
-if the value is less than or equal to 190: Green light
-if the value is between 200-210 : Orange light
-if the value exceeds 210 : Red light

 

Of course, when tuning the same instrument to different standard pitches, all calculations will need to be revised and recalculated.

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

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

 

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.

 


 

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 Gamba, but the 6th string doesn’t perform well

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 still 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 strings (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 .65 m and a frequency of 196 Hz for the ‘g’ note, that it is to say A-440 pitch standard)

The 4th string – the ‘D’ – is instead a wound string; its FL product is around 95 Hz/m (on a scale of .65 m and a frequency for the ‘D’ is of 146.8 Hz)

The principle behind this transition is that a plain 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 work around is using two strings paired in octave).

Working with the 1st string at 225-235 Hz/mt, the 6th course of a Lute will have an FL product of 59-60 Hz/m; the inharmonicity issue is here resolved only using two strings paired with the octave (see Tinctoris 1474 and Virdung 1511) , but this is the lowest limit: under 60 Hz/m the acoustic performance becomes so poor that even a paired octave will not help. Therefore there’s the need to change to a type of strings that will work down til, to the lower limit of 39 Hz/m (wound strings, KF, loaded, Gimped, etc); i.e. a ‘densified’ string.

 

BOWED INSTRUMENTS

On bowed instruments, thanks to the continuous action of the bow on the string, the situation is better: in this case, the lower FL product before to switch to a densified string (wound or loaded) can be considered around 60-70 Hz/m .

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 (scordatura), 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). (1)

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

This is the situation that commonly happens with all those instruments whose FL product of the 1st string is lower than 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 right FL product for the 1st string (i.e. 210-220 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. John Playford 164 etc wrote that the 1st string of Treble, Tenor, consort bass and Lyra viol must be tuned as high as possible.

The 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 well twisted). On the other hand, it’s impossible to employ a plain gut strings for the  C string of the Viola da Braccio (even it is very high twisted) that has a vibrating length of only 38 cm: its FL product is of 47 Hz/m only.

To have a plain gut low c,  the vibrating length of the  Viola da braccio  should ensure an FL product like of the Violin G string: i.e. 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; as to say  the same FL product of the 6th string on the Gamba family).

Vivi felice

Mimmo Peruffo

 

(1) In the case of the Cello, the situation is different: during the 18th century, this instrument never used roped gut strings or loaded strings, in favour of high-twist gut strings instead. In general, cellos worked with higher tensions than those in use on the viola da gamba family: the combination of ‘higher tension and high-twist gut strings’ can negatively affect the sound output and the quality of the bow’s attack. This suggests that when the FL product of the third string is below 70 Hz/m, it is better to use a wound string.