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Why we can't calculate you harps' setup (and why only you can)?

Introduction

Harpists have oftentimes asked us to calculate their new harp’s setups. Unfortunately, the only feasible way to calculate a setup is to physically wield the harp itself, preferably in the presence of the harpist itself.

Why?

As a matter of fact, not only harps have relatively many strings, but they’re also designed in order to get a “self compensated acoustic performance”.

What does “self compensated acoustic performance” mean?

It basically means that, while playing increasingly low frequencies (which are obtained with large strings), the harp’s vibrating length does proportionally increase as a compensatory element, in order to counterbalance the growing inharmonicity (the loss of acoustic performance) of thick strings.

As a matter of fact, when a strings’ diameter increases, the string itself becomes less and less efficient in terms of acoustic properties; but an increase in the vibrating length allows the harp’s strings to become more tense, therefore thinner diameters are needed, as compared to the fixed length setups used in other plucked and bowed instruments (with the only exception of the lyra-guitar and the lutes, where basses have divverent vibrating lengths).

Diameter and vibrating length are inversely proportional.

Given the same note and tension, the thinner a string can get (whether it’s by increasing the vibrating length or increasing its density, or both), the better it will perform acoustically.

To which tension should we calculate the strings?

Considering the difficulty of calculating the correct tension in plucked and bowed instruments – that have fixed lengths – we are not to be surprised by the fact that doing the same ‘table calculation’ on a harp is even more complex, if not impossible.

Preferring a certain tension over another is quite as subjective as how many spoons of sugar one may favour in a teapot. The amount of sugar to be used is an unpredictable measure, especially if a customer has asked for a cup of tea by telephone without specifying his tastes, and expects the product to be delivered. Some customers may prefer a small amount of sugar, others way more: totally unpredictable.

Regarding harps, since vibrating lengths vary for almost each string, in order to get a homogeneous tactile feel on each string, we cannot consider just one tension; we are instead forced to calculate different tensions depending on the vibrating length of every single string. A 38-string harp, for example, will inevitably have 38 different values of tension (in Kg).

One may reasonably question the fact that an increase in the vibrating length implies a consequent increase in the tension. After all, mathematically speaking, we could derive all the diameters starting from just one value of tension, and applying it to all strings.
We have nevertheless to remember that, by pressing the strings, a harpist is examining their tactile feel of tension rather than their precise tension expressed in Kg (that can only be measured using a device, or by calculation).

The tactile feeling depends largely on the strings’ length: given the same tension, a long string is, and will feel, inevitably softer than a shorter one. Consequently, in a setup featuring the same tension in Kg, the lower the frequencies, the softer the strings will feel.

——-

The following is the guiding principle for all musical instruments, whether plucked or bowed: starting from a single string, once an adequate value of ‘tension’ is found (by that it is meant the tactile feel that the player subjectively judges as correct, and can then be converted by calculation into a value of tension, expressed in Kg), that same tactile feel must  then be applied and found on every other string of the set.

In the specific case of a harp, we already introduced the idea that in order to reach the same ‘equal tactile feel’ on strings of different lengths and diameters, increasing tensions should be set as the vibrating length increases.

For example, assuming that a ‘correct’ tactile feel for a string with a vibrating length of 20 cm corresponds to a value of 2.0 kg, the same tactile feel on a longer  and thicker string with a vibrating length of 130 cm could correspond to a value in the range of 8 to 10 kg (values are just to be intended as mere examples).

If the profile curve of a harp’s vibrating lengths (considering the neck/modillon) was ideally similar to the parabolic curve typical of a harpsicord (and if possibly such curve was standardised for all harps), we could easily determine a formula to calculate the correct tension for every string, providing each string a constant equal tactile feeling.

 

Esempio di profilo delle lunghezze vibranti di un clavicembalo
Example of vibrating length profile on a harpsichord

 

Unfortunately, such condition is never met: no harp is actually identical to one another, and the s-shaped necks can have very different curves and customisations, depending on the harp’s models, variations that may occur just for aesthetical reasons (the ideal profile curve of the vibrating lengths could easily follow the one used for the harpsichord, that could be considered as a horizontal harp with mechanical keys instead of fingers).

Also the sound box’s angle of a harp can vary notably, drastically affecting the ratio of the increase of vibrating length between contiguous strings.

“Wartburg harp” – source https://commons.wikimedia.org/wiki/File:Wartburg-Harfe.JPG

Pedal Harp
“Pedal Harp” – source metmuseum.org

As a conclusion, it’s simply impossible to calculate the correct tension for a determined harp without examining it personally, due to its high variableness.
Further information regarding the equal feel/working tension can be found here: Equal tension/ equal feel: some useful information  

Any solution?

Before investigating the viable solution we suggest, first of all it is good to verify the FL product of the highest octave’s strings of the harp.

What is the FL product?

The FL product is a measure which allows to ascertain that a harp was correctly calibrated by a luthier, especially in the case of medieval and Renaissance harp replicas, whose strings have a bad tendency to break, regardless of their quality.

When considering an original historical harp, we can instead identify its original pitch and get to know how it was calibrated. Even the best string would inevitably break if driven out of its limits.

The only remedies for an already bought harp which happens to be incorrectly calibrated are either to use a particularly robust synthetic string (nylon or fluorocarbon) or to calibrate the harp to a pitch that fits the FL’s safety range (the latter option is not always feasible).

 

Why calculate just the FL product of the highest octave’s strings?

Calculating the highest octave’s FL product is the most practical choice available, since in this particular case such strings are subjected to the most extreme conditions in the whole harp. In the following octaves, the FL product is progressively reduced until, on the last bass strings, it even reaches values lower than 70.

 

How to calculate the FL product, and what is its safe range?

The operation is quite simple, and consists in multiplying the string’s frequency (in Hz) by its vibrating length (in metres):

FL = Frequency (Hz) * Vibrating Length (m)

A practical example:

G (pitch 440) = 784 Hz
Vibrating length = 0,23 m (23 cm)
FL product = 784 x 0,23 = 180,32

Once we have obtained the FL product, we have to evaluate its value using the following tables.

For gut strings only

  • FL < 220 : green light – the string won’t break
  • FL between 220 and 230 : yellow light – the string is unstable and there is an increased risk of a breakage, especially in particularly unfavourable environmental conditions, such as high tempreatures or humidity, sweaty hands, etc.
  • FL > 240 : red light – the string is likely to break during tuning or within a few minutes once tuned

  For synthetic strings

  • FL < 240 : green light – the string won’t break
  • FL between 250 and 260 : yellow light – the string is unstable and there is an increased risk of a breakage, especially in particularly unfavourable environmental conditions, such as high tempreatures or humidity, sweaty hands, etc.
  • FL > 260 : red light – the string is likely to break during tuning or within a few minutes once tuned

 

Notice:  it is highly preferable to check the FL product of at least four or five strings of the highest octave trying to identify the one that has the highest FL value, always comparing it to the above tables.

It’s worth noticing that FL values lower than 100 will need the introduction of different types of strings in order to ensure a good acoustical performance (such as wound strings on silk core, loaded gut, roped gut strings, synthetic materials loaded with metal powders, high density polymers, etc.). When the FL product is in the range between 110 and 100, high torsion gut strings may be employed.

Once the usefulness of the FL product is known, it’s easy to understand why in modern pedal harps the ‘zero’ and first octaves make use of Nylon instead of gut: for the strins of these octaves, the FL products easily reach values over 240, with peaks up to 265, making the use of natural material impossible. It has to be said that such harps will also use heavily varnished and extremely rigid gut strings, that work at higher tensions and will give an abrupt increase in tactile feel and a great acoustic difference when passing from the last gut string to the first wound string on metal core.

 

The FL product in historical instruments (or faithful reproductions)

Calculating in advance the FL product is particularly useful, especially when dealing with original instruments or their copies (regardless of whether they are of the sixteenth, seventeenth, eighteenth or nineteenth century).

In fact, we could discover that the FL product of the first octave is too high or too low (by the way, the ‘yellow light’ range that we defined between 225 and 235 has been derived from measurements on original instruments, such as lutes, theorbos and 5-courses guitars,  of which it is assumed that were not modified through time, and whose origin concerns areas and periods whose standard pitch has been established with some certainty: Venetian “mezzo punto”, Roman or French pitch of the seventeenth century, Germany’s Kammerton of mid eighteenth century).

Why?

Because, when calculating the FL product, the reference pitch we use nowadays (e.g. 440 or 415) might not be the same used originally by the luthier when designing his instrument.
Anyway, considering the medium FL value of 230, that, as we said, can be assumed as the typical value used by plucked instruments, we could even get around this incertainity and rediscover the original pitch used by the luthier.

Let’s explain the process with a new example.

Let’s suppose we have a harp whose A string of the highest octave is tuned using the standard pitch of 440 Hz, so that it actually produces 1760 Hz.
Let’s consider that the vibrating length of such string is 13.9 cm (therefore, expressing it in meters, it’s 0,139 m).
We can now calculate the FL product:

1760 Hz x 0,139 m = 244,6 (red range).

Now we can calculate the percentage of frequency reduction needed in order to bring the value of FL back to our reference value of 230:

230 : 244,6 = 0.94

Therefore the frequency of that A string needs to be reduced by 6% in order to work in safe conditions (1.00 – 0.94 = 0.06).

If we apply the same 6% reduction to the 440 pitch we referred to, we obtain 413.6 Hz

440 x 0,94 = 413,6 Hz

This indicates that the original pitch that was considered, when building that specific instrument, was around 413 Hz, therefore a full setup using gut strings will have to consider the baroque pitch of 415 Hz in order to closely comply with the manufacturer’s design conditions.

Using synthetic string, it may be possible to adapt the setup to the standard pitch of 440 Hz, to the pact of naturally adapting the diameters of all strings in order to keep the same tactile feel.

Note to luthiers: if you want to make a historical replica of the harp we used in this example, and calibrating it for a 440 Hz modern pitch, all the harp’s vibrating lengths need to be multiplied by the 0,94 factor.

For more details about the FL product:
What is the FL product?  

 

Available solutions

(notice: in our work we do not deal with all those ancient harps that have the so-called ‘harpoons’, in which case things become more complex both because the harpoons are different from each other and for the special and delicate calibration required to produce the characteristic buzzing).

The solutions we suggest are essentially three.

Solution 1

The first solution is to deliver us a precise list of your harp’s strings diameters, preferably the ones indicated by the luthier who created the instrument, or by someone who happens to possess the same harp model. In this case we will automatically convert the diameter, considering the potential different string’s material.

It is highly recommended to test the new strings once installed, verifying the homogeneity of their tactile feel.

It may be needed to correct some strings which might happen to be too soft or too stretched.
If the string is too soft, it is suggested to raise its tuning by half-tone steps, until the tactile feel gets aligned to the rest of the strings, keeping track of the number of half-tones needed.
If the string is too stretched, the procedure is similar but the tuning should be lowered by helf-tone steps.

To find out the proper corrected diameter:

  • multiply the string’s diameter by 0,944 for each half-tone you had to lower the string to get the right tactile feel;
  • multiply the string’s diameter by 1,06 for each half-tone you had to raise the string to get the right tactile feel.

 

Solution 2

The second solution, despite being more intricate, leads to a more accurate outcome. It’s a bit like reconstructing a ‘tailor-made suit‘ by proceeding through successive ‘steps’.

A notable number of strings of different diameters (ideally between 0.50 and 1.80) is required for the process. Their price shouldn’t be an issue: eventually all strings will be used in the final setup. A well balanced list of strings to test would be with gauge differences of 0.10 (i.e.: 0.50 – 0.60 – 0.70 – 0.80 and so on, up to 1.80); there’s no difference using gut or synthetic strings, as long as they are homogeneous (no mixes using different materials).

The next step is to find the proper gauge for every G note of each octave, starting from the first octave.

We can do it by following these passages.

First of all, on the G of the first octave we install a string that has a thicker diameter than the one supposed to be correct; we therefore tune it until we have the desired tactile feel and tension, whatever note the string will reach.

Let’s presume that we’ve used a 0.50 mm string for the test and that reaching the tuning of an E it already fits the desired tactile feel of tension (that is 3 half-tones lower that the expected G).
How can we get the right gauge to use in that position in order to get a G with the same tactile feel?

It’s easily done: we need to multiply the 0.50 mm diameter by the 0.944 coefficient, and the result is a diameter decrease equivalent to a half tone. If we repeatedly multiply each result by 0.944, we get a decreasing series of diameters correspondent to half tones differences. Since between E and G there are 3 half-tones, three consequential multiplications will give the diameter we were looking for. So:

  • 0.50 x 0.944 = 0.472 (from E to F);
  • 0.472 x 0.944= 0.445 (from F to F sharp);
  • 0.445 x 0.944= 0.42 (from F sharp to G).

In order to obtain the ‘tactile feel’ we required for that string, the proper diameter to use for the G of the first octave will be 0.42 mm.

The process for all other octaves is identical (remembering to start with proper diameters for each octave).

One might now wonder why did we choose to start with a thicker diameter than the expected one, instead of opting for a thinner one?

The answer is quite simple: if the string was too thin, it would have inevitably overcome the safe value of FL product, eventually breaking. This is especially important on the first 2 octaves, where strings work very close to their breaking point (that is with FL products greater than 210).

If during the test the chosen string needs to be tuned higher than the reference G note without having reached the desired tactile feel yet, we suggest restarting the test using a slightly thicker one.

Once we’ve calibrated the first two octaves, the breakage risk is no longer a threat, therefore, if during the test we need to tune the string higher than the G, we can simply note how many half-tones we added to get to the right feel, and then multiply the string diameter by 1.06 (the inverse of 0.944) for each exceeding half-tone.

An example: let’s assume we want to find the proper diameter of a third octave’s G. We may assume a 1.20 mm string will work.

Once reached the tuning of G, we perceive that the string’s tactile feel is too soft, but it feels perfect when tuned to A sharp.
There are three half tones that divide G and A sharp.
If we use the 1.06 coefficient for each of the three half-tones, we’ll get eventually to the correct diameter for the G:

  • 1.20 x 1.06 = 1.272 (from G to G shapr);
  • 1.272 x 1.06= 1.348 (from G sharp to A);
  • 1.348 x 1.06= 1.429 (from A to A sharp).

So the right diameter to use would be 1.43 mm.  

 

Why did we start from the G string?

The question is legitimate.
Of course, the choice of using the G as a reference is a simple suggestion, also considering that it’s an uncolored string (unlike C and F), but it is not compulsory.

The next important step is to identify the right diameter for all D notes.  

 

Why D strings?

In this particular case, D wasn’t chosen arbitrarily, but is a consequence of the fact we first started with the G. In fact, D is the intermediate note between two G of different octaves.

We will proceed as we did on the G, but this time we won’t certainly need to guess which diameter to start with on the D. It is suggested to use a diameter value that is half way between the ones we found for the two Gs.

A quick example: if the lower octave’s G has a 1.36 mm diameter and the higher octave’s G is 0.58, the suggested diameter will be the result of the simple mathematical expression:

(1.36 + 0.58) / 2 = 0.97 mm

It is very likely that such resulting diameter won’t be included in the initial list of available diameters, but one can use the nearest diameter (in our case it could be a 1.00 mm).

The previous procedure can then be performed: tuning by steps of half-tones until the right feel is achieved, and the multiplying by 0.944 or 1.06 if the reached tuning is respectively lower or higher than the D we used as a reference.

Notice: for the D of the first and second octave, you are supposed to follow the same safety rules used for the G, that is using slightly thicker diameters than the expected ones in order not to go over the safe FL product value of 230.  

At this point, we should have found all the correct diameters for the Ds and Gs, given the tactile feel we desired. All we have to do now is ‘connect’ the intermediate diameters of the notes in between, making further empirical tests if needed.

After the setup’s calibration, it is recommended to occasionally adjust the strings, achieving a true ‘tailor-made’ solution.

 

Solution 3

There’s still another way to calibrate a setup.
It consists in choosing a note positioned in the middle of the neck, and installing a diameter we believe could be most likely valid (i.e. supposedly between 1.50 and 0.60 mm).

Let’s assume we install a 0.94 mm gut string.

We should tune it until we reach the tactile feeling we’re looking for.
We should then identify its note and measure its frequency in Hertz: let’s say it’s a C (130.8 Hz) at standard pitch of 440.

Given the vibrating length (0.90 m), the diameter of the string (0.94 mm), the density of the material (for gut we consider 1.3 kg/dm3), we can now calculate the value of tension (in Kg) that corresponds to the tactile feel we want:

   

A practical example:

frequency: 130.8 Hz
vibrating length: 0.90 m
density of the material: 1.3 kg/dm3
string diameter: 0.94

Result of calculation = 5,1 Kg

With this particular vibrating length, the tactile feeling corresponds approximately to 5,1 Kg (it is important to understand that this is only a reference value, because to our tactile feel the difference between a tension of 5.0 kg or 5.2 kg is hardly noticeable).

In order to keep the tacltile feel constant, at lower frequencies a larger diameter would be necessary, and vice versa a thinner diameter for higher frequencies.

Numerous experiments and measurements carried out here at Aquila Corde have shown that we need to apply an extra 0.2 Kg to the tension for every 10 cm increase in the vibrating length of the strings (and remove the same 0.2 Kg for every 10 cm decrease). Using simple proportions, it’s easy to calculate partial tension variations for each increase in vibrating length.

This method allows to correctly calculate every string’s specifics. We suggest nevertheless to find the ones of just two (G and D) or three (G, D and B) strings per octave, and then empirically determine the remaining ones.

Once we have a scheme of tension values for the reference strings (i.e. G, D and B) for all the octaves, the diameter of all the strings can be obtained using the following equation:

   

 

Just to make an example:
Tension: 5.1 Kg
Frequency: 130.8 Hz
Vibrating length: 0.90 m
Density: 1.3 kg/dm3
Result of the calculation: 0.94 mm diameter.  

A final warning: changing model or brand of a certain string will most probably require some adjustments on the diameters. Depending on the producer, for example, gut strings may present different densities, or be more or less stiff. To compensate for these differences, the same procedure described above can be used, raising or lowering the tuning in half-tone steps, in order to recreate the desired tactile feel, and using the coefficients 0.944 or 1.06 to adjust the diameters accordingly (see examples above for the detailed steps). 

In the case of synthetic strings instead, there are three main categories of materials with different density: Nylon (density of 1.04 kg/dm3), Nylgut/Silkgut SH/Sugar (1.30 kg/dm3) and Fluorocarbon (1.78 kg/dm3). When changing materials, a diameter conversion is needed. For example, when switching from Nylon strings to Nylgut/Silkgut SH/Sugar strings, the diameter must be multiplied by 0.91; when changing from Nylgut to Fluorocarbon the coefficient to be used is 1.10.

Considering that Nylgut/Silkgut/Sugar strings usually stretch more than Nylon, our advise is to always use a slightly thicker string; for example, a 1.00 Nylon string would be converted to a 0.91 Nylgut/Silkgut/Sugar, but it is preferred to use the nearest greater diameter available on the market, therefore a 0.94 mm.

Of course, it’s the musician that will evaluate if the diameter that best suits his/her needs will be the 0.91 or 0.94.
As a rule of thumb, it’s always better to recalculate all diameters when there’s a change of string material or producer, and before changing the whole set of a harp it’s advisable to test the tactile feel on one – ore more – diameters, and, if needed, recalibrate using the proper proportions.

 

Wound / loaded bass strings

Wound strings in the past were basically made with a core made of gut overspun with a close winding of “faux silver” (this is how silver-plated copper was called in the eighteenth century). In mid XVIII century, multifilament silk core strings made their appearance, always with silver-plated copper winding.
In some iconographic sources there is also evidence of demi-filé basses on gut core.

     

Nowadays, almost all of them are replaced with close wound silver-plated copper strings with a multifilament nylon core. Their main advantages are improved acoustic performances, more sustain, and a better resistance to traction and more stability to climatic changes.

Characteristics of a wound string

When replicating a wound string, it is common belief that the external diameter is a fundamental parameter, but actually it is substantially irrelevant.

There are almost infinite possible proportions between the string’s core and the diameter of the external metal wire, and such proportions can drastically change the acoustic performance of the string even if the resulting external diameter can be the same.

Just to make an example, an external diameter of 2.0 mm can be composed of a 1.95 mm core and a 0.025 metal wire, or a 1.0 mm core and a 0.5 mm metal wire, but their sound will be completely different!

Therefore, in order to allow a string maker to make an accurate replica of a string, it is essential to:

  1. send the stringmaker a specimen of the original historical string, or
  2. calculate and inform the stringmaker of the gut equivalent of the string.

What is the gut equivalent, and how can it be obtained?

The gut equivalent expresses the gauge, in mm, of a theoretical plain gut string that has the same weight by unit length of the composite wound string, so that, at the same intonation and vibrating length, the gut string and the wound string will therefore have the same working tension.

It can be calculated by measuring the string’s weight and length, and then dividing the weight (in g) by the length (in m); the square root of the resulting number, multiplied by 0.99 will express the gut equivalent.

It’s quite easy to understand that the shorter the measured string, the less accurate will the equivalent gut be.

There’s yet another important measurement that needs to be taken into account: the metallicity index.

In other words, it’s fundamental to understand the weight ratio between the metal and the core of the string, because this will have a direct influence on the final acoustic performance. A high metallicity index is directly connected to a bright, metallic sound, while lower values will be characterized by a warm sound with less sustain. 

Unfortunately, there’s no mathematical formula to determine the best resulting sound, since it is mostly based on personal preferences of the musician, so it will be the stringmaker’s job to give the musician three or four string samples with different metallicity indexes so that the desired timbre and sound can be chosen.

From our experience, the preferred metallicity index range is not that wide, and it can be represented with a Gaussian curve; strings with low values of the index will be more dull and less performing (there is a clear predominance of core mass over the metallic mass), while high values will bring unpleasant metallic sounds.

Following these direct observations and experience, the writer, with a certain amount of imagination, has defind as the ‘range of beauty‘ that specific range of metallicity indexes where almost all musicians will agree that the string performace is acceptable; the extreme values of such range will satisfy those that prefer round and percussive sounds on one side, and bright and metallic sounds on the other side.

Loaded strings

The only information that’s needed is the gut equivalent: therefore all the above suggested rules apply (weigh the strings, measure its entire length, etc.). However, the value of external diameter could be useful in case you want to calculate its specific weight and trace the amount of metal powder that was added to the gut/polymer (in fact, there are low strings that have different percentages of metal charge).

Further info regarding historical loaded strings can be found in this article:

Wound strings for bowed and plucked instruments from the late 17th century to the early 19th century: what do we know?

 

Final notices

At this point, it is appropriate to point out three potential risks.

  • first of all, when calculating the setup of an original historical harp, there’s the risk of inadvertently mounting it with more tension than it could actually bear, because of its age;
  • the second problem could be that one’s ‘tactile feel’ may be based on a personal experience built through the years on a pedal harp, and such experience cannot be automatically applied on other harp types;
  • lastly, one could be tempted not to verify the FL products of the strings in the first octave: never forget to check them!

 

Conclusions

As we have seen, the harp is the only instrument where almost half of its strings operate with an FL product between 180 and 230, conditions that can be found on the first strings only of bowed instruments. Only on medium to lower frequencies the strings work within safety margins.

Given this fact, on the first two higer octaves we strongly suggest to avoid elastic strings and use instead rigid and robust low-torsion strings, that will ensure a great tensile strength, less wearing and less turns around the tuning pins.

Starting from the third octave we suggest to use high torsion strings, that are more elastic, and ensure an improved acoustic performance without the need to use other types, such as wound strings.

 

Being more elastic, a small increase in diameter will be needed in order to keep the tactile feeling constant through all the set.

Before installing the thicker strings, we finally recommend to pre-stretch them in order to avoid excessive turns around the tuning pegs.

 

Vivi felice

Mimmo Peruffo, Giugno 2020


Leopold mozart and his instructions on how to choose a violin’s set of strings

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


6 AND 4 COURSES XVIII CENTURY MANDOLIN SETUP: A FEW CONSIDERATIONS

by Mimmo Peruffo

 Introduction

When faced with the problem of what kind of strings were used on the 18th century Mandolins of six and four courses, the first thing that stands out is the great heterogeneity of these set up. What is really hard to understand is particularly on the 4 course Neapolitan Mandolin: here we find together gut strings; single and twisted metal wires; wound strings on gut/silk.  To complete the already heterogeneous picture, for the 4th course there are also two choices between unison and octave.

Here is the first question: why was it used a gut 1st and not a metal wire like the other courses, when it was then in  use in the 1st half of the nineteenth century?

This question is logic: the average breaking load stress (Breaking Point) of the gut is ‘only’ 34 Kg/mm2, much lower than the average of  iron and bronze of the time, which easily exceeded 100 Kg/mm2.

To understand the reason, we must first start from the mechanical and acoustic behaviour of the string. In this way we will be able to try to figure out what were the guiding criteria used to determine the vibrating lengths of plucked and bowed instruments, including Mandolins.

The strings and their characteristics

Musical strings follow the rules that are summarized in the string equations of Taylor-Mersenne or even called  Hook’s law (although the first to mention it was Vincenzo Galilei around 1580), which relates frequency, vibrating string length, diameter and density of the string.

However, when the gauge of a string increases, another thing is not included in this equation: with the increasing string diameters comes also a progressive loss of its acoustic properties until reaching the point where, over certain gauge, the string  has clearly lost most of its performances. This is caused by the progressive increasing of the stiffness of the string.

This phenomenon is called Inharmonicity: before the appearance of the wound strings (on the second half of the 17th century) it was the main problem with which all the manufacturers of plucked,  bowed and keyboard instruments  had to deal with. (1)

The Inharmonicity  clearly determines a limit to the total number of bass strings that an instrument can have, i.e. the open range.  There is a second problem: a poor elasticity, i.e. a high Elastic Module, also produces an unwanted sharper frequency when pushed on the frets; this phenomenon is particularly noticeable on short vibrating length instruments  (‘pitch distortion’).

The best solutions, in order to keep the Inharmonicity confined and the string sounding still ‘good’, is to limit the diameter increase by mean of some solutions (or, alternatively, keeping a thicker gauge but increasing  the elasticity of the string to reduce the stiffness).

Our main interest is represented by these relationships:

 

-Diameter and vibrating length are inversely proportional

-Diameter and tension are inversely proportional

-Diameter and density are inversely proportional

 

The solutions that, at the same frequency, can contribute to reduce the diameter are the following:

 

1) Reduction of working tension

2) Increasing of the vibrating string length

 

However, there are other implementable actions:

3) Increase the elasticity of the string (does not affect the diameter reduction)

4) Increase the density of the string (affects the diameter reduction)

Point 1 is an exclusive decision of the player: according to the ancients the right string tension (better to call it the right feel of tension)  is when the strings are not too stiff, nor too slack under the finger pressure. There is, however, a lower tension limit otherwise not only you can lose the finger control on the strings but also the acoustic power, its’ fire ‘, along with the increase of what is commonly called’ pitch distortion ‘ due to the fact that the strings are too slack and so, out of control by the performer.

Point 2 depends only by the luthier. This solution was adopted from the far past for the Arps, but latter also for the keyboards,  theorboes/archlutes etc, were the vibrating string length increase, step by step, towards the bottom strings making them, step by step, thinner (proceeding in this way, the Inharmonicity is under  control)

Points 3 and 4 depend only by the strings maker: the appearance of the wound strings in the middle of the seventeenth century can be considered a good example of point 4;  a roped gut string/a very high twist string an example for the point 3.

At the end of the day, the point where a luthier can act is only No. 2, where vibrating length and diameter are inversely proportional (we consider that the performer has already done its job on the choice of the right feel of tension)

In the sixteen, seventieth and (maybe) the first half of the eighteenth century, the problem of string Inharmonicity was a well-known thing for luthiers: it can be seen, for example, from the still existing bowed and plucked instruments, whose vibrating string lengths are all  related to the frequency of the first note and the hypothetic standard pitch: in practice we are speaking of the well knows rule of those times to tune the first string to the most acute possible just before the breakage.

In order to optimize the sound performance of a musical instrument it was therefore followed by the luthiers the rule of using the maximum vibrating length possible for that given treble note indicated by the customer (in other words, in which Country and its related pitch standard the instrument must be then employed) : only in that way all the strings could have the minimum gauge at the right feel of tension  for the benefit of the overall acoustic performance.

However, the vibrating length cannot be increased as desired because of the limit imposed by the breaking load of the 1st string: there is a limit that we call Superior limit

At the same time, it is not possible to increase the amount of bass strings (i.e. increasing of the open range) because there is another boundary called Inferior limit.

In other words, the full open range of a musical instrument is enclosed within these two borders.

The so-called Inferior limit however, using pure gut strings,  begins to heavily manifest when the frequency range  between the 1st  string and the last reach, more or less, two octaves. Only the six course Mandolin, on the two models, comes to this range. Generally speaking, the problem was, however, partially solved after the 2nd mid of the 16th century by the introduction of  a kind of very elastic and/or denser  bass gut strings and then totally solved by the introduction  of  the bass wound strings in the 2nd half of the 17th century. In 2nd  half of the 18th century, the wound strings were probably totally in use.

 

The Superior limit

When a string of any material is progressively stretched between two fixed points (i.e., the vibrating string length), it will at some point reach a frequency where it will, instantly, break (Breaking Point)

In the case of a modern gut string, the average value of this frequency for a vibrating length of one meter is of 260 Hz (actually, after several tests, I have found that the whole range is of 250-280 Hz), which is a slightly low C.

The value of such a limit frequency, known as ‘Breaking Frequency’, is completely independent – as strange as it may seem – from the diameter and this can easily be verified both by mathematics (applying the general formula of the strings) and empirically.

By changing the diameters, the only changing parameter is the tension value always corresponding to the breaking point (i.e. the breaking frequency)

The Breaking Frequency is inversely proportional to the vibrating length at which the string is stretched.

So, if the string length is cut down to a half the frequency doubles and vice versa.

This means that the product between the vibrating length (in m) and the Breaking Frequency (in Hz) is a constant defined as ‘Breaking Index’, or more simply FL product (i.e. Vibrating length x Breaking Frequency),

By introducing the Breaking Index into the string formula considering a unit section of 1 mm2

(that is equal to 1.18 mm in diameter) at 1.0 m of vibrating string length,   at the corresponding breaking frequency value in Hz we obtain (of course) the breaking load stress value of 34 Kg/mm2.  In other terms, a string of 1.18 mm gauge, 1,3 of density, 1.0 mt scale at 34 Kg of breaking tension will reach the limit of 260 Hz.

In short: the breaking point of a modern gut string, according to our practical tests, ranges from 33 to 38 Kg mm2, which is equivalent to a breaking index of 250-280 Hz/m (mean value: 260 Hz/m). (2)

 

Breaking vibrating length

Going back to our main topic, a luthier thinks in opposition to what has been just explained; it is the frequency of the 1st string the first parameter to be fixed when designing a musical instrument such as the Mandolin, Lute, etc etc.

By dividing the Breaking Index for the desired 1st string’s frequency, you will obtain the theoretical vibrating length limit where the string will break when reaching the desired note (Breaking Point):

 

This is a simple proportion:

260: 1 meter = 1st string’s frequency: X       (were X is the vibrating length to be attributed in meters).

In the case of a six courses Mandolin whose first string is a G: 698.5 Hz (18th-century French chorus of 392 Hz) (3) it obtains: 260/698.5 = .37 m

This is therefore the vibrating limit length where we know that the string will break reaching the G (here we are referring to the ancient French pitch standard of 392 Hz).

The choice of vibrating ‘working’ length should therefore consider a prudential shortening of this limit length.

But how much? The more is shortened, more the strings are thicker with the risk of losing acoustic performance.

Prudential Shortening or Working Index

Examining the vibrating string lengths of the plucked and bowed instruments of the tables of Michael Praetorius (Syntagma Music, 1619) made possible to calculate their Working Index and put them in correlation to the gut breaking index. This allowed to understand the security margin adopted in those times (4) (5)

 However, in the various calculations was (unfortunately) Ephraim Segerman taken as reference the average breaking load value -or Breaking Point- of a modern gut string found in literature: 32Kg mm2 (which is equivalent to a breaking index of 240 Hz/m) that is, actually, too low then the reality.

So, this value ca be placed on ‘lower quadrant’ of the range of breaking loads that we have found in today’s commercial strings during our experiments (we will here suggest the average value of a Breaking Point of 34 Kg/mm2, equal to 260 Hz/m of Breaking Index).

Drawing from ‘Syntagma Music’ Michael Praetorius 1619

However, comparing the breaking index of 240 Hz/m with all the other Working Index, he found that the choice of the vibrating working length of the Lute family and some Gambas (Viola Bastarda for example) was about 2-3 semitones below the Breaking Index (and hence also of the theoretical vibrating length that we calculated before).

Considering our example, therefore, the length shortening of two / three semitones would represent the real vibrating length to be adopted (corresponding to a G of 392 Hz): 32.9 / 31.1 cm, values ​​that are included into the measures that are actually found in the six courses Mandolins of the time.

However, there is a concrete evidence of what has been said so far: we have subjected a gut string to a progressively increasing tension (Stress) and measured the related stretching (Strain).

Examining the final Stress/Strain diagram, the initial proportional variation that comes out follows the law of Hook and emerge evidently (also called Taylor / Mersenne).

At a certain point, the proportional variation stops and you reach a condition where the stretching (and therefore the corresponding tension) suddenly rises for small peg’s turns imposed to the string:

 

The string therefore maintains its linearity till about 2-3 semitones from the Breaking Point; beyond this value, it enters the critical phase. This does not follow the phenomenon of the typical yeld of  metals and  Nylon/Nylgut/Fluorocarbon strings. From this point,  gut almost completely loses its ability to stretch itself reaching rapidly its Breaking Point.

It is therefore concluded that the use of the maximum vibrating length can only work in the upper point of the linearity just before that the line start to bent up to reach the final breakage.  The maximum acoustic performance (given by the maximum reduction of the diameter of all strings = maximum control on the inharmonicity) is determined by the fact that the instrument is working on the upper limit of proportionality, just before it changes, and this is exactly two to three semitones from the final exitus, as shows in the graphic.

Such behaviour of the gut string was well known even to the ancients and was therefore applied as one of the basic rules in the design / construction of musical instruments.

For Example, Marin Mersenne was avare of the right proportions that a musical instrument must have (“Harmonie Universelle” 1636, Livre Troisième, Proposition X, 129) :

 

Here there is what Bartoli wrote by the end of XVII century: (6) ‘Una corda strapparsi allora che non può più allungarsi…’ (a string breaks when it cannot stretch furthermore).

 

Daniello Bartoli: ‘Del suono, de’ tremori armonici e dell’udito’ 1679.

On the other hand, it is well-known to everybody the rule of those times of tuning the lute and even some bowed instruments at the highest note and stop immediately before the breakage of the first string: this is the ultimate proof of what we have already showed graphically.

The Lute example

The vibrating lengths that were chosen for some of the old, surviving Lutes of the past sum up valuable information.

The main problem is that in order to make an evaluation you have to use not modified instruments and instruments, whose standard pitches can be determined with a relative certainty. This is the case of some unmodified renaissance venetian lutes, German d minor baroque lutes, French baroque guitars.

Starting from an highly supposed standard pitch (thanks to their origins) and from what emerged in the study of their vibrating string lengths, the research on the various  5 course French guitars ( at the 17th French pitch standard close to  390 Hz) as well  the german 13 course d minor Lutes tuned at the Kamerton of 410-420 Hz (see Baron 1727: Kammerton f note for the 1st) ) and finally including even some surviving renaissance venetian lutes whose scale is of 56-58 cms probably related related to venetian standard pitch of mezzo punto = 460 Kz more or less, has allowed to detect a range of Working Index within 225 and 235 Hz/m with an average of 230 Hz/mt: this can be considered  the Lute Working Index of the past times (theorboes  generally speaking works with a bit more safety;  Some Magno Graill or Buechenberg large theorbo has the vibrating string lengths  around  95 cms; at the Roman pitch standard of 390 Hz/m, the related working Index range is of  210-220 Hz/m) .

We are very close to what we calculated for example from Segerman: 210 Hz/m

If we consider true that the Working Index of these examined original instruments present a safety margin of two or three semitones from the Breaking Point (how we have seen in our Stress/Strain graphic), it is even therefore possible to estimate the average Breaking Point -in Kg- of the Lute 1st strings of those centuries. This can be obtained  by increasing the working index that we have deducted of two or three semitones.From this simple reverse calculation, it is possible to determine that the average breaking load of the gut chanterelles  of the 16, 17 and 18th century would be between 33.7-35.1 Kg/mm2 (that correspond to Breaking Index range of 256-268 Hz/m) in the case of two semitones of safety margin and 35.7-37.3 Kg/mm2 (Breaking Index 273-285 Hz/m) if the safety margin was instead of three semitones.

How we can see, he range of all these values ​​is perfectly in line with that of the current treble Lute gut strings of .36-.46 mm gauge (34-38 Kg / mm2).

Going back to the six-courses Mandolin with a 1st string = G, a prudential shortening of two semitones on the average value of the Breaking Index of 260 Hz/m determines a vibrating length of 32.9 cms; it will be 31.1 cms  if we are considering three semitones down of a safety margin: these are the typical vibrating lengths found in the surviving instruments.

The range of Working Index (the product between the frequency of the first-string x vibrating length in meters) is as follows:

 

Sol ( at the standard pitch 392 Hz);  32.9 cm         31.1 cm

230  Hz/m      217 Hz/m

 

Sol (at the standard pitch of 415 Hz);  32.9 cm         31.1 cm

244 Hz/m    230 Hz/m

 

As can be seen, a 6-courses Mandolin exceeds the typical  Working Index of the surviving Lutes & 5 course Guitars  only if the safety margin  is two semitones whose pitch standard is of 415 Hz.

In the case of the 4 course Neapolitan Mandolin with a vibrating length of 33 cm (the one typical of the Violin) the following is obtained:

Mi (392 Hz reference pitch); 33.0 cm

194  Hz/m

Mi (415 Hz reference pitch); 33.0 cm

205 Hz/m

The conclusion is that both these Working Index are included within the Breaking Index of the gut treble, the 6-courses mandolin in particular works exactly like a Lute while the 4-courses Neapolitan has a lower tension condition on the first string, just like for Violin. The plausible explanation is as follows: while in the 6-courses mandolin the frequency excursion between the first and the last string is two octaves (24 semitones), in the 4 courses this excursion is reduced to 18 semitones. Consequently, in the second case, it is not strictly necessary for the strings working at the highest possible acoustic performance) i.e.  close to the Breaking Point like happens on those instruments that has an open range of two full octaves like the Lute, in order to preserve the acoustic performance of the bottom strings.

However, the starting point is still unsolved: why was not used a metal treble whose sound would be much brighter and more readily available, would have had less wear and tear and even a higher breaking load than the gut? (7)

The breaking load stress of a XVIII century Iron for the Harpsichord can reach up to 100 Kg / mm2. For the old Brass this value is lower but always much higher than the average breaking load stress of the gut.

The explanation is that the highest note is certainly directly proportional to the breaking load but also inversely proportional to the specific weight of the material, which is very high in metals: 7.0 gr / cm3 for iron, 8.5 gr / cm3 for Brass; 1.3g / cm3 only for gut.

From simple calculations, taking into account both the ancient pieces of wire for keyboard instruments discussed on some essays of those times, we can list a series of Breaking Index:

 

Mersenne (8)

Silver:  155 Hz/m

Iron:    160 Hz/m

Brass:  150 Hz/m

The typical high density of metals affects quite strongly the limit of the Breaking Index: an “ancient” steel string with a breaking load of 100 Kg/mm2 for example (which is one of the higher values found among the absolute values ​​of old strings for keyboards pieces), however, has a breaking index of just 178 Hz/m.

This clearly explains why the Battente guitar, fitted with robust metal strings, instead, have a vibrant length limited to only 55-58 cm, while those with the least strong gut strings can reach 68-73 cm (with the same reference chorus). (9)

There has been found a lot of metal strings breaking load stress of the past (10)

Here are some breaking indices found in old metal strings of Spinetta or Harpsichord:

 

‘Old’ harpsicord iron: 158-188 Hz/m; mean 173 Hz/m. (11)

‘Old’ spinet and harpsicord iron: 164-187 Hz/m; mean 175 Hz/m. (12)

Old’ spinet iron from the second half of the 17th century: 159-195 Hz/m; mean 177 Hz/m. (13)

 

Other metals:

‘Old’ copper alloys: 112-138 Hz/m; media 125 Hz/m. (14)

‘Old’ brass: 101-155 Hz/m; media 128 Hz/m. (15)

‘Old’ brass: 148-153 Hz/m; media 150 Hz/m. (16)

It can be easily noticed that the difference between Mersenne data and the average measured values ​​is not particularly relevant.The reason why the Mandolins used the gut for the highest string is therefore clear: they did not have pure metals and/or metal alloys that could reach a breaking index similar to gut (260-280 Hz/m).

Considering the Iron (the metal with the highest breaking index) this would correspond to a breaking load of 145-160 Kg / mm2.

The evidence of the use of gut trebles on the Mandolin is a clear demonstration that strong metal strings were not available, in the course of the XVIII century and even for the first decades of the XIX. A metallic wire with these values would have been employed immediately, as it actually happened between the 16th and 17th centuries and after the 1830. (17)

The Mandolin was therefore inevitably forced to use gut string for the 1st course due to lack of alternatives.

 

Historical sources

 There are few historical sources of XVIII century containing information regarding the string setups of 4 or 6 courses Mandolin; these few are, at the end of the day, only Fouchetti and Corrette. (18) (19)

Let’s see what they wrote and what can be deducted:

Fouchetti

What Fouchetti wrote about the 4-courses Neapolitan mandolin setup, generally speaking, is considered unreliable, if not quite imaginative. A set of strings like those he described appears to be the most bizarre and heterogeneous among those of all the plucked and bowed instruments of his time.

In fact we find, mix together in a just four courses set, gut string, brass wires, twisted brass wires, wound gut/silk strings.

Indeed, this degree of heterogeneity is absolutely amazing. By looking more closely and by making some calculations, we realize that this set include in itself almost the utmost perfection possible for that time both from the mechanical point of view and from the acoustic point of view with very few other possibilities of choice, if we consider what was available in those times, to make strings.

Let’s see why (keeping in mind that the most wanted feature for this instrument was the brightness and the promptness of emission, as it had to imitate the harpsichord): (20)

 

Vibrating lengths: the used sizes confirms with clear evidence, especially on the 6 course Mandolins,  that we are considering an instrument that, like the Lute, had the maximum vibrating length in order to ensure the best acoustic performance.

Here is the set for the Neapolitan 4 courses Mandolin (Fouchetti says nothing about the 6 courses one):

  1. use a Pardessus gut treble
  2. a harpsichord gauge 5 yellow brass
  3. two harpsicord gauges 6 yellow brass twisted together
  4. a light G Violin wound fourth. The core can also be silk. As octave pair you can use a 5-gauge yellow brass like those of the second course. Sometimes the fourth course are installed in unison.

 

 

First string: considering the Range of the working index that we determined the first string must be of gut due to the lack of possible alternatives: Fouchetti suggests an 1st Pardessus string. According to the data provided by De Lalande (21) and other sources, we know that the treble for Pardessus and also Mandolin was made up of two whole lamb guts, and the Violin 1st of three. There are numerous researches (22) (23) (24) that associate three whole lamb-guts with a gauge of .68 to .73 mm.

For simple proportions, the Mandolin/Pardesssus 1st string had a diameter of .56-.59 mm.

-Second string: Fouchetti says you should use a yellow brass wire of gauge 5. The second course’s working index is around 129 Hz/m so the brass wire available (for harpsichord) was not breaking. The use of a Brass string and not of a more robust Iron has only one explanation, of an exclusively acoustic nature: Brass, due to its higher specific weight of iron, is brighter: this fit quite well with the criteria of those time were Mandolin should imitate the sound of the harpsicord.

According to the Cryseul gauges scale, (25) the gauge 5 corresponds to a diameter of about 0.30 mm. (26)

The yellow brass has a specific weight around 8.5 gr/cm3  (red brass: around 8.7 gr/cm3).

Third string

Fouchetti says to take two yellow brass harpsichord of gauge 6 and twisted together. The purpose is clear: the strings twisted together become more elastic, so they minimizing the ‘pitch distortion’ effect on the frets which with a simple single wire would be absolutely evident even for small pressure variations and / or lateral displacements on the string. With a single metal wire, it would also have a considerable difficulty in tuning and keeping it stable over time because even an imperceptible rotation of the tune peg would produce significant variations. By twisting two wires together, the problems listed above are solved; the use of the brass still guarantees the best acoustic performance in terms of tonal brightness and emission power even though it is still a little lower and ‘round’ than those of a single wire.

The gauge 6, always according to the Cryeseul scale, corresponds to .29-.30 mm in diameter. The problem here is to determine the strings twisting degree and the yellow brass behaviours, as Fouchetti says nothing about it.

We can find a solution by realizing different types of twisting and checking the mechanical strength, sound, and especially the resulting working index, and compare it to the working indices of the other courses.

We thus found that two .30 mm brass strings twisted together with a low twisting produced a final string of .39 mm diameter (1.30 times the diameter of the starting wire) or .46 mm (1.54 times the diameter of the starting line) if the twisting ratio is very high.  In this second case, however, we found the sound was far better: the working tension D note of 262 Hz (Parisian pitch  standard of 392 Hz) is around 3.4 Kg.

Fourth string

For the fourth string it was used a  Violin G wound string, but a bit thin than the ordinary (in those times they were done using a average second violin strings as core: we have employed a thin second one) Considering a gut core wound string, it is evident that you lose the characteristic brightness manifested by the three most acute strings.

This problem is greatly mitigated by the fact that a yellow brass octave (and not a gut) string is added, whose obvious purpose was to add brightness to obtain an acoustic alignment with the higher ones. In this course was also made the arrangement in unison but Fouchetti tells us that it seldom used.

The author suggests, alternatively, the use of silk as the core of the fourth, thus anticipating what would then become the standard for the bass of the six-courses guitar of the nineteenth century. By use of silk core the sound became even a bit brighter.

 

Indeed, the use of silk core basses for five-courses guitar had already been described by Juan Guerrero in 1760. (27

 

 

But how was made a wound Violin G strings in those times?

Some sources of those times  wrote that a second string of the instrument was taken for the core and then covered with a thin silver wire or silver copper wire (see Francesco Galeazzi, 1792). The equivalent gut of a string like this, in order to ensure the balanced setup (scaled tension)  for this instrument, ranged from 1.70 to 1.90 mm. Fouchetti writes, however, that this string has to be a bit thinner than ordinary, but how much? We should calculate its equivalent gut having the Mandolin scale, the supposed pitch standard and the gauges of all the other strings (and so, by calculation, the related working tensions)

Here is in practice:  having as vibrating length 33 cm, the diameters and the density of the materials, it is possible to obtain the working indices’ values of all the strings at the supposed Parisian  pitch standard of 392 Hz.

 

1: 5.44 Kg (average tension value between .56-.59 mm diameter = .575 mm)

2: 5.3 Kg (gauge 5 = .34 mm)

3: 3.4 Kg (two gauge 6 low twist = .30 mm)

4: octave: 4.46 Kg (.34 mm)

4: wound bass: the tension should be the same of the  paired octave string: 4.46 Kg

 

Remarks

1) at ‘Parisian’ 392 Hz pitch standard, the working index of the yellow brass wire for the octave of the 4th string is about 115 Hz/m  (122 Hz/m at 415 Hz): a yellow Brass wire can therefore be safely used.

2) The setting presents a scaled tension profile which probably leads to a situation of equal tactile feel if it wasn’t for the third chorus abnormal tension which is quite low. In reality, it is possible to balance the situation if we consider a thinner gauge for the first string realized however always from two guts.

3) Then, in order to have the same working tension of its paired octave, the equivalent gut of the fourth string should be 1.75-1.80 mm: in fact, we have a fourth Violin string that tends to be in the range of the generally definable light tensions.

Conclusion

The setup described by Fouchetti presents almost perfect coherence on the tension values between the various strings and on the acoustic side; due to the careful choice of materials and string types, it achieves the highest performance that results in powerful emission and brightness.

It should be noted, however, that there were not many alternatives: the first string had to be gut, while the fourth had to be a wound gut/silk core string. Most likely, in the second and third courses, iron wires for Harpsichord could be used but this would be at the expense of the brightness (however, it is not expected that wires of this material providing the same working tension of gauge 5 and 6 Yellow brass would be available) as there was no intermediate gauge between n ° 5 and n ° 6.

The octave of the fourth course could be unisons adding another violin G wound string instead of a wire Brass, but here too, there would have been a loss of brightness, a factor which is emphasized by Fouchetti, who points out, as already said, that the Mandolin must imitate the harpsichord and harp.

 

Corrette

By examining the method of Corrette, the first evident thing is that he does not seem to make any novelty as described by Fouchetti for the 4 course mandolin tuned in fifths.

In fact, there are substantial differences and, in our opinion, several errors:

1 course called F: it must be a five courses guitar first string

2 course called G: must be a harpsichord gauge 5

3 course called H and R: R must be a demi file: H says nothing.

4 octave, course called K and I: I is a full yarn, nothing regards the K

 

1) The first gut string is not a treble of Pardessus as for Fouchetti but a 1st string for the 5 course  Guitar: what diameter could it be?

So, let’s focus on solving this problem: we need to know if there are direct references to the number of strips and Guitar strings or at least an indirect reference to another musical instrument.

It is noted that at the moment we have unfortunately no direct reference; instead, there are several references to a well-studied instrument: the Violin.

  1. a) In the Stradivarius Museum there is a drawing on cardboard (drawing no. 375) which shows the description of the necessary strings for the five orders of the Chitarra Attiorbata, which is basically a normal five-string course guitar with five single drone (bordone) added in its length (extended neck)

 

 

  • First and second string (first course): “Questi deve essere compani due cantini di chitara”. (“This must be a couple of two guitar 1st string)
  • Third and fourth strings (second course): “Queste deve essere compane due sotanelle di chitara”. (These must be two guitar 2nd)
  • Fifth and sixth string (third course): “Queste deve essere compane doi cantini da violino grossi”. (These must be a couple of thick violin 1st)

Etc etc. (28)

To solve this problem then we need to know which was the Violin average diameter of those times and what could be called a ‘thick’ treble.

Count Riccati (who was, in addition to a great physicist, a discreet violinist amateur friend of Tartini) around 1740/50 made some interesting measurements on the strings of his Violin: from his calculations we get the size of the treble installed on his Violin: about .70 mm (29)

This estimation is indirectly confirmed by the data provided by the French traveler and astronomer De Lalande -1760 ca. – (30), about the pieces of gut used to make mandolin, violin and double bass string from the famous Abruzzi string – operating in Naples – Domenico Antonio Angelucci and that these proportions have remained strictly constant until the end of the following century, in Italy and in France. (31)

As for the “thick” trebles, let’s consider as reference the thicker Mi and La gauge made from the same number of guts as George Hart suggests. (32) Considering the standardization in the manufacturing process of Violin strings it is then possible to assume that a “thick” three-row guts could be around .73-.74 mm.

Since the third course of this guitar used a violin gut (at the time realized with three guts called “wires”) using simple proportions – maintaining a constant tension – the second course had to consist of two-wire (such as treble of the Mandolin and Pardessus, according to De Lalande) and the first of a gut only, just like the treble of the Lute (33). In theoretical calculations, the ratio between the diameters is equal to the square root of the ratio of the number of wires used; but then we have to deal with the tactile feel of tension that must be homogeneous: two gut wires therefore produce a diameter between .57 and .59 mm

Since with three gut string were obtained an average diameter around 0.70 mm (here we refer expressly to a ‘big’ treble, for example .73 mm, which is consider ‘thick’ by George Hart), considering a set up with the same feel of the Guitar (which, however, leads to a tension of Kg of a scaled profile, conditioning the choice of its gauges), this is what we obtained:

1st course: ~ .44-46 mm (made by a single whole gut).

2nd course: ~ .57-59 mm (made by two piece of gut).

3rd course: ~ .73 mm (tick violin treble: made of three guts). Etc etc.

  1. Corrette :

La guitarre se mont en cinq rangs de cordes, le 1er n’en a qu’un qui se nomme chantarelle, et les quatre autres rangs en ont chacum deux… Il faut observer que les deux cordes du 3me rang et la petite corde a l’octave du 5me rang soient égales en grosseur pas si forte que la chantarelle de violon….  ”.(34)

Corrette himself confirms what was written in the Stradivarian repertoire.

Now that we have a more precise idea of a 5 course guitar gauges, we can go back to the 4 course Mandolin described by Corrette and try to provide the diameters:

 

  1. a) First string: Corrette talks about the guitar 1st The reference starting point to find out the Guitar gut gauges is the third course, which has a gauge equal to a (thick) Violin treble: in order to preserve a even feel of tension between the strings, the 1st then, according to what the author wrote, has to be of about .44 to .46 mm gauge.

 

  1. b) Second string: the harpsichord’s gauge 5 is used. Corrette though does not specify the type of metal; however, the analogy with Fouchetti is consistent this is why we think that he’s talking about yellow brass.

 

  1. c) Third string: Corrette oddly seems to consider each as single string despite on the pentagram you can see that they are unison. Of one, called H, nothing is said of the other, called R, it is written that it is a demifilé without adding any extra consideration: unfortunately, from this statement it is not possible to get anything concrete; we do not know if the strings were both demifilé and there are not any indications on how to make it.

 

  1. d) Fourth string: Corrette says nothing about the octave named K. The bass string called C description is limited to the fact that is a wound string. However, we do not know which core to use (silk or gut). Any way thanks to Fouchetti we know that both the materials were suitable therefore we might guess that, again, it’s a Violin G.

 

Considerations

 

The indications given by Corrette about the 4 course Mandolin are, according to the writer, totally unreliable.

– First course (.44-.46 mm approximately): it would have a working tension of only 3.0-3.2 Kg per string.

-Second course (presumably yellow brass gauge 5, but nothing is specified): it rises to at least 5.3 Kg per string. The gap with the first course is remarkable.

To have a working tension comparable to the second course, the first should use the second-string guitar strings (2 gut = first Mandolin = first Pardessus according to De Lalande) aligning with the Fouchetti.

-Third and fourth chorus: nothing useful can be obtained. If it were not for Fouchetti (which gives a useful comparison) the data provided by Correct would be completely meaningless

 

Six courses Mandolin

 

With the problems already encountered on the 4-string mandolin tuned in fifths, we still inevitably expect issues. In fact, different indications are unfortunately incorrect: some reasoning is needed. Only at the end of this review work the mandolin’s six courses become really doable.

  1. a) First and Second course: Corrette write that the courses called L and M must be Guitar trebles: what does it mean? That he used the guitar treble also for the second course of the Mandolin? Corrette here is very inaccurate. Certainly, they cannot be trebles installed on the second course: there would be a total misalignment in the working tension. We therefore feel that Corrette refers to the first and second course of the Guitar.

 

  1. b) The third course called N: Corrette says to use Harpsichord Gauge 5 but omits to specify the type of metal: however, we think it is the usual Yellow Brass for Harpsichord.

 

  1. c) Fourth course called S: Correct says that this is a demi file string but does not add anything else (silk or gut core? Corrette says nothing in matter)

 

  1. d) Fifth course called P: this is a full wound string but we have no other information: the string in the octave is not mentioned at all.

 

  1. e) Sixth course called Q: This is a full wound string but we have no other information: the string in the octave is not mentioned at all

 

Considerations

Based on the data provided by Corrette, no one today (but also in his time!) is able to draw the entire strings set up; however, it is possible to introduce some reasoning that eventually might solve the enigma:

Let’s start from the only certain data available: the third A note’s course, which correspond to the harpsichord gauge 5. We believe it is of yellow brass (.34 mm).

Using a typical six-courses mandolin average vibrating length, .315 cm, and a presumed Parisian/Roman pitch of 392 Hz, we obtain a working tension of 4.8 Kg.

The first and second courses of the instrument must therefore somehow relate to this value: by installing in these two courses the first and second string of the Guitar (of which we have a more accurate idea thanks to Stradivari’s Violin) the following working tensions are obtained: 3.9-4.3 Kg for the first course and 3.8-3.9 Kg for the second. Compared to the tension value of the third course, there is certainly a non-balanced tension trend yet still functional.

Things are much simpler with the sixth order: as it is a G we can think that can be a fourth violin string as Fouchetti model, whose octave is equal to the gut second (third of the guitar): considering this hypothesis as valid the tension of bass and its paired octave is about 3.9 kg. The paired octave may be the same yellow brass gauges 5 already used for the 3rd course: a gut string would be about .90 mm.

Obtaining the working tension of the first, second, third and sixth courses it is logical to think that the working tension of the fourth and fifth must necessarily be between 4.8 Kg (third course) and 3.9 Kg (fourth course): How can this condition be achieved while fitting in the technological and acoustic correct range?

Fourth course: as we have seen, based on Corrette thinking you should use a demi file string. It is necessary here to consider a working tension range slightly lower than that of the third course but in any case, higher than the theoretically associated range in the fourth course.  The range has to be like this in order to preserve the linearity of the values ​​calculated so far. If we assume that the range is 4.4-4.7 Kg, the following diameters are obtained: 1.10-1.14 mm. These diameters correspond exactly to a third Violin string that was then made in France usually as demi. (35) (36)

 

 

 

Its octave should have a diameter between .55 and .57 mm: the first string for 4 courses Mandolin/2nd Guitar course.

Fifth course: Corrette states that this is a unison and a full wound string. From simple calculations, considering a tension range slightly above that of the fourth course in order to preserve the linearity of the values ​​calculated so far (assuming that the range is 4.1-4.3 Kg) as note B we obtain an equivalent gut of 1.42-1.47 mm in diameter.

The data should be reliable: its octave, at the same tension, varies between .71 and .73 mm in diameter; the third guitar string (i.e, a 1st Violin string).

The problem is its realization, especially if you use a gut- core. During those times, according to our research, metal wires with a diameter of less than about .13- .15 mm. (37) (38) (39) (40) were not produced because they had not available the suitable technology to drawing a thinner metal wires.

In other words, the half-wound string described by the writer was not at all a transition string between a gut string and a wound string but a technological way out and used to avoid the thinner metallic wires shortage problem: in fact, you can find proof in the Metallic Index characteristic of these particular strings, which is similar to that of full-wound strings and not less.

If the core is instead of Silk, which, according to Fouchetti, was used in the Mandolin and then also in the 4 &5 courses of the 5 course guitars and in the XIX-early XX c. guitars.

With the use of silk cores, compared to the gut cores, the relationship between the core and the metal can be unbalanced in favor of the metal, making it possible to have a full-wound string and a brilliant acoustic output (higher Metallicity Index).

It is interesting to note that the equivalent gut and the way of making the close wound strings on silk for the fifth and sixth courses of the six-course Mandolin will then be used respectively for the fourth and fifth string of the six-single string guitar, the one that in 10-15 years will appear on the music scene.

 

 

Concluding, even for this kind of Mandolin Corrette does not allow us to come to certain and plausible conclusion. However, we made a number of arguments that lead to the following set up proposal, based on the few information from Corrette (the only strong point is the indication that the third course uses the gauge 5, from which we can deduct the value of the tension: at this point, the highest course must have a growing tension while the bass ones a degrading tension according to a similar Fouchetti Mandolin profile) and with the support of Fouchetti:

1G: first the Five-courses Guitar = .44-.46 mm in diameter; Average tension: 4.1 Kg per string

2D: Second of the Five courses Guitar = .57-.59 mm in diameter; Average tension: 3.9 Kg per string

3A: gauge 5 Yellow brass for harpsichord = .34 mm diameter; Average tension: 4.8 Kg per string

4E: demi file string (third violin according to French use) = 1.10-1.14 mm equivalent gut; Average tension: 4.0 to 4.5 Kg

5B octave: third of the five-courses guitar = .70-.73 mm in diameter; Average tension: 4.2 Kg

5B: bass: full yarn string on silk core with equivalent gut = 1.42-1.47 mm diameter; Average tension: 4.2 Kg

6G octave: same gauge 5 in yellow brass for harpsichord = .34 mm diameter; Average tension: 3.9 Kg (or a gut string of .88-.91 mm: practically the fourth course of the guitar).

The uncertainty of using octave strings in gut or yellow brass gauge 5 is a matter of relative importance: Fouchetti points out that the use of metal wires or gut strings was a matter of personal taste:    q2

 

 

Practical evidence

Four courses Mandolin: the Fouchetti set

– First course: .56 mm gut gauge string: no mechanical or acoustic problems were found.

-Second course: yellow commercial brass wire for harpsichord diameter .35 mm. The main emerging problem is how to tie it up to the fretboard. Being a very hard harpsichord Brass wire the problem is its fragility when bented. In our case, we solved the problem by making a very long loop so that, when put in tension, it will lock itself by eliminating any string breaking problem at the pegs due to the presence of bend points or over-sharp bumps.

 

 

Third course: we used .30 mm  yellow commercial hard harpsichord brass wire. It is not possible to twist together directly the two wires, being the brass very hard, it tends to break during twisting, until it comes to different twisted degrees along the string. The solution to this problem was to soften just a bit (not totally)  the two wires by heating them to 350 degrees (in this regard we did a number of tests whose final result indicated that the wire has to be heated between 330 and 370 degrees Celsius) for one minute. The wire thus obtains an intermediate degree of hardness, allowing to be bent and still retaining a residual degree of hardness that counteract the yield of the wire under tension.

The degree of twisting of the string is a crucial aspect: if it is very high (high twist) the sound is very bright but it also reduces the tensile strength. With less twist (low twist) the sound is less metallic; you have less sustain but you have a higher tensile stress. In other words, depending on the degree of twisting, you can modulate the desired tone output until you find an acoustic balance between the second and the fourth course.

-Fourth course: following the historical instructions we obtained a Violins wound G whose equivalent gut is of  1.80 mm (slightly lighter second Violin string covered with Silver wire): for the octave it is used a second yellow brass wire same of those of the 2nd course.

Conclusions: The overall acoustic balance of the set was discreetly homogeneous and thus also the feel of tension among the strings (standard pitch of 392 Hz).

 

Six courses Mandolin according to our Corrette interpretation (cherry bark pick)

– First course: .46 mm gut: No acoustic or mechanical problems found

-Second course: .56 mm gut: no acoustic or mechanical problems found

-Third course: .35 mm Yellow Brass wire: the tension feels a little higher than the upper strings; it sounds more brilliant than the second and third course. Working tension:  for a better balance the diameter should be reduced to .33-.34 mm. There is no solution for the brilliant acoustic output. Alternative:  .88 mm gut string: no mechanical problem; acoustically aligned with the first two top courses and with the fourth course

-Fourth  course: two violin 3rd  demifile strings are used: equivalent gut of about 1.15 mm. There were no mechanical problems. The sound was a bit dimmer compared to that of the third course, whenever it is  done using yellow brass instead gut.

– Fifth course: the bass consists of a average XIX century guitar 4th D string wound with silvered copper wire on silk core whose equivalent gut is about 1.40 mm. The octave string is a gut third string of five courses Guitar of .73 mm (see the Stradivari’s information: thick Violin 1st)

– Sixth course: the bass consists of a average XIX century guitar 5th A string wound with a silvered copper wire on silk whose equivalent gut is about 1.80 mm. The octave string is a .88 mm gut gauge equivalent to a 4th string for 5 courses guitar.

What about this setup:

There is no mechanical problem; acoustic and dynamic balance are good also in relation to the fifth course. We tested a yellow Brass wire as octaves for the 6th course resulting in a tonal disequilibrium with the other higher courses.

According to the writer, an experimental set of this type is totally satisfactory.

Critical points revolve around the use of Brass wire in the third course, due to the tonal difference with the others gut’s courses. Likewise, the use of a yellow brass wire as an octave of the sixth courses is unlikely to be successful due to the tonal disequilibrium that occurs. The best balancing set therefore is the one that uses gut strings for the first thre courses and for all the octaves; close wound silk core for the fifth and sixth course and a demifilè wound gut string for the fourth course: however, for this course remains open the experimentation of a silk-type string on silk core, which however so far was not found in the records of the 18th century.

 Conclusions

 Although some 18th century Mandolin methods have survived, when it comes to understanding what kind of strings to use, we have only two available sources: Fouchetti and Corrette.

The data provided by the first one on 4-double strings Mandolins are technically and acoustically consistent: they shape a set whose tension value is within a range of acceptability and homogeneity between the various strings. The strings of the four courses are close to, from the technological and acoustic vision point, almost perfection considering what was available at that time.

Unfortunately, Fouchetti says nothing about the 6-courses Mandolin

 The description provided by Corrette, however, is incomplete and sometimes confusing: it is not possible to directly extract anything usable unless you go through a critically re-elaboration of the provided data like we have done here.

So, if you see how much he wrote in comparison with Fouchetti (in some ways there are interesting overlapping), it is necessary to always taking in account what could or could not be done at the time (in short the technical limits and materials to making strings available) , then it is possible to formulate a concrete proposal even for the Six courses Mandolin.

For the 4 courses Mandolin, therefore, only the Fouchetti data is validated only partially by the Corrette one (the gauge 5 for the second course, for example).

 For the six-courses Mandolin, as we have seen, we can only refer to Corrette: we believe that our elaboration is interesting not only from the acoustic point of view but also. As already said before,  from the point of view of the times available materials.

However, we have a last consideration: Corrette does not clarify whether the six course Mandolin should be played with the plectrum or by fingers like the Lute. We point out that from the values ​​of tension we calculated you could have considerable difficulty playing a six-course Mandolin directly with the fingers. As example, the tension range currently accepted in the Lute today (which is a much larger instrument) is between 2.7 to 3.3 Kg.

The rules of the time are clear and repeated several times in historical documents: longer the scale, thinner the strings (i.e. small strings on smaller Lutes): a Mandolin played with fingers and not with a plectrum with a vibrating length of only 31.5 cm giving a tactile feel of tension similar to the lute should therefore have in proportion a fairly inferior tension, say around 2.0 Kg. This would involve, however, in  a gut string for the 1st course of .31-.33 mm gauge only: this is not possible. In fact, the thinnest, unpolished gauge that comes out from a single lamb gut of a few months of life  – as indicated by ancient sources – is about .40-.46 mm in diameter and produces a higher working tension than those of 2.0 Kg before indicated.

One possible solution (the only that can work out, in my opinion)  is that the six course Mandolin with glued bridge may have been played exclusively by mean of nails. Such a solution would have enabled it to work easily without the plectrum (the nail itself can acts close to a plectrum), with clear and crystal sound and even under considerable working tensions (like in use on the modern classical/flamenco guitars that cannot be used without nails), otherwise objectively difficult to deal only with the fingertip.

 

On the other hand, it is historically known that among the eighteenth-century mandolinists there are also many Theorbo and Archlutes players, who used the nails of their right hand, such as Filippo da Casa. Hard to cut them off for the chance to play the Mandolin while they are, at the same time, playing also the Theorbo.

 

 

I pass the question to all the similar Mandolin players: nevertheless this are the calculations and the result that emerged.

Vivi felice

 

                                                                       NOTES

  • Djilda Abbott – Ephraim Segerman: “Strings in the 16th and 17th centuries”, The Galpin Society journal, XXVII 1974, pp. 48-73.

2) The values ​​obtained in this example are the ones specifically made using the manufacturer technology for trebles, which is used to obtain the maximum tensile strength (and all ‘surface abrasion), as we will see better later on. In other words, in their manufacturing process elasticity is not consider (factor that can be overlooked for these thin strings), factor that is on the contrary consider for all the other kind of that are not used on the first spot: for these strings we only want to achieve the maximum elasticity possible. Elasticity and tensile strength are inversely proportional.

  • To make things easier we decide to use a standard frequency value. The French reference pitch according to the existing studies was included between 385 and 400 Hz. See: Alexander J. Ellis in Studies in the History of Music Pitch: Monographs by Alexander J. Ellis and Arthur Mendel (Amsterdam: Frits Knuf, 1968; New York: Da Capo Press)

Arthur Mendel: “Pitch in western music since 1500: a re-examination”. In -Acta musicologica- L 1978, pp.1-93.

Ephraim Segerman: “On German Italian and French pitch standards in the 17th and 18th centuries”. FOMRHI quarterly no. 30, January 1983, comm.442.

  • Ephraim Segerman: comm 1545 in FOMRHI Quarterly 89, October 1997.
  • Ephraim Segerman: comm 1593  in FOMRHI Quarterly 92, July 1998.
  • Daniello Bartoli: ‘Del suono, de’ tremori armonici e dell’udito’, a spese di Nicolò Angelo Tinassi, Roma 1679, p.
  • Metal strings worked near to their breaking point; see:

-William R. Thomas and J.J.K. Rhodes: “The string scales of Italian keyboard instruments”. The Galpin Society Journal XX -1967, p.48.

-Michael Spencer: “Harpsicord phisics”. The Galpin Society Journal, XXXIV, March 1981, pp. 3-7.

-Ephraim Segerman: “Bulletin Supplement “. FOMRHI quarterly no.39, April 1985, p.11; 1768-Adlung’s statement: “When a harpsicord is strung so that the pitch can be safely raised a semitone, one can be secure…”.

  • Marin Mersenne: “Harmonie Universelle” 1636, Livre Troisiesme, Proposition XII e Proposition XIII, see note7 p.58.
  • Ephraim Segerman: “New Grove DOMI: ES Mo 4: Ca to Ci entries”. FOMRHI quarterly no.43, April 1986, comm.698.

Also:

-Harvey Hope: “Ref J. M. S. remarks on the New Grove Chitarra battente”. FOMRHI quarterly no.43, April 1986, comm. 709.

-Peter S. Forrester: “Citterns and chitarras battente: re. Comm.698, Grove Review”. FOMRHI quarterly no.44, July 1986, comm. 740.

-Ephraim Segerman: “Response to Comms 739 and 740”. FOMRHI quarterly no.44, July 1986, comm.742.

-Peter S. Forrester: “17th c. Guitar woodwork”. FOMRHI quarterly no. 48, July 1986, comm.825.

-James Tyler: “The Early Guitar- A History and Handbook”; Early Music series: 4, Oxford University Press, London 1980. Quoted By by Ciro Caliendo: “La Chitarra battente. Uomini, storia e costruzione di uno strumento barocco e popolare”, Edizioni Aspasia, Aprile 1998, pp.24-25.

  • Cary Karp: “Strings, twisted and Mersenne”. FOMRHI quarterly no.12, July 1978, 137.

Ephraim Segerman & Djilda Abott: “On twisted metal strings and Mersenne’s string data”. FOMRHI quarterly no.13, October 1978, comm. 164.

More:

-Cary Karp: “On Mersenne’s twisted data and metal strings”. FOMRHI quarterly no.14, January 1979, comm. 183.

-Ephraim Segerman: “Mersenne untwisted-a counter-Carp to comm.183”. FOMRHI quarterly no.15, April 1979, comm.199.

 

  • Cary Karp: ”The pitches of 18th Century strung Keyboard Instruments, with Particular Reference to swedish Material, SMS-Musikmuseet Technical Report no.1”, SMS-Musikmuseet, Box 16326, 103 26 Stockholm, Sweden, 1984, 129 pp. See also: “On wire-comms and wire-comm comments”. FOMRHI quarterly no. 11, April 1978, comm. 134. Karp wrote that “ In as much as the lower portion of this range was generated by piano wire…”.
  • Remy Gug: “Abut old music wire”. FOMRHI quarterly no. 10, January 1978, comm.105. Gug wrote that “ Let us first specify that the concerned strings have been taken from instruments used in the XVIIth and XVIIIth centuries: harpsicords, spinets, clavichords, dulcimers”.
  • Marco Tiella: “Problemi connessi con il restauro degli strumenti musicali”, pp.22-23.
  • See note no. 10
  • Ephraim Segerman: “Neapolitans mandolins, wire strengths and violin stringing in late 18th France” . FOMRHI quarterly no.43, April 1986, comm.713. This is the first Segerman known regarding Mandolin 18 Century sets.
  • Gianni Podda: “Prove di trazione e determinazione della tensione di rottura per corde antiche e moderne”, pp.36-38. Atti del seminario per la didattica del restauro liutaio, estate musicale 1981; Premeno.
  • Remy Gug: “Jobst Meuler or the secret of a Nuremberg wire drawer”  FOMRHI quarterly no.51, April 1988, comm 866, p. 29.
  • Giovanni Fouchetti: “Méthode pour apprendre facilement á jouer de la mandoline á 4 et á 6 cordes”. Paris 1771. Reprint: Minkoff, Genève, 1983, p. 5.
  • Michel Corrette: “Nouvelle Méthode pour apprendre à jouer en très peu de temps la Mandoline par Mr. Corrette” Paris 1772.
  • cit 17.
  • Francois De Lalande : “Voyage en Italie […] fait dans les annés 1765 & 1766, 2a edizione, vol IX, Desaint, Paris 1786, pp. 514-9, Chapire XXII “Du travail des Cordes à boyaux…: “ …on ne met que deux boyaux ensemble pour les petites cordes de mandolines, trois pour la premiere corde de violon… ”.
  • Mimmo Peruffo: “ Italian violin strings in the eighteenth and nineteenth centuries: typologies, manufacturing techniques and principles of stringing, “ Recercare”, IX, 1997 pp. 155-203.

Vedere anche: Antoine Germain Labarraque: L’art du boyaudier, Imprimerie de Madame Huzard, Paris 1812, pp. 31-2.

  • Patrizio Barbieri: “ Giordano Riccati on the diameters of strings and pipers, “ The Galpin Society Journal”, XXXVIII, 1985, pp. 20-34: “Colle bilancette dell’oro pesai tre porzioni egualmente lunghe piedi 1 _ Veneziani delle tre corde del Violino, che si chiamano il tenore, il canto e il cantino. Tralasciai d’indagare il peso della corda più grave; perchè questa non è come l’altre di sola minugia, ma suole circondarsi con un sottil filo di rame.“.
  • Ephraim Segerman: “Strings  thorough the ages“, The Strad, part 1, January 1988, pp.20-34”,  52-5, part 2 (“Highly strung“), March 1988, pp.195-201, part 3 ; “Deep tensions“, April 1988, pp.295-9.
  • Cryseul, Géoffrion: “Moyen De Diviser Les Touches Des Instruments à Cordes, Le Plus Correctement Possible…On y Voit La Manière Dont Les Artistes Doivent Considérer Les Loix Qu’Impose Le Tempérment…Et L’on Imagine Un Moyen D’Accorder Les Clavessins.”Paris: Rodez, 1780.
  • http://harps.braybaroque.ie/Taskin_stringing3.htm

In this website you can find interesting comparisons between Cryseul gauges scale and tits relative mm gauges based on numeours scholars opinions.

  • Don Juan Guerrero: “ Methode pour Aprendre a Jouer de la Guitarre”. Paris 1760.
  • Patrizia Frisoli, The Museo Stradivariano in Cremona, “The Galpin Society Journal”, XXIV, July 1971 p. 40.
  • Patrizio Barbieri: op cit 21.
  • De Lalande: op. cit 19.
  • PHILIPPE SAVARESSE: “Cordes pour tous les instruments de musique”, in CHARLES-P.-L. LABOULAYE: Dictionnaire des arts et manufactures, 3rd edition, vol. I, Lacroix, Paris 1865.
  • george hart, The violin and its music, Dulau and Schott, London 1881, pp. 46-7.

Michel Corrette: “Les Dons d’Apollon”. Paris 1763, p. 22, Capitolo XVI

  • Attanasius Kircher: “ Musurgia Universalis sive Ars Magna Consoni et Dissoni in X. Libros Digesta, Roma, 1650, Caput II, p. 476: “…ita hic Romae gravissimam tesdudinis chordam ex 9 intestinis consiciunt, secundam ex 8, & sic usque ad ultimam, & minimam, quae ex uon intestino constat.”.
  • Michel Corrette: “ Les Dons d’Apollon” Paris 1763, p. 22, Capitolo XVI.
  • SEBASTIEN DE BROSSARD: [Fragments d’une méthode de violon], manuscript, ca. 1712, Paris, Bibliothèque Nationale, Rés. Vm8 c.i, fol. 12r (cited in BARBIERI: “Giordano Riccati”, p. 34.
  • JEAN-BENJAMIN DE LABORDE: Essai sur la musique ancienne et moderne, Eugène Onfroy, Paris 1780, livre second, “Des instruments”, pp. 358-9: “Violon […] Ordinairement la troisième et la quatrième sont filées; quelque fois la troisième ne l’est pas” (Violon […]
  • The thinnest Creyseul gauges scale is no. 12, equal to almost 0,15 mm.
  • James Grassimeau : “A musical Dictionary” London 1740.

On this dictionary is clearly written that with the current metallurgic technology only gold, silver, brass and iron wires included between 1/100 inch: 0,50-0,25 mm gauge can be done.  This book is a translation of the Sébastien de Brossard 1703 dictionary.

  • Marco Tiella; information directly gave by the writer, the thinnest diameter found by him in some spinets were around gauge 0.15 mm
  • The clothes of those times could represent an inexplored field of study about metal wire technology: round section metal wires were used for a good part to make complex medival and renaissance clothes decoration. From first examinations of round and flat wire sections turned out that the thinnest gold gauges (the more malleable metal) of those times clothes were around 1/100 till 1/120 inch maximum. This means .12 mm after the starching; an intact wire can reach easily .14-.15 mm gauge.

 

 


VIDEO: Le montature di corda per chitarra del nuovo modello torres tra il 1880 e il 1945

VIDEO: Le montature di corda per chitarra del nuovo modello torres tra il 1880 e il 1945


CD basses installed on theorbo/Chitarrone: why do they sometime break on the 5th?

Why do CD basses installed on theorbo/Chitarrone sometime break (mostly on the 5th course), even if there are no sharp edges and they were installed properly?

 

Tiorba (1882)
Source: Biblioteca de la Facultad de Derecho y Ciencias del Trabajo Universidad de Sevilla / CC BY (https://creativecommons.org/licenses/by/2.0)

The CD strings (from 110 CD up to 220 CD) are designed for renaissance & d minor Lute basses only.

For the long diapasons of medium extensions, the CDL type strings are available.

The thinner CD strings instead (82CD up to 105CD), are designed for 5th courses and octaves on very lower bass strings of renaissance & d-minor lutes.

On theorbo/chitarrone these rules sometime do not work: why?

While on the traditional lute/archlute the 1st string works with the higher FL product (220-230 Hz.m), any kind of theorbo/chitarrone works with an exceptional, longer vibrating string length, far longer than what was calculated for lute, hence the fact that the 1st and 2nd strings are forced to be tuned an octave lower.
For example, while a renaissance lute tuned in A at modern pitch has a scale around 57 cm, a theorbo in A at the same standard pitch has a scale of 85-88 cm (the FL product of the 3rd string b note -that actually is the canterelle- is 220 Hz.m).

In these conditions, the range of the FL products typical of a 5th course of a renaissance Lute (79-81 Hz.m) or the 6th course of a d minor lute (71-74 Hz.m) allows the use of CD strings of the thinner range with no problems, while on a theorbo instead, the FL product raises up to 109-115 Hz-m, and it is too high for strings designed as basses.

In practice, the FL product of a 5th course on a theorbo is the same of a 4th course on a renaissance lute.

This is the explanation of why a CD string installed on the 5th course of a theorbo can sometimes break: the FL product is too high (it’s not a matter of tension; tension has no influence in the FL product).

For more information regarding the FL product, please refer to the following article on our blog.

In conclusion, the proper place for the CD strings is - in the higher position - the 5th course of the renaissance lute, or the 6th , 7th  and 8th fretted theorbo basses. For the 5th course of a theorbo/chitarrone, the right strings to use are NNG or CDL type.

 

Vivi felice