# Early Music Blog

### VIDEO - Le montature storiche per mandolini a 4, 5, 6 ordini e 4,6 corde semplici

## Le montature storiche per mandolini a 4, 5, 6 ordini e 4,6 corde semplici

### VIDEO - L'arpa e il segreto della sua montatura: trucchi, strategie per ottimizzarla al meglio

## L'arpa e il segreto della sua montatura: trucchi, strategie per ottimizzarla al meglio Parte Prima

## L'arpa e il segreto della sua montatura: trucchi, strategie per ottimizzarla al meglio Parte Seconda

### VIDEO - Viola Da Gamba e Corde: documenti e iconografia

## Viola Da Gamba e Corde: documenti e iconografia

### VIDEO - Il segreto dell'esistenza del Liuto: il mondo dietro una lunghezza vibrante

## Il segreto dell'esistenza del Liuto: il mondo dietro una lunghezza vibrante

### VIDEO - Tra il budello intero di agnello e il nostro sintetico f- Reds: il confronto e le ragioni

## Tra il budello intero di agnello e il nostro sintetico f- Reds: il confronto e le ragioni

### 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.

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.

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**

**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**

**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**

**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:

- send the stringmaker a specimen of the original historical string, or
- 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:

## 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:

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 4^{th} 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 1^{st} 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 1^{st} 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 1^{st} 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 2^{nd} 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 2^{nd} 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 1 ^{st} 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 1^{st} 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 = 1^{st} 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:

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).

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 1^{st}) ) 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 1^{st} 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 1^{st} 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 17 ^{th} 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)

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 1^{st} 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)

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

- use a Pardessus gut treble
- a harpsichord gauge 5 yellow brass
- two harpsicord gauges 6 yellow brass twisted together
- 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.

### 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 5^{th} course), even if there are no sharp edges and they were installed properly?

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 5^{th} 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 1^{st} 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 1^{st} and 2^{nd} 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 3^{rd} string b note -that actually is the canterelle- is 220 Hz.m).

In these conditions, the range of the FL products typical of a 5^{th} course of a renaissance Lute (79-81 Hz.m) or the 6^{th} 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 5^{th} course on a theorbo is the same of a 4^{th} course on a renaissance lute.

This is the explanation of why a CD string installed on the 5^{th} 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 5^{th} course of the renaissance lute, or the 6^{th} , 7^{th} and 8^{th} fretted theorbo basses. For the 5^{th} course of a theorbo/chitarrone, the right strings to use are NNG or CDL type.

*Vivi felice*