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

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