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fernandraynaud
Oud Junkie
Posts: 1865
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Location: San Francisco, California
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Yes, of course I understand. But you alone are making that most useful test fixture, and since this is all on the topic of estimating wound string
tensions, is it necessary to open a new thread? If you would be so kind as to also test my approach on your test rig with an open mind, I would really
appreciate it, as I have no such setup.
Since my proposed method begins at one known point supplied by the string-maker and extrapolates, over a narrow range, using the effective density
that fits the known point, I find it quite interesting that the actual measured density of a wound string composite seems to be close to that
reverse-derived in that way.
If we accept that the equation is correct at the point supplied by the manufacturer, the question is just how far we can extrapolate, and how well the
"effective density" helps in predicting other points on the scale/pitch lines.
I eagerly await what you will find with respect to both methods.
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jdowning
Oud Junkie
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As I do not understand why you would want to try to guess the tension of a string by a known manufacturer who can supply that information on request I
feel that I really cannot usefully contribute to the development of your idea about 'effective density'.
On the other hand - if you feel that a string test rig will enable you to progress with your investigation - a rig like the one that I have created is
simple enough and cheap to make yourself. If you (or other interested forum members) cannot figure out how to do it from my posted images and
description of my rig I would be more than happy to provide any further detail on request.
Under the circumstances, however, I do think it would be a good idea if you were to develop your proposal under a new topic on the forum.
As you will have to analyse and compare a multitude of wound strings by various makers (Pyramid alone make over 90 varieties of wound lute strings)
rather than depend upon just one or two tests to convince yourself and others, one way or another, it could take some time and effort (not to mention
the expense of purchasing the strings for testing).
Good luck with your project.
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fernandraynaud
Oud Junkie
Posts: 1865
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Location: San Francisco, California
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Gee, thanks. You don't have to share your toys. A few measurements wouldn't take much time, but obviously too much.
But as to the why, isn't it obvious that many strings come with maybe one tension point defined? If we had a method to extend the data to other
scale-lengths and tuning, the value seems obvious. And if we could perhaps just weigh a piece of string with no tension data supplied, and derive some
tension data from that, isn't the usefulness of that obvious? But OK, ciao.
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jdowning
Oud Junkie
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Location: Ontario, Canada
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Bye bye
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jdowning
Oud Junkie
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Moving on.
The published data for the Pyramid Orange #650 oud string set - subject of this comparison - can be found in the 2012 Pyramid catalogue and from an
'on line' American source - see attached images.
The 'on line' source gives string gauges in Imperial measurement for Arabic tuning C F A d g c' - string length 60 to 62 cm - 'average' string tension
3.7 Kg. for a total of 40.7 Kg at A440.
The Pyramid catalogue data for Arabic tuning C G A d g c' - string length also 60 - 62 cm - is a bit more useful giving string outside diameters as
well as each string tension (which must be an average value given the range of string length) for a total of 45.7 Kg (average 4.15 Kg. at A440)
Comparing the published string diameters:
Converting the Imperial dimensions to the more practical Metric for the online source gives :
1st 0.71 mm (plain nylon)
2nd 0.79 mm (plain nylon)
3rd 0.61 mm (wound copper)
4th 0.74 mm (wound copper)
5th 0.81 mm (wound copper)
6th 1.09 mm (wound copper)
For the Pyramid catalogue data:
1st 0.70 mm (plain nylon)
2nd 0.80 mm (plain nylon)
3rd 0.60 mm (wound copper)
4th 0.73 mm (wound copper)
5th 0.82 mm (wound copper)
6th 1.10 mm (wound copper)
My measurements of a Pyramid #650 Orange string set using a 'Starrett' precision Metric micrometer are as follows:
1st 0.70 mm - 0.72 mm
2nd 0.81 mm - 0.83 mm
3rd 0.63 mm
4th 0.74 mm
5th 0.83 mm
6th 1.10 mm
From these measurements it can be seen that the plain nylon strings are not perfectly cylindrical but are also larger in diameter than the catalogue
values - which will in turn result in higher tensions than published.
The other variations in dimensions may be due to manufacturing tolerance, translation losses between Imperial and Metric measure or differences in
accuracy between my micrometer and Pyramid's.
Calculating the 'equivalent diameters' of the wound strings according to the previously posted formula, for a plain nylon string in each case:
3rd - d pitch - 1.22 mm
4th - A pitch - 1.56 mm
5th - F pitch - 1.85 mm
G pitch - 1.85 mm
6th - C pitch - 2.67 mm
Using the Arto Wikla strings tension calculator for plain nylon 'equivalent diameter' strings at A440 - string tensions for 60 cm to 63 cm string
length range and assuming a density of 1.04 gm/cc for nylon and 8.8 gm/cc for copper are:
3rd - d pitch -3.85 Kg to 4.11 Kg (cf 4.0 Kg avg. Pyramid data)
4th - A pitch - 3.53 Kg to 3.77 Kg (cf 3.8 Kg avg. Pyramid data)
5th - F pitch - 3.13 Kg to 3.34 Kg ( no Pyramid data)
5th - G pitch - 3.94 Kg to 4.21 Kg (cf 4.0 Kg avg. Pyramid data)
6th - C pitch - 3.66 Kg to 3.91 Kg ( cf 3.7 Kg avg. Pyramid data)
So far so good. Next to directly measure string tensions on the test rig for comparison.
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jdowning
Oud Junkie
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The string test rig has been modified so that it is now set up at 10° from the vertical (rather than 45°) in order to minimise any friction at the
lower pulley that would tend to give higher string tension readings than actual.
The following results for my Pyramid #650 set are as follows. As this is a particularly busy time of year for me there has only been time to measure
tensions for 61 cm string length - representing the average values published by Pyramid. Both plain nylon and wound strings have been tested:
1st Nylon c' pitch - 4.5 Kg (cf 4.4 Kg Arto Wikla calculator)
2nd Nylon g pitch - 3.3 Kg (cf 3.2 Kg ditto )
3rd Wound d pitch - 4.0 Kg (cf 4.0 Kg from equivalent diameter calculation)
4th Wound A pitch - 3.8 Kg (cf 3.7 Kg ditto )
5th Wound F pitch - 3.3 Kg (cf 3.2 Kg ditto )
5th Wound G pitch - 4.2 Kg (cf 4.1 Kg ditto )
6th Wound C pitch - 3.8 Kg (cf 3.8 Kg ditto )
So it can be seen that the test and calculated values correspond closely. If anything the test results may a little higher by about 0.1 Kg.
So how do the published Pyramid values compare? For the set advertised as C F A d g c' the average value of tension given is 3.7 Kg per string.
Taking the adjusted test tensions (measured values minus 0.1 Kg) the total tension for the 11 string set is 40.5 Kg which works out to 3.7 Kg per
string.
This result tends to verify that the first and second nylon strings have an average tension of about 4.4 Kg and 3.2 Kg respectively.
This contrasts with the respective average values of 3.9 Kg and 2.8 Kg published in the 2012 Pyramid catalogue (tuning C G A d g c') for the smaller
string diameters of 0.7 mm and 0.8 mm. However using the Pyramid Lute string calculator gives a value from 4.1 Kg to 4.4 Kg and 3.0 Kg to 3.2 Kg
respectively for these diameters for string lengths ranging from 60 cm to 62 cm.
The values for plain nylon lute strings from the attached tension tables, published by Pyramid in the 1970's, give intermediate values to the
above.
It should be born in mind that the Pyramid #650 is a relatively low cost string set compared to the Pyramid custom design string sets costing about 3
or 4 times more so perhaps these apparent discrepancies in tension are due to wider manufacturing tolerances (particularly for the plain nylon
strings) that may may be applicable only to the lower cost strings?
Time permitting some Pyramid wound lute strings that I have in stock will be analysed according to the basic equivalent diameter formula and tension
then verified on the string test rig.
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Brian Prunka
Oud Junkie
Posts: 2918
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Location: Brooklyn, NY
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John, thanks for undertaking this very interesting research.
I confess my math skills a a little rusty, and I am confused by something that I am hoping you could help clarify.
D'addario defines their density by "Unit Weight" aka "Specific Weight", while others use "Linear Density". As far as I can tell, the distinction is
generally that Unit Weight is by volume, not length. I surmise that in an effectively one-dimensional object like a string, that we regard linear
density and unit weight as the same for our purposes? Are there any ramifications to the fact that UW should be in cubic measurements rather than
linear measurements?
D'addario describes their Unit Weight numbers as being in "pounds per linear inch", which sounds like they are really describing linear density.
Also can you clarify to me the role of the constant in the formula supplied by D'addario (clearly based on the Mersenne-Taylor Law)?
T(Tension) = (UW x (2 x L x F)2) / 386.4
(Tension in lbs., UW in lbs./in., L in in., F in Hz)
Where is the 386.4 coming from?
I don't wish to derail your thread now that it is back on track, but you clearly have a lot of experience in this area.
Is there any drawback to calculating the effective linear density using the M-T Law, given T, F, and L info from a manufacturer?
I understand that you are undertaking to extrapolate that information in the absence of such data, so not exactly relevant, but other than
imprecision or inaccuracies on the part of the mfr., such an approach should work, correct? Or am I missing something?
thanks
Brian
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jdowning
Oud Junkie
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Hi Brian
I am not familiar with D'addario strings or the procedures, assumptions and definitions that they use in the design and measurement their strings.
First observation from what you say is that a practical instrument string (of say 60 cm or so in length) is not effectively a one- dimensional object
at all. It has a measurable diameter and volume and so is a three-dimensional object i.e. a smooth uniform homogeneous cylinder for plain strings.
I am not sure but imagine that the concept of 'linear density' originated in the textile industry where accurate measurement of the diameter of a soft
irregular twisted yarn or thread is impossible. Their solution is to 'weigh' a very long length of yarn (1000 to 9000 metres is the standard). For
these lengths the yarn may be considered to be approximately one-dimensional and so the mass per unit length is deemed to be equivalent to 'linear
density'.
It is perhaps possible that a string manufacturer like D'addario uses this definition for convenient measurement of their plain strings in bulk (i.e.
coils of several thousand metres in length)?
For a short instrument string the situation, as I see it, is different in that the mass per unit length is not the same as the density.
Density is an absolute value defined as mass/volume. Therefore, mass per unit length = density per unit length x volume per unit length. The volume =
cross section area of a string x length. so for a unit length volume can be expressed as the cross section area = Pi/4 x D²(Pi being the constant =
3.142). So the mass per unit length, in this case = 3.142/4 x D² x 'linear density'.
You mention that D'addario measure their 'linear density' (Unit Weight) for plain strings in Imperial units 'pounds per linear inch'.
and the formula they use for calculating the tension of strings is:
T = (UW x (2x L x F)²/ 386.4. As you observe this expression is non other than the Mersenne - Taylor law for vibrating strings (smooth, uniform
homogeneous cylinders). But where does the 386.4 constant come from you ask?
Weight = mass x g (where g is the acceleration due to gravity). The acceleration due to gravity varies depending on geographical location on the earth
- but for America and Europe an approximate value of 32.2 ft/sec² in Imperial units is usually considered to be close enough.
D'addario uses Unit Weight as a measure so the unit mass = UW/32.2. However as D'addario also measure lengths in inches (not feet) there being 12
inches to the foot, g becomes 32.2 x 12 = 386.4.
What do you mean by 'effective linear density' applied to the Mersenne - Taylor law which is valid for plain cylindrical uniform homogeneous strings?
The above are my preliminary thoughts. My memory of physics and maths is also pretty rusty so will check out what I have written here for any obvious
errors later on when I can find the time - and add any further observations that may come to mind.
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Brian Prunka
Oud Junkie
Posts: 2918
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Thanks for the response.
The equation you gave:
mass per unit length = 3.142/4 x D² x 'linear density'
Seems to give me strange results when I try using D'addario's values.
For example,
D'addario gives a UW of .00004751 lb/in for a rectified nylon string .041in in diameter.
(.7855) x (.00161 in2) x (.00004751 lb/in) = .00000006273...lb-in
Shouldn't we get lb/in3 instead of pound-inches? So shouldn't we be dividing by diameter instead?
(.7855)x(1/.00161 in2) x (.00004751 lb/in) = .023lb/in3
.023lb/in3 = 641.6Kg/m3
Trying with another string:
UW = .00003078, diameter .033in rectified nylon:
(.7855)x(1/.001089 in2) x (.00003078) = .022lb/in3
.022lb/in3 = 615.5Kg/m3
Now I would think that two rectified nylon strings would have equivalent densities. These are close, but not the same, and are far less than the
1040Kg/m3 that is suggested for nylon by Arto's calculator. I'm sure I did something wrong here, but I am having trouble figuring out what is going
on . . . it's been almost 20 years since physics . . .
D'addario's equation seems to work for its own values but I am having trouble figuring how to relate that to more standard calculations.
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jdowning
Oud Junkie
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O.K let's try this approach.
D'addario weighs a bulk quantity of rectified nylon string (i.e. string that has been ground perfectly round in section) of known length to determine
the weight per inch of the string - this is their Unit Weight value for that particular batch.
Specific Weight is the weight /unit volume this is the 'linear density'
The unit volume of a string (i.e. volume per inch) = Pi x D²/4 = 0.7855 x D² where D is the diameter of the string.
So, for the first example linear density = UW/ 0.7855 D² = 0.00004751/0.00132 = 0.036 lb/cu inch = 1.0 gm/cc or 1000 kg/m³
and likewise for the second example 'linear density' = 0.00003078/0.00086 = 0.036 lb/cu inch = 1000 kg/m³
so the nylon is of the same 'linear density' for both strings.
The slightly low value (compared to 1040 kg/m³) for the nylon may be due to the rectifying process that removes material from the string in order to
make it perfectly round in section at a smaller diameter and so affects (compared to the original plain un-rectified nylon) both the weight per inch
component as well as the diameter component of the 'linear density' relationship.
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Brian Prunka
Oud Junkie
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I see, I was mixing up mass per unit length and mass per volume so instead of :
UW (m/L)= volume[(3.14/4)xD2] x density (m/V)
I was doing:
density = volume x UW
Which of course, makes everything wrong.
The calculation
Density = UW/V
then makes sense.
The confusion comes from the definitions I was getting from Wikipedia, that UW is weight per volume and Linear Density is mass per unit length, where
you are saying it's the opposite in your equation, I think. Right?
Aside, do you have a guess as to what the units of measurement might be in Savarez's linear density figures? All they say is that the wound strings
range in linear density from 70 to 560, but they don't mention the units.
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fernandraynaud
Oud Junkie
Posts: 1865
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I get a density of 995 Kg/m^3 for D'Addario's nylon. Their wound silvered copper on nylon fluff strings have a density between 4700 and 6700 kg/m^3.
Brian, if you download the first spreadsheet in my Oud .. Tensions thread, you can copy the calculation in the densities E column to compute any
density in kg/m^3 from a diameter in mils or mm (D) and linear density (C) (in lbs/inch), by placing them in the appropriate columns, or changing the
cell references in the formula. It's not elegantly expressed, but it works. Just use it in another spreadsheet. The .453.. is the lb->Kg
conversion.
=((C12/(((((D12/2)^2)*3.142)*25.4)/1000000))*0.45359237)*1000
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jdowning
Oud Junkie
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I get a 'linear density' of 0.99627 gm/cc and 0.99068 gm/cc for the two cases respectively which I have rounded up to 1.0 gm/cc for practical
simplicity translating to 1000 kg/m³ if preferred. Close enough for all practical purposes (for these two cases at least).
This seems to be a lower value of density for the grade of nylon often used by string manufacturers for instrument strings that might fall within the
range of 1.04 to 1.08 gm/cc (or 1040 kg/m³ to 1080 kg/m³ if preferred).
Brian - I have no information about Savarez strings but perhaps fernandraynaud (who is currently investigating the possible application of the 'linear
density' concept derived from known data published by some string manufacturers on his 'Ouds ... Tensions' thread) could take the time to look into it
and include the findings on his thread?
Unit Weight (or Specific Weight) is defined as the weight per unit volume of a material (mass per unit volume is the definition of Density) - but
weight per unit volume is sometimes conveniently referred to as 'density'.
D'addario appears to define UW as the weight per unit length (i.e. it is a weight not a 'density') of a string - and so creates some confusion. To
obtain the 'density' of a unit length of string it is necessary then to divide the D'addario UW by the volume of a unit length which is the cross
section area of a string calculated from the outside diameter.
Note that if a homogeneous string is not a perfectly round smooth cylinder then a diameter - which is less than the measured outside diameter - would
have to be determined in order to calculate the density per unit length.
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Brian Prunka
Oud Junkie
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Thanks. I guess to be accurate, it is D'addario who has confused the terms in referring to their measurement as unit weight.
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jdowning
Oud Junkie
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Well I find the terminology confusing but of interest - so thanks for raising the question Brian.
Just to recap, the purpose of this part of the thread is to investigate the mathematical model used to calculate the design of close wound strings
from first principles and to verify how accurate (or otherwise) the formula may be by comparison with published data and measurement of some actual
strings made by a known manufacturer (Pyramid).
This information might prove to be of general interest to those curious about the physics of wound strings or useful to those who might even be
interested in making wound strings (at least as a starting point!). Working in reverse the application may also be of value in assessing and
estimating the tension/pitch characteristics of a wound string or small fragment of string of unknown manufacture (for example see 'Repairing
a"Nahhat" Oud' by Yaron Naor on this forum)
At this point, the results show quite a close correlation with the strings currently examined and tested and there will be more to follow as
opportunity is taken to examine and test more string samples.
The derivation of the wound string formula previously posted makes some simplifying assumptions in order to arrive at an 'equivalent diameter' string
of the same homogeneous density as the core material of the wound string (the component subject to string tension) allowing use of the Mersenne-Taylor
law to calculate string tension. The resulting formula is in agreement with the 'equivalent diameter' formula proposed by other researchers.
A more accurate derivation is to follow - avoiding the simplified assumptions of the basic mathematical model (but still incorporating some
simplification) - in order to evaluate if the resultant, more complicated, formula is worth the extra computational effort in order to achieve a small
increase in accuracy of the final result. A case of diminishing returns perhaps?
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jdowning
Oud Junkie
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In trying to get a better understanding about the design of wound strings I came across the attached brief article "Overspun Strings" from the pages
of FoMRHI published back in January 1978 that I thought might be of general interest. (All of the past articles from FoMRHI, dating from 1975, are now
freely available 'on line').
The author, Dr Cary Karp, is Associate Professor of Organology at Uppsala University, Sweden, past curator of the musical instrument collections at
the Music Museum in Stockholm as well as having been Director of Internet Strategy for the International Council of Museums and member of various
museum related IT development committees and working groups
Dr Karp proposes two formulae for determination of the 'equivalent diameter' used for the design of wound strings. The first is derived from a
simplified mathematical model and the second is a more accurate corrected version.
As Dr Karp does not take the reader through the steps to arrive at the formulae, it has been necessary to figure it out from first principles (always
a good idea when trying to understand a concept!).
The derivation of the first formula, as I understand it, has been previously posted and is currently being investigated to determine how accurate a
design tool it might be.
The first formula assumes that the winding is a series of rings or toroids on the core. The second formula assumes that the winding is wound on spiral
fashion as it would be for a real wound string.
However, is the more accurate formula worth all of the extra computational work compared to the original simplified version?
The author briefly discusses the relative accuracy implications as it applies to the practical manufacture of wound strings as do the authors of the
commentary attached to the article - by the pioneering historical string makers of 'Northern Renaissance Instruments'.
But how the heck did Dr Karp arrive at formula #2?
To follow - for general information - is my Derivation #2 for the corrected 'equivalent diameter' - which, as it happens, agrees with formula #2
proposed by Dr Karp.
Attachment: wound strings comm 102. reduced pdf.pdf (944kB)
This file has been downloaded 439 times
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jdowning
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Derivation #2 for the equivalent string diameter is attached for information.
This model assumes that a single spiral turn of wrapping wire is cut and the two open ends are squeezed together to form a ring of a slightly larger
diameter than the core. This trick allows the modified (slight increase) in the mass of the wire to be substituted in the equation of the first
derivation for the mass of the wire component without affecting the two other components of the equation. the increased diameter of the ring (Dm) is
the hypotenuse of a right triangle one side of length Dw/2 and the other (Dc + Dw). According to the rule of Pythagoras (that the square on the
hypotenuse is equal to the sum of the square of the other two sides):
(Dm)² = (Dw/2)² + (Dc +Dw)²
and so Dm is the square root of the other two components and the length of one turn of wire is 3.142 X Dm.
The cross section area of the wire = 3.142/4 x Dw²
and the volume of one turn = length X cs area
and the mass of one turn of wire = length X cs area X density of wire
all reducing to the corrected 'equivalent diameter' shown in the attached derivation.
This derivation makes the same simplified assumption as the first derivation i.e. that there is no compression of the core or distortion of the wire
(that occurs during manufacture of a string). So core diameter is determined from the outside diameter of the string assumed to be = Dc + 2 Dw.
The correction factor in practice is small and so has little measurable significance until the wire diameter approaches that of the core (as it does
for the 6th course string of the Pyramid #650 orange label oud string set). Otherwise use of the corrected equivalent diameter equation would
generally appear to be an unnecessary complication.
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jdowning
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Note that a string designer when using the equivalent diameter formula must make allowance for the stretching and deformation of the wire and
compression of the core (it is necessary for the wire to be wound onto the core under tension so that it remains tight against the core to avoid
future 'buzzing' of the windings). So the finished diameter of a wound string will always be slightly less than the calculated value. A string maker
must make adjustments (by experimentation) to allow for this.
A further constraint on a string maker is that in practice (in the interests of economy) only a limited range of wire and core diameters will be
available from which to make an appropriate selection of core/wire diameter combinations.
The starting point for a string designer of a wound string would be the determination of a minimum core diameter capable of withstanding the
anticipated string tension without breaking. The stress on the core (not the windings) is the tension divided by the cross section area of the core
(as the wire winding adds mass to a string and does not - or should not - carry a significant proportion of the string tension). The maximum stress
that a string material can take before breaking is defined as its maximum tensile strength which is Tension/ A where A = 3.142/4 x (Dc)² or 0.7855
(Dc)² from which a minimum core diameter can be determined.
The above applies less to the reverse situation where the outside diameter and wire diameter of a finished string are measured directly in order to
determine string tension for a given pitch (using the equivalent diameter equation) as all reductions in diameter of core and wire are already
accounted for. The measured outside diameter of a finished wound string can, therefore, be assumed to be Dc + 2Dw so the resultant calculated tension
should be quite close to the measured tension on a test rig as appears to be the case so far from these trial results.
Note also that only close wound strings are under consideration here. Open wound strings have an application on lutes (plucked with soft finger tips)
but would likely not be practical on ouds or other instruments sounded with a risha, pick or plectrum.
Derivation #2 previously posted may be modified further for open wound string design calculations by increasing the (Dw/2) component of the right
triangle geometry accordingly.
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jdowning
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From a practical point of view, the strength of the wire must also be taken into account by the string maker so that the finer wires (0.1 mm diameter
or less) do not break under the tension of being wound on the core during string manufacture.
Pyramid make some of their smaller diameter wound strings with two strands of wire wound together (side by side) on the core - presumably to avoid
problems of wire breakage?
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jdowning
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Here is another old article from FoMRHI (Comm 163, October 1978) concerning wound string calculations by Eph Segerman and Djilda Abbott of 'Northern
Renaissance Instruments' of Britain.
The authors go into greater mathematical complexity in deriving an equivalent diameter formula that covers both close wound and open wound cases that
is independent of the cross section shape of the wire. As the authors are string makers they also provide some hands on commentary about the
deformation of core and wire observed during the string manufacturing process.
Although a more rigorous approach than the basic formula that is subject of the investigation reported in this thread - the article might be of some
value to those who may be contemplating making their own strings as well as being of general interest to those interested in delving further into the
detail of string design theory.
String diameters in the article are in thousands of an inch so I have added the more convenient metric equivalent.
[file]24232[/file]
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fernandraynaud
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What am I missing here ... the core of most wound strings is not a nylon monofilament, but a bundle of nylon fibers, with a lot of air in between the
strands, more or less squeezed together. So how can we be talking about the core as having a fixed diameter, with the density of a solid nylon
cylinder, and then extending this metaphor to a wound string having a "virtual diameter" of nylon?
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jdowning
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Well there is not 'a lot of air between the strands' - in fact likely none at all - so density is not compromised.
The core is not a fairly loose bundle of fibres with air trapped between but is highly compressed by the winding so much so that the wire stretches
and reduces diameter when a wound string is made (about 3 -6 % 0f original diameter according to N.R.I.) and the string (core) stretches in length
under the compression load (about 0.5% according to N.R.I.). If this were not the case the winding might slip on the core in use causing 'buzzing' due
to a loose winding.
Nylon is a relatively soft material and so will deform under compression loading producing a close packed homogeneous material of uniform density.
Both Nylon floss (and silk filament) used for wound string making are extremely fine with filament diameters that can be 0.015 mm or less in diameter
further ensuring close packing of the core filaments. For example a Pyramid wound #906 lute string measures 0.47 mm outside diameter, wire diameter
0.06 mm diameter so the compressed core diameter of the finished string is 0.35 mm diameter. The wire probably started life with a somewhat larger
diameter before being wound on to the core and stretched under tension.
I have compared the diameter of the wire and nylon filament, side by side, under a microscope and estimate that the filament is about 1/4 to 1/5 the
diameter of the wire.
As previously mentioned in this thread the equivalent diameter formula is just a starting point for the string designer. Factors such as wire
stretching and core compression during manufacture being established empirically (for a particular manufacturing procedure) by actually making strings
and measuring the result.
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fernandraynaud
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Or weighing a piece of the finished string? ... ;-)
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jdowning
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..... or not.
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jdowning
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In order to further evaluate the accuracy (or otherwise) of the basic 'equivalent diameter' formula, some of the 'Pyramid' wound lute strings that I
have in stock were measured, the 'equivalent diameter calculated and then compared with the tension derived from the Mersenne-Taylor law (using the
Arto Wikla calculator), from the 'Pyramid' tension tables, the 'Pyramid' slide rule calculator and measured directly from the string test rig.
'Pyramid' make a range of 91 different wound lute strings covering a range of almost 3 octaves for a given string tension - from the thinnest #809Al
to the heaviest #3272.
The strings subject to testing range from #906 to #1027 covering the middle range of an octave plus 3 semitones for a given string tension. A total of
12 strings have been tested - all copper wound on nylon filament.
Wire diameters measured ranged from 0.06 mm to 0.27 mm and calculated core diameters from 0.35 mm to 0.39 mm.
Calculated 'equivalent diameters' (for monofilament nylon) ranged from 0.88 mm to 2.08 mm assuming a core density of 1.04 gm/cc (or 1040 kg/m³) and
wire density of 8.8 gm/cc (or 8,800 kg/m³).
For information, the most common commercially available wire diameters range from 0.02 mm (metric gauge #2), 0.0254 mm SWG #50 equivalent (Standard
Wire Gauge - British Imperial) and 0.0787 mm #40 AWG equivalent (American Wire Gauge). For the string maker there is likely a practical lower limit to
wire diameter in order to avoid wire breakage during winding the core.
More to follow
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