fernandraynaud
Oud Junkie
Posts: 1865
Registered: 7-25-2009
Location: San Francisco, California
Member Is Offline
Mood: m'Oudy
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Oud string Densities -> Tensions
SUMMARY: once a wound string's density is measured, or derived from even a single data point provided by the maker, it can be plugged into Arto's
Calculator, yielding tensions at different pitches/scale length, just as for plain strings.
It's been often repeated that wound strings are very different from monolithic plain strings, making it impossible to use the standard Mersenne-Taylor
equation, e.g. in Arto's String Calculator, to calculate tensions.
To make a long story short, imagine my surprise when I looked at the tension data and equations that D'Addario provides, and it started looking like
this was just a pessimistic assumption. Moreover, D'Addario routinely calculate tension using string diameter and density alone, with no concern for
the specific composing materials or structure -- and they should know, having the experience of making a huge variety of strings for everything from
ukuleles to concert violins and basses.
We know that unlike plain strings, wound strings don't have the same density at different thicknesses. Each string will have a distinct density,
resulting from different ratios of nylon, copper, silver and air, packed within the outer "micrometerable" cylinder that is the string gauge.
Calculations and comparisons I've tried so far have supported the (surprising) notion that in the end such an assembly probably CAN be treated as if
it were a homogeneous filament of that diameter, made of a hypothetical material having that density. To obtain that density one can derive it from
what data is available, or simply weigh a piece of the string on a milligram scale.
D'Addario specifies "linear density" for strings, in lbs/inch. Having gone through several of their tables and compared to values calculated using
their equation AND Arto's, and even weighing a few string pieces to the milligram, the striking thing has been how well these approaches agree, just
like for plain strings, and well within measurement errors.
D'Addario's equation that yields a tension T in Lbs is
T = (UnitWeight * ((2 * L * F)^2)) / 386.4
Where UnitWeight is in lbs/inch of string, L is scale in inches, F is pitch in Hz.
Unless D'Addario is grossly mistaken, or all the different strings they make all behave one way, entirely differently from other makers' strings, the
density shortcut might well be applicable to all wound strings, without having to mic the winding wire, etc.
I'm putting up this spreadsheet that lists D'Addario's single nylon and silvered copper on nylon filament strings, with the linear lb/inch density for
each, and the Kg/m^3 densities derived from them, for anyone to check or use. These densities plausibly fall between that of the Nylon core and the
Copper wrap of which these strings are made. I am also including a useful spreadsheet of tensions at different pitches from 560 to 630 mm scale, that
I imported from a D'Addario PDF.
These densities work in Arto's to calculate tensions, for different tunings and scale lengths, and these values match the tensions published in
D'Addario's tables. Since I don't have a contraption to check actual tensions on different strings, that's all I can be sure of. The rightmost column
shows a few "effective densities" spot-calculated by using Arto's "in reverse", just to show the agreement, including the one worst match. I can't
rule out the possibility that everything agrees only because D'Addario has generated the tables from their equation, and that measured tensions might
be different.
Some of D'Addario's other published specs on specific strings disagree (equally) with both their own equation and Arto's, though it's within 10%.
I feel reasonably confident at this point that when the density of a string is unknown, and the maker supplies ONE tension measurement, a usable
density can be arrived at, for instance by varying the density parameter in Arto's interface until the tension matches that supplied by the maker for
that scale, gauge and tuning.
The "effective density" derived in that way turns out to match quite closely the actual composite density of the string, and its specified "linear
density". From that point on, the tension at other pitches/scale for that string can be calculated by keeping that density in place in Arto's, and
varying the other parameters.
This technique has been useful to me. I hope other people can go through the steps and either find it of use, or point out any factual or logical
errors.
Attachment: DAddarioStringDensities.xls (25kB)
This file has been downloaded 139 times
Attachment: Daddariosingletensions.xls (38kB)
This file has been downloaded 139 times
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Bodhi
Oud Junkie
Posts: 168
Registered: 2-26-2012
Location: North West England
Member Is Offline
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Here are the attachments as PDF. I find it easier to view them and Adobe is free
Attachment: DAddarioStringDensities.pdf (115kB)
This file has been downloaded 472 times
Attachment: Daddariosingletensionslandscape.pdf (206kB)
This file has been downloaded 220 times
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fernandraynaud
Oud Junkie
Posts: 1865
Registered: 7-25-2009
Location: San Francisco, California
Member Is Offline
Mood: m'Oudy
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Good show!
BTW, as an alternative to deriving the reverse MT equation, I should specify how to scientifically find a density in Arto's - once you plug in the
diameter, scale length and pitch for a data point the manufacturer has defined (say you have 8 Kg tension as a guitar A string). Guess. Try say 4000
kg/cubic meter and see if you get the tension specified by the manufacturer. If too high, go down say to 3500 and try again. Too low? Try 3700. You
can get very good at successively zeroing in to within a few kg in a dozen tries or less. Once you have the density for that string at that gauge,
write it down for later use. Then to get the tension you were after, change the length and pitch to the oud scale and tuning.
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