Violin versus Hardanger fiddle neck

[ezh]

Oh ... about the understrings sharing two hooks among the five understrings:  I did a little bit of repair work on a Harald Lund fiddle.  He set five separated hooks into the front edge of the streng haldar.   Probably glued into holes.   There's a picture in

https://www.dgviolins.com/lund-repair

I did a bit of experiment:  I glued (epoxy) a hook into a piece of wood and hung a 10 kg weight from it.  In two week, no sign of movement.

Dave Golber

 

 

[GoPractice]

Not having had a Hardanger for any length of time, or an instrument with sympathetic strings. Does adjusting the afterlength, of the tailpiece for the primary strings, have a great effect on the overall sound?

On the opposite end, having had a harp guitar for a short while, trying to optimize the string gauges was time consuming and bit pricey. The intonation, tuning back and forth, was key to having the best sound at the time, and lighter tensions were the most practical. 

Is there a chart or resources ( maybe by a manufacturer ) anywhere on the variation of tensions for the sympathetic strings for Hardangers? Does the coating on the steel matter?

[Anders Buen]

On 3/24/2021 at 2:39 PM, ezh said:

I did some calculations, and saw that (to be emphatic) it's impossible to tune a steel string correctly without a fine tuner.   And extremely impossible (   ) to tune a low tension steel string without a fine tuner.   I think fine tuners in the streng haldar is the way to go.  Here's an early version:

http://dgviolins.com/images/Bridge4.jpg

Dave Golber

 

 

Hi Dave!
Must be a pain to find those hooks under the tailpiece. However, I think that heavier tailpieces are better. I tend to add magnets to my light wood tailpieces with only one fine tuner.

Fine tuning the understrings is a bit of both peg use and stretching of the strings using the nail. Sometimes an understring never will end where you want it. And even some instruments work better that way. At larger amplitudes the note probably increases a little. Especially for the lower ones.

Edited August 13, 2021 by Anders Buen
Spelling correction

[ezh]

"Find those hooks"?  You mean getting the string on the hook?  Yes, it's harder than for the usual set up.  But not impossible.   Given that players don't change their understrings very often (they probably change wives or husbands more often than understrings), I figure the extra difficulty is worth it.

Here's the analysis of "impossible to tune an understring with ordinary pegs".   More mathematics than usually here in Maestronet ... but you can read it ...

The question is: why fine tuners for steel strings and not gut strings?  I reason:  When you turn the peg on gut string, some of the turning stretches the string, and some of the turning increases the tension.  With a steel string, the string doesn't stretch much at all, so most of the turning goes into increased tension.  The calculations below show this is true.

The calculations show that

1.  For the same turn of the peg (or fine tuner), change in pitch (in cents) is much higher for steel than gut.

2. Diameter of the string does not change this.

3. The difficulty of tuning is higher for a lower pitched string.

The reason for the strange numbers (F#; 298mm string length, etc):  I work mostly on Hardanger fiddles these days, and these are typical numbers for those instruments.

The calculations were done in Mathematica.   What is below is cleaned up from the Mathematic forms.

Young's modulus of gut = 0.15 * 10^9  from "Mechanical Comparison of 10 Suture Materials ...", Greenwald et al, Journal of Surgical Research 56, 372-377 (1994)

Density of gut = 1.276 from https://www.cs.helsinki.fi/u/wikla/mus/Calcs/wwwscalc.html. I use 1.3
 
(*
EJH Hardanger "E", tuned to F#
0.215mm diameter
Tuner to bridge  42mm
Bridge to nut    298mm
nut to peg        33 mm
------------------
Total stretched length
                 373mm.

Fine Tuner: Long arm 21mm
short arm 9mm
Ratio: 0.43

Thread of screw is 2mx0.4
So full turn of screw advances screw by 0.4mm
So full turn of screw stretches string by 0.4x0.43 = 0.17mm
Of course a fine tuner is capable of much finer adjustment than a full turn of the screw

If diameter of peg at the string is 6mm, then 1/100 turn of the peg is 6 Pi/100 = 0.19mm ...
about the same.  So this 0.17mm is in the neighborhood of the finest possible adjustment of a peg.

------
Actual measurement: Full turn of screw raised pitch 35 to 45 cents *)

(* Steel string, Diameter .215mm, vibrating length 298mm.  Total length 373mm. Tuned to F# *)
(* What happens to pitch if we stretch the whole string by 0.17mm ?               *)

afreq=440.;semitone = 2.^(1/12); fsharpfreq=afreq*semitone ^9 = 739.989
 
(* f=Sqrt[t/(m/l)]/(2*l);  t= tension Newtons;  m= mass of string Kg;  l = length meters *)
(* Solve for t:   t= 4mlf^2      *)

diam = 0.215; (* mm *)
areamm2= (Pi/4)*diam ^2;(*mm^2*)
volmm3 =areamm2*298; (* mm^3 *)
volcc=volmm3/1000;
massgm=volcc*8.; (* Density of steel about 8 *)
masskg=massgm/1000 ;(* kg  *)
t0=4*masskg*.298*fsharpfreq^2  = 56.4937      (* Starting tension in Newtons *)
 
(* t/area = stress; deltalength/length = strain;  stress = ym* strain; ym in Newtons/square meter*)
(* So New t = old t + ym*area*(deltalength/length)   *)

 ymSteel = 200.*10^9; (* Youngs modulus *)

 aream2=areamm2/10^6

 (* The tension goes up  *)
t1=t0+ymSteel*(0.17/373)*aream2   = 59.803

 (* So the  new frequency is *)
f1= Sqrt[t1/(masskg/.298)](1/(2*.298)) =761.354
 
 Log[2,f1/fsharpfreq]*1200  =  49.2768    (* Change in frequency in cents *)

 (* Measured was 40 cents.   So this is pretty good! *)

(* ************************** *)
(* Allow for stretching    373 mm stretches 0.17 mm.  So vibrating mass goes down by *)
factor = 373./(373+0.17) = 0.999544

 (* So freq is *)
f1star=Sqrt[t1/(masskg*factor/.298)](1/(2*.298)) =  761.528

 Log[2,f1star/fsharpfreq]*1200  =  49.6713       (* Change in frequency in cents  *)
(* differs only by half a cent from calculation without stretch factor *)

(*********************************************************************************) 
(* Conclusion: pitch goes up by about 50 cents.                         *)
(* And this is with about 1/100 turn of the peg!                            *)
(* So this is clearly why we need a fine tuner on a steel string                         *)
(* This is also a full turn of the fine tuner screw.  Pretty clear that fine tuner can get within one cent *)
(******************************************************************************) 

 (* What happens if we change the diameter but keep everything else the same?     *)

 diam = 0.430; (* mm *)   (* Double the diameter *)
areamm2= (Pi/4)*diam ^2;(*mm^2*)
volmm3 =areamm2*298; (* mm^3 *)
volcc=volmm3/1000;
massgm=volcc*8.; (* Density of steel *)
masskg=massgm/1000 ;(* kg  *)
t0=4*masskg*.298*fsharpfreq^2   = 225.975    (* Starting tension in Newtons *)
 
(* t/area = stress; deltalength/length = strain;  stress = ym* strain ym in Newtons/square meter*)
(* So New t = old t + ym*area*(deltalength/length   *)

 ymSteel = 200.*10^9; (* Youngs modulus *)

 aream2=areamm2/10^6

 (* The tension goes up  *)
t1=t0+ymSteel*(0.17/373)*aream2  = 239.212

 (* So the  new frequency is *)
f1= Sqrt[t1/(masskg/.298)](1/(2*.298)) = 761.354

 Log[2,f1/fsharpfreq]*1200   = 49.2768   (* Change in frequency in cents *)

 (* ************************** *)
(* Allow for stretching *)
(* 373 mm stretches 0.17 mm *)
(* So vibrating mass goes down by *)
factor = 373./(373+0.17) = 0.999544

 (* So freq is *)
f1star=Sqrt[t1/(masskg*factor/.298)](1/(2*.298)) = 761.528

 Log[2,f1star/fsharpfreq]*1200 =  49.6713  (* Change in frequency in cents *)
      (* differs only by half a cent from calculation without stretch factor *)

(*******************************************************************************)

 (* These are the same change-in-frequency numbers as for the smaller diameter.  So diameter doesn't matter. *)  
(*******************************************************************************)


(* Let's look at gut for the same parameters *)
(* Density is 1.3 ,   Young's modulus is 0.15*10^9  *)

diam = 0.430; (* mm *)
areamm2= (Pi/4)*diam ^2;(*mm^2*)
volmm3 =areamm2*298; (* mm^3 *)
volcc=volmm3/1000;
massgm=volcc*1.3; (* Density of gut *)
masskg=massgm/1000 ;(* kg  *)
t0=4*masskg*.298*fsharpfreq^2  = 36.7209       (* Starting tension in Newtons *)


(* t/area = stress; deltalength/length = strain;  stress = ym* strain,  ym in Newtons/square meter*)
(* So New t = old t + ym*area*(deltalength/length)    *)

 ymGut = 0.15*10^9; (* Youngs modulus *)
 aream2=areamm2/10^6

 (* The tension goes up  *)
t1=t0+ymGut*(0.17/373)*aream2  = 36.7308

 (* So the  new frequency is *)
f1= Sqrt[t1/(masskg/.298)](1/(2*.298)) = 740.089

 Log[2,f1/fsharpfreq]*1200 =  0.233998     (* Change in frequency in cents *)

 (* Allow for stretching *)
(* 373 mm stretches 0.17 mm *)
(* So vibrating mass goes down by *)
factor = 373./(373+0.17) = 0.999544

 (* So new freq is *)
f1star=Sqrt[t1/(masskg*factor/.298)](1/(2*.298)) = 740.258

 Log[2,f1star/fsharpfreq]*1200 = 0.628425 (* Change in frequency in cents *)

(**********************************************************************************)
(*So the same peg or fine tuner manipulation that raises the pitch of a steel string by about 50 cents  raises the pitch of a gut string about 1/2 of one cent  *)
(**********************************************************************************)

(*********************)
(* See if pitch makes a difference *)
(* Highest Hardanger understring is at B.  It is steel. *)
(* Actually nut to peg is longer, but here will use the same number *)

semitone = 2.^(1/12);
bfreq=afreq*semitone ^2

(*  f=Sqrt[t/(m/l)]/(2*l);  t= tension Newtons;  m= mass of string Kg;  l = length in meters*)
(* Solve for t:    t= 4mlf^2     *)

diam = 0.215; (* mm *)     (* Actually 0.22mm for usual string.  But keep same number as "E" string *)
areamm2= (Pi/4)*diam ^2;(*mm^2*)
volmm3 =areamm2*298; (* mm^3 *)
volcc=volmm3/1000;
massgm=volcc*8.; (* Density of steel *)
masskg=massgm/1000 ;(* kg  *)
t0=4*masskg*.298*bfreq^2 =  25.1651    (* Starting tension in Newtons *)

(* t/area = stress; deltalength/length = strain;  stress = ym* strain; ym in Newtons/square meter*)
(* So New t = old t + ym*area*(deltalength/length)       *)

 ymSteel = 200.*10^9; (* Youngs modulus *)

 aream2=areamm2/10^6

 (* The tension goes up by the same number of Newtons as for the "E" string*)
t1=t0+ymSteel*(0.17/373)*aream2  = 28.4744

 (* So the  new frequency is *)
f1= Sqrt[t1/(masskg/.298)](1/(2*.298)) = 525.354

 Log[2,f1/bfreq]*1200 = 106.945
 
 (* ************************** *)
(* Allow for stretching *)
(* 373 mm stretches 0.17 mm,     So vibrating mass goes down by *)
factor = 373./(373+0.17)

 (* So freq is *)
f1star=Sqrt[t1/(masskg*factor/.298)](1/(2*.298)) = 525.474

 Log[2,f1star/bfreq]*1200   = 107.339 (* Change in frequency in cents *)
   (* Allowing for stretching differs only by about half a cent *)

(**************************************************************************)
(* So, for a steel string,  the same peg or fine tuner manipulation that raises the pitch from F# by about 50 cents raises the pitch from B more than 100 cents.  *)
(**************************************************************************)
 

 

 

[Anders Buen]

3 hours ago, ezh said:

"Find those hooks"?  You mean getting the string on the hook?  Yes, it's harder than for the usual set up.  But not impossible.   Given that players don't change their understrings very often (they probably change wives or husbands more often than understrings), I figure the extra difficulty is worth it.

We makers put on strings every now and then. 

[Anders Buen]

3 hours ago, ezh said:

For the same turn of the peg (or fine tuner), change in pitch (in cents) is much higher for steel than gut.

2. Diameter of the string does not change this.

3. The difficulty of tuning is higher for a lower pitched string.

1 is correct. 2 and 3 is not correct. The diameter does indeed have a little influence, as does the diemater of the tuning peg. I have tested reducing the diamter of the pegs for the undertstrings in the pegbox, which works fine, but is problematic when the pegs or holes need to be shaved off a bit over time. Witther gerared pegs are tiredsome to use as are pegs with less diameter.

The highest tuned pitced understring is more difficult to tune than the lower, in my experience. 

[La Folia]

On 3/24/2021 at 8:39 AM, ezh said:

I did some calculations, and saw that (to be emphatic) it's impossible to tune a steel string correctly without a fine tuner.

 

 

 

Yes, it is impossible, but somehow we do it.  It sometimes helps to pull the peg out a little, and we keep trying until we get it by accident.  If that fails, we tug on the string a little.  Actually, 1/100 turn is 3.6 degrees, which is not so impossible.

It would be easier if the pegs were fitted by makers with peg shapers and reamers instead of by beavers, but I've never seen that happen on Hardanger fiddles. 

[ezh]

On 3/26/2021 at 12:48 PM, Anders Buen said:

On 3/26/2021 at 10:36 PM, La Folia said:

 

1 is correct. 2 and 3 is not correct. The diameter does indeed have a little influence, as does the diemater of the tuning peg. I have tested reducing the diamter of the pegs for the undertstrings in the pegbox, which works fine, but is problematic when the pegs or holes need to be shaved off a bit over time. Witther gerared pegs are tiredsome to use as are pegs with less diameter.

The highest tuned pitced understring is more difficult to tune than the lower, in my experience. 

My calculations certainly do not take many fine points into account.  But the first one found 50 cents rise, when I measured about 40 cents.  I think that's a great validation ... especially when the measured 40 cents wandered up to 50 cents.  I'm not surprised that there is "a little effect" from the diameter of the string.  The diameter of the peg of course has a strong effect.   If the peg has half the diameter, then the same turn of the peg tightens the strings by half as much. 

The method of the calculations is verified by the first measurement.  So I tend to believe the other results. 

[ezh]

11 hours ago, La Folia said:

Yes, it is impossible, but somehow we do it.  It sometimes helps to pull the peg out a little, and we keep trying until we get it by accident.  If that fails, we tug on the string a little.  Actually, 1/100 turn is 3.6 degrees, which is not so impossible.

It would be easier if the pegs were fitted by makers with peg shapers and reamers instead of by beavers, but I've never seen that happen on Hardanger fiddles. 

A famous quote "No wonder you are having so much trouble tuning.  The holes are square and the pegs are triangular!"

Yes, 1/100 of a turn is possible ... but that changes the pitch by about 50 cents!

The pegs I put in fit.

 

[Anders Buen]

Often the strings crosss over the peg and «see» a somewhat larger diameter than the peg itself. The pegs are also tapered, so the position on the peg may also matter. 

There is some elastic effects in the tailpiece, from the other strings, the body under tension, a little from the bridge and the local region around there, the neck angle and the resulting forces over the two upper saddles. It is easier to tune a hardanger than a chembalo or spinette. But there are some similarities. Sort of an iterative process.

Tuning the highest string has more influence on the other strings through the force balances than tuning the lowest due to higher tension and probably a higher dT/T.

Pegs only fit perfectly for the climate (RH) they are fitted. The direction of the wood in the pegs and the pegbox is different and the holes are in practice never perfectly round, nor are the pegs.    

[La Folia]

Obviously ezh is right that it's hard to tune the understrings with pegs.  I would just like to point out the experience I've had with pegs.

Pegs fitted by Carl Becker always worked absolutely perfectly, and mine stayed that way for years.  They didn't snap and jerk when they were turned, and they didn't require skill to set them in position.  They were an absolute joy to use.  I think his may be the only shop I have ever seen in which all the pegs on all violins were in that condition.  It's sad that his shop was an exception.

In the Hardanger fiddle world I have seen a lot of pegs that were completely inadequate.  I saw one nice fiddle that had just been prepared for sale by a Hardanger fiddle maker.  The E peg fitted so poorly that someone broke the peg box trying to get the peg to hold.  I saw another fiddle from a very good, well-known maker, with pegs in only slightly better condition.

I can't comment on the practices of HF dealers, because they are few and far between, and I have not seen many fiddles that had recently been in a shop.  What I can say is that I don't remember ever seeing a Hardanger fiddle with good pegs.  Maybe that's not the fault of the shops, but rather because it's hard to find someone to maintain HFs.  Nevertheless, wouldn't it be wonderful if all pegs worked like Carl Becker's pegs?

[Anders Buen]

Even great makers, tools and magic peg paste can´t beat or control the nature laws. 

[La Folia]

Anders, I'm telling you, the best violin makers make really good pegs.  And they stay that way until they eventually wear.

[Salve Håkedal]

A problem (and contradiction) with hardangerfiddles is that decorated pegs on old fiddles are often made of maple. Stained or even painted! Maple is really not suited for pegs, however.

Still, preserving these original pegs is a point, while for most violins, it's not a big issue throwing away worn pegs and fit new ones.

[ezh]

When I see an old fiddle with plain pegs, or completely missing pegs, I say "Oh, good.  I can just put in new pegs".    With old decorated pegs, I can put new ebony shafts on the old decorated heads.   But it's a big deal.

Dave Golber

 

[Anders Buen]

10 hours ago, La Folia said:

Anders, I'm telling you, the best violin makers make really good pegs.  And they stay that way until they eventually wear.

That depends on the climate and how the instruments are stored and used. It is much easier to fit pegs for thicker nylon, perlon or whatever softer core lmaterial, like gut, than it is to get it to work well for steel strings. The only steel string on a violin is the e-string, and it will have a fine tuner. 

Im not a performer. I am fine with a little sueaking pegs or «jump» a little (slip stick motion) because it means it is going to stay where I left it. And I am able to tune the intrument perfectly, just use the time to get the insturment and strings in balance with heat, moisture and all that. The secret lies there, to get the intrument in balance. Not in the workshop for fitting the pegs. 

[ezh]

Take a look at https://www.dgviolins.com/fine-tuners .   (The whole web site section on Hardanger fiddles is https://www.dgviolins.com/hardanger-fiddles)

I've started putting my should-be-patented finer tuners for understrings as well as playing strings on the tailpiece.  I think they are great!

Dave Golber

[ezh]

Sigh ... the web site is different now, so all those references are bad.  The pictures I was referring to are in https://www.dgviolins.com/number-18

But I'm now doing some stuff with understring fine-tuners that I really like.  See https://www.dgviolins.com/fine-tuners

And all my HF stuff is at

https://www.dgviolins.com/hardanger-fiddles

Dave Golber

 

[La Folia]

Hi, Dave.  The first links come through fine.  Not bad!  Those may well be a good solution for the understrings. 

Some people use planetary gears, but they may take a little extra skill to change wiry understrings.  Personally, I'm resigned to just living with the devil wooden pegs I know.  Mine are not great, but they're manageable. 

[Marty Kasprzyk]

On 3/26/2021 at 8:47 AM, ezh said:

"Find those hooks"?  You mean getting the string on the hook?  Yes, it's harder than for the usual set up.  But not impossible.   Given that players don't change their understrings very often (they probably change wives or husbands more often than understrings), I figure the extra difficulty is worth it.

Here's the analysis of "impossible to tune an understring with ordinary pegs".   More mathematics than usually here in Maestronet ... but you can read it ...

The question is: why fine tuners for steel strings and not gut strings?  I reason:  When you turn the peg on gut string, some of the turning stretches the string, and some of the turning increases the tension.  With a steel string, the string doesn't stretch much at all, so most of the turning goes into increased tension.  The calculations below show this is true.

The calculations show that

1.  For the same turn of the peg (or fine tuner), change in pitch (in cents) is much higher for steel than gut.

2. Diameter of the string does not change this.

3. The difficulty of tuning is higher for a lower pitched string.

The reason for the strange numbers (F#; 298mm string length, etc):  I work mostly on Hardanger fiddles these days, and these are typical numbers for those instruments.

The calculations were done in Mathematica.   What is below is cleaned up from the Mathematic forms.

Young's modulus of gut = 0.15 * 10^9  from "Mechanical Comparison of 10 Suture Materials ...", Greenwald et al, Journal of Surgical Research 56, 372-377 (1994)

Density of gut = 1.276 from https://www.cs.helsinki.fi/u/wikla/mus/Calcs/wwwscalc.html. I use 1.3
 
(*
EJH Hardanger "E", tuned to F#
0.215mm diameter
Tuner to bridge  42mm
Bridge to nut    298mm
nut to peg        33 mm
------------------
Total stretched length
                 373mm.

Fine Tuner: Long arm 21mm
short arm 9mm
Ratio: 0.43

Thread of screw is 2mx0.4
So full turn of screw advances screw by 0.4mm
So full turn of screw stretches string by 0.4x0.43 = 0.17mm
Of course a fine tuner is capable of much finer adjustment than a full turn of the screw

If diameter of peg at the string is 6mm, then 1/100 turn of the peg is 6 Pi/100 = 0.19mm ...
about the same.  So this 0.17mm is in the neighborhood of the finest possible adjustment of a peg.

------
Actual measurement: Full turn of screw raised pitch 35 to 45 cents *)

(* Steel string, Diameter .215mm, vibrating length 298mm.  Total length 373mm. Tuned to F# *)
(* What happens to pitch if we stretch the whole string by 0.17mm ?               *)

afreq=440.;semitone = 2.^(1/12); fsharpfreq=afreq*semitone ^9 = 739.989
 
(* f=Sqrt[t/(m/l)]/(2*l);  t= tension Newtons;  m= mass of string Kg;  l = length meters *)
(* Solve for t:   t= 4mlf^2      *)

diam = 0.215; (* mm *)
areamm2= (Pi/4)*diam ^2;(*mm^2*)
volmm3 =areamm2*298; (* mm^3 *)
volcc=volmm3/1000;
massgm=volcc*8.; (* Density of steel about 8 *)
masskg=massgm/1000 ;(* kg  *)
t0=4*masskg*.298*fsharpfreq^2  = 56.4937      (* Starting tension in Newtons *)
 
(* t/area = stress; deltalength/length = strain;  stress = ym* strain; ym in Newtons/square meter*)
(* So New t = old t + ym*area*(deltalength/length)   *)

 ymSteel = 200.*10^9; (* Youngs modulus *)

 aream2=areamm2/10^6

 (* The tension goes up  *)
t1=t0+ymSteel*(0.17/373)*aream2   = 59.803

 (* So the  new frequency is *)
f1= Sqrt[t1/(masskg/.298)](1/(2*.298)) =761.354
 
 Log[2,f1/fsharpfreq]*1200  =  49.2768    (* Change in frequency in cents *)

 (* Measured was 40 cents.   So this is pretty good! *)

(* ************************** *)
(* Allow for stretching    373 mm stretches 0.17 mm.  So vibrating mass goes down by *)
factor = 373./(373+0.17) = 0.999544

 (* So freq is *)
f1star=Sqrt[t1/(masskg*factor/.298)](1/(2*.298)) =  761.528

 Log[2,f1star/fsharpfreq]*1200  =  49.6713       (* Change in frequency in cents  *)
(* differs only by half a cent from calculation without stretch factor *)

(*********************************************************************************) 
(* Conclusion: pitch goes up by about 50 cents.                         *)
(* And this is with about 1/100 turn of the peg!                            *)
(* So this is clearly why we need a fine tuner on a steel string                         *)
(* This is also a full turn of the fine tuner screw.  Pretty clear that fine tuner can get within one cent *)
(******************************************************************************) 

 (* What happens if we change the diameter but keep everything else the same?     *)

 diam = 0.430; (* mm *)   (* Double the diameter *)
areamm2= (Pi/4)*diam ^2;(*mm^2*)
volmm3 =areamm2*298; (* mm^3 *)
volcc=volmm3/1000;
massgm=volcc*8.; (* Density of steel *)
masskg=massgm/1000 ;(* kg  *)
t0=4*masskg*.298*fsharpfreq^2   = 225.975    (* Starting tension in Newtons *)
 
(* t/area = stress; deltalength/length = strain;  stress = ym* strain ym in Newtons/square meter*)
(* So New t = old t + ym*area*(deltalength/length   *)

 ymSteel = 200.*10^9; (* Youngs modulus *)

 aream2=areamm2/10^6

 (* The tension goes up  *)
t1=t0+ymSteel*(0.17/373)*aream2  = 239.212

 (* So the  new frequency is *)
f1= Sqrt[t1/(masskg/.298)](1/(2*.298)) = 761.354

 Log[2,f1/fsharpfreq]*1200   = 49.2768   (* Change in frequency in cents *)

 (* ************************** *)
(* Allow for stretching *)
(* 373 mm stretches 0.17 mm *)
(* So vibrating mass goes down by *)
factor = 373./(373+0.17) = 0.999544

 (* So freq is *)
f1star=Sqrt[t1/(masskg*factor/.298)](1/(2*.298)) = 761.528

 Log[2,f1star/fsharpfreq]*1200 =  49.6713  (* Change in frequency in cents *)
      (* differs only by half a cent from calculation without stretch factor *)

(*******************************************************************************)

 (* These are the same change-in-frequency numbers as for the smaller diameter.  So diameter doesn't matter. *)  
(*******************************************************************************)


(* Let's look at gut for the same parameters *)
(* Density is 1.3 ,   Young's modulus is 0.15*10^9  *)

diam = 0.430; (* mm *)
areamm2= (Pi/4)*diam ^2;(*mm^2*)
volmm3 =areamm2*298; (* mm^3 *)
volcc=volmm3/1000;
massgm=volcc*1.3; (* Density of gut *)
masskg=massgm/1000 ;(* kg  *)
t0=4*masskg*.298*fsharpfreq^2  = 36.7209       (* Starting tension in Newtons *)


(* t/area = stress; deltalength/length = strain;  stress = ym* strain,  ym in Newtons/square meter*)
(* So New t = old t + ym*area*(deltalength/length)    *)

 ymGut = 0.15*10^9; (* Youngs modulus *)
 aream2=areamm2/10^6

 (* The tension goes up  *)
t1=t0+ymGut*(0.17/373)*aream2  = 36.7308

 (* So the  new frequency is *)
f1= Sqrt[t1/(masskg/.298)](1/(2*.298)) = 740.089

 Log[2,f1/fsharpfreq]*1200 =  0.233998     (* Change in frequency in cents *)

 (* Allow for stretching *)
(* 373 mm stretches 0.17 mm *)
(* So vibrating mass goes down by *)
factor = 373./(373+0.17) = 0.999544

 (* So new freq is *)
f1star=Sqrt[t1/(masskg*factor/.298)](1/(2*.298)) = 740.258

 Log[2,f1star/fsharpfreq]*1200 = 0.628425 (* Change in frequency in cents *)

(**********************************************************************************)
(*So the same peg or fine tuner manipulation that raises the pitch of a steel string by about 50 cents  raises the pitch of a gut string about 1/2 of one cent  *)
(**********************************************************************************)

(*********************)
(* See if pitch makes a difference *)
(* Highest Hardanger understring is at B.  It is steel. *)
(* Actually nut to peg is longer, but here will use the same number *)

semitone = 2.^(1/12);
bfreq=afreq*semitone ^2

(*  f=Sqrt[t/(m/l)]/(2*l);  t= tension Newtons;  m= mass of string Kg;  l = length in meters*)
(* Solve for t:    t= 4mlf^2     *)

diam = 0.215; (* mm *)     (* Actually 0.22mm for usual string.  But keep same number as "E" string *)
areamm2= (Pi/4)*diam ^2;(*mm^2*)
volmm3 =areamm2*298; (* mm^3 *)
volcc=volmm3/1000;
massgm=volcc*8.; (* Density of steel *)
masskg=massgm/1000 ;(* kg  *)
t0=4*masskg*.298*bfreq^2 =  25.1651    (* Starting tension in Newtons *)

(* t/area = stress; deltalength/length = strain;  stress = ym* strain; ym in Newtons/square meter*)
(* So New t = old t + ym*area*(deltalength/length)       *)

 ymSteel = 200.*10^9; (* Youngs modulus *)

 aream2=areamm2/10^6

 (* The tension goes up by the same number of Newtons as for the "E" string*)
t1=t0+ymSteel*(0.17/373)*aream2  = 28.4744

 (* So the  new frequency is *)
f1= Sqrt[t1/(masskg/.298)](1/(2*.298)) = 525.354

 Log[2,f1/bfreq]*1200 = 106.945
 
 (* ************************** *)
(* Allow for stretching *)
(* 373 mm stretches 0.17 mm,     So vibrating mass goes down by *)
factor = 373./(373+0.17)

 (* So freq is *)
f1star=Sqrt[t1/(masskg*factor/.298)](1/(2*.298)) = 525.474

 Log[2,f1star/bfreq]*1200   = 107.339 (* Change in frequency in cents *)
   (* Allowing for stretching differs only by about half a cent *)

(**************************************************************************)
(* So, for a steel string,  the same peg or fine tuner manipulation that raises the pitch from F# by about 50 cents raises the pitch from B more than 100 cents.  *)
(**************************************************************************)
 

 

 

The Hardanger's string length between the nut and bridge is only 298mm?

[Salve Håkedal]

Approximately so, yes. It's not a strict standard. I.e. my fiddles have neck stop at 112mm and body stop at ~185mm. (String length close to 300mm.)