Tuning pin comparisons

[Phillip L Ford]

Ed,
I expanded a little bit on your numbers.

I took a couple of specific strings.  I hope the numbers that I picked at random
are somewhat realistic.
I will compare a .272 in. diameter pin to a .286 in. diameter pin as you did.

String 1

Frequency 4000 hz
Wire diameter .031 in.
Total length of string from tuning pin to hitch pin 10 in.

For the .272 in. pin.  At center of wire circumference = .952 in.  which gives
.00264 in./degree of pin rotation

For the .286 in. pin.  At center of wire circumference = .999 in. which gives
.00277 in./degree of pin rotation

Assuming a 6 degree pin rotation (by the way, why did you assume this? - it seems
quite a bit to me).

Total strain on wire for .272 in. pin = 6 x .00264 = .01584 in.  Assuming steel
wire load increase on wire to give this strain is 35.9 lb.

Total strain on wire for .286 in. pin = 6 x .00277 = .01662 in.  Load increase on
wire to give this strain is 37.6 lb.

First thing is the larger pin causes a 4.7 % greater load increase in the wire for a
6 degree rotation.  That doesn't seem insignificant.

Second, if we assume the original load was 150 lbs. then the frequency should
increase by the square root of the ratio of the new load to the original load.  So,

For .272 in. pin new frequency is (185.9/150)*1/2 x 4000 = 4453 hz

For .286 in. pin new frequency is (187.9/150)*1/2 x 4000 = 4473 hz

So, the larger pin would cause the string to be 20 hz sharper for a 6 degree rotation.
That seems noticeable to me.

String 2

Frequency 100 hz
Wire diameter = .043 in.
Total length = 60 in.

For .272 in. pin.  At wire center circumference = .990 in. which gives
.00275 in./degree of pin rotation.

For .286 in. pin.  At wire center circumference = 1.034 in. which gives
.00287 in./degree of pin rotation.

Following the same procedures as above I end up with

Larger pin causes 4.2% greater increase in tension for 6 degree rotation.

For .272 in. pin new frequency = 103.9 hz

For .286 in. pin new frequency = 104.1 hz

So, the difference for 6 degrees of rotation is .2 hz.  Noticeable?  Maybe, since
on these strings you are trying to make very slight changes because even a
very small mismatch is quite audible.  So, apart from other considerations about
pin size, it seems possible that even a small reduction in pin size may affect the
degree of control available to the tuner.

Phil F

On Fri, 25 Jan 2002 11:45:02  
 A440A wrote:
>Greetings, 
>   I have just finished Paul's article this month in the journal.  
>Interesting views can be found on all aspects of tuning pins, since we all 
>spend so much time with them,  and I am wondering what the numbers can tell 
>me. Specifically, the difference in tuning control between sizes of pins.  I 
>don't think that the increased radius of the larger pin is a significant 
>factor.   I find (admittedly, I am NO math whiz, so if I have missed a step, 
>please disregard everything that follows), that by determining the 
>circumferance of two different sizes of pins, and then relating that to 
>degrees of movement, the differences begin to seem academic. 
>    Here is how it looks to me: 
>
>   A pin that is .272" in diameter has a cir. of .85408".  This equates to 
>.00237" per degree of rotation.  A .286" pin, by the same calculation has 
>.0025" per degree of rotation. This means that the larger pin will move the 
>string approx.  .0001" more per degree of rotation.  If we consistantly move 
>pins by increments of 6 degrees in fine tuning, then the difference in pin 
>size accounts for maybe .0006" (that is 6 ten-thousandths!!) difference in 
>string length being pulled around the pin.   ( I have omitted the 1/2 string 
>diameter from the circumferance equations,since that is a variable on a per 
>string basis, though increasing the diameters of the two calculations would 
>further reduce the percentage difference between them). 
>   Since difference as it relates to tuning is based on changing the tension 
>per degree of rotation, and tension/pitch relationships are functions of the 
>square,  I have to ask just how much difference can be found from  .0006" of 
>string movement, at the pin?  I don't think it would be a discernable 
>quantity. Others?
>Regards, 
>Ed Foote RPT
>