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Piano Tuning Pin Rotation: A Physics Problem

  1. Mar 29, 2008 #1
    Hello everyone, I am new here and so I apologize in advance
    for the many forum faux pas I am likely to commit.

    I am not particularly well educated in the field of physics,
    but I am a piano tuner by trade, and we work with issues every
    day that are basically problems of physics and engineering, and
    some us tend to be interested in the theoretical side of things.
    A discussion on our trade specific mailing list has hit a dead
    end, and so I find myself here at your collective mercy.

    Here is my question:

    Imagine a length of music wire affixed at one point to a unmoving
    anchor point, and to the other coiled around a tuning pin secured
    in a unmoving pinblock.

    When the tuning pin is rotated by a tuning "hammer" (wrench)
    The termination points remain the same, but the tension increases,
    causing the pitch to rise (obviously)

    Disregarding (for the moment) all factors of flex of the anchor
    points, and also disregarding any friction points, how would I
    calculate relationship between the rotation of the tuning pin and
    the pitch of the string?

    The known factors are as follows:

    String length: 972mm (38.267")
    String diameter: 1.09mm ( .043")
    String density: 7900 kg/m3 [is that right for high grade steel?]
    String tension: 86.583kg (190lbs)
    Tuning pin diameter: 7.01mm (.276")
    Note Pitch: 174.61hz (Note F-33)

    (I cannot guarantee the accuracy of these figures,
    so feel free to point out any errors.)

    So, I rotate the pin say, 0.5 degrees-
    How does that translate into pitch change?

    How would I calculate this on my own?

    Thanks so much in advance for your time.


    Kurt
     
    Last edited by a moderator: Aug 14, 2010
  2. jcsd
  3. Mar 29, 2008 #2

    Danger

    User Avatar
    Gold Member

    Welcome to PF, Kurt.
    Wow, what a question to bring up! I'm not the one to answer it, but I certainly look forward to seeing the responses from the guys who know what they're talking about.
    I suspect that the specific composition of the wire would be a major factor, as well as the ambient temperature. The number of turns of the wire around the peg would probably make a difference as well, since that would in essence change the properties of the peg.
     
  4. Mar 29, 2008 #3
    On a piano tuning pin, there may be 4 coils, but they lie along side each other- the wire
    never overlaps itself. The termination point will always be flush against the side of the
    tuning pin.

    The extra coils may indeed cause the pin to act stiffer, but we are ignoring flex or twist
    at the moment.



    [kurt]
     
    Last edited: Mar 29, 2008
  5. Mar 29, 2008 #4

    pam

    User Avatar

    You have to know the elasticity of the wire, given by what is called "Young's modulus", and make a calculation. But the accuracy with which you can measure the angle of turn is so much less than you could measure with beats from a tuning fork (or even by ear) , that the formula would be useless.
     
  6. Mar 29, 2008 #5

    Agreed. I guess what I am looking for is some idea of how tiny the pin movements need to
    be to get a really perfect note.
     
  7. Mar 29, 2008 #6
  8. Mar 29, 2008 #7

    Q_Goest

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    Science Advisor
    Homework Helper
    Gold Member

    Hi Kurt,
    Do you really need to know how the rotation of the pin translates to pitch change? If so, you need to determine the change in tension due to the stretching of the string, because the pitch is a function of the string tension. Here's a couple references to string pitch based on tension:
    http://www.cs.helsinki.fi/u/wikla/mus/Calcs/wwwscalc.html
    http://hyperphysics.phy-astr.gsu.edu/Hbase/waves/string.html

    To determine the change in tension due to rotation of the pin, you need to determine how much the string stretches. As pam points out, you need the modulus of the string. Steel is roughly 30,000,000 psi (change to metric) which is generally represented by E. So you have:
    S = F/A
    or: F = S A
    Where:
    S = Stress in the string
    F = Force (tension) on the string in Newtons
    A = cross sectional area of the string

    And

    S = e E
    Where:
    e = string stretch (mm/mm)
    E = Young’s Modulus

    Combining:
    F = S A = e E A

    F is the tension in the string. Use that in the calculator to determine pitch. Play with this a bit and with the calculator (see attached above) to see if you can get what you’re looking for.
     
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