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#648834 - 12/26/01 10:30 PM Piano string size and length request
ayu_bearz Offline
Junior Member

Registered: 12/12/01
Posts: 9
Hello,
I'm a student working on a math project on pianos. In one section of the project I have decided to do it on the length and size of piano strings and how they relate to produce the sound.

eg. if the string is longer, the sound is more.......if the string is thicker, the sound is more.......

I need the measurements of how thick and how long the strings are from the lowest pitch to the highest pitch.
I would need the information in order to be able to draw some kind of conclusion of piano strings and how they relate to math.
Can someone plz help me and give me the information.

Thanks \:\)

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#648835 - 12/27/01 12:08 PM Re: Piano string size and length request
EricL Offline
Full Member

Registered: 06/04/01
Posts: 140
Loc: Upstate NY
It is not clear from your post what level you are at, and whether the project you are working on is a high school or college math project.

In any event, the equation that relates the different parameters including the effect of inharmonicity is

f_n=n*f1*(1+J*n^2)

where

n = frequency (or mode) number
f1 = fundamental frequency = sqrt(T/(4*pi*d))/(L*r)
J = inharmonicity factor = (pi^3*r^4*E)/(8*T*L^2)

in which

pi = 3.1416
T = string tension
L = string length
r = string radius
d = density of string material
E = modulus of elasticity of string material

Because this message board is not set up for equations, I have used the following symbols in the above equations

sqrt means square root
* means multiplication
/ means division
^ means power
_ means subscript

If you ignore inharmonicity (by setting J=0), the equation reduces to

f_n = n*f1

which is the equation for the n-th natural frequency of an ideally flexible string supported at both ends by frictionless hinges found in a number of physics textbooks.

Also, strictly speaking the above equations apply only to unwound strings. However, for practical purposes, one can use the above equations for wound strings provided that effective E, effective d, and effective r are used in the equations.

Hope this helps.

Eric

[ December 27, 2001: Message edited by: EricL ]

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#648836 - 12/27/01 02:00 PM Re: Piano string size and length request
SteveY Offline
1000 Post Club Member

Registered: 11/06/01
Posts: 1820
Loc: NJ
whooooosh!!!
(the sound of the last post going over my head)
_________________________
PianoWorld disclaimer: musician, producer, arranger, author, clinician, consultant, PS2 aficionado, secret agent...

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#648837 - 12/28/01 12:49 AM Re: Piano string size and length request
ayu_bearz Offline
Junior Member

Registered: 12/12/01
Posts: 9
um......ok.....
i'm a 10th grade student.
haha, i don't think i understand the equation above......

um.....is there no such written measurements for every single piano string in a standard piano that someone has and that someone can give me?!

thanks anyway, i will try and understand the equation, but maybe perhaps you can but it in a more simple and easy version? so i can understand~~ thanks!!

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#648838 - 12/28/01 01:06 PM Re: Piano string size and length request
EricL Offline
Full Member

Registered: 06/04/01
Posts: 140
Loc: Upstate NY
 Quote:
Originally posted by ayu_bearz:
um......ok.....
i'm a 10th grade student.
haha, i don't think i understand the equation above......[/b]


If this is a high school math project, you definitely don't need to go into such depth. In any event, I do want to commend you on your interest in this subject. You have an inquisitive mind, and if you continue your hard work in school, I am sure you will be very successful in the future.


 Quote:
um.....is there no such written measurements for every single piano string in a standard piano that someone has and that someone can give me?![/b]


Unfortunately, the answer to your question is NO. Pianos come in different sizes. Even for a given size, different manufacturers design their pianos differently. Because of the different scale designs in different pianos, there is no such thing as 'standard' string scale.


 Quote:
maybe perhaps you can but it in a more simple and easy version? so i can understand~~ thanks!![/b]


Okay, to make the problem more tractable, let's make some (very grossly) simplifying assumptions. If we ignore the inharmonicity effect, the (higher) harmonics effect, the (flexible) anchorage effect, the multiple stringing effect, and the impedance effect, we can use the equation for f1 (the fundamental vibrating frequency of a perfect string) in my earlier post to help explain things. The equation is reproduced as follows:

f1 = sqrt ((T/(4*pi*d))/(L*r)

Now, let's replace the density d in the above equation by the term m/(pi*r^2), where m is the mass per unit length of piano wire and (pi*r^2) is the cross-sectional area of the piano wire. The equation can now be written as

f1 = sqrt(T/m)/(2*L)

Knowing that pitch is related to the fundamental vibrating frequency f1 (the higher the value of f1, the higher the pitch, etc.), we can make the following conclusions:

1. Higher pitch can be achieved by increasing the tension T in the string, reducing the mass per unit length m of the string, or decreasing the string length L.

2. Lower pitch can be achieved by decreasing the tension T in the string, increasing the mass per unit length m of the string, or increasing the string length L.

In a real piano, higher pitch (for the notes in the upper register of the piano) is achieved by using smaller diameter piano wire (i.e., by reducing m) AND shorter string length (i.e., by reducing L). Lower pitch (for the notes in the middle register of the piano) is achieved by using larger diameter piano wire AND longer string length. As one goes down to the bass, the theoretical length required for the strings to produce the specific pitch is so long that it is not realistic to use just plain wire. As a result, in this region the lower frequency is achieved primarily by increasing the mass per unit length of the wire by wounding copper wire over the steel wire.

The only thing that contradicts the above conclusions is the tension in the wound strings is actually higher (not lower as inferred from the equation) than the unwound (or plain) strings. This is because these wound strings will not give out a good tone if they are not pulled tight.

Lastly, if you are really interested, the string length of the middle C (C4) is usually around 62.5cm (24.5 in.) and the string length of the highest C (C8) is about 5cm (2 in.) long. The strings in the bass and lower tenor sections vary widely depending on the size of the piano and its scale design.

I hope this helps, but bear in mind that the above is a (very) simplified explanation of a (very) complex problem.

Eric

[ December 28, 2001: Message edited by: EricL ]

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#648839 - 12/30/01 09:39 AM Re: Piano string size and length request
ayu_bearz Offline
Junior Member

Registered: 12/12/01
Posts: 9
Hello~

Thanks for everyone that have given answers to my question.....i have understand more about piano strings..and i will try my best to complete that section of my math project on strings
But I'm sure as i do other sections of my project i will come across OTHER questions about piano~
and that time i guess i will come back here again and get answers from ALL u KIND people \:\)

THANKS!!

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