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.

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.

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 ]