More on piano strings

I feel that some important details must be elaborated in regards to piano string tensions. First, let us consider the following model of using a hanged weight to tune a string made from an available tensile material:

|________ L _______
O O|
|
|
W

where the vibrating lenght of the string (L) is the portion between the two bridges, producing a certain frequency due to pull of W.

If we agree that W is equal to T in every way, the equation for calculating the tension was:

4 F^2 L^2 Mu
W= __________

n^1

And the relationship for Mu (mass per unit lenght) was:

Mu = Pi * r^2 * rho.

By increasing the weight, we would be introducing a longitudinal perturbation to the tensile medium, which would result in an increase in frequency as demonstrated below:

=========
=========
=========
=========
=========
|
|
|
W

=========
=
=========
=
=========
=
=========
=
=========
|
|
|
2W

This demonstration assumes a uniform microscopic modular structure of the string viewed from the side, where the additional force, by drawing the modules apart, results in a lower Mu within the vibrating L.

In other words, an increase in tension directly causes the density within the given lenght to decrease, where, after reaching a certain limit, the Mu of a certain cross-sectional area will rapidly drop to zero and the string will snap.

Of course, the strenght of the material chosen (measured by the relationship for Mu) is a determining factor in this.

However, aside from this longitudinal perturbation, the hammers of the piano introduce an additional force known as the transversal perturbation. This effect is demonstrated below:

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The illustration attempts to show that a string struct by a hammer will travel a short distance in either direction throughout the course of its oscillation. This effect is the magnitude, in other words, the amplitude of sound.

A first order approximation that disregards this effect would require that W, L, F, and Mu are all taken as constant throughout.

A second order approximation might lead one to assume W and Mu to be constant while the lenght varies inversely proportional to F, so that fff (forte fortessimo) hammer-strikes would lower the frequency.

In contrast, a complete model would require that all the variables change when the transversal perturbation is introduced into the medium. This means, that the string not only elastically extends in lenght during oscillation, but also, that the additional stretch lowers its density considerably.

An auditory experiment shows that fff hammer-strikes differ from a ppp hammer-strikes by a barely perceptible amount. The louder passages are thus heard to be higher in frequency, leading us to believe that the density decreases much more rapidly than lenght increases for oscillations with great amplitude.

If a given piano string is calculated to break at about 2W (in other words, 2T), a vectoral equivalance shows that:

(2T)^2 = T^2 + X^2

where T is the horizontal vector of tension, while X is the perpendicular (transversal) vector of tension required to break the string.

Thus,

(2T)^2 = T^2 + X^2

4(T^2) = T^2 + X^2

X^2 = 3 (T^2)

X= sqr {3 (T^2)}

X= (sqr 3) * (sqr T^2)

X= (sqr 3) T

X= 1.73 T

All of this means, that a hammer strike delivering an additional force equivalent to 0.73 T will break the string just as 2W does.

The angle of oscillation can thus be calculated:

2T
/ |
/ |
/ | X
/ |
/ V
----------------->
T

cos alpha = 1/2 = 60 degrees.

It can be seen that the string must vibrate with an angle of 60 degrees in reference to its (horizontal) plane before it can be made to snap.

However, this calculation assumes a homogenous structure and does not consider the uniform density of the material, dislocation of its atomic components and metal fatigue, all of which would eventually cause the string to break after just a few degrees of displacement.

In short, there is much more to strings than just the radii.

Cordially,
Ozan Yarman

----- Original Message -----
From: Cris Forster
To: Tuning
Sent: 30 Mayıs 2005 Pazartesi 19:02
Subject: [tuning] Aspects of restringing and retuning pianos

Dear Fellow Builders,

Good morning!

I am offering the following article to those of you who may have
been wondering why the science of restringing pianos and the art
of retuning pianos is not a well-defined subject.

>From a technical perspective, the most difficult aspect of
restringing a piano is to understand the interlinked relationships
between variables. An introduction into this subject requires
three initial steps.

(1) Consider the following equation for calculating the tension of a
string:

T = 4 * F^2 * L^2 * Pi * r^2 * rho

where F is the frequency, in Hz; L is the vibrating length of the
string, in meters; r is the radius of the string, in meters; and rho is
the mass density of the stringing material, in kg/m^3.

This equation includes the following subequation:

M/u.l. = Pi * r^2 * rho

where M/u.l. is the mass per unit length of the string.

For a quality piano, typical properties for middle G, or the G above
middle C, are

F = 392.0 Hz
L = .489 meter (19.25 in.)
r = .000483 meter (.019 in.)
rho = 7833 kg/m^3 for spring steel [ .00073299 (lbf * s^2)/in^4 ]

so that,

M/u.l. = 3.1416 * .000483^2 * 7833 = .00574 kg/m

and,

T = 4 * 392.0^2 *.489^2 * 3.1416 * .000483^2 * 7833

T = 843.8 Newtons (189.3 pounds-force)

(2) To calculate a string's break strength, we must know the
tensile strength of the stringing material. For spring steel, tensile
strength values range between 300,000 psi -- 400,000 psi, or
between 2.07 * 10^9 Pa -- 2.76 * 10^9 Pa. (Pa short for Pascals.)
An average value of 350,000 psi is equivalent to 2.41 * 10^9 Pa.

First, calculate the string's cross-sectional surface area:

S = Pi * r^2 = 3.1416 * .000483^2 = 7.329 * 10^-7 sq. m

Now calculate the string's break strength by multiplying the
surface area times the average tensile strength of spring steel:

Break Strength = (7.329 * 10^-7) * (2.41 * 10^9) = 1766.3 N

Finally, determine what percentage 843.8 N is of 1766.3 N. Divide
the smaller number by the larger, and multiply the quotient by 100:

(843.8 / 1766.3) * 100 = 48

Since most piano strings are tensioned between 40% and 50% of
their break strength, 48% is within the desired range.

(3) Finally, we must evaluate the inharmonicity of this string.

The quote below is from my manuscript _Musical Mathematics: A
Practice in the Mathematics of Tuning Instruments and Analyzing
Scales_.

To read the Table of Contents of this work, please logon at:

http://www.Chrysalis-Foundation.org/musical_mathematics.htm

Here is an excerpt from the beginning of Chapter 4:

******************************

"On any given string, a true harmonic series could only occur if the
string were perfectly flexible. Since all strings exhibit varying
degrees of _stiffness_, the flexible string model no longer applies.
Stiffness causes the modes to vibrate at frequencies considerably
higher than suggested by Equation 4.1. For this reason, we call
the sharp mode frequencies of stiff strings inharmonic frequencies
[ IF(n) ], a term that refers to non-integer multiples of the
fundamental frequency."

******************************

To calculate the IF(n) requires a dimensionless stiffness
parameter ( J ). There are two different types of equations for J:
one includes a tension ( T ) variable; the other includes a
frequency variable ( F ). In the former equation, J is directly
proportional to D^4, where D is the diameter of the string. In the
latter equation, J is directly proportional to D^2. Therefore, as the
diameter or the thickness of the string increases, the stiffness
parameter increases as well. This tells us that for thin strings the
upper modes of vibration are more in tune than for thick strings.
For the above-calculated G-392.0 Hz string, J = .0002088. If we
substitute this value into an equation for IF(n), and if we then
calculate the inharmonic frequency of the second mode of
vibration, the result is IF(2) = 784.49 cps. In a perfectly flexible
string, F(2) = 784.00 cps. Consequently, the second mode
sounds 1.083 cents sharp. Similarly, the third mode sounds 2.888
cents sharp, and the fourth mode sounds 5.415 cents sharp.

(( _Musical Mathematics_ contains detailed equations for
calculating J and IF(n). If you would like to place an advance
order for this book (scheduled for publication by the Chrysalis
Foundation Press in Spring of 2006), please logon at

http://www.Chrysalis-Foundation.org

to learn more about this publication and the ordering process.
Thank you. ))

For pianos tuned in equal temperaments, inharmonic mode
frequencies (within conventionally accepted limits) pose no
significant problems. Since the tones of these scales (with the
exception of the "octave") are tuned to irrational frequency ratios,
and since in stiff strings the upper modes of vibration sound
irrational frequencies, there exists no instantly recognizable
conflict.

However, when conventionally strung pianos are tuned in just
intonation, where all the tones are tuned to rational frequency
ratios, intonational problems instantly become audible.

To read a description of my Just Keys piano, which I restrung
three times and retuned four times in 1990, please logon at:

http://www.Chrysalis-Foundation.org/instruments.htm > Just Keys

In pianos, the bass and treble bridges are glued to the
soundboard. Since they are not moveable, one has no choice in
determining the vibrating lengths of piano strings. Therefore, once
a scale has been decided, the diameters of strings constitute the
most critical physical variables in restringing pianos.

To summarize:

(1) Tension is directly proportional to r^2. (2) Break Strength is
directly proportional to r^2. (3) J is either directly proportional to
D^4, or to D^2.

The science of restringing and the art of retuning pianos are
critically determined by either the radii or the diameters of strings.

OK. Now, let's open the lid of this Bösendorfer Imperial,
take out all the strings, and get it right the first time!

Sincerely,

I-am-an-instrument-in-need-of-tuning

AKA

Cris Forster, Music Director
www.Chrysalis-Foundation.org