Feeling the tension (2): Calculating the tension on a piano string

In the last post, I promised that I would explain how to calculate the tension on a piano string, so I'm going to do that now. To do this, we will need to know three things:

• The speaking length of the piano string - that is the length between the capo bar and the upper bridge pin;
• The diameter of the wire - this should ideally be measured with a micrometer to get an accurate reading, since a small error may make a significant difference to the calculation; and,
• The exact pitch of the note (in Hertz or cycles per second).

With the pitch of the note, there are two ways to approach the matter - if you are so minded, you could use a chromatic tuner to ascertain the precise pitch of the note, or you could simply work on the assumption that the piano is at standard pitch (A above Middle C = 440Hz) which very often will be the case (though it is not uncommon for older pianos to be at a lower pitch).

A table of theoretically correct frequencies for each note on a piano can be found here. It should be noted that, on any well-tuned piano, the actual frequencies of notes outside the middle octaves may deviate somewhat from this, because of octave "stretching" - which makes the piano sound much better. Notes in the bass may be slightly flat of the "theoretically correct" frequency, those in the treble slightly sharp.

The Mersenne Equation 

At this point, enter the hero of our story - Marin Mersenne (1588-1648), who was a French monk, theologian, scientist, mathematician and "Renaissance Man", particularly noted for his contribution to acoustic theory.

One of Mersenne's most famous mathematical concepts was the "Mersenne Prime", namely prime numbers with the form:

Where n is a whole number. In fact, the six largest known prime numbers (at the time of writing) are all Mersenne Primes - this is due to the fact that they are easier to test mathematically than other prime numbers.

But on with the acoustic theory - the equation that concerns us is this one:

The Mersenne Equation:

Where F is the fundamental frequency of the note, L is the length of the string, T is the amount of tension on the string, and µ is the mass of the string per unit length. Put another way, this explains what are called Mersenne's Laws, viz., that the frequency is inversely proportional to the length, proportional to the square root of the tension and inversely proportional to the square root of the mass per unit length. This result is called Mersenne's equation (sometimes known as the Mersenne-Taylor equation or the Ideal String equation). The reason for the second alternative name is that the formula assumes that the string is perfectly flexible (zero stiffness) which is not the case in reality, and for that reason it is not quite perfect, but it will still provide an extremely good estimate for piano strings. (As a point of interest, one Galileo Galilei, who was a frequent correspondent of Mersenne's, also worked out the same thing, but Mersenne gets the credit because he demonstrated it experimentally).

We then need a set of units of measurement that will work. I won't go into the reasons for this, but if the following units are used, then the calculation will be correct:

• F (frequency) in Hertz (Hz)
• L (length) in metres (m)
• T (tension) in Newtons (N)
• µ (mass per unit length) in kilogrammes per metre (kg/m)

(There are other combinations of units that will work).

We then need to rearrange the equation above, because we are trying to calculate T, the tension. This gives the following result:

Before we can proceed any further, we also need a formula for µ, the mass per unit length of the string, which is as follows:

Where:

• π is the mathamatical constant pi
• d is the diameter of the wire (in metres)
• ρ is the density of the material from which the wire is made (in kg/m³) 

A sensible value for the density of high-grade steel used in piano wire is 7.85 g/cm³, which is 7,850 kg/m³ in the units we need to use. For the highest bass strings, according to this website, a value of 7.4 g/cm³ (7,400 kg/m³) is appropriate, ranging gradually down to 6.9 g/cm³ (6,900 kg/m³) for the thicker double-wound strings at the very bottom; this is because, although copper (used in the windings) is more dense than steel, the wound strings include a significant amount of air in the column.

Using the calculation in practice

This process can be demonstrated in practice using the middle C string of a Yamaha U1 upright piano, as follows:

I have removed the action of the piano to allow the string to be measured with a rule. In this case I haven't taken out the celeste rail (the piece of felt at the top) - this needs to come out for tuning to allow access to the pins. The top of the speaking length is the capo bar which is a ridge just underneath the pressure bar (the silver-coloured bar with the screws in it). I measured the length of the piano string with a rule and its diameter with a micrometer (seen in photo). Different strings on the same note will always have the same speaking length.

So you can see a little more clearly, here's a picture showing the pins, the pressure bar and the capo bar just below it without my arm in the way. On grand pianos, the end of the speaking length may be on the underside of the frame for some of the strings.

This shows part of the frame below the level of the keyboard. Each string passes across the bridge (the piece of wood sticking up in the middle of the photo), which transfers the vibration of the strings to the soundboard behind it. There are two pins attaching each string to the bridge - the upper one is the bottom end of the speaking length (in most cases there are three strings per note). In this case, the  middle C strings pass over the bridge towards the top right of the photo, behind the bass strings.

The speaking length of the middle C string is 0.665m and the diameter is exactly 1mm (0.001m).

Using the formula:

d² = 0.001² = 0.00001 m²
ρ = 7,850 kg/m³ (remembering, ρ = 7,850 kg/m³ for the steel strings or between 7,400 kg/m (upper bass copper-wound strings) and 6,900 kg/m³ (lower bass strings).

Multiplying up, we get µ = 0.006165 kg/m (i.e. one metre of the string weighs 6.2 grammes).

Then with the formula:

We get:

µ = 0.006165 kg/m
F = 261.626 Hz (the pitch of Middle C in equal temperament when the piano is at standard pitch of A = 440 Hz).
So F² = 68448.2
L = 0.665m
So L² = 0.442225

And the overall equation has the result T = 746 N. The result we get is measured in Newtons, which is a scientific unit of force - as any physicist will tell you, a kilogramme (or a pound) is a unit of mass, not force. However, a kilogramme force can be defined as the downward force exerted by a mass of one kilogramme in the gravitational field at the earth's surface. We can get this figure by dividing our result (746 N) by the physical constant g = 9.81m/s² which is the rate of acceleration of an object in freefall towards the earth.

This gives us our final result of 76.0 kilogrammes force. Assuming the tension on all 218 strings of the piano is roughly the same (as it almost certainly will be on a modern instrument), we can calculate that there will be approximately 16.6 tonnes of total pressure on the cast-iron frame.

One interesting point here is that Samuel Wolfenden, in his "Treatise on the Art of Pianoforte Construction" written in 1916, gives a set of model dimensions for a piano scale, in which the diameter of wire used on the middle C string is 1mm and the length is 0.688m. If this piano were tuned to A = 440 Hz, it would require a higher string tension of 81.4 kilogrammes force, but bearing in mind that Wolfenden was actually aiming for an older pitch standard then in use of A = 435 Hz, it can be seen that the modern U1 uses remarkably similar string dimensions and tensions to those that would have been employed on a high-quality piano from 100 years ago.

There is actually a great deal more that can be said about piano scale design, but as this post has already got fairly long and technical, I'll save that for another occasion.