Stringing theory

The project has now reached an important milestone: it's time to string the instrument.

Choosing the appropriate string diameters (gauges, to use the technical term) will have a significant impact on the sound of the resulting instrument. If one looks at the wide range of available gauges and string materials, the question immediately arises: how do you know which string material to use, and which gauge to choose for each note?

The first question is a little easier to deal with. Historically, harpsichords have been strung with iron, yellow brass and red brass. Some of earliest Italian harpsichords appear to have been strung with iron: a practically zero-carbon iron, high in phosphorus, which produces a strong yet flexible wire. If true, this choice of string material would have yielded an overall pitch level about a fourth lower than brass-strung instruments. Iron strings need a longer scale than the short scale typical of Italian instruments, so if they are used on an Italian harpsichord, the effect is as if the notes on the keyboard have all shifted leftward to longer strings. As musical requirements changed, it appears that the early instruments were converted to brass stringing, which also brought them more or less into the range of pitches familiar to modern players of Baroque music.

Yellow brass (a 70%-30% copper-zinc alloy) is the most suitable material for this harpsichord project. Red brass (a 90%-10% copper-zinc alloy) is scarcely used in Italian harpsichords, though some modern makers find it useful on a few of the very lowest notes.

Any consideration of a stringing plan needs to remember the following points, which entered into the picture back in the design phase:

1. The string scale cannot be too short because then, at the chosen pitch level, the strings will be too slack and will sound strange.
2. On the other hand, if the string scale is too long, the strings will break as they are tuned up to the chosen pitch level.

In striking an effective balance between these two factors, the harpsichord maker sets the operating tension of the string band. If the string scale is well designed, the chosen pitch level will require that all the strings be tuned to within a few semitones of their breaking point. The instrument will sound good and the problems above will be avoided.

A handy thing to do, early in the design process, is to calculate the tension of each string to see if there will be any problems with the chosen scale. The tension is calculated from the string diameter, length, pitch, and density of the string material as follows:

T=(ρπ/g)(fld)²

T=tension, in kg

ρ=density of the string material, in kg/m³

g=the gravitational constant, 9.8 m/s²

f=pitch frequency, in Hz

l=string length, in m

d=string diameter, in m

For yellow brass, the wire makers give ρ=8536 kg/m³. The string length comes from a Pythagorean scale based on c''=273 mm (except below c, where the strings foreshorten). The frequency of each note also follows a Pythagorean scale, meaning that neighbouring notes differ by 1/12 octave. Diameters depend on what the wire makers produce; most of them make (in inches) 0.008, 0.009, 0.010, 0.011, 0.012, 0.013, 0.014, 0.016, 0.018, 0.020, 0.022 and so on.

With this information, I created a spreadsheet that shows the tension on any note of the compass for any chosen diameter of wire. Note that wire gauges in this chart are in millimetres, not inches:


The orange line identifies the point at which the Pythagorean scaling stops, so the top portion of the chart does not accurately represent the bass string lengths, which are actually shorter than Pythagorean. Therefore the real string tension in this region is smaller than what is shown.

Upon reading the chart, an interesting phenomenon quickly becomes apparent: wires of the same gauge actually have the same tension, irrespective of pitch or length, provided they fall within the Pythagorean part of the scale below the orange line. This is no coincidence: in the equation above, the frequency increases by the same factor that the string length decreases, as one goes from left to right across the instrument, so the changes effectively cancel each other out.

The breaking point of the wire has to be known to determine if any of the tensions are excessive. The wire maker provides this information in the form of a tensile strength figure: the stress (tension/unit area) at which the material breaks, in PSI or MPa. Centuries ago, the old makers would have determined this empirically using a monochord. They set a specific length for the wire and cranked it up to a chosen pitch, noting whether it broke before it got there.

However, the breaking stress is not directly useful. A harpsichord wire is subject to additional stresses and friction as it passes around bridge pins, nut pins and tuning pins. As the wire bends, it is subject to compression on the inside of the curve and tension on the outside, experiencing a level of stress some 15-20% higher than in the straight sections. So a wire cannot safely be tuned just a little below its mathematical breaking point, because it will encounter stresses greater than the breaking point in several places. On top of that, an additional safety factor of about 20% must be included to guard against swings in humidity or clumsy tuning, both of which can increase the overall tension. The maximum "safe stress" is therefore about 1.4 times less (2 × 20%) than the breaking stress, which corresponds to a decrease of several semitones in the highest pitch the string can safely sustain.

Moreover, although it is true that thicker wire is capable of bearing a greater tension, one cannot solve the problem of an excessively long scale by putting on "stronger" (i.e. thicker) wire. A thicker wire will require increased tension to reach the same level of pitch as a thinner wire, and that extra tension will cancel out the greater strength of the thicker wire. In fact, the rule of thumb is that wires of various gauges will basically break at the same pitch level, regardless of their diameter, as long as they are of the same metallurgy.

After all this discussion, the question still remains: which gauges are used? The tension chart only shows that the wires won't break at the chosen pitch level. One could actually draw the odd conclusion that the entire harpsichord could be strung with a single gauge of wire. Of course, this is nonsensical. Most people appreciate the fact that a thin wire naturally produces a higher pitch and a thick one a lower pitch, and on any stringed instrument the diameters certainly increase as one descends the compass. The truth is that, as the great pioneering harpsichord maker Frank Hubbard said, "One strings by ear, not by physics". The wire must be chosen to produce a good overall sound and balance the tonal qualities of the various regions of the instrument's compass. And how to do that is the subject of the very next post...