Stringing practice

After all the verbiage in the previous post, the question still begs to be answered: how does one actually string a harpsichord?

The old makers often used a system of numerical progression. For Italian harpsichords with a scale of about c''=280 mm, they used a simple rule of thumb like this, starting from the top down:

10 wires of #10 gauge
9 of #9
8 of #8
...

and so on.

There is a certain numerological elegance to this system, and possibly that was part of its appeal. From a practical standpoint, as one descends the compass, the string gauges change more frequently. This makes sense because the sounding length of the strings changes rapidly as the bridge curvature straightens out, which the progressive stringing system takes into account.

Gauge numbers are sometimes found inked or stamped onto the wrestplanks of old harpsichords, showing which gauges were used and where they changed. The numbers in German and Italian harpsichords correspond to the old Nürnburg gauge system, which had at least 10 different diameters. Based on measurements of surviving wire fragments, the closest modern equivalent diameters are:

#10 = 0.008" = 0.20 mm
#9 = 0.009" = 0.23 mm
#8 = 0.010" = 0.25 mm
#7 = 0.011" = 0.27 mm
#6 = 0.012" = 0.30 mm
#5 = 0.013" = 0.33 mm
#4 = 0.014" = 0.36 mm
#3 = 0.016" = 0.40 mm
#2 = 0.018" = 0.46 mm
#1 = 0.020" = 0.52 mm

Note that the U.S. and metric units are not exact conversions of each other (for example, 0.020"=0.508 mm, not 0.52 mm). The chart is, as stated, a list of the closest available modern diameters.

Our knowledge of the Nürnberg gauges is complicated by the fact that exacting measurements of surviving wire are skewed by centuries of corrosion. Another significant issue is the gradual increase over time in the diameter of historical wire as the holes in the draw plates wore out and got larger. Draw plates were extremely valuable—literally worth their weight in silver—and wire makers were not anxious to dispose of them just because the wire was getting a tiny bit thicker. So, at best, the old gauge system represents a range of diameters instead of a single precise number.

I decided to use this system of numerical progression in stringing my own harpsichord, with one caveat. Several modern makers report that better results are obtained by stringing one gauge heavier, which means 10 wires of #9 and so on. I've adopted this modification as well.

Given the 50-note range of my keyboard, it should be clear that the stringing will end in the bass without having employed all 10 gauges shown above. An instrument with exactly 4 octaves will use 7 gauges. I'll need one more because of my extra low note. The extra pair of strings at the top (which provide c''' at A=440 Hz) are strung with #9 gauge but are not counted as part of my overall tally.

If you look at my tension chart in the previous post, you'll see two columns off to the right where I mapped out gauges by numerical progression, one column starting with #10 gauge, the other with #9. The equivalent gauge numbers are also listed horizontally just below each metric diameter along the top.

The very last wire for the note GG/BB needs a little extra thought. Since it is a third lower than the keyboard key assigned to it, I'm going to try stringing it in 0.56 mm/0.022" red brass. This is pretty thick stuff, but I have a German harpsichord here at home that has a similar GG string length, and it's strung that way. I'll find out whether that's a good idea once the instrument is up to pitch.

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...

Making the bridge and nut

The bridge and nut define the sounding lengths of the strings in a stringed instrument. On a harpsichord, both usually have a similar if not identical appearance, and their position on the instrument determines the name: the bridge is glued to the soundboard, while the nut is glued to the wrestplank's top surface. A bridge, therefore, conducts string vibrations into the soundboard while a nut, together with the heavy pinblock, reflects vibrations back towards the bridge.

There is occasionally an exception to this: some harpsichords have their nut placed on a small piece of free soundboard, meaning that the wrestplank ends in front of the nut instead of continuing all the way to the register gap. This is called a "hollow wrestplank", but is not a concern in the Trasuntino design, which places the nut on the wrestplank.

The shape of the bridge reflects the scaling choices made by a harpsichord's designer, and the exact curvature is a function of how closely the design adheres to or departs from a theoretically just scale. I have already indicated that the Trasuntino follows a just scale through much of its range, so all but the lowest octave of the bridge follows a Pythagorean scaling curve. The shape of the nut cannot be too radical as it must fit within the confines of the wrestplank. Nuts range from perfectly straight to slightly curved; typically they are closer to the gap in the treble. The Trasuntino's nut is unusual in being parallel to the front of the wrestplank.

Unlike most other stringed instruments, the harpsichord's strings do not touch the wooden bridge and nut surfaces without first bending around a metal bridge/nut pin. If a harpsichord string were to touch wood first, the tone would be feeble and choked. Therefore the cross-section of the bridge and nut must have a little slope or hollow on the edge facing the sounding portion of the string. This provides a little clearance for the string and prevents it from touching a wooden surface before it reaches the pin.

Hardwoods are the material of choice for bridges and nuts, and walnut is appropriate for Italian and certain other harpsichords (although, interestingly enough, the original Trasuntino actually has a cypress bridge). I started by obtaining a walnut offcut a bit longer than 6 feet. Since this is nearly the length of the finished bridge and I still need a nut of about half that length, I made sure my walnut was of a width that allowed me to work on both its edges, thereby producing two identical pieces. All operations were carried out at the router table.

The first step was to establish the height and width of the bridge: 13 x 8 mm, respectively. I did this by cutting out a centred groove with a straight-cutting bit, making the groove 13 mm deep and adjusting its position until the remaining material was 8 mm thick:


Next I used a portion of a router bit designed for decorating small items like jewellery boxes to cut a profile that included a cove and groove:


The cove will provide the string clearance I discussed earlier, while the groove will receive the brass pins. I deepened this groove slightly with a 1.6 mm straight-cutting bit, and then chamfered the back edge with a 25-degree chamfering bit:


This chamfer makes it easier for the strings to slope downwards on their way to the tuning pins and hitch pins.

All that remains is to cut these parts free at the table saw.

Laying out the bentside

At this point in time, the baseboard has three final edges: left, front and right. The curvature of the bentside and the tail angle remain to be determined.

Instruments built in the Italian tradition arrived at the proper bentside curvature by simply drawing a curve parallel to the soundboard bridge at a constant distance to the right. So determining the layout of the bentside is equivalent to deciding the bridge curvature, except that the bridge curvature must be highly accurate, while minor variances in the bentside are probably fine as long as everything looks good.

The bridge curvature is not too difficult to determine, as the string lengths through a large part of the compass are based on what is termed Pythagorean scaling. In a nutshell, this means that a string one octave lower than a given string is exactly twice as long, while a string an octave higher is half as long. This might sound quite obvious were it not for the fact that stringed instruments can be made without following this rule exactly. If you wanted to use a lower string less than twice as long, you could choose thicker wire to compensate. Likewise, if a higher string were more than twice as long, additional tension would bring it to the proper pitch.

Other string lengths within the octave are determined using simple whole-number ratios derived from the overtone series. Some examples:

a major 2nd = 8:7
a minor 3rd = 6:5
a major 3rd = 5:4
a perfect 4th = 4:3
a diminished 5th = 7:5
a perfect 5th = 3:2
a major sixth = 5:3
a minor 7th = 7:4
a perfect octave = 2:1 (obviously)

With these ratios, it is in fact possible to specify the length of just one string, fill in the remainder of the octave as above, then start doubling and halving string lengths from there. The one string so specified is called the scale and is usually c above middle c (c'', in Helmholtz notation). The original Trasuntino has a scale of c''=276 mm, which my friendly harpsichord maker advised me to shorten slightly to c''=273 mm. This provides a safety margin in case of major swings in humidity, as well as some protection against "ham-fisted tuning", as he put it (I'm pretty sure he wasn't talking about me). A marginally shorter scale means the instrument doesn't need to be pulled up to quite as high an operating tension as the original, as all the strings are a bit shorter. The home pitch will be A=415 Hz, with the keyboard transposing one semitone to the right.

Instead of the simple ratios above, one can approximate a Pythagorean string scaling fairly well by calculating the string lengths according to an equal-tempered scale. In that case, one multiplies successively by the twelfth root of 2 to find the next lower string, and divides by the twelfth root of 2 to find the next higher string. That's what I've done in making my calculations.

One thing to be aware of is that Pythagorean scaling in Italian instruments usually doesn't determine the entire compass, especially in the bass: if it did, the instrument would have to be quite long to accommodate the bass strings. Often the scaling is Pythagorean from the very top down to c below middle c (c), or, less often, for still another octave (C). The scaling must then alter, in any case, because most instruments have a joint in the bass bridge where it goes off to the left at a sharp angle for the last couple of notes.

My drawing of the Trasuntino has string lengths for all c and f# notes (f# is right in the middle of octave). The scale is pretty close to Pythagorean, but not bang-on. This could be due to the fact that the instrument has been rebuilt several times in the course of its existence. The manner of placing the bridge on the soundboard—it just gets bent to shape by hand and glued down—might introduce minor discrepancies as well. In any case, I calculated a Pythagorean scaling down to c quite easily, then spent the rest of the day thinking about the bass strings and their departure from this pure scale. Eventually I put down some provisional numbers based on the shrinking octave ratios in the bass (it goes to 1.922, 1.712 and so on).

Here is the procedure for laying out the curvature of the bridge on the bottom. First, I clamped one of my registers along the pencil line showing the back of the wrestplank:


This line is at an 8 degree angle and of course the register slot spacing is correct only when tilted to this angle. I put the register on edge because the bottom of each slot provides a convenient edge for my pencil to make a tick mark.

I didn't just lay the register down anywhere along this line. In looking at the drawing, I saw that the leftmost string was 30 mm from the left edge of the instrument. I decided to put my first string at 35 mm, because I'm making my instrument a bit wider. Before committing to this, I located where the first and last strings would be with respect to the leftmost and rightmost slots (about 5 or 6 mm to the side, roughly), to make sure they didn't get placed inconveniently. When everything looked good, I put a pencil tick at 35 mm and aligned the left edge of the first register slot with it before clamping.

After that, it was simply a matter of counting along the slots and putting tick marks on each c and f#. This took only a minute:


Next, I laid my home-made T-square along the baseboard and aligned the left edge with each tick mark in turn. Using a tape measure, I measured out the lengths of the c and f# strings I had already calculated and put a second tick mark on the baseboard in line with the first:


The T-square has a small nail that lines up with the nut location to hold the tape measure. This nail position is fixed because the Trasuntino's nut is straight and perpendicular to the left side. In any other instrument, the nail would have to be repositioned every time the T-square was moved.

Once all the measurements were made, I drove a 1.5" finishing nail into each second tick mark location. Then I bent a long strip of door stop composite (some sort of plastic; very cheap and a nice balance between stiff and flexible) along the nails and held it in place with spring clamps:


And that is how the bridge curvature looks. The actual bentside will be about 4" away from this. I set my dividers to 4" and got a sense of what this might look like by lightly running the dividers along the strip without making a mark. When the time comes to mark for real, I'll run a small engineer's square along the strip and place a compass set to 4" along the edge. This will accurately transfer the curve.

I should point out that the bridge position in the extreme bass is not accurately shown by the plastic strip. The furthest nail is properly placed, but the bridge does not run in a straight line to it. Instead, slightly behind the second-last clamp, the bridge keeps curving to the right and then hooks left to meet the nail.

The tail position is straightforward: at the point where the actual bentside width shrinks to 185 mm (the same width as the bass end of the wrestplank, which is unlikely to be a coincidence), the bentside ends and the tail goes off to the left at a 40-degree angle.