The Statistical Analysis of the Lateral Pin Spacings in Some Neapolitan and Flemish SeventeenthCentury Harpsichords
Figure 1  The singlemanual harpsichord attributed to Onofrio Guarracino, RCM 175, Royal College of Music, and the anonymous doublemanual harpsichord, Antwerp, 1617, ravalé by Francois Blanchet, 1750, and by Barberini and Hoffman, 1786, Paris, private property, London.
Introduction
The way a maker marks out the lateral spacing of the bridgepins, nut pins and registerslots in a harpsichord, spinet or virginal is perhaps one of the most characteristic features of his work. By lateral pin spacing I mean the distance which separates the strings in a direction perpendicular to their length. In most, but not all, instruments the strings in the tenor, alto and treble are parallel to the spine of the instrument. However, in the bass the strings on the bridge are usually angled away from the spine at the end furthest from the player. The reason for this is clearly to make extra space between the bass end of the bridge and the spine side of the instrument. In order to improve the quality of the extreme bass notes it is necessary to make the area of soundboard around the bass end of the bridge as large and flexible as possible. Angling the strings away from the spine so that the bass end of the bridge is distanced from the spine and spine liner achieves the required area and flexibility in the bass part of the soundboard. This therefore means that the lateral bridgepin spacing is different in the bass from that in the rest of the compass.
Normally the bridge and nutpins are very regularly spaced and so an analysis of their spacing lends itself naturally to statistical methods giving a result of high accuracy and low error. What I would like to show here is that the value of the lateral pin spacing can, in turn, be used to establish the maker’s workshop unit of measurement to a high degree of accuracy. Normally the unit of measurement used in a maker’s shop is close to the unit of measurement used in the city or region in which he worked. But the high degree of accuracy of the determination of the unit of measurement using the lateral pin spacing statistical method enables the work of two or more makers working in the same city or region to be distinguished from one another. Indeed it is likely that the workshop unit of measurement determined in this way is even more characteristic of a maker’s work than the mouldings he used since there is no evidence to show that these were not ‘bought in’ to individual workshops by several different makers from a common supplier.
For example, Stefano Bolcioni, Francesco Poggio and Bartolomeo Cristofori were all working in Florence together during the same period. All of them used a value of the soldo used in Florence close to 27.56mm. However an analysis even of the case measurements shows that each of them used a value of the soldo slightly different from the other and characteristic of the individual maker. The difficulty with an analysis of the case measurements is that the unit of measurement derived in this way has a rather large error and this makes a positive distinction between workshops in the same centre rather difficult. This difficulty is solved using an analysis of the lateral bridge and nutpin spacing because of the high accuracy resulting from a statistical analysis of the lateral pin position measurements.
To my knowledge the analysis of the lateral bridgepin, nutpin and register slot spacings using statistical methods has not so far been done by anyone else. What I hope to show in this paper is how this new and powerful method is carried out, how the method can be used to distinguish one maker’s work from another, and also how it can be used to identify an individual maker.
Before beginning on an example of the use of these statistical analyses I would like to review some simple mathematical concepts.
The concept of the slope of a straight line graph
A simple concept, fundamental to the analysis of the bridge and nutpin spacing in a harpsichord, spinet or virginal is that of the slope of a straight line graph. The slope is defined simply as the change in the y coordinates of the beginning and the end of a line divided by the change in the x coordinates of the beginning and end of the straight line.
In the straightline graph shown in Figure 2 below the slope of the graph is given by
Hence if this were a graph of the pin spacing of an instrument, then the slope would give a spacing of 13.83. The 36 notes in three octaves of strings with this spacing would therefore have a width of
and this is a typical 3octave span suitable for human hands of average size.
Figure 2  The slope and intercept of the line y = mx + b
This graph and this example show how the slope of a graph could be used as a measure of the lateral pin spacing. What I would like to do now is to apply this method to a real instrument.
Lateral
pin spacing in the 1651 singlemanual harpsichord by Onofrio Guarracino
belonging to Prof. Andrea Coen in Rome.
A good example of the use of the lateral pin spacing which both helps to understand how a typical 17^{th} century harpsichord builder worked, and also how the lateral pin spacing can be used as one of the characteristic workshop features of a particular maker is the 1651 singlemanual harpsichord by Onofrio Guarracino belonging to Prof. Andrea Coen in Rome. This instrument possesses many of the characteristics of Neapolitan construction which I won’t repeat here. Neither will I describe all of the features of the instrument and its measurements for the purposes of this discussion.
The instrument is signed “[Hon]ofrius Guarracino fecit 1651” in ink on the top surface of the c^{3} keylever behind the balance point (see Figure 3 below). The first part of the signature is covered over by a nonoriginal piece of cloth covering the tail part of the keylever on which the jacks rest, and the number ‘45’ is written to the right of this near the balance point.
Figure 3 – Photograph showing the signature on the top of top c^{3} keylever.
Singlemanual Italian harpsichord by Onofrio Guarracino, Naples, 1651
Property of Prof. Andrea Coen, Rome
The soundboard rosette is made of parchment and wood (probably pear?) and is of the usual Neapolitan ‘wedding cake’ style of construction in four different levels each of 2 or 3 layers of parchment and wood.
Figure 4 – Two photographs showing a plan view (left) and a side view (right) of the soundboard rosette
Singlemanual Italian harpsichord by Onofrio Guarracino, Naples, 1651
Property of Prof. Andrea Coen, Rome
The case is constructed in the usual Neapolitan style with the baseboard raised about 3½mm above the edges of the case sides and the lower outer moulding. The nameboard does not have a central excavated portion as is more usual with Guarracino and many other Neapolitan instruments but has both the top and bottom mouldings added on. There are carved figures at either end of the keywell representing Neptune riding a hippocampus. Also like many Neapolitan instruments and all of the instruments of Guarracino that I know the keyboard withdraws from the instrument like a drawer, rather than being lifted up vertically (see Figure ).
Figure 5 – Photograph showing the keyboard partly withdrawn, drawer fashion, from the instrument
Singlemanual
Italian harpsichord by Onofrio Guarracino, Naples, 1651
Property of Prof.
Andrea Coen, Rome
The determination of the unit of measurement used to
construct this instrument
Like virtually all other Italian builders, Guarracino began the process of construction by measuring out the baseboard. This he did using simple units of the Neapolitan oncia. It is therefore the measurements of the baseboard that are used in the determination of the unit of measurement used in the design and construction of this instrument. The dimensions of the baseboard and of the case sides without the top moulding are given below:
Length of spine side: 1848
. Width at the front: 684½
Width at cheek: 684½
Length of cheek side: 501½
Length of oblique tail side: 464½
Component of the tail, parallel to spine: 403½
Component of the tail, perpendicular to spine: 232½
Tail angle: 30º
Height of the case sides: 205
Table 1  Dimensions of the baseboard and case height in millimetres
Singlemanual
Italian harpsichord by Onofrio Guarracino, Naples, 1651
Property of Prof.
Andrea Coen, Rome
The procedure of determining the unit of measurement used to construct this harpsichord begins most easily with the measurement of the angle of the tail. This angle is 30º and sin 30º = ½, suggesting that the side opposite the tail angle and the length of the tail itself have lengths in the ratio of 1:2. A process of elimination by trial and error shows that the ratio that gives an oncia which applies to the other dimensions of this instrument is that of . Measurement of these two sides in mm enables the calculation of an approximate estimate of the size of the oncia which can then be applied to the other measurements of the baseboard, keyboard, wrestplank, string scalings, and all of the other parts and design features of the instrument including the lateral pin spacing. A summary of the measurements of the baseboard and case height in once are given in the following table:
Measurement Measurement Length of
in mm in local unit oncia in mm
Tail length: 464½ = 21½ once Þ 21.61
Tail side opposite angle: 232½ = 10¾ once Þ 21.61
Spine side: 1848 = 85½ once Þ 21.61
Baseboard width: 684½ = 31 once Þ 21.62
Baseboard cheek (short side): 501½ = 23¼ once Þ 21.57
Case sides height: 206 = 9½ once Þ 21.58
Total 3937 = 182 once Average: 21.61mm ± 0.013mm
Table
2 Baseboard measurements used to calculate
Guarracino’s unit of measurement
Singlemanual
Italian harpsichord by Onofrio Guarracino, Naples, 1651
Property of Prof.
Andrea Coen, Rome
These measurements are shown in the diagram on the next page where the actual measurements of the baseboard in millimetres are shown on the left, and the measurements in Guarracino’s unit of measurement are shown on the right. According to most reliable sources, in Naples the palmo, divided into 12 units, had a length of 262.01mm[1]. Hence the oncia had a length of:
The difference between this and the oncia found here is only of the order of 1% and therefore there is very good agreement between the value found here and the textbook value of the Neapolitan oncia.
I have, however, measured a number of other instruments made by Onofrio Guarracino. From these measurements it seems clear that Guarracino had his own workshop unit of measurement which is somewhat different from the ‘textbook’ value of the Neapolitan oncia found above. I have been able to calculate an average value of the oncia used by Guarracino in his workshop[2] as 21.61mm so that the palmo would be:
This value of the oncia is only 0.05% different from that found here and is a very good indication that this instrument is indeed by Guarracino!
Figure 6  Dimensions of the baseboard in mm (left) and in units of the Neapolitan oncia (right)
Singlemanual
Italian harpsichord by Onofrio Guarracino, Naples, 1651
Property of Prof.
Andrea Coen, Rome
Lateral spacing of the bridge pins and strings:
Although it is now disposed with 2x8', the instrument originally had only one set of strings and these seem to have been the present long set as is usual. The distance from the spine of each of the bridge and nut pins was measured and these are given in the table below:
Note 
Long 8' bridge pin 
Long 8' nut pin 
c^{3} 
640½ 
646 
b^{2} 
626½ 
632 
b^{I2} 
611 
618½ 
a^{2} 
599 
603½ 
g^{T2} 
585 
589½ 
g^{2} 
571½ 
575 
f^{T2} 
557 
562 
f^{2} 
543½ 
549. 
e^{2} 
531 
536 
e^{I2} 
517 
522 
d^{2} 
503 
509 
c^{T2} 
490 
496½ 
c^{2} 
475½ 
482½ 
b^{1} 
462 
466½ 
b^{I1} 
448 
452 
a^{1} 
434 
439 
g^{T1} 
421 
424½ 
g^{1} 
408 
410½ 
f^{T1} 
394 
398 
f^{1} 
380 
385 
e^{1} 
365½ 
372 
e^{I1} 
352½ 
358½ 
d^{1} 
339 
343 
c^{T1} 
325½ 
329½ 
c^{1} 
311 
315 
B 
297 
303 
b^{I} 
284 
289 
A 
271 
275½ 
g^{T} 
257 
261½ 
G 
244 
247 
f^{T} 
230 
233 
F 
216½ 
219 
E 
203 
205 
e^{I} 
190 
192 
D 
176 
178 
c^{T} 
163 
164½ 
C 
148 
150 
B 
136 
136 
B^{I} 
121 
121½ 
A 
109½ 
106½ 
E/G^{T} 
95½ 
93 
G 
81 
79 
D/F^{T} 
69 
64 
F 
56½ 
50 
C/E 
43 
35½ 
Table 3  Distance of the bridge and nut pins from the spine in mm
Singlemanual
Italian harpsichord by Onofrio Guarracino, Naples, 1651
Property of Prof.
Andrea Coen, Rome
It should be
noted from the measurements in Table 3 that for the notes
around tenor B to tenor c, the strings are parallel  that is the distance of
the nut and bridge pins from the spine are equal. Below tenor c the nut pins are nearer to the
spine than the corresponding bridge pins and above tenor c the nut pins are
further from the spine than the bridge pins.
This is the first clear indication that the
strings are angled away from the spine in order to increase the area of freely
vibrating soundboard area around the bass end of the bridge. Indeed for this instrument the strings are
angled away from the cheek as well to increase the area of soundboard around
the treble end of the bridge. These distances
are plotted below in Figure .
The pin spacing of both the bridge and of the
nut (see Figure 7) falls into two distinct sections.
Above the note tenor c, the pins have one regular spacing relative to
one another, whereas below tenor c they have another, slightly different
spacing. This difference is indicated by
the slight but significant change in the slope of the line that joins the
plotted points above and below the note tenor c.
The
slight change in the spacing of the bass bridge pinning shown in Figure below is the result of Guarracino’s desire to improve the quality of
the lowest bass notes by moving the bass end of the bridge away from the spine
side of the instrument as discussed above.
Therefore, the strings have a different relative spacing above and below
the played note tenor c.
Figure
7 
Graph of the lateral spacing of the
bridge pins from the spine
Singlemanual
Italian harpsichord by Onofrio Guarracino, Naples, 1651
Property of Prof.
Andrea Coen, Rome
Clearly
if the bass bridge pins are moved away from the spine, the strings also move
away from the spine along the line of the jacks at the register. This would have the effect of pushing the
strings towards the jacks and would require shorter quills there. In order to avoid this problem the strings
were moved away from the jacks by positioning the bass nut pins further apart
for the notes below tenor c. This is
shown in Figure 7 below where there is again a small but significant change in the slope of the
line joining the points above and below tenor c. The difference is that, in this case, the
spacing of the nut pins is greater below the note tenor c than it is
above tenor c.
Figure
8 
Graph of the lateral spacing of the nut
pins from the spine
Singlemanual
Italian harpsichord by Onofrio Guarracino, Naples, 1651
Property of Prof.
Andrea Coen, Rome
Here,
in order to calculate the average lateral spacing of the bridge and nut pins in
each part of the compass, the analysis has been based on a statistical method
slightly more complicated, but giving a much higher accuracy than simply using
the coordinates of the beginning and end of each of the lines. In this type of analysis the measured points
have been fitted to a straight line using a regression analysis and the method
of least squares. Each section of the
graphs shown in Figure 6 and Figure 7 can be represented mathematically by a simple, straightline equation
of the form y = mx + b like that shown in Figure above. By using each of the points on the graph and
by fitting the best possible straight line to all of the points in each section
of the graph, the regression analysis gives the best average slope, and greatly
increases the accuracy of the determination of this slope. Hence a very accurate value of the lateral
pin spacing was obtained for each of different sections of each of the two
graphs.

Bass section below tenor c  Middle and treble
section above tenor c 


m  b  m  b 
Bridge: 
13.18±0.8%  168.2  13.648±0.07%  179.4 
Nut: 
14.31±0.3%  193.3  13.74 ± 0.1%  178.9 
Table 4  The values of the constants m and b in the straightline equation y = mx + b found from the regression analysis of the bridge and pin positions for the bass and treble sections
Singlemanual
Italian harpsichord by Onofrio Guarracino, Naples, 1651
Property of Prof.
Andrea Coen, Rome
Of
interest in this analysis are the regular spacings of the strings, given by the
constant m, for the nut and for the
bridge and for each section of the pinning above and below the note tenor
c. The percentage error calculated
during the regression analysis procedure has been given here for each of the
values of m, but not for b which is not of interest in this
particular analysis. The value of m for the sections of the pinning in the
bass below tenor c are clearly quite different and have a larger percentage
errors (since there are far fewer measurements to contribute to the averages)
from the treble bridgepin and nutpin spacing. The values of m
for the pinning above tenor c are very nearly the same, but differ
significantly from one another. Indeed the
difference between them is greater than the error for each and so they need to
be considered separately. In other words
the middle and treble strings have a different spacing at the bridge end from
that at the nut end and are therefore not exactly parallel to one another. In this case it is clear that they have a
slightly greater spacing at the nut than they do at the bridge.
I
now want to consider each of the lateral pin spacings separately in order to
see how each of them is related to the unit of measurement used by
Guarracino. First of all let’s consider
the average middle and treble bridgepin
spacing expressed in terms of Guarracino’s workshop oncia found above[4].
This gives
(Equation 1)
Although at first sight this spacing seems rather strange
and arbitrary, this is very close to the spacing that would be obtained if
Guarracino had spaced exactly 19 notes in one palmo:
(Equation
2)
The close agreement (error =
0.02%!) suggests
very strongly that Guarracino spaced out the middle and treble bridge pins so
that 19 of each occupied a space of one palmo. The difference is smaller than the
experimental error and so is a good indication that this interpretation is
correct.
In
the bass the bridgepin spacings have to
be analysed separately from those found in the treble. Here, the bass bridgepin spacing, again expressed in terms
of Guarracino’s workshop oncia, is:
(Equation
3)
This is very close to the spacing that would be obtained if
Guarracino had spaced 9 notes in exactly 5½ once (error = 0.3%):
(Equation
4)
and this, in turn, suggests that Guarracino spaced out the bass bridge pins so
that 9 pins occupied a space of 5½ once.
The reader is reminded at this point that there are exactly 9 notes from
C/E to tenor c, and it is precisely these 9 notes that are spaced out in 5½ once.
The
analysis of the lateral nutpin spacing follows along similar lines. The tenor, alto and treble nutpin spacing is
13.74mm/string. This gives:
(Equation 5)
The close agreement (error =
0.09%) is within the calculated experimental error and suggests that
Guarracino spaced out the middle and treble nut pins so that 11 strings
occupied a space of 7 once.
Again
the bass the nutpin spacings have to be analysed separately from those found
in the treble. Here, the bass nutpin spacing, again expressed in terms of
Guarracino’s workshop oncia, is:
(Equation
6)
This is very close to the spacing that would be obtained if
Guarracino had spaced out the 9 notes of the bass short octave in exactly 6 once
(error = 0.7%) instead of the 5½ once
already found above for the bass bridge spacing.
On
the other hand the simplicity and elegance of the way the pins were spaced on
the bridge and nut is striking. Use of
the value of the 21.61mm oncia found from the analysis of the case
dimensions suggests how the instrument’s pin spacing was designed and
carried out. However, this value of the oncia,
measured from the case dimensions, does not have the same high accuracy as that
to which the value of the bridgepin
spacings have been calculated. Indeed
the value of the oncia used by Guarracino to mark out the sticks he used
to position the bridge pins can be calculated by assuming an exact spacing of
19 notes in 12 once for the middle and treble bridgepin spacing and of exactly 9 notes in
5½ once for the bass bridgepin
spacing. For the middle and treble
bridge pins:
(Equation
7)
or
(Equation
8)
And for the bass bridge pins:
(Equation
9)
or
(Equation
10)
The two values of the oncia calculated above are in agreement with one another to within the statistical error calculated using the regression analyses. The value calculated here for Guarracino’s oncia = 21.607mm±0.015mm is the most accurate value of his workshop oncia calculated so far by any means of analysis. The high accuracy of this result also makes it possible to say definitively that Guarracino’s workshop unit of measurement was not the ‘text book’ value[5] of the Neapolitan oncia = 21.834mm.
The lateral pin spacing, given in terms of the unit of measurement used by Guarracino is summarised in Table 5 below

Bass section below tenor c  Middle and treble
section above tenor c 
Bridge: 
9 notes in 5½ once  19 notes in
12 once 
Nut: 
9 notes in 6 once  11
notes in 7 once 
Table 5  Lateral string spacing expressed in terms of Guarracino’s unit of measurement
Table 6 below shows the unit of measurement calculated using the case measurements and the lateral string spacing for a number of Guarracino instruments. The measurements for the 1651 Guarracino belonging to Andrea Coen are highlighted in green at the top of the table. These have a very low error. The virginals below this have errors in the calculated unit of measurement that are somewhat higher because of the smaller measurements involved and the smaller number of strings in a virginal which can be used in the determination of the lateral string spacing. However they all form a consistent set of analyses for instruments spanning the 41year period of Guarracino’s activity, and they all show that Guarracino’s workshop unit of measurement was consistently different over this long period from the ‘text book’ unit of measurement for Naples.
The instruments highlighted in orange at the bottom of Table 6 are all instruments which have numerous features of Neapolitan harpsichords in general and of the workshop of Onofrio Guarracino in particular. The woods used, the method of marking out, the order of construction, the geometry of the tail, and the sectional shapes of a number of the mouldings used are all similar. But in addition to these the workshop unit of measurement calculated for these instruments is the same unit close to 21.61mm and is within the experimental error for each determination of the unit of measurement to that normally used by Guarracino. This, to me, strongly suggests that these five anonymous instrument have all been correctly attributed to Guarracino.
oncia

Error 
Error 


Instrument

mm 
mm 
% 
Method 
1651 Guarracino harpsichord, Coen, Rome 
21.61 
0.02 
0.093 
Case measurements 
1651 Guarracino harpsichord, Coen, Rome 
21.607 
0.015 
0.069 
Lateral string spacing 
1663 Guarracino virginal, Tagliavini, Bologna 
21.608 
0.015 
0.069 
Case measurements 
1663 Guarracino virginal, Tagliavini, Bologna 
21.61 
0.13 
0.602 
Lateral string spacing 
1668 Guarracino virginal,
Museo Naz., Rome 
21.61 
0.02 
0.093 
Case measurements 
1668 Guarracino virginal,
Museo Naz., Rome 
21.58 
0.13 
0.602 
Lateral string spacing 
1677 Guarracino virginal,
Museo Naz., Rome 
21.664 
0.015 
0.069 
Case measurements 
1689 Guarracino virginal, Roger Mirrey, London 
21.62 
0.015 
0.069 
Case measurements 
1692 Guarracino virginal,
Museo Naz., Rome 
21.61 
0.015 
0.069 
Case measurements 
1692 Guarracino virginal,
Museo Naz., Rome 
21.58 
0.05 
0.232 
Lateral string spacing 
n.d. ‘Guarracino’ harpsichord, Gemeentemuseum 
21.649 
0.04 
0.185 
Case measurements 
n.d. ‘Guarracino’ harpsichord, RCM 175, London 
21.637 
0.019 
0.088 
Case measurements 
n.d.
‘Guarracino’ harpsichord, Giulini, Milan 
21.612 
0.011 
0.051 
Case measurements 
n.d. ‘Guarracino’ harpsichord, MMA, New York 
21.65 
0.03 
0.125 
Case measurements 
n.d. ‘Guarracino’ harpsichord, BMFA, Boston 
21.60 
0.03 
0.13 
Case measurements 
Average: 
21.616 
0.010 

Table
6 – Calculation of Guarracino’s averaged workshop unit of
measurement
A comparison of the lateral bridge and nutpin spacing of the 1638 Ioannes Ruckers and an anonymous FrancoFlemish harpsichords
Figure
9  The 1638 Ioannes Ruckers harpsichord, Russell
Collection, Edinburgh (left)
and the anonymous FrancoFlemish ravalement harpsichord, Antwerp, 1617 (private
ownership, London).
These two instruments, although now physically very different, do have many features in common. The 1638 Ioannes Ruckers harpsichord is in almost unaltered condition since the time when it was first made in Antwerp. In particular, the pinning of the bridges and nuts has never been changed. Because of this the lateral string spacing and the lateral pinning of the bridges and nuts can be used as a kind of standard for the Ruckers’ workshops. The second instrument has been much altered, but the original pinning of the bridges has never been changed. In a paper in the Early Keyboard Journal[6] I have been able to show that it was originally a nonaligned or a socalled ‘transposing’ harpsichord and that, despite the fact that this instrument bears a genuine Ioannes Ruckers in the soundboard, it was made in Antwerp, although it is not by a member of the Ruckers family.
The lateral bridgepin spacings of these two instruments are shown in the graphs of Figure 9 and Figure 10 below. Comparing the lateral bridgepin spacing of the 1638 Ioannes Ruckers harpsichord with that of the Guarracino harpsichord analysed above, it can be seen that, despite the enormous differences in the construction style, string scalings, and unit of measurement used to design the instruments, the lateral string spacings of the mid and treblestring spacings are very similar: 13.86mm and 13.84mm for the 1638 Ioannes Ruckers harpsichord and 13.684mm for the 1651 Guarracino harpsichord. The determining factor here is obviously the stretch of the human hand: no matter where the instruments are made and designed and what the unit of measurement used is, the lateral string spacing and the octave span of the keyboard is, in the end, determined by the size of the human hand.
_{}
Figure 10  The lateral string spacing of the 8' and 4' bridge pins
1638 IR
doublemanual harpsichord, Russell Collection of Early Keyboard Instruments
Figure 11  Lateral 8' and 4' bridge pin positions
Anonymous doublemanual FrancoFlemish harpsichord, Antwerp/France, c.1620/1756
Before going on I would like to point out that the change in the slope in the graph of the 1638 Ioannes Ruckers harpsichord shown inFigure 10 occurs at the uppermanual note G^{T} (or at the lowermanual note c^{T}). Given that the instrument was designed on the basis of the lowpitch lower manual keyboard, this is the same note at which the stringing changed from yellow brass to iron. Indeed for all models of clavecimbel listed in Douwes, the change from yellow brass stringing to iron stringing occurred between the notes c^{T} and d. The change in the slope of the graphs of the FrancoFlemish harpsichord (the pitches in the graph are one semitone to high relative to the original pitch of the strings of the upper manual because the maker would have had to overcome the van Blankenburg problem[7]) occurs much higher up in the compass at the lowermanual note f. This suggests that for the maker of the FrancoFlemish harpsichord there was no relationship between the brass to iron transition and the change in the lateral spacing of the bass and treble bridge and nut pinning.

Bass section  Middle and treble section 
1638 IR 8' bridge: 
10.01 mm/note  13.86 mm/note 
1638 IR 4' bridge: 
11.43 mm/note  13.84 mm/note 
FrancoFlemish 8' Bridge: 
12.67 mm/note  14.18 mm/note 
FrancoFlemish 4' Bridge: 
12.89 mm/note  14.17 mm/note 
Table 7  Slopes of the graphs of lateral bridge spacing from Figure 9 and Figure 10
Table 7 shows clearly that the lateral string spacings of the pins on the bridge and nut of the 1638 IR and the FrancoFlemish harpsichord are totally different. Nonetheless it is interesting to look at their similarities. In both instruments the spacing of the middle and treble pins are the same for the 8' pins and the 4' pins. This is entirely to be expected so that the 8' and 4' strings run parallel to one another with a constant relative spacing. In each case therefore an average was taken of the middle and treble section lateral spacings.
Although not shown here, the errors in the results for each of the 8' and 4' slopes of the bass sections of the FrancoFlemish harpsichord are large enough that the errors overlap to the extent that they are not significantly different from one another. Therefore it is necessary to consider the 8' pins to have the same spacing as the 4' pins.
In the bass the situation is somewhat different. Here, because of the smaller number of pin positions being analysed, the error in the slopes is considerably greater. Nonetheless, the slopes of the bass sections of the 8' and 4' bridges of the 1638 IR are significantly different and therefore need to be analysed separately. On the other hand in the FrancoFlemish harpsichord, although the difference in the slope of the bass section of the 8' and 4' differ by about 1.7% the statistical error in the determination of these values using the regression analysis is even greater than this showing that there is no significant difference between these two. They were therefore averaged in the analysis below.
Just as the spacing of the strings was determined in units of Guarracino’s oncia, the spacings in Table 7 can also be determined in units of the Antwerp duim. In my work on the instruments of the Ruckers family I found that there seemed to be two workshop units of the duim used in the Ruckers workshops. These were the large duim = 25.88mm and the small duim, used for the measurements of the case height and string scalings = 25.48mm. Using the latter value to calculated the average middle and treble lateral string spacing of the 1638 IR harpsichord shows that:
In other words Ruckers spaced out the middle and treble notes so that each octave took up a space of 6½ duimen.
The other measurements in Table 7 can be calculated in terms of the Antwerp duim in a similar way, and the results of these calculations are shown below in Table 8.
Bass
section Middle and treble section
1638 IR 8' bridge: 5 notes in 2 duimen 12 notes in 6½ duimen
1638 IR 4' bridge: 5 notes in 2¼ duimen 12 notes in 6½ duimen
FrancoFlemish 8' Bridge: 22 notes in 11 duimen 20 notes in 11 duimen
FrancoFlemish 4' Bridge: 22 notes in 11 duimen 20 notes in 11 duimen
Table
8 
Lateral string spacing expressed in terms of the Antwerp duim
Table 8 above shows that the makers who made these two instruments, although both worked in the same centre and used the same duim, they thought about the lateral string spacing in a totally different way. Ioannes Ruckers considered each octave of strings in the treble part of the compass and made each octave of strings occupy a space of 6½ duimen. In the bass the 8' bridge pins are spaced very closely together in order to move the bottom bass bridge pin a long distance from the spine and spine liner. But here the 4' strings have a different bass lateral spacing so that the 4' strings end up being slightly closer to the spine, but are closely parallel to the 8' strings. This is a very sophisticated refinement and is an indication of the care and attention that the Ruckers paid to this aspect of the stringband layout.
In the FrancoFlemish harpsichord the middle and treble lateral bridgepin spacing is based not on the octave span, but on the voet. In this case the maker spaces out the treble strings so that there are 20 notes in one voet. In the bass the strings are spaced so that there are 22 notes in 1 voet (= 11 duimen), so that the lateral spacing of the 4' bridge pins is just ½ duim. However, in this case the lowest 8' bridge pin and the lowest 4' bridge pin are the same distance from the spine. The 8' and 4' strings therefore cannot possibly be parallel. This is not a problem as the spacing of the 8' and 4' strings is important only where they pass the jacks. This is, in turn, determined by the nutpin spacing which was not recorded here because the original nut is now missing.
The main
thrust of my argument is, however, that the makers of these two instruments
thought about the lateral string spacing in two totally different ways. Because of this there is clearly no
possibility that the FrancoFlemish harpsichord was made in the Ruckers’
workshops. The characteristic way that
each maker uses his own workshop unit of measurement, that each changes the
note at which the bass and treble spacing changes, and the characteristic way
that each designs the lateral string spacing in his instruments underlines the
importance and usefulness of this method of analysis. Examination by me of numerous other
instruments shows that the same lateral stringspacing analysis can be applied
to instruments in all of the different workshops, regions and schools of
harpsichord building throughout Europe.
It shows that this is a valid and important part of the authentication
and attribution of instruments as well as to an understanding of how
harpsichord makers throughout all of Europe thought and worked.
 begun in Zanzibar, March 2006, finished in Edinburgh, April 2006
 © Grant O’Brien 2006
[1]
See: Horace Doursther, Dictionnaire universel des poids et mesures anciens
et modernes, (M Hayer, Brussels, 1840) and Hercule Cavalli, Tableaux
comparatifs des mesures, poids et monnaies modernes et anciens…, (Paul
Dupont, Paris, 2/1874). Both of
these authors give the same value for the Neapolitan oncia.
[2] See the section concerning the Royal College of Music Neapolitan harpsichord on my website http://www.claviantica.com/Publications_files/Cristofori_Naples.htm for a calculation of the average unit of measurement used by Guarracino.
[3] In Table 4, the errors are the standard deviation errors calculated during the regression analysis for each of the results. These are expressed here as a percentage rather than as an absolute value.
[4] See the results calculated above.
[5] Hercule
Cavalli, Tableaux comparatifs des
mesures, poids et monnaies modernes et anciens…, (Paul Dupont, Paris,
2/1874) gives a value of the Neapolitan oncia of 21.813mm; Cavalli (op.
cit.), Ephraim Chambers, ‘Measures’, Ciclopædia:
or An Universal Dictionary of Arts and Sciences, Vol. 2 (London,
1728; 4/1741; 5/1743), Horace Doursther, Dictionnaire universel des poids et mesures
anciens et modernes, (M Hayer, Brussels,
1840), Barnaba Oriani, Istruzione su le misure e su i pesi che si
usano nella Repubblica Cisalpina, (Milano, 1891) and Luigi Pancaldi, Raccolta
ridotta a dizionario di varie misure antiche e moderne coi loro rapporti
alle misure metriche…, (Sassi, Bologna, 1847) all give a value of 21.834mm
and Giovanni Croci, Dizionario universale
dei pesi e delle misure in uso presso gli antichi e moderni con ragguaglio ai
pesi e misure del sistema metrico, (The Author, Milan, 1860) and Doursther
(op. cit.) both give a value of 21.835mm.
[6] See: Grant O’Brien, ‘An analysis of the origins of a large FrancoFlemish doublemanual harpsichord. Would a Ruckers by any other name sound as sweet?’, Early Keyboard Journal, 22 (2004): 4990.
[7]
See my book Ruckers. A harpsichord
and virginal building tradition (Cambridge: Cambridge University Press, 1990) p. 209, 213, 214,
216, 217, 232, 249, 254, 255 where the van Blankenburg problem is discussed in
detail.