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THE GEOMETRY OF EARLY ENGLISH VIOLS
Ben Hebbert
Authors note: The subject of viol geometry is quite a
large one, and although it was pleasing to describe broad
aspects of it in general terms at the DartingtonConference, a literal transcription would be of very
little use to the reader, and would appear to show flaws
in places where the discussion did not permit a fuller
explanation. Moreover this is a subject that continues to
develop and I hope to be able to produce a more definitive
work at some point in the future. Therefore in lieu of a
literal transcription, I have provided some general points
and a few nice pictures as a taste of things to come.
In presenting this edited version of my Dartington talk, I
should first like to express my appreciation to theMetropolitan Museum of Art. Although I have been studying
English viols since the middle of the 1990s, and have been
scribbling circles and lines over photographs for at least the
last eight years, the museums award of a curatorial fellowship
within the department of Musical Instruments in 2005-2006
allowed me the facilities and time to be able to research this
subject to the fullness that it deserves.
The origins of this project are as an offshoot of my doctoral
thesis at Oxford University (which was still being finished at
the time of the conference), entitled The London Music Trade1500-1725. This is neither the time nor place to discuss this
research in detail, except to say that much of the work led to
a re-evaluation of the status of early instrument makers, from
the popular mythologies that characterises them as humble
artisans working for an obstinate love of a particular art
form, to regarding them both as craftsmen integrated into the
sophisticated systems of court and aristocratic patronage and,
in the late seventeenth century, as manufacturers of luxury
goods whose power as entrepreneurs allowed them to maintain
workshops and retail outlets in the most sought after locations
in London.
There are other studies of geometry that I should give credit
to. Michael Heale published a short paper on the geometry of
English viols in the Galpin Society Journal in 1989, in which
he used systems of circles and rectangles in order to
illustrate some rudimentary properties of some of the viols
that he had restored. He had already realised some of the
design mechanisms that are fundamental to this interpretation
of geometry, and I was privileged to enjoy many long
discussions about his ideas, covering his kitchen table with
photographs and drawings in the few years before he passedaway.
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Although much criticised when it was first published, Kevin
Coatess Geometry Proportion and the Art of Luthiery (Oxford,
1985) is a particularly praiseworthy milestone in this field.
Although he was unable to find a successful mechanism to
explain phenomena within the instruments that he examined, hewas able to draw attention to the presence of intentional and
unintentional features within the design of many instruments of
the sixteenth and seventeenth century that indicated a
geometrical methodology behind how they were made. I sincerely
doubt that any of the more recent works on geometry would have
been achievable if it wasnt for Coatess contribution.
It remains to make mention of Franois Denis recent work
Traite de Lutherie(2006). By the time that this was available
to the public, I had already broken the back of my research
into viol geometry. Understanding from others the brilliant andcompelling nature of his work, my choice of action was to
ignore it until mine was completed in order not to be
influenced by a study of Italian Renaissance ideas, and in the
future I look forward to meeting him, reading his work, and to
discovering how different my methodology is to his. It is self-
evident that viols and violins follow different proportional
schemes. The objectives of both the maker and consumer were
both radically different, as the violin was generally made for
public performance, and the viol for private use. The aim of my
research was to make exclusive use of English theoretical texts
in order to build a tool box of ideas to apply to exclusivelyEnglish instruments, effectively to examine grammar school
textbooks of Shakespeares time in order to understand where
the philosophical priorities for a viol maker with a grammar
school education lay.
English viols are not only obviously made to a set of
geometrical rules, but because both the instrument and its
repertoire had a symbolic meaning within the sixteenth and
early seventeenth century as representative of noble learning
it further follows that any such ideas would have been an
expectation of the clientele who bought these instruments, andtherefore that the rules for the geometry of viols, whilst
conforming to philosophical ideals, would be simple enough to
be transmitted from maker to customer in order to fulfil their
expectations about how the instrument was made.
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Further evidence of the bespoke nature of viol making comes
from the fact that although bass viols by a particular maker
may all be of a recognisable shape, there is no uniformity in
size. In effect they are geometrically congruent, rather like a
set of Russian dolls. The back lengths of a sample of twenty-
six bass viols by the most prolific English maker, Barak Norman
(made between 1689 and 1723) is given in the following graph,and shows that although the designations of lyra viol,
division viol and consort bass existed on some level to
define small, medium and large instruments. The reality is that
for whatever reason, there is no pattern to the sizes of his
instruments which exist at every size imaginable from 620 mm to
730mm.
Therefore, English makers certainly did not use moulds (and
abundant further evidence beyond the scope of this talk
supports this contention) and were making instruments
individually for the specific needs of the clientele.
Barak Norman is, in fact, a problematic example for this
present discussion because he is the last significant English
viol maker working within this tradition. There are fundamental
differences between the culture, times and clientele of the
William and Mary period, and the courts of Queen Elizabeth and
James I (analogous to explaining Andrea Amatis achievements by
using examples of Stradivaris making). However, so few
instruments survive by any single maker from the earlier period
that it would be impossible to provide a graph such as this
with any real meaning. That said an identical disparity of
560
580
600
620
640
660
680
700
720
740
1697:
1
1698:
1
1697:
2
1696:
1
1696:
5
1692:
2
1692:
1
1689:
1
1693:
1
1712:
2
1723:
1
1700:
1
1702:
1
1712:
1
1713:
1
1714:
1
1718:
1
1711:
1
1718:
2
1699:
1
1713:
3
1713:
2
1697:
3
1703:
1
1718:
3
1696:
2
LengthofBack(millimetres)
Bass Viols by Barak Norman
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measurements extends throughout all 300 or so surviving English
viols.
A note about units of measurement
is licit before continuing. As
far as I am aware, there are noinstruments that can be described
in terms of units of measurements
that were familiar in London
during the period in question. My
hypothesis is that every bass
viol was made according to a unit
of measurement derived from the
body of the person for whom it
was made, much like the bespoke
service of buying a fitted suit
from a tailor. The instrument wasthen generated from this unit of
measurement so that the musical
instrument shared the same
proportion as its player, and
likewise the music was also in
the same proportion. This
explains why the graph of Barak
Normans instruments shows the
same variety of sizes that one would encounter if a graph was
made of the heights of twenty-six randomly chosen members of
the BVMA. This casual discussion is not the place to draw outthe philosophical context of the period in question, but it
fits logically within the neo-Pythagorean ideals of the time
that are manifested in Robert Fludds divine monochord from
Utiriusque Cosmi (Oppenheim, 1617-19), in which Divine
proportion as shown by the harmonic properties of the string of
a monochord accorded to the same super-particular ratios in
which everything created by God including the relative
positioning of the planets could be explained. The idea that a
bespoke instrument meant that the player, his instrument and
the music he played were all created from the same divine
proportion was especially pleasing because the three-in-onenature of it resonated well with ideas of the Holy Trinity
which had special importance since music and human form were
both manifestations of divine form.
The following example is the first instrument that I
successfully found a geometrical scheme for and is in a private
collection in London. It is a small bass viol, probably what we
should call a lyra viol. The instrument is not labelled, but
it was made in London and I give it a putative date of 1580-
1620 based on stylistic concordances with other instruments
that are more reliably dated. The reason that I studied itfirst is because the shape is a little out of the ordinary.
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Following Michelangelos dictum that the artist should have
compasses in his eyes, I reasoned that the most perfectly
pleasing instrument should be the one with the best possible
proportion. As we shall see, this can be rendered through a
very simple proportional pattern, thus providing an easy
introduction to the more complex ideas of proportion that areencountered later on in this paper.
One final note to the reader is to explain how the geometrical
constructions relate to the actual instrument. Instruments that
have survived for three or four hundred years are distorted to
some extent, and photographs have parallax problems that
further distort the image. Moreover, Jacobean viols were
probably never made with quite the same precision as a Ford
Mondeo and the saggy bottom found on the lower bouts appears
to be a consistent and intentional aesthetic feature.
I select a photograph where I am satisfied that the dimensional
quality is acceptable, and I trace the outline of the body and
sound holes from it warts and all. I then superimpose my
circles, lines and shapes onto the instrument and make a sketch
that fits it best (slightly wobbly and un-geometrical). I then
remove all evidence of the original instrument and reconstruct
the geometry properly and symmetrically. In reading the
following images, all outlines and sound-holes are exactly as
they are found on the photographs, and all geometrical
constructions have been corrected to be exactly symmetrical and
exactly proportional. I let the reader be the judge.
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In figure 1.1 I have constructed
two rectangles of the ratio 2:3 and
positioned them one on top of the
other the upper rectangle rotated
through 90. This means that the
largest dimensions of theconstruction are 3 units wide, and
5 units long. Two circles have also
been constructed whose diameter is
3 units. There are numerous
numerical ways of describing how
the centre of the circle is located
it is easier to simply state that
they are contained by the box
construction. The upper circle is
important because it shows the
position of the top of the body.
In figure 1.2 I have reduced the
size of the upper circle by the
ratio 5:6 in order to provide
the curve for the upper bouts.This ratio is important in music
because it is the minor third,
the closest that two notes can
be to one and other before
becoming discordant. I have also
applied the same ratio to the
body length in order to mark the
position of the fold in the back
of the instrument.
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In figure 1.3 I have drawn a
line through the intersection of
the upper and lower circles.
Using the radius of the upper
circle I have constructed a new
rectangle of the ratio 2:3 whichwill control the position of the
c-bouts and the soundholes.
This rectangle is positioned so
that it is intersected by the
large upper rectangle.
The rectangle is divided into 5
parts, and is positioned
vertically at the point 2:3
(repeating the principals infig. 1.1 in miniature).
Figure 1.4 shows the outline of
the instrument superimposed onto
the proportional form. Theoutline is taken from a
photograph, and therefore may
have its own problems of
distortion to add to the fact
that the instrument itself is
something like 400 years old.
From the top, we see that the
upper bouts closely follow the
circle, and end where it
intersects the c-bout boxes. Theinner most point of the c-bout
curve is the intersection of the
two circles, and the bottom outer
corner of the same rectangle
provides the precise location of
the bottom corners.
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Figure 1.5 shows the positioning
of the sound-holes relative to
the inner edge of the c-bout
rectangles. The height of the
sound holes is dictated by the
bottom of the same rectangle,and by the intersection of the
two circles.
Finally figure 1.6 shows these
calculations against the
original photograph of theinstrument. This instrument has
been brought to you using the
super-particular ratios 2:3 and
5:6.
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One of the major problems with this
method of design is the ease with
which a maker would be able to
convert a given unit into other
proportional measurements, since
neither an arithmetical method not ageometrical one would be
particularly effective. A simple
solution for a witty viol maker
armed with a pair of dividers would
be to have an elaboration of the
following scratched into his
workbench, thus providing a swift
way of negotiating ratios.
To convert the ratio 2:3 the viol
maker would simply set his dividersto the first unit of measurement, and place them against the
line marked 2. He would then expand the dividers up to the line
marked 3, thereby successfully negotiating an otherwise
difficult transformation.
There appear to have been several approaches to viol design inthe early period, of which the above example is just one.
During the talk at Dartington, I brought to light another viol,
in which the numerological interpretations of the geometry that
was evident suggested that it was made in a Roman Catholic
context, and that there was a strong case to give it a date
consistent with the reign of Queen Mary I. Given that this is a
preliminary discussion, and given that the numerological
significance of the instrument would require several pages of
detail, I hope that a more significant paper on this instrument
will be available to the public in the future, and apologise to
the reader in the meantime.
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It makes sense that the geometrical
process involved in viol design was a
kind of performance by the viol maker
to the buyer. The type of geometry is
simple enough to be rendered quickly
and effectively, and it is unlikelythat viol makers would have gone to
such lengths to make instruments of
such varied size unless they had good
reason for it. Moreover the number of
instrument makers who were using the
same tool-box of ideas in the early
seventeenth century indicates that
there were no secrets to this
formula.
Numerology and mysticism was hugelytied up with Protestant perceptions
of the Catholic faith. Sir Thomas
Tresham (father of one of the
Gunpowder plotters) had built a triangular hunting lodge at his
estate in Rushton between 1593 and 1598 (see picture), as a
bold statement designed to evoke every possible numerological
representation of his Catholic faith. Wary of possible
interpretations of any geometric or proportional scheme, viol
makers had adapted their work by about 1600 in order to place
the entire design in a certifiably Pythagorean, and therefore
secular context.
Pythagoras, shown in a detail from the frontispiece to
Anathasia Kirchers Musurgia Musicalis (Rome, 1640) was
credited by humanists as the inventor of music, having passed
by a blacksmiths forge, and observing that hammers of different
weight sounded at different pitches when struck against ananvil. This led to the discovery of the mathematical
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relationship of pitches and ultimately to the creation of the
monochord, the discovery of music, and paved the way to the
discovery of divine proportion the ratios that define the way
that viols were designed. He also, completely separately
discovered things about triangles. In this engraving he is
differentiated from Jubal, the biblical inventor of music,because he is holding his theorem in his hand. Likewise, the
method used by Henry Jaye, Henry Smith, John Hoskins, William
Turner, and the second John Rose all place Pythagoras Theorum
at the heart of their design.
The first action of the viol maker is to strike a line across
the centre of the viol whose length is the primary unit
(measurement a in the figure below) probably the natural
hand-span of the player, but there is little way of telling
with certainty what this should be. From this line the viol
maker constructs two equilateral triangles. As we shall see thehorizontal ends of the diamond become critical for sound-hole
placement. The upper and lower apexes become the centres of the
circles that create the bouts. Critically it is the height of
the triangle which is used as the basis for transformations to
create the upper and lower bouts, and not the primary unit. The
transformation from the use of one unit to the other is
explicable by Pythagoras Theorum.
The Hypotenuse of a Right Angle Triangle can by calculated by
Hypotenuse= Length+Width
Therefore if the length of one side of an equilateral triangle
is known, the height of the triangle can also be known (figure
2.1 below).
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In figure 2.2 measurement b is
expanded by the ratio 5:6 in order
to give the radius of the circle.
The lower circle gives the shape ofthe lower bouts, and the upper
circle controls the uppermost point
of the body. The lowest point of
this circle is also the bridge
position. (Note that the neck length
is such that the string is 4c long).
In figure 2.3 the primary unit is
used to generate a rectangle of the
ratio 2:3 which is intersected
horizontally by the centre line, to
give the dimensions of the sound-
holes. A horizontal line that is
double the width of the rectangle is
drawn along its lower edge. The
extremities of this line give thepositions of the bottom corners.
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In figure 2.4 the rectangle
controlling the sound holes is rotatedby 90 and positioned at the end of
the line controlling the bottom
corners.
The rectangle is bisected vertically,
and horizontally by the ratio 2:3 (as
in the previous example). This
provides the framework for the c-
bouts.
In figure 2.5 the upper bouts are
reduced by the ratio 5:6 providing
a complete framework by which the
outline can by modelled.
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In figure 2.6 the basic outline and
sound-hole position is rendered thus.
In the diagram below we see that the
concentric circles on the upper bouts
can be extended through the ratio 5:6
and provide a framework for the
construction of the tulip pattern at
the centre of the instrument. Although
not illustrated here, the tulip
pattern can also be contained in the
same box that describes the c-bouts,
and is positioned with the apex of thetriangle at 5/6 of the height
The instrument in this example is a
Henry Jaye bass viol from 1619
(Dietrich Kessler Collection)