The Geometry of Early English Viols-libre

8/10/2019 The Geometry of Early English Viols-libre

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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)

Documents

The Geometry of Early English Viols-libre