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The Art of the Violin Design
Construction as One Aspect of Musical Instrument
The Geometric Designing of the Violin
The Reconstruction of the Stardivarius
Method of Violin Design
This book is a general conclusion of observations and reflections about the regularities and principles of the design; or to be more specific, the creation of the bowed, stringed instruments of the Italian classical tradition. They are examined here with regard to the peculiarities of their design and workmanship in the musical culture of the Italian Renaissance. Hereby I will investigate not only specific stringed instruments but also common laws of design. Following is the work that contains an investigation into the inherent logic of violin design.
Proceeding from the specific character of my work I have used different methods which are applied to modern art criticism (music, painting, architecture) as well as other spheres of human artistic activity. For me the basic principle is modality, which I use as a particular method to design the musical instruments. It lets us demonstrate the pithy logic of violin construction, which has been shaped over a long evolution and the basic purport of which is the search for a natural correlation between the aesthetics of construction and the aesthetics of sound. Namely, the logic of the design (not the sum of the different elements and their technology), side by side with acoustics are the important elements that enable the luthier to see the process of the creation of the sound, and to devise a geometric scheme for the design of the violin.
It is clear that any design scheme will be unsatisfactory if it is not guided by acoustic regularities, i.e. by the sound as a final result of the luthier art. In my research I have chosen a method that helps me balance both the shape of the instrument with the sound it creates and to expound a semantic basis for the violin as a construction, and to display it as something integral, like something specific, but, in contrast to, a generalized model of the instrument that is modeled from an idealized designed.
Construction as One Aspect of Musical Instrument
We use construction to refer to different things. Musicians associate it with the composition and arrangement of music. Construction is always used by engineers and inventors at their practice. In all cases, this word has to do with the subject of human activity.
In my work, construction means both the object, i.e. the violin, and the process of the instrument design that includes such the concepts as the realisation of an idea and its outcome. If the idea is a concept or a mental impression of the future instrument, then the achievement of the idea includes two processes: the realisation of the idea by drawing the whole instrument on paper and the realisation of the idea by making one. So the result will appear twice: as the drawing and the instrument itself. In addition, the realisation of the idea divides into intermediate stages that have their own results. Every previous stage determines the next one, which makes an algorithmic line of the process of the musical instrument design.
Every instrument, including the violin, is an artificial object that different natural phenomenon and forms leave their marks on. Handicrafts are similar to nature because the formative process resembles that seen in nature. In the abstract the astonishing similarity between natural objects and artificial objects suggests the idea that all artificial objects are crystallized from a material of nature by the conscious activity of men. In other words, an artificial object is a result of the transformation of a natural one.
There are principal differences between natural and artificial objects. If the first exists in or caused by nature then the second is made or produced by human beings after planning or design.
It is noticeable to everybody who studies the violin art that instruments that came into existence in the Renaissance have a similarity with other works of art. It can be observed even at first acquaintance the presence of a certain uniting principle, the subordination to the common basis of the artistic thinking. For example, the concentricity of composition, i.e. a comprehension of the work of art as the finished unity in which all components are completely submitted to the whole, and the perfect system of proportions, demonstrates this unifying principle.
Of course these common principles are shown in every kind of art in accordance with the specificity of 'building materials' and artistic language. So in the art of the luthiers the perfect system of proportions is valuable not in itself, but for the acoustic and design purposes. Moreover it is very important to highlight that luthiers of the Renaissance were in search of sources of the beautiful not in speculative models, but in real life, i.e. they created the form of the instrument from the combination of tangible object. Thus a creative work of the luthiers of the Renaissance was based on the careful study of nature, and the violin was formed according to principles of natural life as an integral and conformed work of art.
Therefore I have to discover the regularities that were the bases of the great Italians' creative work and to re-create not only the violin itself anew, but also the algorithm of the creative process and the way of the luthier's thinking. The complication of the raised task is evident. The creative process is not so much an obligation to conform to rules and programs, but a freedom to break and revise them so that new rules create a new inevitability.
It is found, that the subject-matter of a work of art cannot precede its creation. The work of art reveals itself to an author only during its creation. Thus in the art of violin design the creative process is so paradoxical that, creating a new instrument, the luthier creates every time a new algorithm.
The creation of a violin is a scientific and artistic work. Cognition and comprehension of the violin as a musical instrument is similar to the cognition of any phenomenon and associates itself with two basic processes: assimilation and accommodation. Assimilation is the process whereby an individual interprets reality in terms of his own internal model of the world based on previous experience; whereas, accommodation is the process of changing that model by developing the mechanisms to adjust to reality.
These two processes, being projected in the history of the development of the scientific study of the musical instrument (the instrumentology) as well as in the history of musicology on the whole, go stage by stage. The Renaissance can be considered as the first stage of the development of an idea about the violin (the practical work of the Italian luthiers) when the theory of the violin design was studied esoterically, i.e. it was explored within a school and passed on to pupils verbally.
Proceeding from the proposition that the practical and theoretical creative work of the luthiers was in syncretic unity with art, science, philosophy and religion, one can say that the luthiers were sufficiently informed in the sphere of exact sciences. In the 17th century the disintegration of the syncretic principle led to the decline of the luthiers' Golden Age, the last specimens of which were gone in the 18th century (A.Stradivari, Guarneri del Gesù and others).
Having no chance to be apprenticed to the great luthiers, makers simply copied their instruments in the hope of repeating their sonority. On this tide of universal respect but misunderstanding of the principles of the violin design (made by the great specimens of this kind of art), the Academy of Arts and Sciences at Padua announced a competition for the best work about the violin design. Antonio Bagatella (1726-1806), author of Regulation for Constructing Stringed Instruments (1782), obtained a prize. This publication created astonishment among all European connoisseurs and luthiers of his day, and was quoted frequently through subsequent ages.
Of course every investigator did his bit to this methodology of the violin design, but all of them kept the very main sign by which one can determine their idol: the use of compasses for drawing, or rather copying, the outline of the instrument. Basically these luthiers copied the Amatis' violins, not the Stradivaris' or the Guarneris', whose instruments were copied from the 19th century. And one of the earliest works about violin acoustics, Memoir on the Construction of Stringed Instruments (Paris, 1818) by Felix Savart (1791-1841), is related to this time. He completely rejected the traditional violin form, supposing the acoustic of the stringed instrument to be independent of its shape. Savart's 'trapezoid-violin' was tested and compared with Cremonas, eulogized as being superior, etc., but subsequently found no endorsement from soloists. Practically all innovators who cardinally changed the shape of the violin suffered the same fate.
So the second stage of the development of the idea about the violin can be called a period of search for new methodological bases.
The third stage, taking place in our time, is that basic methodological schools bound up with acoustics or with the search for the rational construction of the musical instrument, or with physico-chemical processing, or with other practical and theoretical features of the violin design, have already formed. And the subsequent study of the 'secrets' of the violin as an acoustical phenomenon takes the path of the interaction of these methods, i.e. it reverts to its 'syncretic past', but now on a new phase of the dialectical development.
If the luthiers had worked like a technical designer, i.e. firstly with an idea and a working drawing, then the making of parts, their arrangement and, finally, the adjusting of the finished articles, then their creative work would be characterized as technological. And after analyzing the instruments themselves by different methods, determining the physical data of their parts, even simply copying them, one could emulate their tone. The same causes produce the same effects, don't they? But size-for-size copies do not work because the frequencies are determined not only by the dimensions and construction of the instrument but also by the wood's mechanical properties, which vary from sample to sample.
There are also the elements of artistic work in the luthier's activity, concerning not only the outward appearance of the instrument, but its tone, which are impossible to appraise by any modern apparatuses, but only directly by man. So the quality of a violin's sound is a result of both the luthier's sophisticated hearing and his scientific knowledge.
In the primary stage of a violin construction the technological problem is most important, whereas in the process of work the main role gradually passes to the artistic outcome. Proceeding from this the luthier's creative work must be considered from both points of view: science and art.
The problem of the development of the violin in the historical plane as well as its design must be the basis for the scientific research of the modern instrumentologist. As is generally known this development involves the progressive changes in size, shape, and function of an object during its historical existence. In the process of the development in the intermediate stages the object's condition has a certain characteristic, which I name the modus. In its application to the violin the modus is the configuration, design and sound atmosphere of the instrument.
As to the development of the violin in the historical plane, this issue is only explored to a certain extent, because of the limitations of historical and archaeological documents and finds.
It is impossible to restore the methodology of the violin design of the Italian classical schools in full, although the Civic Museum of Cremona contains a collection of moulds, drawings, sketches, templates and original studies by A.Stradivari. But the working drawing, displaying the way of thinking of the great luthiers, is absent. The whole of this set, which was used for the design of the different parts of the instrument, comprises only a number of the copies of a principal drawing which, if it only existed, would throw light on the 'mystery' of the creative process of Stradivari and other luthiers of that time.
I shall not dispute whether such a drawing really ever existed, but too few investigators have made an attempt to re-create it. In these works one can see the centuries-old interest of people in the underlying mathematical regularities of the arts.
The question of mathematical prerequisites in the beautiful, and the role of mathematics (specifically geometry) in the arts stirred the ancient Greeks and Babylonians. One can even assume that mathematics and the arts came into existence almost at the same time in view of the religious and philosophical searches of man, and that there are close and varied connections between mathematics and the arts.
The role of mathematics in laying bare the secrets of the arts has been traced in the creative work of such people as Pythagoras, Vitruvius, Albrecht Dürer, Leonardo da Vinci and Thomas Hobbes. The enormous importance of geometry was not only relevant to the above-named artists and architects, but to great luthiers too. Unfortunately, in contrast to the former, the luthiers did not leave any theoretical propositions about their work.
If we discuss the geometry of the violin, then the question is: What can we assume the basis of its design - the aesthetic principle (beauty, elegance) or to be the physical one (acoustics, mechanics)? The borderland between scientific and the artistic work turned out to be a rather impassable obstacle for the mutual assimilation of two different worlds lying on opposite sides: the world of scientific notions and the world of artistic images. In the scholarly and scientific study of the musical instrument, employing geometry to build a bridge between these two worlds proves to be difficult. Numerous popular methods of the geometrical analysis of the string instruments, which were made by great luthiers, have no acoustic substantiation and, what is more, the aesthetic advisability of such methods gives rise to doubts. Various parts of the violin are drawn with compasses by the mere selection of radii, which rather looks like copying, than a search for logical regularity.
Of course at all times both the architects and the engineers used compasses and a ruler when making the working drawing. And it is small wonder, as basically straight lines and arcs are used in the constructions. But, for example, when designing aircraft, high-speed cars, radar, etc., the circles of the compasses are not a great help. This kind of construction can be designed only with some mathematical curve. Our task is finding such a curve, one that would be up to the requirements of the violin design, i.e. it has to be elegant and must allow for the curvatures of the instrument and needs to meet the criterion of acoustic properties and their projection. As the character of the curvature of the whole of the instrument is invariable, we must use only one kind of mathematical curve, which we can increase or decrease, according to the given parts of the violin. In other words, we must find such a standard module, which when scaled up or down can be used to design any stringed and bowed instrument.
By analyzing different mathematical curves, I come to the conclusion that there exists only one curve that is up to the requirements of the violin design. It is the Cornu spiral or clothoid (C and S are the so-called Fresnel integrals) (Figure 1), important in optics and engineering.
Clothoids have been used in engineering design for many years. In the past the spirals have been found manually by draftsmen. This was a tedious process, which I did myself twenty years ago in 1981 solving the problem of violin design for the first time. It is much easier to use a computer to draw and calculate the position of the clothoids. The design curve of a violin will be made up of segments of clothoids joined in such a way that the curvature is continuous throughout.
The clothoid has the following parametric representation in Cartesian coordinates:
where the scaling factor a is positive, the parameter t is non-negative.
Figure 1: The Cornu spiral.
A curve parametrized by an arclength such that the radius curvature is inversely proportional to the parameter at each point is a Cornu spiral. In contrast to another spirals the clothoid has this very important property: the radius curvature starts from infinity and aspires towards zero, continually approaching its asymptote (centre of volute), while the curvature aspires to the ideal form - to the circle.
The curve of the violin outline is formed by joining segments of clothoids. In all cases it is necessary to solve a nonlinear equation (I used the scale Tool of the computer software) to find the scaling factor a. The angle of rotation of the tangent in each spiral will be found empirically.
The other important factor in the violin's geometrical design is the use of proper proportions. Throughout the ages, designers and architects have attempted to establish ideal proportions. The numerically simple ratios 1:2; 2:3; 3:4; 4:5; 3:5, etc., were considered the preferable proportions, but the most famous of all axioms about proportion was the golden division (1.6180339...), established by the ancient Greeks. According to this axiom, a line should be divided into two unequal parts, of which the larger is to the smaller as the whole is to the larger. At a mathematical expression I shall present it as μ
To construct two finite straight lines in the ratio of the golden division is very easy. Let ABEF be a square (Figure 2).
Figure 2. The golden rectangle.
Let the point D to divide AF in half. If AD = DF, then BD is the hypotenuse of the right-angled triangle with the ratio of catheti (the other two sides) as 1:2. Therefore, by the Pythagoras' Theorem, the length of the hypotenuse is √5. The ratios of the sides of this triangle are very simple: AD/AB = 1/2, BD/AD = √5/1, BD/AB = √5/2. And therefore:
If μ = 1.6180339..., then 1/μ = (√5-1)/2 = 0.6180339... .
If BEM is an arc of a circle with centre D, then AM/AB = μ In that way we can construct the finite lines which will be longer than the original line in the ratio of the golden division.
The rectangle ABPM, having the side AM = μAB, is called a golden rectangle. ABEF is a square, and we observe that the rectangle FEPM is also a golden rectangle, since EF = μFM. If we now take this rectangle FEPM, and mark off a square EPTS from it, the remaining rectangle FSTM also will be golden, and we can continue this process as long as we want. In this way we can construct the lines which will be shorterthan the original line in the ratio of the golden division.
If the proportion μ or 1/μ is found by the simple expedient of working out a problem of the golden rectangle, the proportion 2/μ or μ/2 (very important in our work too) is determined by the next method: with radius DA draw the arc of a circle centre D to cut BD produced at N. Then BD is divided in the 2/μ ratio at N. To prove this, we note that DN = 1 and NB = √5 - 1. If μ = (√5 + 1)/2 and 1/μ = (√5 - 1)/2 and 2/μ = (√5 - 1), then NB/DN = 2/μ.
With radius BN draw the arc of a circle centre B to cut AB produced at K. Then AB is divided in the golden section at K. AB and AM can be the sides of the golden triangle. The golden triangles are constructed by the following method (Figure 3):
Figure 3. The golden triangles.
We see that golden triangle ABC is divided into three golden triangles AEC, ADE and DBE, the sides of which are: AD = DE = EC = 1; DB = BE = AE = AC = μ; AB = BC = 1 + μ = μ². Another golden triangle, having the angles 90º and 54º and 36º with the ratio of 5:3:2, is very interesting too. In this right-angled triangle the ratio of the big cathetus to the hypotenuse is μ : 2 = cos 36º, hence the formula which binds the golden section and π:
μ = (√5+1)/2 = 2 cos π
The geometry of the Great Pyramid of Khufu at Giza is a golden triangle too (Figure 4).
Figure 4. The Great Pyramid of Khufu at Giza.
If cos 51.82729º = 1/μ, then AD/AB = 0.6180339..., AB/AD = 1.6180339...
One can divide all A.Stradivari's creative work into a few periods:
1) From 1666 to 1688 Stradivari had worked after the Amati model. From 1689 he experimented with the large model by N. Amati and enlarged it some more.
2) In 1692 Stradivari had created the 'elongated' model of a violin.
3) In 1698 he had returned to the Amati model, working on the model by Antonio and Hieronymus Amati.
4) From 1705 to 1725 Stradivari worked with his own original model. It was his best period of creative work.
5) From 1725 to 1737 - the last years in Stradivari's creative work - one can see his declining powers, which can be attributed to his old age.
It would be true to suppose that the alterations of Stradivari violin forms, which are being retraced in the span of his creative work, have an acoustic substantiation. If the curvature of the internal mould of the Amati violin has a guitar-shaped form, the Stradivari has a form as if its lines are affected at the middle C-bout. Also, Stradivari has altered the arching and thickness of both the belly and the back and has revised the proportions between different parts of the instrument.
But the process of the development of the violin, which started in the 16th century, was completed only at the beginning of the 19th century, when the violin was modernized by replacing the neck, the fingerboard, the bridge, the bass-bar and the soundpost. So the Baroque violin is only an approximation of the ideal proportionality of its different parts as well as of the whole of the instrument. And the sound of these instruments is being defined by us now when they have the 'modernised' neck, fingerboard, etc., whereas in the 18th century they sounded differently.
The most apparent modification in sound comes from the different strings of present instruments. Here a new material for the production of strings has become stronger and longer because of the new long neck, and the bridge has been raised, which increases the string strain on the table. And, what is more, the eighteenth-century pitch, in general, may be taken only as a¹= 422.5 Hz (according to the pitch of Handel's English tuning-fork, which still exists). Therefore in the days of Tartini the strain on the strings was 29 kg, whereas it is now 90. As resistance to this strain the original bass bar has been replaced by one longer and stronger. The sum effect of these alterations was to develop the optimum sonority of which the instrument was capable.
Hence one can say that the great Italians did not create the present violin sound, which supposedly has passed ahead beyond its time (they could not even imagine how their instruments could sound after 'modernization', you know), but they simply made the violin body with the rich potentialities which did not reveal itself in 18th century in full.
Aforesaid changes in physical design started from about the beginning of the 19th century. It was a second stage of the development of the violin. The paradox was that luthiers, modernizing the old violins and making the new one, disregarded the traditional methods of the violin design and did not study the instrument as well as the pupils of the old time did it. And the 'secrets' of the Italian violins were gone.
We have some problems when we try to make an appraisal of the quality of the instrument's sonority. Nowadays two methods exist: subjective, based on the hearing of the investigator, and so-called objective, when special devises are used. The investigator has a wide range of powerful analytical techniques at his command now. These instruments take the frequency of sound and its intensity. The instruments, which measure the sound pressure of the individual harmonics of the complex tone, analyze its spectrum.
Because timbre is a quality of auditory sensations produced by the tone of a sound wave, the timbre of the particular sound depends not only on its wave form, which varies with the number of overtones, or harmonics that are present, their frequencies, and their relative intensities, but on the some subjective peculiarity of our auditory sensation too. The fact is that every harmonic, which is heard by a musician, is a compound tone, consisting of an objective overtone, which can be fixed with the sound spectrograph, and subjective resultant tones, which only occur in our consciousness because of interaction between the objective overtones. One of them (difference tone) is a low one tallying with the difference between the two vibration numbers, and the other of them (summation tone) is a high one, but a very much fainter one, tallying with their sum. And so, it is hard to describe the timbre of any instrument with objective and subjective components only with analytical techniques; this process also needs a personality appraisal of the musician.
Let us examine two spectrograms. The spectrum of (Figure 5) the force exerted by a bowed string at the bridge, and (Figure 6) the sound radiated by violin playing the same note (open G-string). As we see (Figure 5), the amplitudes of harmonics gradually decrease from first to last that is quite naturally. The spectrogram of the violin sound, which is heard by man, would be like the first one, if only our brain could draw such pictures. But really (Figure 6) the violin insufficiently radiates both the first (G3) and the second (G4) harmonics.
Figure 5. The spectrum of the force exerted by a bowed G-string at the bridge.
Figure 6. The spectrum of the sound radiated by the violin playing the open G-string.
Strange as it may seem, but D5 sounds louder then other harmonics. The fact that we hear the fundamental tone (the first harmonic) as loudest is a function of our brain which creates and adds the amplitudes of the difference tones to the actually sounded harmonics. Because the difference between frequencies of the adjacent harmonics always is equal to the frequency of the fundamental tone, the insufficient amplitude of the first harmonic is compensated by the difference tones of all adjacent pairs of the Harmonic Series.
Now I want to dwell on the highly interesting moment connected with the note D5 and length of a violin. Firstly I calculate the wavelength of the D5. In dry air (at 0 C and a sea-level pressure of 1013.25 millibars) the speed of sound is 331.29 m/second. If the frequency of A4 is 440 Hz, then open G string has 195.5 Hz. Hence the third harmonic (D5) has 587.7 Hz. Dividing the speed of sound by 587.7 I find the wavelength of the D5:
33129 cm / 587.7 Hz = 56.3706 cm.
Getting ahead of my statement (details will be in the next chapter); I produce my calculations of the length of a violin. As initial value I use πcm. (3.14159265...cm), which is the first term of the progression. If the golden division is the common prime factor, the seventh term of the progression will be 56.3735 cm, which is the length of the whole instrument.
Because the spectrum of the sound radiated by the violin is inadequate to the timbre, which is heard by us, it is naturally to ask, 'Is it possible to divine the sound of the musical instrument, working at the acoustic of the parts?'
If the final results were dependent on the sum of the timbre of the different violin parts, this problem would be worked out by merely tuning them up according to the certain principle, copying some great instrument. But really all is far more intricate.
Sound is produced when a vibrating surface interacts with the surrounding air. As the large, lightweight plates (the belly and back) moves forwards and backwards, the surrounding air pressure is increased and decreased. These pressure variations speed away from the source as sound waves at 331.29 m per second. The sound waves, traveling from the outside and inside of the plates, differ in phase by 180º.
If the violin body was a single table, i.e. in free air, it would be like a fish out of water. To see why a bare belly sounds bad, consider Figure 7.
Figure 7. Why a bare belly is inefficient at low frequencies.
The plus signs represent an increase in pressure as the belly moves against the air; the minus sign, a decrease (a). When air from the high-pressure side of the belly mixes with air from the low-pressure side, sound cancellation occurs. At high frequencies, the sound is directional, so little mixing occurs; however, for frequencies at which the wavelength is long compared to the size of the belly, the waves can curve back around the belly so that the out-of-phase waves mix (b). One of the basic requirements of a violin body is to block this unwanted mixing of out-of-phase waves (c).
As a violin body has small holes (the ff-holes), the air in the body retains its ability to act like a spring, while the air in the f-holes acts like another diffuser. This air diffuser vibrates in phase with some frequencies and out of phase at others (Figure 8). So sound is created not only by the motion of two plates but also by the air being squeezed resonantly in and out of the ff-holes. This system acts as a resonator, properly called a Helmholtz resonator. The frequency of resonance for any Helmholtz resonator is determined by the compliance of the air in the container and the mass of the air in the hole.
Figure 8. How air moves at different frequencies. At some frequency, the f-hole air moves in phase with the belly (a). At another frequency, the f-hole air moves out of phase with that of the belly (b).
The violin radiates the wide spectrum of the sound. Owing to the shape of the violin body, the phases of the harmonics are altered, when they go out of the ff-holes. It is conducive to the subtraction and addition of its amplitudes. One of the peculiarities of a Helmholtz resonator is that the sound that is radiated from a hole does not vary with the size of the hole if it has the round form (like in a guitar). When the hole has the form of a slit (like in a violin) the radiated sound varies appreciably through the ratio of length and width of the hole. So, the narrow hole is used by the luthiers for adjusting the sound quality.
Since the ffs with the internal volume of air in the violin body form the resonance system, it is very important for a luthier to check the correlation between these two volumes of air. The balance is achieved by the increase or reduction of the volume of air into the instrument's body so thereby changing the parameters of the ffs. The configuration of the body has no small importance.
Certainly, I remember about the nature of arcs of the belly and the back, their thicknesses and adjustment; both the whole boards and their separate areas. However it is not possible to define exhaustively what work is necessary to be conducted with all the details of an instrument to get the Italian timbre. Any attempt to limit the class of considered phenomena by a type of an equation or an enumeration of some physical characteristics usually brings about failure, an example will always be found that will not go into the accepted scheme.
The use of probabilistic-statistical methods of study in the field of violins (study of the Chladni patterns, the laser interferograms, thicknesses and tones of separate areas of the belly and the back and a great deal of other concerns including holograms and voiceprints) reveals the effect of a total action of unambiguous dynamic laws.
The wave processes, occurring in the system of body-ffs-outside air, have a complex nature and must be described by different systems of equations. However, for the understanding of the most important phenomena, occurring in the given system (interference, diffraction, reflection and refraction, dissipation and etc.) there is no need to analyze the source, generally speaking, complex systems of equations. The simple effects, as a rule, are described by simple and universal mathematical models.
The violin body is a closed space for the sound field (while for this explanation the ff-holes have no importance). In the closed space the sound waves, repeatedly reflecting from borders, form the complex field of the air's oscillatory moving, which is defined not only by the characteristics of the sound source (in the violin body the belly and the back are these sources), but also by the geometric form and sizes of the space, and the ability of the borders of the space to reflect, miss and absorb the acoustic energy. The picture of the wave processes, occurring in the violin body, gets complicated by the presence of the ff-holes.
Because of its small volume the body of a violin cannot be diffusive, so the sound waves of its field are coherent and there are the stable phenomena of interference in it. As a result of that the secondary sources of the sound waves, which are located between the actual sources of the waves (the belly and the back), appear in a certain point of space in the violin body (the Huygens-Fresnel' principle). Due to its contours the body of a violin forms this secondary source in the region of the ffs.
On the output from the body through the ffs the sound wave is changed into the wave pencil. Sometimes this pencil can be considered as a ray, whose behavior is described by the laws of the geometric optics. However the spread of the real wave pencils is different from the behavior of the rays. The reason for this difference is due to the phenomena of diffraction.
We cannot get the exact and mathematically correct decision of the diffraction of the sound wave when it passes through the ffs, since this will entail greater difficulties: the very complex form of the screen (the belly) and not less complex form of the slot (the ffs). So the good ear for music of the luthiers is very important for the determination of the quality of the sound, passing through the ffs. But if we take into consideration only the good ear of the luthiers, we must finish cutting the ffs after the instrument was assembled? It was done by A.Stradivari whose ffs never agree with the intended drawing on the inner face of the belly.
It is hardly probable that great masters worried about the external aesthetics of the ffs more than about the acoustics of the instrument. Many masters, including Guarnerius del Jesu, cut the ffs crudely enough in general, then stopped to consider if contented with the sound knowing that a drastic 'correction' could harm the sound quality. Precisely such a work method is substantiated with the ffs by modern theoretical physics (the Kirchhoff's method), which proves that when the wave passes through a screen with a hole, its spectrum is enlarged.
The width of the angular spectrum is defined by the attitude of a wavelength to sizes of the hole and dependent upon the direction of the spreading wave, falling on the screen. The last remark pertains to distance between the ffs. To tell the truth, the wider the ffs are located on the belly, the clearer the lower harmonics stand out and the violin speaks in a bass voice. If for a violin such an effect can be considered as a defect, then for a viola and a cello a deeper sound with shortened model is possible only, when the ffs are located wider than on the big model. In other words, the low timbre of an instrument depends on the wide location of the ffs more than on the size of the instrument's body. This principle was understood by all luthiers of the old time and sons of A.Stradivari had well assimilated this rule, which their father conceived and carried out his own instruments.
The Museum of Cremona contains A.Stradivari's drawing of the central part of a cello with the scheme for the location of the ffs. On the back of this sheet of paper his sons, Francesco and Omobono, have repeated the same design with a modification of the measurement and the placing of the ffs for the shortened model of the cello. In their variant the distance between ffs is increased in contrast with the variant of their father by approximately 15 mm. Shortening the model, Francesco and Omobono tried to maintain the depth of sound of Antonio's cello.
The Geometric Designing of the Violin
The design phase is largely theoretical. Drawing upon the general fund of violinmaking knowledge and my own research, I produce a mathematical model of a violin that I think will meet all of the specifications to study the violin design. My simpler simulation performed by personal computer consists of geometric models. More advanced simulation, such as that that emulates the dynamic behavior of this acoustical system, is usually performed on powerful workstations or on mainframe computers. This simulation can be useful in enabling observers to measure and predict how the functioning of an entire system may be affected by altering individual components within that system.
I used patterns of the clothoid to draw the outline of the violin. The clothoid was drawn in Adobe Illustrator with Spiral tool by the co-ordinates referred to below (Table 1). This Table is made up at the relative dimensions (a = 1). It is necessary to multiply these dimensions by the clothoid's scaling factor to draw any given clothoid.
s X Y R
0.00 0.0000 0.0000
10 1000 0005 3.1831
20 1999 0042 1.5915
30 2994 0141 1.0610
40 3975 0334 0.7958
0.50 4923 0647 6366
60 5811 1105 5305
70 6597 1721 4547
80 7228 2493 3978
90 7648 3398 3537
1.00 7799 4383 3183
10 7638 5365 2894
20 7154 6234 2653
30 6386 6863 2449
40 5431 7135 2274
1.50 4453 6975 2122
60 3655 6389 1989
70 3238 5492 1872
80 3336 4509 1768
90 3945 3733 1675
2.00 4883 3434 1592
Table 1. The co-ordinates of the clothoid.
I have already mentioned sketches by A.Stradivari and emphasised that they were only a number of the copies of that principal drawing by which one can retrace the way of his thinking. And his drawing of the violin scroll is not an exception.
Side by side with this sketch I will analyse scrolls by both A.Stradivari and other Italian luthiers.
By analysing the outline of a scroll for the violin I can conclude that it was drawn with two curves: The logarithmic spiral or Bernoulli spiral (Figure 9) and The Cornu spiral or clothoid.
Figure 9. The Bernoulli spiral.
I draw this spiral in Illustrator with Spiral tool by next parameters:
In the process of analysis of different scrolls I will use different initial radii and their decay.
An algorithm of the geometric reconstruction of Stradivari's sketch of a violin scroll is shown in Figure 10. I begin with two parallel lines AB and ED, which are the tangents to the scroll; the first extended meets the surface of the neck. The Bernoulli spiral for development of the scroll from B to O has parameters: radius = 16 mm, decay = 85%, segments = 11. The clothoid's scaling factors for the development of other parts of the scroll are 106, 58 and 51. OC/AD = 82.25 mm/50.83 mm = μ.
Figure 10. The geometric reconstruction of Stradivari's sketch of a violin scroll.
Figure 11. The geometric reconstruction of A.Stradivari's scroll of the violin, 1715.
In Figure 11 one can see that the disposition and sizes of the clothoids are identical with the previous reconstruction. Here and further I have added one more clothoid a-50, which shapes a tail of the scroll. OC/AD = μ. The Bernoulli spiral is slightly different from the previous one, and its parameters are visible in Figure.
Although the scroll of the 'Emperor' violin was made with the same pattern to the previous one, its outline is slightly different.
The methods of violin analysis, which are chosen by me and which demand superimposing drawings and photos of the whole instrument as well as its different parts, have one shortcoming: a photo cannot reproduce a geometrically accurate outline of an instrument without some distortion. It is clearly visible in the next example (Figure 13), where I analyze A.Stradivari's scroll of the violin photographed from both sides. Here one can see not only the difference between the sides made by Stradivari, but the optical distortion too.
In Figure 14 we can see that the outline of the pegbox has a different shape. Now the clothoid a-110 for the development of the upper part of the pegbox begins its movement from the line AB to the volute, repeating the curvature of the box. The back of the pegbox is drawn with clothoid a-65, which touches with the clothoid a-102, whereas in the previous example such a junction was impossible.
Figure 12: The geometric reconstruction of A.Stradivari's scroll of the 'Emperor' violin, 1715.