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The Art Of The Violin Design

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  • Аннотация:
    Английская версия моей книги "Искусство скрипичного конструирования"
Предисловие
Скрипка, вопросы истории происхождения
Скрипка как конструкция
Геометрия скрипки
Реконструкция страдивариевского метода построения скрипки
Символизм скрипки
Скрипка и череп
The Art Of The Violin Design
Улучшение звука скрипки
Реставрация деформированной деки
Установка пружины в скрипке
  
  
  
  
  
  
  Sergei Muratov.
  The Art of the Violin Design
  Illustrated by Sergei Muratov
  Copyright(C) 2002-2015 by Sergei Muratov.
  All Rights Reserved
  No part of this book may be reproduced,
  stored in a retrieval system, or transmitted by any means,
   electronic, mechanical, photocopying, recording,
  or otherwise, without written permission from the author.
  ISBN: 1-4033-7460-0 (e-book)
  ISBN: 1-4033-7461-9 (Paperback)
  []
 
  The Art of the Violin Design
  
  By
  Sergei Muratov
  
  Contents
  Preface
  Chapter one
  Construction as One Aspect of Musical Instrument
  Chapter two
  The Geometric Designing of the Violin
  Chapter three
  The Reconstruction of the Stardivarius
  Method of Violin Design
  Conclusion
  Bibliography
  
  Preface
  
  
  
   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.
  
  
  Chapter one
  
  
  
  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:
  
   (AD+BD)/AB=(√5+1)/2= 1.6180339...
  
   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.
  
  
  Chapter two
  
  
  
  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.
  
  
  The scroll
  
   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.
  
  
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  Figure 9. The Bernoulli spiral.
  
   I draw this spiral in Illustrator with Spiral tool by next parameters:
  
  radius = 16.5 mm, decay = 86%, segments =11, rotate = 90º.
  
   Each segment is a quadrant.
   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.
  
  
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  Figure 12: The geometric reconstruction of A.Stradivari's scroll of the 'Emperor' violin, 1715.
  
  
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  Figure 13. The geometric reconstruction of A.Stradivari's scroll of the violin drawn from both sides.
  
  
   []
  Figure 14. The geometric reconstruction of A.Stradivari's scroll of the violin, 1689.
  
  
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  Figure 15. The geometric reconstruction of Guarnerius del Jesu's scroll of the violin, 1725.
   In my opinion Guarnerius del Jesu's scroll of the violin (Figure 15) is nearly ideal. I like, for instance, that all the clothoids begin from lines, which are a framework of my construction where I use the additional line GH, which is parallel to AB and CD. Moreover, the distance between all three lines is equal, i.e. AC = CG = 50.8 mm, and their sizes are in simple proportional relations. The main clothoid a-100 describes the outer face of the scroll. Just the same clothoid, which begins from line HG, determines the building of the rear sides of the pegbox. For drawing the higher part of the pegbox I use the clothoid a-113. Here it begins directly with one of the lines of the framework. The violin scrolls of Guarnerius with their configuration are closer to the Bernoulli spiral, than the scrolls of Stradivari.
   In general, the configuration of the violin scroll of Guarnerius del Jesu noticeably differs from Stradivari's one. It reads easily compared to the mismatched scroll outlines with clothoids which are had with the violins of Stradivari, and Guarnerius.
   I emphasize that the clothoid only helps me to describe the configuration of different parts of a violin and do not confirm that the luthiers of the past used the clothoid as well as I do.
  
  
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  Figure 16. The geometric reconstruction of Guarnerius del Jesu's scroll of the violin, 1730-30.
  
   I had already noticed that the curvature of the higher part of the pegbox is described by the clothoid, disposed as toward the volute, and in inverse direction. And though the rotation of the tangent in the main clothoid a-100, which describes the volute, has another angle, the second clothoid a-100, beginning from the line HG, is tangential to the first.
  
  
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  Figure 17. The geometric reconstruction of Guarnerius del Jesu's scroll of the violin, 1733.
  
  
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  Figure 18. The geometric reconstruction of Guarnerius del Jesu's scroll of the violin, 1733.
  
  
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  Figure 19. The geometric reconstruction of Guarnerius del Jesu's scroll of the violin.
  
  
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  Figure 20. The geometric reconstruction of Guarnerius del Jesu's scroll of the violin, 1735.
  
  
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  Figure 21. The geometric reconstruction of Guarnerius del Jesu's scroll of the violin, 1740-41.
  
  
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  Figure 22. The geometric reconstruction of Diuseppe Guadagnini's scroll of the violin.
   The violin scroll (Figure 22) of Diuseppe Guadagnini (1736-1805) is very interesting geometrically. The volute has only 9 segments. But one can see how beautifully the clothoids are disposed on the drawing.
  
  
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  Figure 23. The geometric reconstruction of Stradivari's pattern of viola scroll.
  
  
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  Figure 24. The geometric reconstruction of the scroll of the 'Medicea viola' by A.Stradivari.
  
  
   []
  Figure 25. The geometric reconstruction of the scroll of the 'Paganini' viola by A.Stradivari', 1731.
  
  
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  Figure 26. The geometric reconstruction of the scroll of the 'Gore Booth' cello by A.Stradivari', 1710.
  
   In spite of the greater difference of the scrolls, which we see in instruments of the Italian masters, all of these share a similar nature of construction and, to greater or smaller degrees, repeat the sizes and proportions which I have adduced above.
   Clearly it is very easy to make a drawing of a violin scroll with the help of the clothoid and the Bernoulli spiral. With the desire to create our own original form of the scroll we need to use these two spirals, varying the rotation of clothoids with different angles and drawing the Bernoulli spiral with a different degree of reduction of the radius. The main condition for the production of new variants always must be the logic in the algorithmic building of the whole drawing; but an aesthetic value of spirals and proportions will help the artist in this difficult function.
   Because the scroll is three-dimensional, we need to analyze the successive widths of the back of it. A.Stradivari has left a drawing of the rear sides of the scroll, showing the geometric proportions of the successive widths of the pegbox (Figure 27).
  
  
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  Figure.27. Reconstruction of the A.Stradivari's sketch for the back of the violin scroll.
  
   This geometry is not difficult. The tail of the pegbox is outlined with the radius of the compasses approximately at 12-13 mm, and the width of the finest place of the scroll is approximately 11mm. The greater difficulty is presented in the successive widths of the volute (Figure 28) from the narrowest place (the point M) to the broadest one in the centre of the volute (the point K), which correspond to the similar points in Figure 15. Here I present the successive widths of the volute in their radii.
  
  
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  Figure 28. The successive widths of the volute.
   
 



  0        0         5.5
________________________________

  1        5.9       5.53

  2       11.8       5.65

  3       17.7       5.84

  4       23.6       6.10

  5       29.5       6.44

  6       35.4       6.85

  7       41.3       7.32

  8       47.2       7.85    

  9       53.1       8.42

 10       59.0       9.03

 11       65.9       9.66

 12       71.8      10.29

 13       77.7      10.90

 14       83.6      11.46

 15       89.5      11.94

 16       95.4      12.30

 17      101.3      12.53

 18      107.2      12.61

 19      112.1      12.75

 20      118.0      13.17

 21      123.9      13.87

 22      130.8      14.85

 23      136.7      16.09

 24      142.6      17.53

 25      148.5      18.99

 26      154.4      20.17

 27      160.3      20.63
______________________________
  Table 2.
  
  
   As can be seen from the table, the narrowest place of the scroll is 11 mm, but the broadest one is 41.26 mm. Of course these precise measurements are not axiomatic; one can use other sizes, leaving the nature of the successive widths of the volute similar with my drawing.
  
  
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  Figure 29. A Stradivari: the violin, 1702.
   As already seen in the example of the violin scroll, for its construction I use the golden division, which I defined by the relation between the height of the scroll (50.83 mm) and the distance between the upper nut of the neck and the centre of the volute (82.25 mm). The size 50.83 mm is deduced by multiplying the number π (3.14159...) by 1.6180339 cm (golden division - μ).
   Thereby, for the design of the violin I will use two moduses: the number π for revealing the sizes of the main parts of an instrument and the clothoid for the drawing of its outline. As the modulor I will use the golden division, its derivatives and relations 1/2; 2/3; 3/4; 4/5; 3/5; 5/8; etc.
   Henceforth, I will define the main sizes of the violin as a geometric progression of the number π in the proportional attitude of the golden division. In Figure 29 this progression is defined by the following lengths: AB = 82.25mm; BC = 133.08mm; AC = 215.33mm; CD = 348.41mm; AD = 563.74mm. 31.4 mm is the height of the bridge and the height of the ribs.
   A note should be taken that CD (348.41 mm) not the length of the body of the instrument, but the length of the mould which helps to assemble the ribs. The point E (the internal V notches) divides CD into two lengths in the following ratio:
  
  2ED : CE = μ(1.6180339...) .
  
   One can find these lengths as follows:
  
  CE = CD: (μ : 2 + 1) = 192.6 mm;
  ED = CD: (2μ - 1) = 155.81 mm.
  
   Usually, the length of a neck is measured from the upper nut to the edge of the belly, so I will define it as the length AB, solving the right-angled triangle ABC, where AC is the hypotenuse. In Stradivari's time the neck was applied with a slight backward inclination. In the given picture of the modified A.Stradivari's violin (Figure 30) this angle CAB is 7.5º.
   AB = AD (133.08 mm) - BD (3.5 mm - the distance between the edge of the belly and the block) = 129.58 mm, then AC = AB/cos 7.5º = 130.7 mm
  
  
   []
  Figure 30. The neck of the modified violin of A.Stradivari.
  
  THE BODY OF THE VIOLIN
  
   For the geometric analysis of the violin body I begin with the design of the ff-holes. I have chosen this starting point because it determines the size and position of the centre bout, the so-called waist of the violin.
   As was shown above (Figure 29), we find the location of the internal V notches of the ffs on the line E. Then the measure of the instrument (the distance between the internal notches of the ffs and the upper edge of the belly) is 192.6mm + 3.5mm = 196.1mm when the length of the body is 355.41mm (348.41mm + 3.5mm + 3.5mm). Below we adduce the summary table of the length and the measure of the instruments, made by different masters of Italy (the table 3).
   v 
____________________________________________________
Master	             Date    Length,mm	Measure,mm
____________________________________________________
A.Stradivari	     1686	354	195
	             1688	358	193
	             1688	357	198
	             1700	358	197
	             1705	351	190
	             1706	356	196
	             1707	356	196
	             1708	353.5	195
	             1708	362	196
	             1710	358	197
	             1711	359	195
	             1716	354.5	192
	             1718	350	188
	          no date	354	196
	          after 1727	357.5	195
	          after 1727	356.5	195
	             1736	358	198
N.Amati	             1658	356	195
	             1663	354.5	197
	           no date	353	196
	             1678	351	190
A.&J.Amati	     1628	352	196
	             1629	351	195
Guarnerius del Jesu "Ysayë"	357.5	196
	             1742	350	188
A.Guarnerius	     1665	353	193
_______________________________________________

  Table 3.
  
  THE ff-HOLES
  
   I will begin the analysis of sizes, configuration and locations of the ffs with drawings which were made by A.Stradivari (Figure 31). The lines and arcs of a circle, which we see in the drawing, are only an orientation for carrying the ffs from the paper on to the belly. We don't know how Stradivari drew the ffs, though he has left the wooden patterns of ffs for all models of his instruments. And they are only the copies of a drawing which, if it only existed, would throw light on the 'mystery' of the creative process not only of Stradivari, but also of other luthiers of the past.
   I shall demonstrate my study of the ffs design on the several drawings "step by step". I think it is of very suitable to primary familiarization with a train of my thought. The further examples, describing both the ffs of A.Stradivari and other masters, I shall already give only on one drawing.
  
  
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   Figure 31. Reconstruction of the A.Stradivari's sketch for the position of the ff holes in violins (the "G" model).
  
  
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  Figure 32. The geometric reconstruction of the ff holes in violins (the "G" model).
  
   First, I have elaborated the diameters of the eyes (10 mm for lower eyes and 7.5 mm for upper one) and radius of arc (with the centre in the lower eye), which gets through the centre of the upper eye and point E (61mm). Hereinafter I have drawn two parallel lines A¹A² and F¹F², which are coincident to the lines of the bridge and the narrowest place of the violin's centre bouts. The lines AB and A¹B¹ , which are also parallel to each other and perpendicular to the previous lines, are the tangent to the centre bouts and internal surfaces of the lower eyes.
   With radii 38mm and 61mm draw the arcs of a circle centre E, going through the centers of the upper and lower eyes accordingly. The smaller circumference also goes through the centre of the ffs and touches with the line F¹F². The ratio between both circumferences is: 61mm/38mm = 1.61, very close to the golden section.
   Since the ratio between both circumferences is much close to the golden division, I try to place the isosceles golden triangle (the angle at base is 72º) on my drawing. Two corners are rested in the centers upper and lower eyes, but the third one - on the point E. Certainly, one can see that the corners of the triangle do not touch the correspondent points exactly, but this may be explained by the discrepancies of Stradivari's drawing at compasses, whereas I used the computer.
   I have added three lines: HJ, LM and PN. The rectangle GHJK has the sides in ratio 3/2, since 103.7mm/69.1mm = 1.5 and LMNP is a square with side 38.8mm.
   I draw the ff-holes with three clothoids: a-22; a-29.3; a-39.1. Since the sizes of the eyes have a ratio 3/4, the sizes of the clothoids have the same ratio:
  
  22/29.3 = 3/4; 29.3/39.1 = 3/4; 22/39.1 = 9/16, i.e. 3²/4².
  
   The positioning of the clothoids is clearly visible on the drawing. The clothoids a-22 (the inner face of the upper connecting arm of ffs) and a-29.3 (the outer face of the lower connecting arm of ffs) are inscribed in the circle of the eyes; a-39.1 begins from the internal V notch, touches the lines HJ and is inscribed in the a-29.3.
   Stradivari connected the upper eye to the lower one by means of paper templates, which reproduced the ffs. Often in the cutting he did not follow the outline completely. It is interpreted by both the acoustic task (when the master works at the ffs, correcting the sound quality) and a peculiarity of the paper template. It consists of three parts: a long body and two small tails. The template has a rather broad crosspiece between these parts, but in the cutting of the ffs a cut remains very narrow in this place. In Figure 33 I have shown this difference by unbroken and dotted lines.
   Below I adduce the analysis of the Stradivari's ffs, made from photographs of his violins. I have the same problems with optical distortion, as before when analysing the violin scroll, but the main geometric ideas, it seems to me, will be made obvious.
  
  
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  Figure 33. The ff-holes.
  
  
   []
   Figure 34. A.Stradivari: detail of the belly of the violin, 1702.
  
   In Figure 34 I use only two clothoids: a-25 for the upper eye and a-37 for the lower one. It is noticeable that the line of the waist (F) is a tangent to the small circle (r = 36.5mm) and crosses the greater circle (r¹ = 62.8 mm) at the point of the narrowest place of the violin's centre bouts. The diameters of the eyes are 10 mm and 6 mm.
   In Figure 35 I analyze the central part of the "Emperor" violin. As in the previous violin, the line F¹F² is a tangent to the small circle (r = 38 mm) and intersects the greater circle (r¹ = 62.8 mm) at the narrowest place of the violin's centre bouts. The small circle (r = 38 mm) gets through the upper eyes, and left f-hole, but it only touches the right f-hole. This can be explained by asymmetrical ffs, rather than optical distortion, since both lower eyes are symmetrical. The golden triangle rests one of its own angles to the point E, but other two slightly do not comply with the centers of eyes. I drew the ffs with two clothoids: a-28 and a-34.7. And an though analysis of the patterns with corners will be adduced later, I have taken this opportunity to show the location of the clothoids in the centre bout (C-bout), which are absolutely symmetrical and described by the clothoids a-47.7 and a-71.7.
  
  
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  Figure 35. The geometric reconstruction of A.Stradivari's ffs of the 'Emperor' violin.
  
   []
   Figure 36. The geometric reconstruction of A.Stradivari's ffs of the violin.
  
  
   Figure 36 shows the analysis of another Stradivari violin. Here, as we see, the centre bouts are asymmetrical. The lines of both bouts so different that I drew them with different clothoids, whose sizes are seen on the drawing.
   The geometry of these ffs is completely subordinated to a law of the golden division, for instance: two circles with radii 65.38 mm and 40.4 mm (65.38/40.41 = 1.618...), the golden triangle PES with sides 65.38 mm and 40.41 mm and the golden rectangle PNMV with sides 31mm and 50mm. The ffs were drawn with two clothoids: a-31 and a-37.
  
  
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  Figure 37. Guarnerius del Jesu: detail of the belly of the violin, 1733.
  
  
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  Figure 38. The geometric reconstruction of A.Stardivari's ffs for a viola.
  
   In Figure 37 I have analyzed the violin of Guarnerius del Jesu by the same method. I used the golden rectangle with sides 31.4mm and 50.8 mm and the golden triangle, but as can be seen from drawing, not all of their angles have complied with the centres of the eyes. I have tried to dispose the clothoids on the right f-hole unsuccessfully - they obviously do not comply with the lines of the f-hole. On the left f-hole, instead of a clothoid, I have drawn the circles (D = 17.6 mm and 22.5 mm), which comply with the f-hole enough well.
  
  
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  Figure 39. The geometric reconstruction of A.Stardivari's ffs for a cello.
  
  THE VIOLIN PATTERN
  
   The term violin pattern refers the internal part of the body of the instrument, comprising the ribs and all six blocks.
   Because the composite position of the corners of the pattern is always co-coordinated with the scroll in the violins of A.Stradivari, in my first drawing (Figure 41), I have shown the whole instrument. Additionally this is only a theoretical scheme. Its use is for the purposes of geometric analysis.
   There are many versions about the role of the violin waist, the majority of these concerning its practical functions, allowing a performer to play the violin with a bow comfortably. I hold the opinion, that the waisted musical instruments must be considered as a combination of two resonators united together. In all times masters made musical instruments both with one, and with two resonators, and it did not matter whether it was plucked musical instrument or a bowed one.
   Below I produce the drawings of three plucked instruments with two resonators (Figure 40).
  
  
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  Figure 40. The plucked musical instruments. 1. The South Indian vina, which has two resonators made from a pumpkin. 2. The sitar, which has two resonators made from a pumpkin. 3. The waisted tar of Turkey.
  
   As a rule the resonators are made from different sizes: the lower one is greater, but the upper one is smaller. Hereby, I will consider the violin body as a waisted double resonator. Moreover, the upper and lower bouts are of two bulbs united together.
   When analyzing the violin moulds of A.Stradivari I will use clothoids of different scales proceeding from the sizes of the moulds themselves. The sizes and location of the widest places of the lower and upper bouts are defined by both the scales of the clothoids, and by the angle of their rotation. For the lower bout this angle is KMP, where MP is the tangent (X) of the clothoid, but for the upper bout this angle is JST, where ST is also the tangent of the clothoid.
   The location of the lower corners of the violins (to put it more exactly, corner blocks, rather then the belly itself) is found at the intersection of three lines: G¹A¹, which connect the widest place of the lower bout with the eye of the scroll; MD, being the tangent (X) of the clothoid; GH is some straight line, having some angle with CD. For the upper corners these lines are G¹A¹, ST and FH accordingly.
   I will draw these corners with small clothoids, which must touch the proper bulb and get through points T or P.
  
  
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  Figure 41. The scheme of analysis of geometry of a violin.
  
   The geometric reconstruction of Nicolo Amati's mould 'MB' for a violin (Figure 42). The length of the pattern (CD) is 343 mm, the width of the upper bout (F¹F¹) is 155 mm, the width of the lower bout (G¹G¹) is 193.2 mm, and the waist is 101.5 mm. The two widths are in the ratio of 5 to 4 (G¹G¹/F¹F¹ = 5/4).
   The point M, a root of the lower bulb, divides the length of the pattern into two lengths in the ratio of 2/μ(1.2360678...), then MD = 189.6 mm and CM = 153.4 mm.
   Having constructed the lower bulb I used the clothoid a-200. The root of the clothoid lies at the point M, touches the C-bout and is inserted in the lower bout of the mould up to the end block. Moreover, the angle EMP = 58º.
   Similarly one constructs the opposite side of the bulb. The area around the end block is completed by the clothoid a-130 so that it is a tangent to the horizontal line D, and segments of both clothoids a-130 and a-200 are joined in such a way that the curvature is continuous throughout.
   As it can be seen from the drawing, the lower bout was composed symmetrically.
   The upper bout is drawn with the clothoid a-185. Here I do not find any mathematical regularity in the position of the point S, but will only indicate that CS = 176 mm, and SD = 167 mm. The angle of the rotation of the clothoid, i.e. JST = 51.83º. I have already indicated that this angle lies in the base of the Great Pyramid and is used when building the golden triangle.
   I finish the design of the upper bout with the clothoid a-96.5
   As can be seen from the drawing, the upper bout is not symmetrical at all. If the left clothoid lies in the point S by its root, the right one is shifted to the right a little and has the smaller angle of rotation (51º).
   Before analyzing the geometric position of the corners, we must obtain the points F and G, which correspond to the widest places of the upper and lower bouts accordingly. I do it by a simple measurement and define that CF = 64 mm and GD = 67 mm. The length FG (212 mm) and the length of the pattern (343 mm) are in the ratio of the golden division.
   The line, connecting the widest place of the lower bout with the eye of the scroll, goes through the lower and upper corners of a violin pattern (G¹A¹). I have only to draw another two lines to indicate the exact location of these corners.
   Draw the line from the point F through the upper corner and get FH, which is at an angle of 54º to CD. Remember this angle participates in building of the golden triangle. The third line, which is the tangent of the clothoid a-185, passes very close to point T.
   The lower corner is found by a similar way, where the tangent of the clothoid a-200 gets through the point P precisely and the line GH makes an angle of 51.83º with CD.
   The configuration of the corners and C-bout is drawn with the clothoids a-66 and a-44, which are in the ratio of 3/2.
   If the lines TJ and PK through the corners divide the length of the violin pattern into three parts, I get three lengths in the next ratios: CJ(117.5 mm)/JK(90.2 mm) = 4/3; JK/KD(135.3mm) = 2/3. The centre of circle, getting through all corners lies in point S; diameter = 185 mm.
  
  
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  Figure 42. The geometric reconstruction of Nicolo Amati's pattern 'MB' for a violin.
  
   The geometric reconstruction of A.Stradivari's pattern "PG"-1689 for a violin (Figure 43). The length of the pattern (CD) is 348 mm, the width of the upper bout (F¹F²) = 161 mm, the width of the lower bout (G¹G²) is 200 mm, and the waist is 103 mm. Moreover G¹G²/ F¹F² = 1.24 (that is close to the ratio of 5/4 = 1.25).
   The point M, a root of the lower bulb, divides the length of the pattern into two lengths in the ratio of 2/μ (1.2360678...), then MD = 192.4 mm and CM = 155.6 mm.
   Having constructed the lower bulb I used the clothoid a-200. The root of the clothoid lies in the point M, touches the C-bout and is inserted in the lower bout of the mould up to the end block. Moreover, the angle KMP = 60º. Remember that the angle 60º occurs in the right-angled triangle with angles of 90º, 60º and 30º in the ratios of 3:2:1.
   Similarly one constructs the opposite side of the bulb. The area around the end block is completed by the clothoid a-128.5 so that it is a tangent to the horizontal line D, and segments of both clothoids a-128.5 and a-200 are joined in such a way that the curvature is continuous throughout.
   As can be seen from the drawing, the lower bout was composed symmetrically.
   The upper bout is drawn with the clothoid a-185. CS = 176 mm (corresponding to the previous mould of N. Amati) and SD = 172 mm. Since in this pattern the widest place of the upper bout is more, than that of Amati's, the angle of the rotation of the clothoid is increased, i.e. the angle JST = 54º. I finish the design of the upper bout with the clothoid a-94.3.
   As can be seen from the drawing, the upper bout is symmetrical.
   I obtain the points F and G by simple measurement and define that CF = 66.1 mm and GD = 75.3 mm. As it can be seen from this construction, Stradivari has enlarged the length of the pattern by increasing the lower bout in size. The length of the pattern (348 mm) and the length FG (206.6 mm) are in the ratio of 1.68.
   The line, connecting the widest place of the lower bout with the eye of the scroll, goes through the lower and upper corners of the violin pattern (G¹A¹).
   Draw the line from the point F through the upper corner and get FH, which makes an angle of 54º with CD. The third line, which is a tangent of the clothoid a-185, goes through the point T exactly.
   The lower corner is found by a similar way, where the tangent of the clothoid a-200 goes through the point P exactly, and the line GH makes an angle of 54º with CD.
   The configuration of the corners and C-bout are drawn with the clothoids a-66 and a-44, which are in ratio of 3/2.
   If the lines TJ and PK through the corners divide the length of the violin pattern into three parts, I get three lengths in the next ratios: CJ(121.3 mm)/JK(86.5 mm) = 7/5; KD(140mm)/JK = μ (golden division). The centre of circle, getting through all corners lies in point W; diameter = 188 mm.
  
  
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  Figure 43. The geometric reconstruction of A.Stradivari's pattern 'PG' for a violin.
  
  
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  Figure 44. Superimposing the drawing of Nicolo Amati's pattern 'MB' for a violin (dashed lines) and the drawing of A.Stradivari's mould 'PG' for a violin.
  
   The geometric reconstruction of A.Stradivari's pattern "SL"-1691 for a violin (Figure 45). The length of the pattern (CD) is 350 mm, the width of the upper bout (F¹F²) = 154.5 mm, the width of the lower bout (G¹G²) is 195.5 mm, and the waist is 100 mm. Moreover G¹G²/ F¹F² = 1.27 (that is close to the ratio of 5/4 = 1.25).
   The point M, a root of the lower bulb, divides the length of the pattern into two lengths in the ratio of 2/μ (1.2360678...), then MD = 193.5 mm and CM = 156.5 mm.
   Having constructed the lower bulb I used the clothoid a-202. The root of the clothoid lies at the point M, touches the C-bout and is inserted in the lower bout of the mould up to the end block. Moreover, the angle KMP = 58.7º (that is slightly less than 60º).
   Similarly one constructs the opposite side of the bulb. The area around the end block is completed by the clothoid a-128.5 so that it is a tangent to the horizontal line D, and segments of both clothoids a-128.5 and a-200 are joined in such a way that the curvature is continuous throughout.
   As can be seen from the drawing, the lower bout is composed symmetrically.
   The upper bout is drawn with the clothoid a-185.4. CS = 178 mm, and SD = 172 mm. Since in this pattern the upper bout is a little more extended than that of the previous pattern, the angle of rotation of the clothoid is decreased, i.e. the angle JST = 51.83º. I finish the design of the upper bout with the clothoid a-100. As it can be seen from the drawing, the upper bout is composed symmetrically.
   I obtain the points F and G by simple measurement and define that CF = 65 mm and GD = 73.4 mm. The length of the pattern (350 mm) and the length FG (211.6 mm) are in the ratio of 1.68.
   The line, connecting the widest place of the lower bout with the eye of the scroll, goes through the lower and upper corners of the violin pattern (G¹A¹).
   Draw a line from the point F through the upper corner and get FH, which makes an angle of 54º with CD. The third line, which is the tangent of the clothoid a-185, goes through the upper corner of the pattern exactly.
   The lower corner is found by a similar way, where the tangent of the clothoid a-202 goes through the point P exactly, and the line GH makes an angle of 51.83º with CD.
   The configuration of the corners and C-bout are drawn with the clothoids a-66 and a-44 (which are in ratio of 3/2).
   If the lines TJ and PK through the corners divide the length of the violin pattern into three parts, I get three lengths in the following ratios:
  
  CJ(118.3 mm)/JK(90 mm) = 4/3; JK/KD(141.7mm) = 0.635.
  
   If we compare this pattern with the previous one, it is obvious that Stradivari has enlarged the length of the pattern by increasing the centre part. The centre of circle, getting through all corners lies in point S; diameter = 188 mm.
  
  
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  Figure 45. The geometric reconstruction of A.Stradivari's pattern "SL"-1691 for a violin.
  
  
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  Figure 46. Superimposing the drawing of A.Stradivari's pattern 'PG' for a violin (dashed lines) and the drawing of A.Stradivari's mould 'SL' for a violin.
  
   The geometric reconstruction of A.Stradivari's pattern "B"-3/6/1692 for a violin (Figure 47). The length of the pattern (CD) is 353.5 mm, the width of the upper bout (F¹F²) is 154.5 mm, the width of the lower bout (G¹G²) is 194.8 mm, and the waist is 102 mm. Moreover G¹G²/ F¹F² = 1.26 (that is close to the ratio of 5/4 = 1.25).
   The point M, a root of the lower bulb, divides the length of the pattern into two lengths in the ratio of 1.36, then MD = 203.8 mm and CM = 149.7 mm.
   Stradivari again enlarges the previous model by increasing the lower bout. Having constructed the lower bulb I used the clothoid a-214. Moreover, the angle KMP = 55.6º much less than 60º, since the bout is extended downwards.
   Similarly one constructs the opposite side of the bulb. The area around the end block is completed by the clothoid a-130 so that it is a tangent to the horizontal line D, and segments of both clothoids a-130 and a-214 are joined in such a way that the curvature is continuous throughout.
   As can be seen from the drawing, the lower bout was composed a little asymmetrically.
   The upper bout is drawn with the clothoid a-184.5. CS = 181.3 mm, and SD = 172.2 mm. Since in this pattern the upper bout is a little extended in contrast with the previous pattern, the angle of the rotation of the clothoid is decreased, i.e. the angle JST = 50.5º. I finish the design of the upper bout with the clothoid a-94. As can be seen from the drawing, the upper bout was composed asymmetrically.
   I obtain the points F and G by simple measurement and define that CF = 66.2 mm and GD = 76.5mm. The length of the pattern (353.5 mm) and the length FG (210.8 mm) are in the ratio of 1.68.
   The line, connecting the widest place of the lower bout with the eye of the scroll, goes through the lower and upper corners of the violin pattern (G¹A¹).
   Draw the line from the point F through upper corner and get FH, which makes an angle of 54º with CD. The third line, which is the tangent of the clothoid a-194.5, goes through the upper corner of the pattern exactly.
   The lower corner is found by a similar way, where the tangent of the clothoid a-214 goes through the point P exactly, and the line GH makes an angle of 51.83º with CD.
   The configuration of the corners and C-bout are drawn with the clothoids a-60 and a-43 (which are in ratio of 7/5).
   If the lines TJ and PK through the corners divide the length of the violin pattern into three parts, I get three lengths in the next ratios: CJ(119 mm)/JK(89.6 mm) = 4/3; KD(144.9mm)/JK = μ.
   If we compare this pattern with previous one, it is seen that Stradivari has enlarged the length of the pattern by increasing the lower bout. The centre of circle, getting through all corners lies in point W; diameter = 185 mm.
  
  
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  Figure 47.The geometric reconstruction of A.Stradivari's pattern "B"-3/6/1692 for a violin.
  
  
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  Figure 48. Superimposing the drawing of A.Stradivari's pattern 'SL' for a violin (dashed lines) and the drawing of A.Stradivari's mould "B"-3/6/1692 for a violin.
  
  
   The geometric reconstruction of A.Stradivari's pattern "B"-6/12/1692 for a violin (Figure 49). The length of the pattern (CD) is 347.5 mm, the width of the upper bout (F¹F²) is 154 mm, the width of the lower bout (G¹G²) is 195 mm, and the waist is 102 mm. Moreover G¹G²/ F¹F² = 1.26 (that is close to the ratio of 5/4 = 1.25). In this mould Stradivari retains the general width of the previous model, but makes it shorter, approaching in size to the pattern 'PG'.
   The point M, a root of the lower bulb, divides the length of the pattern into two lengths in the ratio of 1.32, then MD = 197.9 mm and CM = 150.2 mm.
   Having constructed the lower bulb I used the clothoid a-207.5. Moreover, the angle KMP is increased, in contrast with the previous pattern, up to 56.7º
   Similarly one constructs the opposite side of the bulb. The area around the end block is completed by the clothoid a-129 so that it is a tangent to the horizontal line D, and segments of both clothoids a-129 and a-207.5 are joined in such a way that the curvature is continuous throughout.
   As can be seen from the drawing, the lower bout was composed a little asymmetrically.
   The upper bout is drawn with the clothoid a-185. CS = 176.3 mm, and SD = 171.8 mm. Since in this pattern the upper bout is not extended as much as in the previous pattern, the angle of the rotation of the clothoid is increased, i.e. the angle JST = 51.83º. I finish the design of the upper bout with the clothoid a-94.8. As can be seen from the drawing, the upper bout was composed asymmetrically too.
   I find the points F and G by simple measurement and define that CF = 64.7 mm and GD = 73.6mm. The length of the pattern (347.5 mm) and the length FG (209.8 mm) are in the ratio of 1.66.
   The line, connecting the widest place of the lower bout with the eye of the scroll, goes through the lower and upper corners of the violin pattern (G¹A¹).
   Draw the line from the point F through the upper corner and get FH, which makes an angle of 54º with CD. The third line, which is the tangent of the clothoid a-185, goes through the upper corner of the pattern.
   The lower corner is found by a similar way, where the tangent of the clothoid a-207.5goes through the point P, and the line GH makes an angle of 51.83º with CD.
   The configuration of the corners and C-bout are drawn with the clothoids a-66 and a-44 (which are in ratio of 3/2). The centre of circle, getting through all corners lies in point S; diameter = 185 mm.
   If the lines TJ and PK through the corners divide the length of the violin pattern into three parts, I get three lengths in the following ratios:
  
  CJ(117.1 mm)/JK(88.2 mm) = 4/3; KD(142.7mm)/JK = μ
  
  
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  Figure 49.The geometric reconstruction of A.Stradivari's pattern "B"-6/12/1692 for a violin.
  
  
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  Figure 50. Superimposing the drawing of A.Stradivari's pattern "B"-3/6/1692 for a violin (dashed lines) and the drawing of A.Stradivari's mould "B"-6/12/1692 for a violin.
  
   The geometric reconstruction of A.Stradivari's pattern "S"-1703 for a violin (Figure 51). The length of the pattern (CD) is 345 mm, the width of the upper bout (F¹F²) is 155 mm, the width of the lower bout (G¹G²) is 195 mm, and the waist is 100 mm. Moreover G¹G²/ F¹F² = 1.26.
   The point M, a root of the lower bulb, divides the length of the pattern into two lengths in the ratio of 1.22, then MD = 189.7 mm and CM = 155.3 mm.
   To construct the lower bulb I use the clothoid a-199.3. Moreover, the angle KMP, in contrast with the previous pattern, is increased up to 59º.
   Similarly one constructs the opposite side of the bulb. The area around the end block is completed by the clothoid a-128.5 so that it is a tangent to the horizontal line D, and segments of both clothoids a-128.5 and a-199.3 are joined in such a way that the curvature is continuous throughout.
   As can be seen from the drawing, the lower bout was composed symmetrically.
   The upper bout is drawn with the clothoid a-185. CS = 175.9 mm, and SD = 169.1 mm. The angle of the rotation of the clothoid, i.e. the angle JST = 51.83º. I finish the design of the upper bout with the clothoid a-94.3.
   As can be seen from the drawing, the upper bout was composed little asymmetrical.
   I obtain the points F and G by simple measurement and define that CF = 62 mm and GD = 70 mm. The length of the pattern (345 mm) and the length FG (213.2 mm) are in the ratio of μ.
   The line, connecting the widest place of the lower bout with the eye of the scroll, goes through the lower and upper corners of a violin pattern (G¹A¹).
   Draw a line from the point F through the upper corner and get FH, which makes an angle of 51.83º with CD. The third line, which is a tangent of the clothoid a-185, gets through the upper corner of the pattern.
   The lower corner is found by a similar way, where the tangent of the clothoid a-199.3 goes through the point P, and the line GH makes an angle of 51.83º with CD. Thereby, for the first time the triangle FGH is seen to be isosceles and it completely repeats the geometry of the Great Pyramid.
   The configuration of the corners and C-bout are drawn with the clothoids a-60 and a-42. The centre of circle, getting through all corners lies in point S; diameter = 185 mm.
   If the lines TJ and PK through the corners divide the length of the violin pattern into three parts, I get three lengths in the following ratios:
  
  CJ(119.5 mm)/JK(88 mm) = 1.36; KD(137.5mm)/JK = 1.56.
  
  
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  Figure 51. The geometric reconstruction of A.Stradivari's pattern "S"-1703 for a violin.
  
  
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  Figure 52. Superimposing the drawing of A.Stradivari's pattern "B"-6/12/1692 for a violin (dashed lines) and the drawing of A.Stradivari's mould "S"-1703 for a violin.
  
   The geometric reconstruction of A.Stradivari's pattern "P"-1705 for a violin (Figure 53). The length of the pattern (CD) is 348 mm, the width of the upper bout (F¹F²) is 161 mm, the width of the lower bout (G¹G²) is 200 mm, and the waist is 102 mm. Moreover G¹G²/ F¹F² = 1.24.
   The point M, a root of the lower bulb, divides the length of the pattern into two lengths in the ratio of 1.22, then MD = 190 mm and CM = 158 mm.
   In the construction of the lower bulb I use the clothoid a-200. Moreover, the angle KMP, in contrast with the previous pattern, is increased up to 60º.
   Similarly one constructs the opposite side of the bulb. The area around the end block is completed by the clothoid a-128.4 so that it touches the horizontal line D, and segments of clothoids a-128.4 and a-200 are joined in such a way that the curvature is continuous throughout.
   As can be seen from the drawing, the lower bout was composed symmetrically.
   The upper bout is drawn with the clothoid a-182. The point S divides the length of the pattern in half so CS = SD = 174 mm. The angle of the rotation of the clothoid, i.e. the angle JST = 55º. I finish the design of the upper bout with the clothoid /b>a-106.
   As can be seen from the drawing, the upper bout was composed a little asymmetrically.
   I obtain the points F and G by simple measurements and define that CF = 65mm and GD = 73 mm. The length of the pattern (348 mm) and the length FG (210mm) are in the ratio of 1.657.
   The line, connecting the widest place of the lower bout with the eye of the scroll, goes through the lower and upper corners of the violin pattern (G¹A¹).
   Draw the line from the point F through the upper corner and get FH, which makes an angle of 54º with CD. The third line, which is a tangent of the clothoid a-182, goes through the upper corner of the pattern.
   The lower corner is found by a similar way, where the tangent of the clothoid a-200 goes through the point P, and the line GH makes an angle of 54º with CD. Thereby, the triangle FGH can be seen as isosceles again, but with other angles that correspond to the golden triangle.
   The configuration of the corners and C-bout are drawn with the clothoids a-66 and a-44 (which are in ratio of 3/2). The centre of circle, getting through all corners lies in point W; diameter = 190 mm.
   If the lines TJ and PK through the corners divide the length of the violin pattern into three parts, I get three lengths in the following ratios:
  
  CJ(120 mm)/JK(90 mm) = 1.36; KD(138mm)/JK = 1.53.
  
  
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  Figure 53. The geometric reconstruction of A.Stradivari's pattern "P"-1705 for a violin.
  
  
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  Figure 54. Superimposing the drawing of A.Stradivari's pattern "S"-1703 for a violin (dashed lines) and the drawing of A.Stradivari's mould "P"-1705 for a violin.
   The geometric reconstruction of A.Stradivari's pattern "G"-1715 for a violin (Figure 55). The length of the pattern (CD) is 354 mm, the width of the upper bout (F¹F²) is 161.5 mm, the width of the lower bout (G¹G²) is 201 mm, and the waist is 103 mm. Moreover G¹G²/ F¹F² = 1.245.
   The point M, a root of the lower bulb, divides the length of the pattern into two lengths in the ratio of 1.27, then MD = 198 mm and CM = 155.9 mm.
   Having constructed the lower bulb I used the clothoid a-207.7. Moreover, the angle KMP is 59º.
   Similarly one constructs the opposite side of the bulb. The area around the end block is completed by the clothoid a-124.5 so that it is tangential to the horizontal line D, and segments of both clothoids a-124.5 and a-207.7 are joined in such a way that the curvature is continuous throughout.
   As can be seen from the drawing, the lower bout was composed symmetrically.
   The upper bout is drawn with the clothoid /b>a-185. The point S divides the length of the pattern in half so CS = SD = 177 mm. The angle of the rotation of the clothoid, i.e. the angle JST = 54º. I finish the design of the upper bout with the clothoid a-124.5.
   As can be seen from the drawing, the upper bout was composed a little asymmetrically.
   I obtain the points F and G by simple measurement and define that CF = 65.6 mm and GD = 73.2 mm. The length of the pattern (354 mm) and the length FG (215.5 mm) are in the ratio of 1.64.
   The line, connecting the widest place of the lower bout with the eye of the scroll, goes through the lower and upper corners of the violin pattern (G¹A¹).
   Draw the line from the point F through the upper corner and get FH, which makes an angle of 54º with CD. The third line, which is a tangent of the clothoid a-185, goes through the upper corner of the pattern.
   The lower corner is found by a similar way, where the tangent of the clothoid a-207.7 goes through the point P, and the line GH makes an angle of 51.83º with CD.
   The configuration of the corners and C-bout are drawn with the clothoids a-66 and a-44 (which are in ratio of 3/2). The centre of circle, getting through all corners lies in point S; diameter = 190 mm.
   If the lines TJ and PK through the corners divide the length of the violin pattern into three parts, I get three lengths in the following ratios:
  
  CJ(121.4 mm)/JK(89.2 mm) = 1.36; KD(144.3mm)/JK = μ
  
  
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  Figure 55. The geometric reconstruction of A.Stradivari's pattern "G"-1715 for a violin.
  
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  Figure 56. Superimposing the drawing of A.Stradivari's pattern "P"-1705 for a violin (dashed lines) and the drawing of A.Stradivari's mould "G"-1715 for a violin.
  
  
  The summary table of the main sizes of the violin patterns
  of N. Amati and A.Stradivari.
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  Table 4.
  
   Above I introduce the summery table of the main sizes of the violin patterns of N.Amati and A.Stradivari (the table 4). The upper row of letters represents the marks made by them on their own moulds. "MB" is the mould of N.Amati; "B¹" is the mould of Stradivari made on 3/6/1692; "B²" is the mould of Stradivari made on 6/12/1692.
   When I spoke of the proportional relations between parts of a violin (Figure 29), I pointed to the size of the pattern equal to 48.4 mm. A.Stradivari has this size in his two moulds "PG" and "P". Moreover the mould "P" was made in 1705, when Stradivari begins to create his best instruments.
   In these moulds the ratios between the width of the lower bout and the width of the upper bout are also equal and approach proportions of 1.2360678 = 2/μ. If we connect the centers of the bouts with the corners, we will get an isosceles triangle with angles at its base = 54º. This is only in the patterns "PG" and "P", but in the rest the angle in the lower bouts is 51.83º. Only one pattern "S" has an isosceles triangle with an angle of 51.83º.
   The proportional relations between the sizes of upper, centre and lower parts of the violin patterns (CJ, JK, KD) are basically repeated from pattern to pattern. In four patterns "PG", "B¹", "B²" and "G" the ratio of the lower part (KD) to the central one (JK) is the golden division; although in the rest this proportion is less, but close to μ.
   In five patterns 'MB ", "SL", "B¹", "B²" and "P" the ratio of the upper part (CJ) to the central is 4/3, in the rest this proportion is, or is close to 7/5.
   Only in two patterns "MB" and "S" the base (FG) of a large triangle, whose sides go through the corners of the violin pattern, have the ratio of the golden division with the general length (CD), in the rest this proportion is either a little more or less.
   The lower bouts of all patterns irrespective of their sizes are drawn with the clothoid a-200 or close to it, but the upper bouts are drawn with the clothoid a-185 or close to it. The angle of the rotation of the lower clothoid is, or is close to 60º. In four patterns "MB", "SL", "B²" and "S" the upper clothoid is rotated at an angle of 51.83º, in the pattern "B¹" this angle is a little less, but in the rest this angle is, or is close to 54º. In the first three patterns the point M divides CD into two lengths in the ratio of 2/μ = 1.2360678, in the rest this ratio is either a little more or less.
  ***
   But now, using the method, which I have used when analysing the violins of Italian luthiers, I will build my own model of the instrument, which will differ from instruments already known by us, but will use the main principles of violin design. It will give me the possibility to show the creative method of approach of a luthier in the art of the violin design. Whilst maintaining the cardinal principles of design, the master is free enough in his creative activity to make a unique and inimitable musical instrument in best traditions of the great masters of Italy.
   Designing the upper and lower ovals of violin (Figure 57) I begin from the vertical line CD = 348.4 mm, which is calculated from the geometric progression of number π in the ratio of the golden division. The point M is obtained by dividing CD into two lengths MD = 192.6 mm and CM = 155.8 mm in the ratio of 2/μ.
   The tangent of the clothoid a-200 of the lower bout makes an angle of 60º with CD (i.e. the angle KMP = 60º) and rests its root to point M. The widest place of the lower bout is 200 mm and removed from the lower edge at a distance of 75.3 mm (GD).
   The upper bout is built with the clothoid a-185, whose tangent makes an angle of 54º with CD (i.e. the angle JST = 54º) and rests its root to point S, dividing CD in half (i.e. CS = SD = 174.2 mm). The widest place of the upper bout is 160 mm and removed from the upper edge at a distance of 64.1mm (CF).
  
  
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  Figure 57. Designing the upper and lower ovals of violin.
  
   The ratio between the two widths is 5/4, i.e. 200/160 = 5/4. The areas around the end blocks are completed by the clothoids a-91 and a-130 accordingly.
   Hereinafter I connect the edges of widest place of the lower bout with the eyes of the scroll, as I did earlier, and mark the cross points of these lines and the tangents of the clothoid with the letters N, L, P and T. Practically I already have marked the positions of the upper and lower corners of the violin pattern by these points. Let us connect these points with the centres of the upper (F) and lower (G) bouts accordingly: the angles FGH and GFH = 54º.
   The length FG (209 mm) and the length of the pattern (348.4 mm) are in the ratio of 3/5. Let us recall that in the past the proportions connected with the Fibonacci series: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 and so on, were used often. This set of numbers is formed by adding two previous numbers, the ratio between which approaches the golden division. So if 3/5 = 0.6 and 55/34 = 1.6176, then 144/89 = 1.618.
   Before trying to draw the C-bouts, it is necessary to obtain the size and position of the waistlines of the violin, which are defined by the sizes and position of the ffs. So we go to Figure 58.
   In Figure 58 I have shown the terminating stage of the modeling of the violin pattern. The positioning of the point E, through which the line of the bridge and internal V notches of the ffs are passed, is determined by finding the point of balance of the finished belly. With same length of belly, but with the different ratio between the widths of the upper and lower bouts, this point is displaced upward or downward according to this ratio. At the length of a belly of 35.5 mm and the ratio between the widths of the bouts of 5/4 the measure of instrument (the distance between the internal notches of the ffs and the upper edge of the belly) is approximately 196 mm. If an upper bout is slightly wider, and therefore heavier, this point is positioned upward and the measure can be 193-194 mm. On the contrary, at a narrow upper bout (or a wide lower bout that is same) the measure can get to 198 mm.
   If the length of pattern is 348.4 mm, the Point E must be at a distance of 192.6 mm from the upper end-block. I find this point by dividing the length of pattern into two lengths in the ratio of 2/μ = 1.2360678. Then CE = 192.6 mm and ED = 155.8mm.
   The following stage concerns the selection of the ffs' model and size. Each master is free to choose what he prefers: to copy the ffs of the great masters or to design one's own. I draw your attention to my own model of the ffs, completely built with the golden division, the main ideas of which I have demonstrated already when analyzing the ffs of the great Italian masters. On the horizontal line through the point E I build the golden rectangle, were the smaller side is perpendicular to the horizontal line and 10π mm (31.415 mm) in length, while the greater side is 50.83 mm and bisected at the point E. On the drawing this rectangle is marked by the letters a, b, c and d.
   If we connect the points E and b, the length Eb can become the base of an isosceles triangle with an angle at the base of 72º, which is identified as a golden triangle. This triangle rests its own angles at the point E, in the centre of the upper eye (the point b) and in the centre of the lower eye (the point e) respectively.
   The length Eb, as a hypotenuse of the right-angled triangle aEb, is 40.41mm. Then Ee = Eb x μ = 65.38 mm.
  
  
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  Figure 58. Terminating stage of modeling of the violin pattern.
  
   Draw a circle centre E and radius Eb (r - the small radius), and another circle, centre E and radius Ee (R - the big radius). Let the small circle and CD intersect at f. Draw the horizontal line through f. Let this line and big circle intersect at g and h. I define the length of gh, which is the waistline of violin pattern, as follows:
  
  gh = 2√(R²- r²) = 2√(65.38 mm²- 40.41 mm²) = 102.8mm.
  
   Draw the circles centre b (radius of 3.75 mm for the upper eye) and centre e (radius 5 mm for the lower eye).
   The further work of building the ffs and the C-bout with clothoids is similar to my analyses of the instruments of the Italian masters and is easy to read from the drawing. Here I have used the clothoids a-40 for the lower eyes and a-30 for the upper eyes, but the C-bout was drawn with the clothoids a-71 and a-44.
  
  THE VIOLAS AND CELLOS PATTERNS
  
   A.Stradivari made his first mould of a viola contralto in 1672. There was only one instrument made from this form. All the following instruments came from the mould of 1690 made for the viola contralto Medicea. On that day, October 4th 1690, Stradivari made two forms: the first mould was for the contralto, and the second one - for the tenor. The geometric proportions of violas are noticeably differ from the violins' one. To trace the logic of the master, in changing the proportional relations between the parts of each instrument, I will do similar collations to those I did with his violins. When comparing the drawing of the viola moulds with drawings of violin patterns I will reduce them to the comparable sizes.
   In Figure 60 I have superimposed the drawing of the viola mould of 1672 with the drawing of the violin pattern "MB". Here one can see that Stradivari has enlarged the volume of the lower bout, leaving a diminished C-bout. If we compare this pattern with those looked at previously, it can be seen that Stradivari has enlarged the length of the pattern by increasing the lower bout.
   The geometric reconstruction of A.Stradivari's pattern for a viola contralto, 1672 (Figure 59). The length of the pattern (CD) is 403 mm, the width of the upper bout (F¹F²) is 184 mm, the width of the lower bout (G¹G²) is 241 mm, and the waist is 124 mm. Moreover G¹G²/ F¹F² = 1.31. The point M, a root of the lower bulb, divides the length of the pattern into two lengths in the ratio of 1.3, then MD = 228 mm and CM = 175 mm. Having constructed the lower bulb I used the clothoid a-238.5. The root of the clothoid lies at the point M, touches the C-bout and is inserted in the lower bout of the mould up to the end block. Moreover, the angle KMP = 60º.
   Similarly one constructs the opposite side of the bulb. The area around the end block is completed by the clothoid a-148 so that it is a tangent to the horizontal line D, and segments of both clothoids a-238.5 and a-148 are joined in such a way that the curvature is continuous throughout. As can be seen from the drawing, the lower bout was composed a little asymmetrically.
   The upper bout is drawn with the clothoid a-213.6, the root of which does not lie on the line CD. Since in this pattern the widest place of the upper bout to the waist is less than that of a violin, the angle of rotation of the clothoid is decreased, i.e. the angle JUS¹ = 54º. I finish the design of the upper bout with the clothoid a-108.9.
   As can be seen from the drawing, the upper bout is symmetrical. I obtain the points F and G by simple measurements and determine that CF = 76.7 mm and GD = 85 mm. The length of the pattern (403 mm) and the length FG (241.3 mm) are in the ratio of 1.67. The line connecting the widest place of the lower bout with the eye of the scroll goes through the lower and upper corners of the viola pattern (G¹A¹). Draw the line from the point F through the upper corner and get FH, which makes an angle of 54.5º with CD.
   The lower corner is found by a similar way, where the tangent of the clothoid a-238.5 gets close to the point P, and the line GH makes an angle of 51º with CD.
   The configuration of the corners and C-bout are drawn with the clothoids a-69.1 and a-51.5, and are in ratio of 1.34.
  
  
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  Figure 59. The geometric reconstruction of A.Stradivari's pattern for a viola contralto, 1672.
  
  
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  Figure 60. Superimposing the drawing of Amati's pattern "MB" for a violin (dashed lines) and the drawing of A.Stradivari's mould for a viola contralto, 1672.
   If the lines TJ and PK through the corners divide the length of the violin pattern into three parts, I get three lengths in the following ratios:
  
  CJ(139 mm)/JK(96 mm) = 1.45; KD(168 mm)/JK = 7/4 KD/CJ = 6/5.
  
   The mould for a viola contralto (1690) was made by A.Stradivari after the mould 'PG' for a violin. So, in Figure 69 I have compared the proportions of these moulds.
   The geometric reconstruction of A.Stradivari's pattern for the viola contralto of 1690 (Figure 61). The length of the pattern (CD) is 403 mm, the width of the upper bout (F¹F²) is 177 mm, the width of the lower bout (G¹G²) is 233 mm, and the waist is 118 mm. Moreover G¹G²/ F¹F² = 1.32. The point M, a root of the lower bulb, divides the length of the pattern into two lengths in the ratio of 1.37, then MD = 233 mm and CM = 170 mm.
   Having constructed the lower bulb I used the clothoid a-244. The root of the clothoid lies at the point M, touches the C-bout and is inserted in the lower bout of the mould up to the end block. Moreover, the angle KMP = 58º.
   Similarly one constructs the opposite side of the bulb. The area around the end block is completed by the clothoid a-145 so that is a tangent to the horizontal line D, and segments of both clothoids a-244 and a-145 are joined in such a way that the curvature is continuous throughout.
   As can be seen from the drawing, the lower bout was composed a little asymmetrically.
   The upper bout is drawn with the clothoid a-184, the root of which does not lie on the line CD. Since in this pattern the widest place of the upper bout to the waist is less than that of a violin, the angle of the rotation of the clothoid is decreased, i.e. the angle FF²S¹ = 41.3º. I finish the design of the upper bout with the clothoid a-104.4.
   As can be seen from the drawing, the upper bout is symmetrical.
   I obtain the points F and G by simple measurement and determine that CF = 73 mm and GD = 90mm. The length of the pattern (403 mm) and the length FG (240 mm) are in the ratio of 1.68.
   The line connecting the widest place of the lower bout with the eye of the scroll goes through the lower and upper corners of the viola pattern (G¹A¹).
   Draw the line from the point F through the upper corner and get FH, which makes an angle of 54º with CD.
   The lower corner is found by a similar method, where the tangent of the clothoid a-244 gets close to the point P, and the line GH makes an angle of 51º with CD.
  The configuration of the corners and C-bout are drawn with the clothoids a-72.7and a-48.4, which are in ratio of 3/2.
   If the lines TJ and PK through the corners divide the length of the violin pattern into three parts, I get three lengths in the following ratios:
  
  CJ(134 mm)/JK(98 mm) = 7/5; KD(171 mm)/JK = 1.74; KD/CJ = 1.28.
  
  
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  Figure 61. The geometric reconstruction of A.Stradivari's pattern for a viola contralto of 1690.
  
  
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  Figure 62. Superimposing the drawing of the pattern "PG" for a violin (dashed lines) and the drawing of A.Stradivari's mould for a viola contralto of 1690.
  
  
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  Figure 63. Superimposing the drawing of the pattern for a viola contralto of 1690 (dashed lines) and the drawing of A.Stradivari's mould for a viola contralto of 1672.
  
   The geometric reconstruction of A.Stradivari's pattern for the tenor viola of 1690 (Figure 64). The length of the pattern (CD) is 468 mm, the width of the upper bout (F¹F²) is 207 mm, the width of the lower bout (G¹G²) is 257 mm, and the waist is 137 mm. Moreover G¹G²/ F¹F² = 1.24.
  
  
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  Figure 64. The geometric reconstruction of A.Stradivari's pattern for the tenor viola of 1690.
  
  
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  Figure 65. Superimposing the drawing of the pattern "PG" for a violin (dashed lines) and the drawing of A.Stradivari's mould for a tenor viola of 1690.
  
  
   The geometric reconstruction of A.Stradivari's pattern for the 'Duport' cello (Figure 66). The length of the pattern (CD) is 749 mm, the width of the upper bout (F¹F²) is 340 mm, the width of the lower bout (G¹G²) is 435 mm, and the waist is 220.5 mm. Moreover G¹G²/ F¹F² = 1.28.
   The point M, a root of the lower bulb, divides the length of the pattern into two lengths in the ratio of 1.38, then MD = 434.4 mm and CM = 314.6 mm.
   Having constructed the lower bulb I used the clothoid a-458 Moreover, the angle KMP is 57.3º.
   Similarly one constructs the opposite side of the bulb. The area around the end block is completed by the clothoid a-272.7 so that it is a tangent of the horizontal line D, and segments of both clothoids a-272.7 and a-458 are joined in such a way that the curvature is continuous throughout.
   As can be seen from the drawing, the lower bout was composed symmetrically.
   The upper bout is drawn with the clothoid a-392.6. The point S divides the length of the pattern in half so CS = SD = 374.5 mm. The angle of the rotation of the clothoid, i.e. the angle JST = 54º. I finish the design of the upper bout with the clothoid a-220.
   As can be seen from the drawing, the upper bout was composed a little asymmetrically.
   I obtain the points F and G by simple measurement and determine that CF = 140 mm and GD = 161mm. The length of the pattern (749 mm) and the length FG (448 mm) are in the ratio of 1.67.
   The line connecting the widest place of the lower bout with the eye of the scroll goes through the lower and upper corners of a violin pattern (G¹A¹).
   Draw the line from the point F through the upper corner and get FH, which makes an angle of 53.5º with CD. The third line, which is a tangent of the clothoid a-392.6, goes through the upper corner of the pattern.
   The lower corner is found by a similar method, where the tangent of the clothoid a-458 goes through the point P, and the line GH makes an angle of 49.5º with CD.
   The configuration of the corners and C-bout are drawn with the clothoids a-116 and a-85, and are in ratio of 1.36.
   If the lines TJ and PK through the corners divide the length of the cello pattern into three parts, I get three lengths in the following ratios:
  
  CJ(258 mm)/JK(173 mm) = 3/2; KD(318 mm)/JK = 1.84.
  
  
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  Figure 66. The geometric reconstruction of A.Stradivari's pattern for the 'Duport' cello.
  
  THE ARCHES OF THE BELLY AND THE BACK
  
   The arches of the belly and the back are surfaces. Before giving the rules of building the arches of the belly and the back we must define what a surface is. The surface is a continuous extent having only two dimensions (length and breadth, without thickness), whether plane or curved, finite or infinite. The position of the point on it is defined by two surface coordinates.
   The curved surfaces are subdivided into regular, graphic, topographical, gravitational and others.
   Before defining what type of surfaces we shall rank the arches of the belly and the back, address to the history of this question.
   Many researchers of the creative activity of Italian luthiers developed the topographical diagrams of the plates' arches to study them. The topographical diagram is marked with contour lines, joining points of equal height of the curvature. Such a method of analysis of the violin arches is very good in many events: it can be used for the study of the deformations occurring in the belly and the back and it can be used during the making of new plates as a particular case of checking on the symmetry of the curvature.
   A.Stradivari himself has left absolutely other method of building of arches. His sixths (one longitudinal and five transversal guides) give the rule of building for the arches of the belly and the back and they are an analytical instrument for the study of the different arches.
   Proceeding from the nature of these sixths, the surface of a belly or a back can be defined as a graphic surface, generated by moving a variable line (five transversal guides) along a fixed direction (the sixth longitudinal guide) in accordance with the results of calculations, satisfying assessed conditions. As the conditions in this instance there are a typical curvature of the longitudinal guide, the horizontal plane and the outline of the basis of the plate. Five transverse lines, as different positions of the variable generator, together with a sixth longitudinal line as the guide, form the linear framework of the surface of the belly or the back.
   Though the nature of the arches of the different instruments of A.Stradivari is variable (because of particularities in the wood), nevertheless it is geometrically analyzable and can be expressed by a formula, i.e. the curvature of the arches can be described by the clothoid. Then the generator, and guide of these surfaces will be regular curved lines and the surface of the belly or the back will be a regular curvilinear surface with a variable generator. The variability of a generator is assigned by the outline of the plate, the curvature of the longitudinal guide and the plane of basis.
   The initial position of the generator is defined by the geometric centre of the belly or the back. Proceeding from the specifics of the belly and the back, these have different curvatures. In Figure 67 I have presented the transverse section of the violin's belly (a) and the back (b) of A.Stradivari. As can be seen from the drawing, the different positions of the clothoid on the belly and on the back show the different nature of their curvature. So the top of the belly has the greater radius of curvature, but closer to edges the radius is decreased. At the back is quite the reverse - the smaller radius is located on the top, but closer to edges the radius is increased.
   The longitudinal curve (or guide, as defined mathematically) of the belly repeats the nature of the curvature of the generator (Figure 68, a) i.e. in both cases the clothoid is disposed from the centre to the edge.
   That the guide of the back (Figure 68, b) has a more complex curvature than what is clearly seen from the drawing (of which I have shown only half). If we dispose the clothoid in the same direction, as with the belly, the clothoid is deviated from the guide far enough from the edge, showing hereunder a rather big radius of curvature of the back's guide beside the edges that correspond to the nature of the generator's curvature. Thereby, the smallest radius of the back's curvature on the longitudinal line is located not at its geometric centre (as the generator has), but at the centre of the upper and lower bouts.
  
  
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  Figure 67. The transverse section of the violin's belly (a) and the back (b) of A.Stradivari.
  
  
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  Figure 68. The longitudinal section of the violin's belly (a) and the back (b) of A.Stradivari.
  
   In Figure 69 I have drawn the sixths of curvature of the violin belly, which are placed: A)- at centre of the C-bout; B) - corresponding with the upper corners; C) - at the maximum width of the upper bout; D) - corresponding with the lower corners; E) - at the maximum width of the lower bout; F) - corresponding with the upper half of the longitudinal section; G) - corresponding with the lower half of the longitudinal section.
   The distance between the vertical lines, dividing the central guide (A) into even areas, is 6 mm; the height of the arch in the centre is 15.5 mm; the width of the belly here is 112 mm.
   The distance between the upper corners (B) is 150 mm, and the height of the arch in this place is 14.8mm.
   The widest place of the upper bout (C) is 167 mm whilst the arch here is 12.0 mm.
   The distance between lower corners (D) is 179 mm, and the height of the arch in this place is 15.3mm. The widest place of the lower bout (E) is 207 mm whilst the arch here is 13.0mm. The longitudal curve (FG) is symmetrical and on my drawing it is divided in half. The distances between the vertical lines in the guides B, C, D, E, F, and G correspond to the points of their intersection with the lines of the topographical diagram. One can see the location of the sixths on the belly in Figure 71, where for greater clarity I have combined the topographical diagram with the graphic framework. The numbers on the drawing indicate the heights of the arch of the belly.
   The topographical diagram of the belly is shown in Figure 70. The numbers on the diagram indicate the distances between lines.
   In Figure 72 I have drawn the sixths of curvature of the violin back, which are placed: A) - at the centre of the C-bout; B) - corresponding with the upper corners; C) - at the maximum width of the upper bout; D) - corresponding with the lower corners; E) - at the maximum width of the lower bout; F) - corresponding with the upper half of the longitudinal section; G) - corresponding with the lower half of the longitudinal section.
   The distance between the vertical lines, dividing the central guide (A) into even areas, is 6 mm; the height of the arch in the centre is 14.8 mm; the width of the back here is 112 mm. The distance between the upper corners (B) is 150 mm, and the height of the arch here is 13.4mm.
   The widest distance of the upper bout (C) is 167 mm and the height of the arch here is 10.1 mm. The distance between the lower corners (D) is 179 mm, and the height of the arch here is 14.2mm. The widest distance of the lower bout (E) is 207 mm and the height of the arch here is 11.0 mm. The longitudal curve (FG) is symmetrical and on my drawing it is divided in half, and I have added the button to the upper half of the back.
   The distances between the vertical lines in the guides B, C, D, E, F, and G correspond to the points of their intersection with the lines of the topographical diagram. One can see the location of the sixths on the belly in Figure 81, where for greater clarity I have combined the topographical diagram with the graphic framework. The numbers on the drawing indicate the heights of the arch of the belly.
   The topographical diagram of the belly is shown in Figure 73. The numbers on the diagram indicate the distances between lines. Hereinafter in Figures from 75 to 88 I show the drawings of arches of violas and cellos.
  
  
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  Figure 69. The sixths of the violin belly.
  
  
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  Figure 70. The topographical diagram of the violin belly.
  
  
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  Figure 71. Combining the topographical diagram of the violin belly with the sixths.
  
  
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  Figure 72. The sixths of the violin back.
  
  
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  Figure 73. The topographical diagram of the violin back.
  
  
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  Figure 74. Combining the topographical diagram of the violin back with the sixths.
  
  
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  Figure 75. The sixths of the viola belly.
  
  
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  Figure 76. The topographical diagram of the viola belly.
  
  
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  Figure 77. Combining the topographical diagram of the viola belly with the sixths.
  
  
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  Figure 78. The sixths of the viola back.
  
  
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  Figure 79. The topographical diagram of the viola back.
  
  
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  Figure 80. Combining the topographical diagram of the viola belly with the sixths.
  
  
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  Figure 81. The fifths of the cello belly.
  
  
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  Figure 82. The longitudinal section of the cello belly.
  
  
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  Figure 83. The topographical diagram of the cello belly.
  
  
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  Figure 84. Combining the topographical diagram of the cello belly with the sixths.
  
  
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  Figure 85. The fifths of the cello back.
  
  
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  Figure 86. The longitudinal section of the cello back.
  
  
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  Figure 87. The topographical diagram of the cello back.
  
  
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  Figure 88. Combining the topographical diagram of the cello back with the sixths.
  
  * * *
  
   The afore-cited analysis of the scrolls, ffs and moulds of the stringed and bowed instruments allows me to draw a conclusion about the goal-directed usage of some of the proportional relations and geometric angles by great masters. There is only the need to settle the problem of clothoid usage by them, in certain patterns, for the drawing of the contour of the instrument.
   Neither A.Stradivari, nor fellow masters used the clothoid in the manner I have. Quite simply, they could not draw with the precision we can nowadays.
   But they could shape it from a string or some other suitable material.
   This conclusion is confirmed by the fact that a variety of curved lines, made by Stradivari are still extant them very close to or exactly modeled on this remarkable mathematical curve, the clothoid. The small inexactnesses are explained by the fact that Stradivari did not strive for perfect accuracy when making the drawing; additionally, the string cannot be as accurate a method as that which can be produced with help of a computer.
   In the next chapter I will try, as far as possible, to reproduce the method of Stradivari for the drawing of all contoured curved lines of the violin, including the scroll, ffs and the pattern.
  
  Chapter three
  
  
  The Reconstruction of the Stardivarius Method of Violin Design
  
   In this chapter I will try to show a method of violin design using compasses and a ruler to determine sizes and proportions only; and a way of drawing contour lines with the help of strings (a method that could have been used by olden day masters).
   I already noted that my designs were produced with the help of two modules: the number π and the clothoid.
   At the turn of the 16th century, when such prominent masters as Stradivari and Guarnerius were working, mathematics and geometry had reached a stage of rather high development, and there was decimal calculus around the world, but the inchwas the fundamental unit of measurement.
   It is impossible to talk about the size of one inch at that time with sufficient accuracy because it has only been since 1959 that the inch has been defined officially as 2.54 centimeters. This unit derives from the old English unce, or ynche, which in turn came from the Latin unit uncia, which was "one-twelfth" of a Latin foot, or pes. The old English ynche was defined by King David I of Scotland about 1150 as the breadth of a man's thumb at the base of the nail. During the reign of King Edward II, in the early 14th century, the inch was defined as "three grains of barley, dry and round, placed end to end lengthwise. Sometimes the inch has even been defined as the combined length of 12 poppy seeds.
   If at various times the grains or the thumb were used for quickly finding this size, it is important to search for the true origin of the inch in geometric proportions. I mean the number π and the golden ratio: 1 in. = πμ/2 cm = 2.5416 cm.
  Thereby, the number π is connected with the British inch and the modern metre by means of the golden division. One can say that the inch, which was used by the ancient masters, was approximately, as I have indicated in my formula.
   Consequently, the ancient constructors of violins did not use the number π as a module, but one inch instead. Thereby, the main sizes of violin can be defined as a geometric progression of the British inch in the proportional attitude of the golden division. In Figure 29 this progression can be defined by the following lengths: AB = 82.25mm (2in,μ BC = 133.08mm (2inμ²); AC = 215.33mm (2inμ³); etc.
   I remind the reader that as the modulor I used the golden division and numerically simple ratios. Concerning the numerically simple ratios, their usage is very simple and does not require any special explanations. In Figure 2 I had demonstrated the calibration of lengths in the ratio of the golden division.
   Now I begin to design the violin with the help of spirals, using simplest ways, which could use the luthiers of the past.
  
  THE SCROLL
  
   Let us begin with the Bernoulli spiral. We can see that because of the small size of the violin scroll, it is unreasonable to use of the method of drawing known to Vitruvius [Marcus Vitruvius Pollio, an author of the De architectura libri decem (Ten Books on Architecture)]. The degree of reduction of the arcs of the circumferences (decay) in his method does not help us to solve our problems.
   For the drawing of the scroll I offer to use the scale compasses (Figure 89), one variety of which was used by the masters for the fretting out of guitars, lutes, mandolins and so on.
   If one pair of arms has, for instance, a length of 4 inches (101.6 mm), then the other pair must be shorter by a percentage chosen by the luthier himself.
  
  
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  Figure 89. The scale compasses.
  
   I have chosen the 15% reduction of radius and proceed to draw the violin volute (Figure 90).
   A spiral is constructed as a series of circular arcs. I begin by drawing a quadrant of the circular arc ab centre O and radius of 16 mm with the spread of the big pair of compasses. Then I turn the compasses over and with the spread of the small pair of compasses draw a quadrant with a circular arc bc centre x, which rests on the line ob. The new radius of 13.6 mm has only 85% of the original radius (16 mm). Spread the big pair of compasses an equal distance to this radius and draw a quadrant with a circular arc cb centre y with the spread of the small pair of compasses.
   Next arc de with the centre at point z also is a quarter of circumference.
   The further process of drawing the volute completely repeats the previous one and is easily understood from the drawing. Naturally we can use the compasses with another percentage reduction of the radius (as I did when analyzing the violin scrolls).
   The quality of the volute is dependant on the accuracy of the present drawing. If I made my drawing with the help of a computer with a higher degree of accuracy, than masters of the past, who using only compasses and a ruler, certainly allowed for some inaccuracy, which, of course influenced the final result.
   When I analyze the violin scrolls, certain inexactness in the building of the Bernoulli spiral by some Italian masters is striking. This inexactness was not so much the effect of carelessness as non-observance of the main rule for designing the Bernoulli spiral: each new arc must be reduced by a constant percentage.
  
  
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  Figure 90. The phased drawing of the volute.
  
  
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  Figure 91. The volute of Pietro Guarneri I.
  
   Using the scale compasses, one can allow a certain inaccuracy, but in very small limits. But if we draw the volute with the usual compasses and produce a reduction 'by eye', then the spiral although it will be rather graceful (Figure 91), cannot be identified as a genuine Bernoulli spiral. The parameters of this spiral are as following: arc ab - radius of 27.5 mm; bc - 22.8 mm; cd - 16 mm; de -14.6 mm; ef - 11.3 mm; fg - 9.5 mm; gh - 7.3 mm; hi - 6.3 mm; ij - 5.4 mm; jk - 4.1 mm; kl - 3.9 mm; lm - 3.1 mm; mn - 2.5 mm. The percentage reduction of the radius from the greater radius is as follows: 82.8%, 70%, 91%, 77%, 84.3%, 77.4%, 85.3%, 87%, 75.9%, 94%, 80.1%, 80.1%. The average factor is 82%.
   Return to Figure 90. On the last eleventh drawing of my construction the centre of the volute, which will serve us for the further construction of the violin scroll, is already seen distinctly. I have only to draw three parallel lines: through the centre of the volute, above the volute through point a and the lower line, which goes through the point a at a distance of 2 inches (50.8 mm) from the line c. The centerline is limited by the vertical line at the distance of 2inμ from the centre of the volute.
   The next stage of constructing the scroll is concerned with the drawing of the curved lines, which I have done with the help of the clothoid in the First Chapter.
   To this effect I use the springy string, which can take the shape of a clothoid when we roll it. Using the rolled string shaped as a loop (Figure 92) I continue to draw the outer face of the volute. Certainly, we can vary not only the size, but also the configuration of the curvature, created by a curled string, forcing the string with the thumb of the left hand towards the nut.
  
  
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  Figure 92. Modeling the external curvature of the volute.
  
   I model the rear sides of the peg box, which has an S-form, in one step, holding the string in the way shown in Figure 93. As can be seen, the curled string exactly copies the sidebar of the scroll. It would not be out of place to observe that the nature of the curvature depends in full measure on the way of holding the string, on the distance between hands, on the pressure of the right thumb, etc. So the violin scrolls, drawn in this way, can be different (with a fluent change of the radius), but cannot be attained with the use of compasses alone.
  
  
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  Figure 93. Modeling the rear sides of the peg box.
  
   The upper part of the peg box is also modeled with the help of a curled string, which can be held by a master in the way shown in Figure 101. Even A.Stradivari's scrolls have a different curvature in this part of the peg box. A smaller radius is used in the centre, and closer to the volute or to the upper nut. So the nature of the curvature is completely dependent upon the method of holding the string, when the thumb of the right or left hand plays the leading role. In my photograph, for instance, the right thumb forces the string, creating the smallest radius of the curvature closer to the nut.
  
  
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  Figure 94. Modeling the upper part of the peg box.
  
  THE VIOLIN PATTERN
  
   The violin moulds, made by A.Stradivari, have several round holes, which help to assemble the ribs. They correspond with all six blocks. But there are other holes in the moulds, which were changed into a semicircular shape when cutting the spaces for the upper and lower blocks. In some moulds such semicircular cuts are placed at the centre of the space (for instance, the model B, 6/12/1692), some moulds have two semicircular cuts (the model PG, 1689); one has three cuts (the model P, 1705). I deduce the reason for these cuts, was that they were former holes; i.e. auxiliary holes when modeling the contour of the pattern.
   When modeling the violin scroll with the help of a springy string, I held the latter with both hands. As a variant, I have offered to fasten one end of the string by any known way. Now I will show one of the possible variants of such fastening.
   In the hole, drilled in the board of the willow wood for a future pattern, on the axis of symmetry the violin peg is placed in such a way that the small hole for the string is at the surface of board exactly. Spring a string from the peg's hole and roll it to determine the curved shape of the bout. Then trace the curvature of the string with a pencil. As can be seen in the photograph, the string repeats the contour of the clothoid completely (Figure 95).
  
  
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  Figure 95. Modeling of the upper bout of the violin pattern from the hole on the axis of symmetry.
  
   It will be useful to notice that the correct angle of rotation of the peg, holding the string, is very important in this matter. Rotating the peg to one side or another, I hereby change the curvature of the string and the form of the future pattern too. In my case a string passes through the peg at an angle of 54º to the axis.
   We see that the position of these holes was attained by experiment and not affixed formula. On the Stradivari's moulds we see the residuum of these holes of different sizes: from deep cuts (the model B-3/6/1692) to small cuts (the model SL), and also their total absence. This is explained by the different distance between the hole and the edge of pattern, whereas the depth of cuts for the blocks is always 15 mm.
   On the next photograph (Figure 96) I investigate the possibility of modeling the upper bout, placing the peg into the hole, which is not located on the axis of symmetry.
  
  
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  Figure 96. Modeling of the upper bout of the violin pattern from the hole not on the axis of symmetry.
  
   As can be seen, this works with greater success, because in this case the curvature of the string includes the area of the upper bout, something the clothoid and the previous experiment with the string did not do. In both cases the string repeats the contour of the pattern exactly, only right now it passes through the peg at an angle of 60º to the axis.
   The left side of the pattern is built as well as the right one. Some asymmetry between the left and right sides is discernible because it is practically impossible to repeat this procedure with such a flexible thing, as string. This asymmetry that does not spoil the general appearance of the instrument at all, in fact it even contributes certain individuality to the final contour of the violin.
   In Figure 97 I demonstrate the modeling of the lower bout of the violin pattern. This procedure does not differ from the previous work with the upper bout. The angle at which the string passes through the peg is 62º.
   To finish the modeling of bouts at the area of the upper and lower blocks is not laborious with the use of the same string (Figures 98, 99).
  
  
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  Figure 97. Modeling of the lower bout of violin pattern.
  
  
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  Figure 98. Finishing of the modeling of upper bout.
  
  
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  Figure 99. Finishing of the modeling of lower bout.
  
   The next stage is the determination of the position of the upper and lower corners of the pattern. Connect the edges of the widest place of the lower bout and the volute of the scroll with a straight line. Then the lines, passing through the centres of the bouts at certain angles (as I defined this in the First Chapter), intersect the first line. In the intersections of these lines drill the holes for the pegs in such a way that the edge of the peg touches the cross point, and the line through the centre of the bout crosses the hole strictly at the centre (Figure 100).
  
  
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  Figure 100. The position of the hole for the peg when modeling the C-bout.
  
   The pegs are fixed in such a position that their holes are compliant with the lines, which pass through the centres of the bouts. The string is placed in the pegs and pulled through them up to the desired size of the waist (Figure 101). Though in this case the string did not repeat the contour of Stradivari's pattern, such C-bouts (the same curvature of the upper and lower corners, and a rounded centre of the waist) are found in the violins of different masters, as with those of Andrea Amati. I will name this method: modeling with a free string.
  
  
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  Figure 101. Modeling of the C-bout of the violin pattern with a free string.
   I model the Stradivari's C-bout by another way, limiting the excessive concavity of the string inside of the pattern with help of a stop (Figure 102). Since this stop exerts pressure on the string, not at the centre, but a little above, the curvature of the upper corner will be bigger than the curvature of the lower corner. I will name this method: modeling with a restricted string.
  
  
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  Figure 102. Modeling of the C-bout of the violin pattern with a restricted string.
  
   I model the outer face of the corners with the help of the same pegs, directing the string along other curved lines, which must touch the bulbs of the corresponded bout (Figures 103).
  
  
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  Figure 103. Modeling the outer face of the corner.
  
  
  THE FF-HOLES
  
  
   In Figures 104, 105, 106 and 107 I show four stages of modeling of the ffs. Here I have placed the pegs into the eyes, but from the back of the board only in order not to overlay the drawing of the f-hole by the head of the peg. The string, which I use for the modeling of the f-hole, is thinner than in previous cases.
   As to the strings which helped me to draw the violin, I used a piano string for the modeling of large objects: the bulbs of the bouts and the scroll; a guitar string for the C-bout and a violin string for the ffs. In short, the smaller the object of modeling, the thinner the string used.
  
  
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  Figure 104. Modeling of the upper part of the f-hole.
  
  
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  Figure 105. Modeling of the lower part of the f-hole.
  
  
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  Figure 106. Modeling of the inner part of the f-hole's arm.
  
  
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  Figure 107. Modeling of the external part of the f-hole's arm.
  
  THE ARCHES OF THE BELLY AND THE BACK
  
   When speaking of the arches of the belly and the back of A.Stradivari's violins, we noted that to date two methods of construction of their buildings and geometric analysis is known. First, using the sixths of the curvature and of course the topographical diagrams. Both of these ways are very good for copying, but completely unsuitable for primary modeling. Even my mathematical descriptions of the arches of the belly and the back, although suitable for the modeling of a new violin, do not answer the most important question: "How was the primary modeling of the arches of a violin produced by the great luthiers of 16th -18th centuries?"
   After a series of experiments with the string, which has helped me to create the hypothesis about the designing of the outline of the instrument's body with the help of some flexible material, and after finding the mathematical relationship between a curled string and the clothoid, for a long time I searched for anything similar regarding the arches. In Reality, neither Amati nor Stradivari, nor other luthiers have left us any traces of this creative process. Common sense tells us that the sixths of the curvature, made by Stradivari, are nothing more than auxiliary instruments, used by the master for the transfer of the curvature of some arch on the real violin plate. I hold the view, that there was a certain object, identical to the arch of the plate, conceived by the master, which must have been copied in wood by him alone. No doubt he created this object himself, to meet the requirements of each new idea. It is obvious that this object, unlike the string, had a constant hard form, which allowed him to use its measurement easily.
   I do not think that Amati and Stradivari created the form of the violin arch intuitively, using merely the talent of a sculptor and an aesthetic vision of the beautiful. In the first place they were concerned with sound, and so, engineering problems for them rose above pure aesthetics. Thus, it is possible to conjecture that firstly they created some model of a future arch, but afterwards they copied it with the help of the sixths. My suggestion is that this model wasn't made of wood. To reach this conclusion I can list several more or less cogent arguments:
  - they considered that it is impossible to do the model of wood;
  - if it were easy to make of wood, that we do not need the model, because it is possible to make the arch of the plate directly;
  - the material for the making of the model must be flexible and springy as was the string for the design of the violin outline;
  - the physical force, which changes the form of material, must be natural and directed to its whole surface to obtain a high-quality construction from an engineering perspective;
  - after the completion of the model the material must lose its springy qualities and become enough hard as a matter of convenience to make a copy of it.
   In my experiments I tried a great deal but the results of those attempts will remain outside this book. For now I will show only the final result of my studies. In the section The Arches of the Belly and the Back I listed several types of curvilinear surfaces with the generator of the variable type: regular, graphic, topographical, gravitational and others? I referred to all, except the gravitational one. Now it is time to turn our attention to it.
   Make a frame by cutting out the hole in the form of the violin plate without corners from some board (Figure 108) and glue a knitted fabric around the frame. By wetting the horizontally disposed frame with water, you will find it will sag under the weight of its own gravity exactly in the shape of the violin arch. This demonstrates that the violin plate can be seen to be a result of the gravitational surface.
  
  
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  Figure 108. The frame for modeling of the arches of the plates.
  
   This soft construction can be stiffened with plaster of Paris (gypsum). Of course, if the wet plaster is spread too much, then the fabric will sag too deeply, more than the required 11 mm. Additionally, unequal spreading of the plaster on the surface of the fabric will not create the smooth surface of an arch, unless it is spread equally with very fluid consistency. So, I dilute the plaster to the consistency of milk, and spread it on the fabric with the wide and soft brush (Figure 109).
  
  
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  Figure 109. Modeling of the arches.
  
  
   The difficulty in properly fulfilling this work is akin to the varnishing of the ready violin. In practice, it is necessary to cover the fabric with about 5-6 layers of the gypsum milk to get more or less a hard construction. Any incidental bulges in the layers can change the nature sag of the fabric and disturb the arch of the model. Spreading the fluid gypsum on the second and even the third layer must be very careful, since it is possible to break the still thin gypsum shell and to spoil all the work. If an attempt is made to spread a layer of thicker gypsum (for instance with the consistency of sour cream) on a yet not hardened shell with one or two layers, then we risk spoiling the configuration of the arch, because it is very difficult to spread the thick gypsum on the whole surface evenly.
   When the fabric becomes rather stiff and already can not be bent, the gypsum with the consistency of sour cream is spread in several layers in the same way (Figure 110). I finish the work by flooding the model with the gypsum up to the edges (Figure 111).
  
  
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  Figure 110. Spreading the layer of thicker gypsum.
  
  
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  Figure 111. Flooding the model with the gypsum up to the edges.
  
   We know that the configuration of the arch of the belly differs from the arch of the back. If we spread the gypsum absolutely evenly in the first layers, then we will get the arch of the belly (Figure 112). For obtaining the arch of the back it is necessary to spread the first layer unevenly, i.e. thicker in the centre and thinner on the edges. The greater sag in the centre, because of the excess of gypsum, will give us the necessary arch of the back (Figure 113).
  
  
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  Figure 112. Plaster model of the violin belly.
  
  
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  Figure 113. Plaster model of the violin back.
  
   In general, the first layer of gypsum forms the future arch, and the resilience of the wet fabric allows forming the arch smoothly, without "big nubs" and other roughness. But if the second layer will be spread unevenly, then the excess gypsum will bend the weak fine first layer more and the arch will be uneven.
   It is known that A.Stradivari had been defining the height of each arch before the making of a violin and the sixths of the curvature prove it. How does one form the arch of the model with a given height for the intended violin? We know that his sixths do not yet show the curvature at the edges of the instrument, because this was done after the purfling is inlaid. So before the purfling is inlaid the height of the arch in the centre of the plate will be about 11 mm disregarding the thicknesses of the edges. And therefore it is necessary to allow for the sagging of the fabric to such a depth during the fixing of the first layer of gypsum. If the depth of sagging fabric is not enough, then it is necessary to spread a second layer of gypsum milk, when the first layer is not yet dry. It is necessary to repeat the procedure until the sagging fabric will meet the intended size. Technically these layers when completed will be considered as the first layer. The second and the next layers must not be spread until the previous layer is dry.
   If we work with the model of the back, then we increase the sagging by spreading additional gypsum only along the vertical central line, more in the centre and less on the edges.
   But if after spreading the first layer of gypsum we get too much sagging of the fabric, then it is necessary to correct one of two things: either reduce the consistency of the gypsum milk (and replace the fabric), or change the stretching fabric on the frame. In the last case two variants are possible:
  - replace the fabric (as in the previous case) or
  - if the glue is not yet dry, it is possible to tighten the fabric.
   In the last case, possible slanting of the fabric will influence the general quality of the work, and so, the stretching of the fabric during its fixing to the frame is better.
   One can make a copy of the model using both ways: making the sixths of curvature and drawing the topographical diagram.
   It seems to me that the arches of the belly and the back, built in such a way, with their smooth and plastic lines of curvature meet the requirements of the distribution of strain in them perfectly. Certainly, in the process of working with specific wood a master can change the height of the arches mildly, but the general nature of the curvature must be the same as we have determined in the model. And A.Stradivari did it like this, in those violins which have not been built by according to his sixths of curvature absolutely exactly. These arching guides only helped Stradivari to operate correctly, reserving the right of partial changes for him to adjust according to the acoustic qualities of the wood.
   The frame is made with due regard for the desired width of the trough along the edges of the plates, since this curvature at the edges of the instrument will be done only after the purfling is inlaid. In my case I used the outline of the first internal curved line, shown at Figure 70 for the belly and at Figure 73 for the back.
   One can make the sixths of the curvature of the arches in this way: to make cross sections of the plaster model along lines, intended for copying (Figure 114), to place those segments of the arch on a wooden plate, to trace a sidebar of the curvature with a pencil and to cut out the patterns according to this drawing.
   As this way is the simplest way of copying the plaster model, then A.Stradivari probably used it. If for some violin forms he made not only fifths, but also the sixth longitudal pattern, then the last was made before making cross sections of the plaster model. Common sense tells us that we can expect that A.Stradivari did not save these plaster segments.
   In Figures 115 and 116 I have demonstrated the coincidence of the drawings of the fifths of the belly and the back with their plaster models.
  
  
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  Figure 114. The cross sections of the plaster model.
  
  
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  Figure 115. Superposing the drawings of the fifth and the photos of the cross sections of plaster model of the violin belly.
  
  
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  Figure 116. Superposing the drawings of the fifth and the photos of the cross sections of plaster model of the violin back.
  
  Conclusion
  
   I have finished the geometric analysis of stringed and bowed instruments of the great Italian masters and hope that the presented considerations will help other masters get to know the creative process in the design of these instruments better. This, in turn, will inspire them in the quest of making their own models, which will possess the particularities of the style of the great Italians and still have the feature of individuality, which a modern master possesses.
   Practically on each page of my book I fought with a wish to begin to prove the advantage of my method in comparison with the investigations of other authors.
   The fact that different methods of study of the same instruments yield different results is absolutely natural. Anyway, all theories, known to the present-day luthiers, reflect the real characteristics of the violin geometry with a greater or smaller degree of approximation and are equally right from the standpoint of the abstract feature of its proportions, but they are incompatible with a reconstruction of the method of working of the old-time Italian masters.
   Comparing different methods, in the first place a researcher turns his attention to such qualities of a theory as its systematic character, fullness, accuracy of results and versatility (i.e. wide range of using). But often it is not enough to confirm that this system of proportions of the instrument was the result of methods, which the master himself used. Rather, it is a secondary, side result.
   How do we find the methods which were used by the luthier, when he was designing violins? The fact is that the working of the master leaves not only final results, but also direct traces of the ways by which it was made. The most observable of them is seen in the different sort of 'mistakes' they made along the way.
   A mistake itself can be accidental, but its nature depends upon the intention, when it was made. Let us take, for example, the asymmetry of the upper or lower bout of some violin moulds of Stradivari. What can we see? The configuration of the left and the right curved lines is the same absolutely, but they have no common axis of symmetry. It can happen either when using the pattern of the whole curved line between the block and the corner, which have been placed both times a little bit differently, or when using the springy string, which although it resembles the configuration of the clothoid in all cases, can not repeat itself as a mirror image exactly.
   Since Stradivari did not use special patterns for drawing the outline of the violin's body, then nothing else is left, but to suppose, he used the springy string, which does not retain its form after having been used. The use of the compasses would give another kind of distortion, which is unlike that described above.
   We can see the nature of the mistakes in designing the curled line with the compasses in the example of designing the violin scroll, when a mistake in the determination of the radius of the following arc of the Bernoulli spiral changes the nature of the curved line itself.
   There was a time when I tried to analyze the violin scroll by the clothoid, but it was unsuccessful. It is possible, certainly, to find the scroll, which could be described by the clothoid, but it would be a rather occasional event, not a regular one.
   Failure overtook me when using the Archimedes' spiral, which after C.F.Sacconi everybody considered as central for the development the scroll. Below I adduce two drawings of the Archimedes' spiral: 1st - drawn by the hand of C.F.Sacconi, and so, not quite exact; and 2nd - reconstructed with the help of a computer with the sufficient degree of accuracy.
   The particularity of the Archimedes' spiral is that distance between the whorls of the spiral is alike everywhere, but such a case exists extremely seldom and only in some violin scrolls. This figure shows the montage of two drawings: A.Stradivari's violins (reconstruction of the 1715 violin pattern by clothoids according to my method) and the skull (VP Vorobiev and RD Sinelnikov. Atlas of Human Anatomy. M.-L.: Medgiz. 1946 .-T.1). It is noticeable that the upper oval and the corners of the violin coincide with the outlines of the skull. Lower nasal conchs are very similar to ffs. It should be recalled here that the sound of a violin is approaching the timbre of a human voice, which is more dependent on head and nasal resonators.
  
  
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  Figure 117. The Archimedes' spiral.
  
   If this analysis may seem unconvincing to whomever and he/she will want to correct or even completely change the modus operandi, then I will say: "Good luck!" If such new studies will be logical and motivated enough, I am ready to consider these forthcoming opinions.
  
  During the research for justification of a certain character of the violin ovals and other geometric relationships of the violin body, I found a great similarity between the violin contour and a human skull.
  
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  Bibliography
  
  
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  JALOVEC, K. - Enzyklopädie des Geigenbaues. Prag, 1965, 2 voll.
  JALOVEC, K. - Deutsche und Österreichische Geigenbauer. Prag, 1967.
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  MOECKEL, O. - Die Kunst des Geigenbauers. Berlin, 1954.
  NICHOLOSON, J. - Design and plans for the construction and arrangements of the new model violin. London, 1880.
  OAKES, W.W. - Construction of the violin. (In: Music, A monthly Magazine, Chicago, 1897).
  PANCALDI, C. - Progresso italiano nella costruzione del violino operato da Antonio Gilbertini da Parma. Palermo, tip. Maddalena.
  PELUZZI, E. - Tecnica costruttiva degli antichi liutai italiani. Firenze, 1978.
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