Diseño de un Violin

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Английскаяверсиямоейкниги"Искусствоскрипичногоконструирования"

SergeiMuratov

SergeiMuratov

TheArtOfTheViolinDesign

ПредисловиеСкрипка,вопросыисториипроисхожденияСкрипкакакконструкцияГеометрияскрипкиРеконструкциястрадивариевскогометодапостроенияскрипкиСимволизмскрипкиСкрипкаичерепTheArtOfTheViolinDesignУлучшениезвукаскрипкиРеставрациядеформированнойдекиУстановкапружинывскрипке

SergeiMuratov.TheArtoftheViolinDesignIllustratedbySergeiMuratovCopyright(C)2002-2015bySergeiMuratov.AllRightsReservedNopartofthisbookmaybereproduced,storedinaretrievalsystem,ortransmittedbyanymeans,electronic,mechanical,photocopying,recording,orotherwise,withoutwrittenpermissionfromtheauthor.ISBN:1-4033-7460-0(e-book)ISBN:1-4033-7461-9(Paperback)TheArtoftheViolinDesign

BySergeiMuratov

ContentsPrefaceChapteroneConstructionasOneAspectofMusicalInstrumentChaptertwoTheGeometricDesigningoftheViolin

ChapterthreeTheReconstructionoftheStardivariusMethodofViolinDesignConclusionBibliography

Preface

This book is a general conclusion of observations and reflections about the regularities andprinciplesofthedesign;ortobemorespecific,thecreationofthebowed,stringedinstrumentsoftheItalianclassicaltradition.Theyareexaminedherewithregardtothepeculiaritiesoftheirdesignandworkmanship in the musical culture of the Italian Renaissance. Hereby I will investigate not onlyspecificstringedinstrumentsbutalsocommonlawsofdesign.Followingistheworkthatcontainsaninvestigationintotheinherentlogicofviolindesign.

Proceeding from the specific character ofmywork I have used differentmethodswhich areapplied to modern art criticism (music, painting, architecture) as well as other spheres of humanartisticactivity.Formethebasicprincipleismodality,whichIuseasaparticularmethodtodesignthemusicalinstruments.Itletsusdemonstratethepithylogicofviolinconstruction,whichhasbeenshapedovera longevolutionand thebasicpurportofwhich is thesearchforanaturalcorrelationbetweentheaestheticsofconstructionandtheaestheticsofsound.Namely,thelogicofthedesign(notthesumofthedifferentelementsandtheirtechnology),sidebysidewithacousticsaretheimportantelements that enable the luthier to see the process of the creation of the sound, and to devise ageometricschemeforthedesignoftheviolin.

It is clear that any design scheme will be unsatisfactory if it is not guided by acousticregularities,i.e.bythesoundasafinalresultoftheluthierart.InmyresearchIhavechosenamethodthat helpsme balance both the shape of the instrumentwith the sound it creates and to expound asemanticbasisfortheviolinasaconstruction,andtodisplayitassomethingintegral,likesomethingspecific,but,incontrastto,ageneralizedmodeloftheinstrumentthatismodeledfromanidealizeddesigned.

Chapterone

ConstructionasOneAspectofMusicalInstrument

Weuseconstructiontorefertodifferentthings.Musiciansassociateitwiththecompositionandarrangementofmusic.Constructionisalwaysusedbyengineersandinventorsattheirpractice.Inallcases,thiswordhastodowiththesubjectofhumanactivity.

Inmywork,constructionmeansboththeobject,i.e.theviolin,andtheprocessoftheinstrumentdesignthatincludessuchtheconceptsas therealisationofanideaanditsoutcome. If the idea isaconceptoramentalimpressionofthefutureinstrument,thentheachievementoftheideaincludestwoprocesses:therealisationoftheideabydrawingthewholeinstrumentonpaperandtherealisationofthe ideabymakingone.So theresultwillappear twice:as thedrawingand the instrument itself. In

addition,therealisationoftheideadividesintointermediatestagesthathavetheirownresults.Everypreviousstagedeterminesthenextone,whichmakesanalgorithmiclineoftheprocessofthemusicalinstrumentdesign.

Everyinstrument, includingtheviolin, isanartificialobject thatdifferentnaturalphenomenonand forms leave their marks on. Handicrafts are similar to nature because the formative processresembles that seen in nature. In the abstract the astonishing similarity betweennatural objects andartificialobjectssuggeststheideathatallartificialobjectsarecrystallizedfromamaterialofnaturebytheconsciousactivityofmen.Inotherwords,anartificialobjectisaresultofthetransformationofanaturalone.

There are principal differences between natural and artificial objects. If the first exists in orcausedbynaturethenthesecondismadeorproducedbyhumanbeingsafterplanningordesign.

Itisnoticeabletoeverybodywhostudiestheviolinartthatinstrumentsthatcameintoexistencein the Renaissance have a similarity with other works of art. It can be observed even at firstacquaintancethepresenceofacertainunitingprinciple,thesubordinationtothecommonbasisoftheartisticthinking.Forexample,theconcentricityofcomposition,i.e.acomprehensionoftheworkofart as the finished unity in which all components are completely submitted to the whole, and theperfectsystemofproportions,demonstratesthisunifyingprinciple.

Of course these common principles are shown in every kind of art in accordance with thespecificityof'buildingmaterials'andartisticlanguage.Sointheartoftheluthierstheperfectsystemofproportionsisvaluablenotinitself,butfortheacousticanddesignpurposes.MoreoveritisveryimportanttohighlightthatluthiersoftheRenaissancewereinsearchofsourcesofthebeautifulnotinspeculativemodels,butinreallife,i.e.theycreatedtheformoftheinstrumentfromthecombinationoftangibleobject.ThusacreativeworkoftheluthiersoftheRenaissancewasbasedonthecarefulstudyofnature,andtheviolinwasformedaccordingtoprinciplesofnaturallifeasanintegralandconformedworkofart.

Therefore I have to discover the regularities thatwere the bases of the great Italians' creativeworkandtore-createnotonly theviolin itselfanew,butalso thealgorithmof thecreativeprocessand the way of the luthier's thinking. The complication of the raised task is evident. The creativeprocessisnotsomuchanobligationtoconformtorulesandprograms,butafreedomtobreakandrevisethemsothatnewrulescreateanewinevitability.

Itisfound,thatthesubject-matterofaworkofartcannotprecedeitscreation.Theworkofartrevealsitselftoanauthoronlyduringitscreation.Thusintheartofviolindesignthecreativeprocessissoparadoxicalthat,creatinganewinstrument,theluthiercreateseverytimeanewalgorithm.

Thecreationof aviolin is a scientific andartisticwork.Cognitionandcomprehensionof theviolin as amusical instrument is similar to the cognitionof anyphenomenon and associates itselfwith twobasicprocesses:assimilationandaccommodation.Assimilation is theprocesswherebyanindividual interprets reality in terms of his own internal model of the world based on previousexperience; whereas, accommodation is the process of changing that model by developing themechanismstoadjusttoreality.

Thesetwoprocesses,beingprojectedinthehistoryofthedevelopmentofthescientificstudyofthemusicalinstrument(theinstrumentology)aswellas in thehistoryofmusicologyonthewhole,gostagebystage.TheRenaissancecanbeconsideredasthefirststageofthedevelopmentofanideaabouttheviolin(thepracticalworkoftheItalianluthiers)whenthetheoryoftheviolindesignwasstudiedesoterically,i.e.itwasexploredwithinaschoolandpassedontopupilsverbally.

Proceedingfromthepropositionthat thepracticalandtheoreticalcreativeworkoftheluthierswas insyncreticunitywithart, science,philosophyand religion,onecansay that the luthiersweresufficiently informed in the sphere of exact sciences. In the 17th century the disintegration of the

syncreticprinciple led to thedeclineof the luthiers'GoldenAge, the lastspecimensofwhichweregoneinthe18thcentury(A.Stradivari,GuarneridelGesùandothers).

Havingnochancetobeapprenticedtothegreatluthiers,makerssimplycopiedtheirinstrumentsinthehopeofrepeatingtheirsonority.Onthistideofuniversalrespectbutmisunderstandingoftheprinciplesoftheviolindesign(madebythegreatspecimensofthiskindofart),theAcademyofArtsandSciencesatPaduaannounceda competition for thebestworkabout theviolindesign.AntonioBagatella(1726-1806),authorofRegulationforConstructingStringedInstruments(1782),obtainedaprize.ThispublicationcreatedastonishmentamongallEuropeanconnoisseursandluthiersofhisday,andwasquotedfrequentlythroughsubsequentages.

Ofcourseeveryinvestigatordidhisbittothismethodologyoftheviolindesign,butallofthemkepttheverymainsignbywhichonecandeterminetheiridol:theuseofcompassesfordrawing,orrathercopying,theoutlineoftheinstrument.BasicallytheseluthierscopiedtheAmatis'violins,nottheStradivaris'ortheGuarneris',whoseinstrumentswerecopiedfromthe19thcentury.Andoneofthe 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 thetraditional violin form, supposing the acoustic of the stringed instrument to be independent of itsshape. Savart's 'trapezoid-violin' was tested and compared with Cremonas, eulogized as beingsuperior,etc.,but subsequently foundnoendorsement fromsoloists.Practicallyall innovatorswhocardinallychangedtheshapeoftheviolinsufferedthesamefate.

So the second stageof thedevelopmentof the idea about theviolin canbe called aperiodofsearchfornewmethodologicalbases.

The third stage, taking place in our time, is that basicmethodological schools boundupwithacousticsorwiththesearchfortherationalconstructionofthemusicalinstrument,orwithphysico-chemical processing, or with other practical and theoretical features of the violin design, havealreadyformed.Andthesubsequentstudyofthe 'secrets'oftheviolinasanacousticalphenomenontakesthepathoftheinteractionofthesemethods,i.e.itrevertstoits'syncreticpast',butnowonanewphaseofthedialecticaldevelopment.

If the luthiers had worked like a technical designer, i.e. firstly with an idea and a workingdrawing, then the making of parts, their arrangement and, finally, the adjusting of the finishedarticles, then their creativeworkwould be characterized as technological.And after analyzing theinstrumentsthemselvesbydifferentmethods,determiningthephysicaldataoftheirparts,evensimplycopying them,onecouldemulate their tone.Thesamecausesproduce thesameeffects,don't they?But size-for-size copies do not work because the frequencies are determined not only by thedimensionsandconstructionof the instrumentbutalsoby thewood'smechanicalproperties,whichvaryfromsampletosample.

There are also the elements of artistic work in the luthier's activity, concerning not only theoutwardappearanceoftheinstrument,butitstone,whichareimpossibletoappraisebyanymodernapparatuses,butonlydirectlybyman.Sothequalityofaviolin'ssoundisaresultofboththeluthier'ssophisticatedhearingandhisscientificknowledge.

In the primary stage of a violin construction the technological problem is most important,whereas in theprocessofwork themain rolegraduallypasses to theartisticoutcome.Proceedingfromthistheluthier'screativeworkmustbeconsideredfrombothpointsofview:scienceandart.

Theproblemofthedevelopmentoftheviolininthehistoricalplaneaswellasitsdesignmustbethe basis for the scientific research of the modern instrumentologist. As is generally known thisdevelopment involves the progressive changes in size, shape, and function of an object during itshistoricalexistence.Intheprocessofthedevelopmentintheintermediatestagestheobject'sconditionhasacertaincharacteristic,whichInamethemodus.Initsapplicationtotheviolinthemodusisthe

configuration,designandsoundatmosphereoftheinstrument.Astothedevelopmentoftheviolininthehistoricalplane,thisissueisonlyexploredtoacertain

extent,becauseofthelimitationsofhistoricalandarchaeologicaldocumentsandfinds.ItisimpossibletorestorethemethodologyoftheviolindesignoftheItalianclassicalschoolsin

full, although theCivicMuseumofCremona contains a collection ofmoulds, drawings, sketches,templates and original studies by A.Stradivari. But the working drawing, displaying the way ofthinkingof thegreat luthiers, isabsent.Thewholeof thisset,whichwasusedfor thedesignof thedifferent parts of the instrument, comprises only a number of the copies of a principal drawingwhich,ifitonlyexisted,wouldthrowlightonthe'mystery'ofthecreativeprocessofStradivariandotherluthiersofthattime.

I shall not disputewhether such a drawing really ever existed, but too few investigators havemadeanattempttore-createit.Intheseworksonecanseethecenturies-oldinterestofpeopleintheunderlyingmathematicalregularitiesofthearts.

The question of mathematical prerequisites in the beautiful, and the role of mathematics(specificallygeometry)intheartsstirredtheancientGreeksandBabylonians.Onecanevenassumethatmathematicsandtheartscameintoexistencealmostatthesametimeinviewofthereligiousandphilosophicalsearchesofman,andthattherearecloseandvariedconnectionsbetweenmathematicsandthearts.

TheroleofmathematicsinlayingbarethesecretsoftheartshasbeentracedinthecreativeworkofsuchpeopleasPythagoras,Vitruvius,AlbrechtDürer,LeonardodaVinciandThomasHobbes.Theenormous importanceofgeometrywasnotonly relevant to theabove-namedartistsandarchitects,but to great luthiers too. Unfortunately, in contrast to the former, the luthiers did not leave anytheoreticalpropositionsabouttheirwork.

Ifwediscussthegeometryoftheviolin,thenthequestionis:Whatcanweassumethebasisofitsdesign- theaestheticprinciple (beauty,elegance)or tobe thephysicalone (acoustics,mechanics)?Theborderlandbetweenscientificandtheartisticworkturnedouttobearatherimpassableobstaclefor themutualassimilationof twodifferentworlds lyingonopposite sides: theworldof scientificnotions and the world of artistic images. In the scholarly and scientific study of the musicalinstrument,employinggeometry tobuildabridgebetween these twoworldsproves tobedifficult.Numerouspopularmethodsofthegeometricalanalysisofthestringinstruments,whichweremadebygreatluthiers,havenoacousticsubstantiationand,whatismore,theaestheticadvisabilityofsuchmethods gives rise to doubts. Various parts of the violin are drawn with compasses by the mereselectionofradii,whichratherlookslikecopying,thanasearchforlogicalregularity.

Of course at all times both the architects and the engineers used compasses and a rulerwhenmakingtheworkingdrawing.Anditissmallwonder,asbasicallystraightlinesandarcsareusedintheconstructions.But,forexample,whendesigningaircraft,high-speedcars,radar,etc., thecirclesof the compasses are not a great help. This kind of construction can be designed onlywith somemathematicalcurve.Ourtaskisfindingsuchacurve,onethatwouldbeuptotherequirementsoftheviolindesign,i.e.ithastobeelegantandmustallowforthecurvaturesoftheinstrumentandneedstomeetthecriterionofacousticpropertiesandtheirprojection.Asthecharacterofthecurvatureofthewholeoftheinstrumentisinvariable,wemustuseonlyonekindofmathematicalcurve,whichwecanincreaseordecrease,accordingtothegivenpartsoftheviolin.Inotherwords,wemustfindsuchastandardmodule, which when scaled up or down can be used to design any stringed and bowedinstrument.

Byanalyzingdifferentmathematicalcurves,Icometotheconclusionthatthereexistsonlyonecurvethatisuptotherequirementsoftheviolindesign.ItistheCornuspiralorclothoid(CandSaretheso-calledFresnelintegrals)(Figure1),importantinopticsandengineering.

Clothoidshavebeenusedinengineeringdesignformanyyears.Inthepastthespiralshavebeenfoundmanually by draftsmen.Thiswas a tedious process,which I didmyself twenty years ago in1981solvingtheproblemofviolindesignfor thefirst time.It ismucheasier touseacomputer todraw and calculate the position of the clothoids. The design curve of a violinwill bemade up ofsegmentsofclothoidsjoinedinsuchawaythatthecurvatureiscontinuousthroughout.

TheclothoidhasthefollowingparametricrepresentationinCartesiancoordinates:

wherethescalingfactoraispositive,theparametertisnon-negative.

Figure1:TheCornuspiral.

AcurveparametrizedbyanarclengthsuchthattheradiuscurvatureisinverselyproportionaltotheparameterateachpointisaCornuspiral.Incontrasttoanotherspiralstheclothoidhasthisveryimportant property: the radius curvature starts from infinity and aspires towards zero, continuallyapproaching its asymptote (centreof volute),while the curvature aspires to the ideal form - to thecircle.

The curve of the violin outline is formed by joining segments of clothoids. In all cases it isnecessarytosolveanonlinearequation(IusedthescaleToolofthecomputersoftware)tofindthescalingfactora.Theangleofrotationofthetangentineachspiralwillbefoundempirically.

Theother important factor in theviolin'sgeometricaldesign is theuseofproperproportions.Throughout the ages, designers and architects have attempted to establish ideal proportions. Thenumericallysimpleratios1:2;2:3;3:4;4:5;3:5,etc.,wereconsideredthepreferableproportions,butthemostfamousofallaxiomsaboutproportionwasthegoldendivision(1.6180339...),establishedbytheancientGreeks.Accordingtothisaxiom,alineshouldbedividedintotwounequalparts,ofwhichthelargeristothesmallerasthewholeistothelarger.AtamathematicalexpressionIshallpresentitasμ

Toconstructtwofinitestraightlinesintheratioofthegoldendivisionisveryeasy.LetABEFbeasquare(Figure2).

Figure2.Thegoldenrectangle.

LetthepointDtodivideAFinhalf.IfAD=DF,thenBDisthehypotenuseoftheright-angledtrianglewiththeratioofcatheti(theothertwosides)as1:2.Therefore,bythePythagoras'Theorem,thelengthofthehypotenuseis√5.Theratiosofthesidesofthistriangleareverysimple:AD/AB=1/2,BD/AD=√5/1,BD/AB=√5/2.Andtherefore:

(AD+BD)/AB=(√5+1)/2=1.6180339...

Ifμ=1.6180339...,then1/μ=(√5-1)/2=0.6180339....IfBEMisanarcofacirclewithcentreD, thenAM/AB=μ In thatwaywecanconstruct the

finitelineswhichwillbelongerthantheoriginallineintheratioofthegoldendivision.TherectangleABPM,havingthesideAM=μAB,iscalledagoldenrectangle.ABEFisasquare,

andweobservethattherectangleFEPMisalsoagoldenrectangle,sinceEF=μFM.IfwenowtakethisrectangleFEPM,andmarkoffasquareEPTSfromit,theremainingrectangleFSTMalsowillbe

golden,andwecancontinuethisprocessaslongaswewant.Inthiswaywecanconstructthelineswhichwillbeshorterthantheoriginallineintheratioofthegoldendivision.

If the proportionμ or 1/μ is found by the simple expedient ofworking out a problemof thegoldenrectangle, theproportion2/μorμ/2 (very important inourwork too) isdeterminedby thenextmethod:withradiusDAdrawthearcofacirclecentreDtocutBDproducedatN.ThenBDisdividedinthe2/μratioatN.Toprovethis,wenotethatDN=1andNB=√5-1.Ifμ=(√5+1)/2and1/μ=(√5-1)/2and2/μ=(√5-1),thenNB/DN=2/μ.

WithradiusBNdrawthearcofacirclecentreBtocutABproducedatK.ThenABisdividedinthegoldensectionatK.ABandAMcanbethesidesofthegoldentriangle.Thegoldentrianglesareconstructedbythefollowingmethod(Figure3):

Figure3.Thegoldentriangles.

WeseethatgoldentriangleABCisdividedintothreegoldentrianglesAEC,ADEandDBE,thesidesofwhichare:AD=DE=EC=1;DB=BE=AE=AC=μ;AB=BC=1+μ=μ².Anothergoldentriangle,havingtheangles90ºand54ºand36ºwiththeratioof5:3:2,isveryinterestingtoo.Inthisright-angledtriangletheratioofthebigcathetustothehypotenuseisμ:2=cos36º,hencetheformulawhichbindsthegoldensectionandπ:

μ=(√5+1)/2=2cosπ

ThegeometryoftheGreatPyramidofKhufuatGizaisagoldentriangletoo(Figure4).

Figure4.TheGreatPyramidofKhufuatGiza.

Ifcos51.82729º=1/μ,thenAD/AB=0.6180339...,AB/AD=1.6180339...

OnecandivideallA.Stradivari'screativeworkintoafewperiods:1)From1666to1688StradivarihadworkedaftertheAmatimodel.From1689heexperimented

withthelargemodelbyN.Amatiandenlargeditsomemore.2)In1692Stradivarihadcreatedthe'elongated'modelofaviolin.3) In 1698 he had returned to the Amati model, working on the model by Antonio and

HieronymusAmati.4)From1705to1725Stradivariworkedwithhisownoriginalmodel.Itwashisbestperiodof

creativework.5)From1725to1737- the lastyears inStradivari'screativework-onecanseehisdeclining

powers,whichcanbeattributedtohisoldage.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 theinternalmouldoftheAmativiolinhasaguitar-shapedform,theStradivarihasaformasifitslinesareaffectedat themiddleC-bout.Also,Stradivarihasaltered thearchingand thicknessofboth thebellyandthebackandhasrevisedtheproportionsbetweendifferentpartsoftheinstrument.

But the process of the development of the violin, which started in the 16th century, wascompletedonlyatthebeginningofthe19thcentury,whentheviolinwasmodernizedbyreplacingtheneck, the fingerboard, thebridge, thebass-barand thesoundpost.So theBaroqueviolin isonlyan

approximation of the ideal proportionality of its different parts as well as of the whole of theinstrument. And the sound of these instruments is being defined by us now when they have the'modernised'neck,fingerboard,etc.,whereasinthe18thcenturytheysoundeddifferently.

The most apparent modification in sound comes from the different strings of presentinstruments. Here a new material for the production of strings has become stronger and longerbecauseofthenewlongneck,andthebridgehasbeenraised,whichincreasesthestringstrainonthetable.And,whatismore,theeighteenth-centurypitch,ingeneral,maybetakenonlyasa¹=422.5Hz(accordingtothepitchofHandel'sEnglishtuning-fork,whichstillexists).ThereforeinthedaysofTartini the strain on the strings was 29 kg, whereas it is now 90. As resistance to this strain theoriginalbassbarhasbeenreplacedbyonelongerandstronger.Thesumeffectofthesealterationswastodeveloptheoptimumsonorityofwhichtheinstrumentwascapable.

Hence one can say that the great Italians did not create the present violin sound, whichsupposedly has passed ahead beyond its time (they could not even imagine how their instrumentscould sound after 'modernization', you know), but they simplymade the violin bodywith the richpotentialitieswhichdidnotrevealitselfin18thcenturyinfull.

Aforesaidchangesinphysicaldesignstartedfromaboutthebeginningofthe19thcentury.Itwasasecondstageofthedevelopmentoftheviolin.Theparadoxwasthatluthiers,modernizingtheoldviolinsandmakingthenewone,disregardedthetraditionalmethodsoftheviolindesignanddidnotstudytheinstrumentaswellasthepupilsoftheoldtimedidit.Andthe'secrets'oftheItalianviolinsweregone.

We have some problemswhenwe try tomake an appraisal of the quality of the instrument'ssonority.Nowadays twomethodsexist:subjective,basedon thehearingof the investigator,andso-called objective, when special devises are used. The investigator has a wide range of powerfulanalytical techniques at his command now. These instruments take the frequency of sound and itsintensity. The instruments, which measure the sound pressure of the individual harmonics of thecomplextone,analyzeitsspectrum.

Because timbre is aqualityofauditorysensationsproducedby the toneof a soundwave, thetimbreoftheparticularsounddependsnotonlyonitswaveform,whichvarieswiththenumberofovertones,orharmonics thatarepresent, their frequencies, and their relative intensities,buton thesomesubjectivepeculiarityofourauditorysensationtoo.Thefact is thateveryharmonic,whichisheardbyamusician, isacompound tone, consisting of an objective overtone,which canbe fixedwiththesoundspectrograph,andsubjectiveresultanttones,whichonlyoccurinourconsciousnessbecauseof interactionbetween theobjectiveovertones.Oneof them(difference tone) isa lowonetallyingwith thedifferencebetween the twovibrationnumbers, and theotherof them (summationtone)isahighone,butaverymuchfainterone,tallyingwiththeirsum.Andso,itishardtodescribethe timbre of any instrument with objective and subjective components only with analyticaltechniques;thisprocessalsoneedsapersonalityappraisalofthemusician.

Let us examine two spectrograms. The spectrum of (Figure 5) the force exerted by a bowedstring at the bridge, and (Figure 6) the sound radiated by violin playing the same note (open G-string).Aswesee(Figure5),theamplitudesofharmonicsgraduallydecreasefromfirsttolastthatisquitenaturally.Thespectrogramoftheviolinsound,whichisheardbyman,wouldbelikethefirstone,ifonlyourbraincoulddrawsuchpictures.Butreally(Figure6)theviolininsufficientlyradiatesboththefirst(G3)andthesecond(G4)harmonics.

Figure5.ThespectrumoftheforceexertedbyabowedG-stringatthebridge.

Figure6.ThespectrumofthesoundradiatedbytheviolinplayingtheopenG-string.

Strangeas itmayseem,butD5 sounds louder thenotherharmonics.The fact thatwehear thefundamentaltone(thefirstharmonic)asloudestisafunctionofourbrainwhichcreatesandaddstheamplitudesofthedifferencetonestotheactuallysoundedharmonics.Becausethedifferencebetweenfrequenciesof theadjacentharmonicsalwaysisequal tothefrequencyof thefundamental tone, theinsufficient amplitude of the first harmonic is compensated by the difference tones of all adjacentpairsoftheHarmonicSeries.

NowIwanttodwellonthehighlyinterestingmomentconnectedwiththenoteD5andlengthofaviolin. Firstly I calculate thewavelength of theD5. In dry air (at 0 C and a sea-level pressure of1013.25millibars)thespeedofsoundis331.29m/second.IfthefrequencyofA4is440Hz,thenopenGstringhas195.5Hz.Hencethethirdharmonic(D5)has587.7Hz.Dividingthespeedofsoundby587.7IfindthewavelengthoftheD5:

33129cm/587.7Hz=56.3706cm.

Gettingaheadofmystatement(detailswillbeinthenextchapter);Iproducemycalculationsofthe length of a violin.As initial value I useπcm. (3.14159265...cm),which is the first term of theprogression.Ifthegoldendivisionisthecommonprimefactor,theseventhtermoftheprogressionwillbe56.3735cm,whichisthelengthofthewholeinstrument.

Because the spectrumof the sound radiatedby theviolin is inadequate to the timbre,which isheardbyus,itisnaturallytoask,'Isitpossibletodivinethesoundofthemusicalinstrument,workingattheacousticoftheparts?'

If the final resultswere dependent on the sum of the timbre of the different violin parts, thisproblemwouldbeworkedoutbymerelytuningthemupaccordingtothecertainprinciple,copyingsomegreatinstrument.Butreallyallisfarmoreintricate.

Sound is produced when a vibrating surface interacts with the surrounding air. As the large,lightweightplates(thebellyandback)movesforwardsandbackwards,thesurroundingairpressureisincreasedanddecreased.Thesepressurevariationsspeedawayfromthesourceassoundwavesat331.29mpersecond.Thesoundwaves,travelingfromtheoutsideandinsideoftheplates,differinphaseby180º.

Iftheviolinbodywasasingletable,i.e.infreeair,itwouldbelikeafishoutofwater.Toseewhyabarebellysoundsbad,considerFigure7.

Figure7.Whyabarebellyisinefficientatlowfrequencies.Theplus signs represent an increase inpressureas thebellymovesagainst theair; theminus

sign,adecrease(a).Whenairfromthehigh-pressuresideofthebellymixeswithairfromthelow-pressure side, sound cancellation occurs. At high frequencies, the sound is directional, so littlemixingoccurs;however,forfrequenciesatwhichthewavelengthislongcomparedtothesizeofthebelly,thewavescancurvebackaroundthebellysothattheout-of-phasewavesmix(b).Oneofthebasicrequirementsofaviolinbodyistoblockthisunwantedmixingofout-of-phasewaves(c).

Asaviolinbodyhassmallholes(theff-holes),theairinthebodyretainsitsabilitytoactlikeaspring,whiletheairinthef-holesactslikeanotherdiffuser.Thisairdiffuservibratesinphasewithsomefrequenciesandoutofphaseatothers(Figure8).Sosoundiscreatednotonlybythemotionoftwoplatesbutalsobytheairbeingsqueezedresonantlyinandoutoftheff-holes.Thissystemactsas

aresonator,properlycalledaHelmholtzresonator.ThefrequencyofresonanceforanyHelmholtzresonator isdeterminedbythecomplianceof theair in thecontainerandthemassof theair in thehole.

Figure8.Howairmovesatdifferent frequencies.At some frequency, the f-hole airmoves inphasewiththebelly(a).Atanotherfrequency,thef-holeairmovesoutofphasewiththatofthebelly(b).

Theviolinradiates thewidespectrumof thesound.Owingtotheshapeof theviolinbody, thephasesoftheharmonicsarealtered,whentheygooutoftheff-holes.Itisconducivetothesubtractionandadditionofitsamplitudes.OneofthepeculiaritiesofaHelmholtzresonatoristhatthesoundthatis radiated from a hole does not varywith the size of the hole if it has the round form (like in aguitar).Whentheholehastheformofaslit(likeinaviolin)theradiatedsoundvariesappreciablythrough the ratio of length andwidth of the hole. So, the narrow hole is used by the luthiers foradjustingthesoundquality.

Sincetheffswiththeinternalvolumeofairintheviolinbodyformtheresonancesystem,itisveryimportantforaluthiertocheckthecorrelationbetweenthesetwovolumesofair.Thebalanceisachieved by the increase or reduction of the volume of air into the instrument's body so therebychangingtheparametersoftheffs.Theconfigurationofthebodyhasnosmallimportance.

Certainly, I rememberabout thenatureofarcsof thebellyand theback, their thicknessesandadjustment; both the whole boards and their separate areas. However it is not possible to defineexhaustivelywhatwork isnecessary tobeconductedwithall thedetailsofan instrument toget theItaliantimbre.Anyattempttolimittheclassofconsideredphenomenabyatypeofanequationoranenumerationofsomephysicalcharacteristicsusuallybringsaboutfailure,anexamplewillalwaysbefoundthatwillnotgointotheacceptedscheme.

Theuseofprobabilistic-statisticalmethodsofstudyinthefieldofviolins(studyoftheChladnipatterns, the laser interferograms, thicknessesandtonesofseparateareasof thebellyandthebackandagreatdealofotherconcernsincludinghologramsandvoiceprints)revealstheeffectofatotalactionofunambiguousdynamiclaws.

Thewaveprocesses,occurringinthesystemofbody-ffs-outsideair,haveacomplexnatureandmust be described by different systems of equations. However, for the understanding of the mostimportant phenomena, occurring in the given system (interference, diffraction, reflection andrefraction,dissipationandetc.) thereisnoneedtoanalyzethesource,generallyspeaking,complexsystems of equations. The simple effects, as a rule, are described by simple and universalmathematicalmodels.

Theviolinbodyisaclosedspaceforthesoundfield(whileforthisexplanationtheff-holeshaveno importance). In the closed space the soundwaves, repeatedly reflecting fromborders, form thecomplexfieldoftheair'soscillatorymoving,whichisdefinednotonlybythecharacteristicsofthesoundsource(intheviolinbodythebellyandthebackarethesesources),butalsobythegeometricformandsizesofthespace,andtheabilityofthebordersofthespacetoreflect,missandabsorbtheacousticenergy.Thepictureofthewaveprocesses,occurringintheviolinbody,getscomplicatedbythepresenceoftheff-holes.

Becauseofitssmallvolumethebodyofaviolincannotbediffusive,sothesoundwavesofitsfield are coherent and there are the stable phenomena of interference in it. As a result of that thesecondarysourcesofthesoundwaves,whicharelocatedbetweentheactualsourcesofthewaves(thebelly and the back), appear in a certain point of space in the violin body (the Huygens-Fresnel'

principle).Duetoitscontoursthebodyofaviolinformsthissecondarysourceintheregionoftheffs.

On theoutput from thebody through the ffs the soundwave is changed into thewave pencil.Sometimes this pencil can be considered as a ray,whose behavior is described by the laws of thegeometricoptics.Howeverthespreadoftherealwavepencilsisdifferentfromthebehavioroftherays.Thereasonforthisdifferenceisduetothephenomenaofdiffraction.

Wecannotgettheexactandmathematicallycorrectdecisionofthediffractionofthesoundwavewhenitpassesthroughtheffs,sincethiswillentailgreaterdifficulties:theverycomplexformofthescreen(thebelly)andnot lesscomplexformof theslot(theffs).Sothegoodearformusicof theluthiersisveryimportantforthedeterminationofthequalityofthesound,passingthroughtheffs.Butifwetakeintoconsiderationonlythegoodearoftheluthiers,wemustfinishcuttingtheffsaftertheinstrumentwasassembled?ItwasdonebyA.Stradivariwhose ffsneveragreewith the intendeddrawingontheinnerfaceofthebelly.

Itishardlyprobablethatgreatmastersworriedabouttheexternalaestheticsoftheffsmorethanabouttheacousticsoftheinstrument.Manymasters,includingGuarneriusdelJesu,cuttheffscrudelyenough in general, then stopped to consider if contented with the sound knowing that a drastic'correction'couldharmthesoundquality.Preciselysuchaworkmethodissubstantiatedwiththeffsbymodern theoretical physics (theKirchhoff'smethod), which proves that when the wave passesthroughascreenwithahole,itsspectrumisenlarged.

Thewidthoftheangularspectrumisdefinedbytheattitudeofawavelengthtosizesoftheholeand dependent upon the direction of the spreading wave, falling on the screen. The last remarkpertains todistancebetween the ffs.To tell the truth, thewider the ffs are locatedon thebelly, theclearer the lowerharmonicsstandoutand theviolinspeaks inabassvoice. If foraviolinsuchaneffectcanbeconsideredasadefect,thenforaviolaandacelloadeepersoundwithshortenedmodelispossibleonly,whentheffsarelocatedwiderthanonthebigmodel.Inotherwords,thelowtimbreof an instrument dependson thewide locationof the ffsmore thanon the size of the instrument'sbody.ThisprinciplewasunderstoodbyallluthiersoftheoldtimeandsonsofA.Stradivarihadwellassimilatedthisrule,whichtheirfatherconceivedandcarriedouthisowninstruments.

TheMuseumofCremonacontainsA.Stradivari'sdrawingofthecentralpartofacellowiththescheme for the location of the ffs. On the back of this sheet of paper his sons, Francesco andOmobono,haverepeatedthesamedesignwithamodificationofthemeasurementandtheplacingoftheffsfor theshortenedmodelof thecello. In theirvariant thedistancebetweenffs is increased incontrastwith thevariantof their fatherbyapproximately15mm.Shortening themodel,FrancescoandOmobonotriedtomaintainthedepthofsoundofAntonio'scello.

Chaptertwo

TheGeometricDesigningoftheViolin

The design phase is largely theoretical. Drawing upon the general fund of violinmakingknowledgeandmyownresearch,IproduceamathematicalmodelofaviolinthatIthinkwillmeetallof the specifications to study the violin design. My simpler simulation performed by personalcomputer consists of geometricmodels.More advanced simulation, such as that that emulates thedynamic behavior of this acoustical system, is usually performed on powerfulworkstations or on

mainframecomputers.This simulation canbeuseful in enablingobservers tomeasure andpredicthowthefunctioningofanentiresystemmaybeaffectedbyalteringindividualcomponentswithinthatsystem.

Iusedpatternsoftheclothoidtodrawtheoutlineoftheviolin.TheclothoidwasdrawninAdobeIllustratorwithSpiraltoolbytheco-ordinatesreferredtobelow(Table1).ThisTableismadeupattherelativedimensions(a=1).Itisnecessarytomultiplythesedimensionsbytheclothoid'sscalingfactortodrawanygivenclothoid.

sXYR0.000.00000.000010100000053.183120199900421.591530299401411.061040397503340.79580.50492306476366605811110553057065971721454780722824933978907648339835371.00779943833183107638536528942071546234265330638668632449405431713522741.50445369752122603655638919897032385492187280333645091768903945373316752.00488334341592

Table1.Theco-ordinatesoftheclothoid.

Thescroll

IhavealreadymentionedsketchesbyA.Stradivariandemphasisedthattheywereonlyanumberof the copies of that principal drawing bywhich one can retrace theway of his thinking.And hisdrawingoftheviolinscrollisnotanexception.

SidebysidewiththissketchIwillanalysescrollsbybothA.StradivariandotherItalianluthiers.By analysing the outline of a scroll for the violin I can conclude that it was drawnwith two

curves:ThelogarithmicspiralorBernoullispiral(Figure9)andTheCornuspiralorclothoid.

Figure9.TheBernoullispiral.

IdrawthisspiralinIllustratorwithSpiraltoolbynextparameters:

radius=16.5mm,decay=86%,segments=11,rotate=90º.

Eachsegmentisaquadrant.IntheprocessofanalysisofdifferentscrollsIwillusedifferentinitialradiiandtheirdecay.AnalgorithmofthegeometricreconstructionofStradivari'ssketchofaviolinscrollisshownin

Figure10. Ibeginwith twoparallel linesABandED,whichare the tangents to thescroll; the firstextendedmeetsthesurfaceoftheneck.TheBernoullispiralfordevelopmentofthescrollfromBtoOhasparameters:radius=16mm,decay=85%,segments=11.Theclothoid'sscalingfactorsforthedevelopmentofotherpartsofthescrollare106,58and51.OC/AD=82.25mm/50.83mm=μ.

Figure10.ThegeometricreconstructionofStradivari'ssketchofaviolinscroll.

Figure11.ThegeometricreconstructionofA.Stradivari'sscrolloftheviolin,1715.

In Figure 11 one can see that the disposition and sizes of the clothoids are identicalwith thepreviousreconstruction.HereandfurtherIhaveaddedonemoreclothoida-50,whichshapesatailofthe scroll. OC/AD = μ. The Bernoulli spiral is slightly different from the previous one, and itsparametersarevisibleinFigure.

Althoughthescrollofthe'Emperor'violinwasmadewiththesamepatterntothepreviousone,itsoutlineisslightlydifferent.

The methods of violin analysis, which are chosen by me and which demand superimposingdrawingsandphotosofthewholeinstrumentaswellasitsdifferentparts,haveoneshortcoming:aphotocannotreproduceageometricallyaccurateoutlineofaninstrumentwithoutsomedistortion.Itisclearlyvisibleinthenextexample(Figure13),whereIanalyzeA.Stradivari'sscrolloftheviolinphotographedfrombothsides.HereonecanseenotonlythedifferencebetweenthesidesmadebyStradivari,buttheopticaldistortiontoo.

InFigure14wecanseethattheoutlineofthepegboxhasadifferentshape.Nowtheclothoida-110forthedevelopmentoftheupperpartofthepegboxbeginsitsmovementfromthelineABtothevolute,repeatingthecurvatureofthebox.Thebackofthepegboxisdrawnwithclothoida-65,whichtoucheswiththeclothoida-102,whereasinthepreviousexamplesuchajunctionwasimpossible.

Figure12:ThegeometricreconstructionofA.Stradivari'sscrollofthe'Emperor'violin,1715.

Figure13.ThegeometricreconstructionofA.Stradivari'sscrolloftheviolindrawnfrombothsides.

Figure14.ThegeometricreconstructionofA.Stradivari'sscrolloftheviolin,1689.

Figure15.ThegeometricreconstructionofGuarneriusdelJesu'sscrolloftheviolin,1725.InmyopinionGuarneriusdelJesu'sscrollof theviolin (Figure15) isnearly ideal. I like, for

instance,thatalltheclothoidsbeginfromlines,whichareaframeworkofmyconstructionwhereIusetheadditionallineGH,whichisparalleltoABandCD.Moreover,thedistancebetweenallthreelinesisequal,i.e.AC=CG=50.8mm,andtheirsizesareinsimpleproportionalrelations.Themainclothoida-100describestheouterfaceofthescroll.Justthesameclothoid,whichbeginsfromlineHG, determines the building of the rear sides of the pegbox. For drawing the higher part of thepegboxIusetheclothoida-113.Hereitbeginsdirectlywithoneofthelinesoftheframework.Theviolin scrolls of Guarnerius with their configuration are closer to the Bernoulli spiral, than thescrollsofStradivari.

Ingeneral,theconfigurationoftheviolinscrollofGuarneriusdelJesunoticeablydiffersfromStradivari'sone.ItreadseasilycomparedtothemismatchedscrolloutlineswithclothoidswhicharehadwiththeviolinsofStradivari,andGuarnerius.

IemphasizethattheclothoidonlyhelpsmetodescribetheconfigurationofdifferentpartsofaviolinanddonotconfirmthattheluthiersofthepastusedtheclothoidaswellasIdo.

Figure16.ThegeometricreconstructionofGuarneriusdelJesu'sscrolloftheviolin,1730-30.

I had already noticed that the curvature of the higher part of the pegbox is described by theclothoid, disposed as toward the volute, and in inverse direction. And though the rotation of thetangentinthemainclothoida-100,whichdescribesthevolute,hasanotherangle,thesecondclothoida-100,beginningfromthelineHG,istangentialtothefirst.

Figure17.ThegeometricreconstructionofGuarneriusdelJesu'sscrolloftheviolin,1733.


Figure19.ThegeometricreconstructionofGuarneriusdelJesu'sscrolloftheviolin.


Figure21.ThegeometricreconstructionofGuarneriusdelJesu'sscrolloftheviolin,1740-41.

Figure22.ThegeometricreconstructionofDiuseppeGuadagnini'sscrolloftheviolin.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 aredisposedonthedrawing.

Figure23.ThegeometricreconstructionofStradivari'spatternofviolascroll.

Figure24.Thegeometricreconstructionofthescrollofthe'Mediceaviola'byA.Stradivari.

Figure 25.The geometric reconstruction of the scroll of the 'Paganini' viola byA.Stradivari',1731.

Figure26.Thegeometricreconstructionofthescrollofthe'GoreBooth'cellobyA.Stradivari',1710.

In spite of the greater difference of the scrolls, which we see in instruments of the Italianmasters,alloftheseshareasimilarnatureofconstructionand,togreaterorsmallerdegrees,repeatthesizesandproportionswhichIhaveadducedabove.

Clearlyitisveryeasytomakeadrawingofaviolinscrollwiththehelpoftheclothoidandthe

Bernoullispiral.Withthedesiretocreateourownoriginalformofthescrollweneedtousethesetwospirals,varyingtherotationofclothoidswithdifferentanglesanddrawingtheBernoullispiralwith a different degree of reduction of the radius. Themain condition for the production of newvariantsalwaysmustbethelogicinthealgorithmicbuildingofthewholedrawing;butanaestheticvalueofspiralsandproportionswillhelptheartistinthisdifficultfunction.

Becausethescrollisthree-dimensional,weneedtoanalyzethesuccessivewidthsofthebackofit.A.Stradivarihasleftadrawingoftherearsidesofthescroll,showingthegeometricproportionsofthesuccessivewidthsofthepegbox(Figure27).

Figure.27.ReconstructionoftheA.Stradivari'ssketchforthebackoftheviolinscroll.

Thisgeometryisnotdifficult.Thetailofthepegboxisoutlinedwiththeradiusofthecompassesapproximatelyat12-13mm,and thewidthof the finestplaceof thescroll isapproximately11mm.The greater difficulty is presented in the successive widths of the volute (Figure 28) from thenarrowest place (the pointM) to the broadest one in the centre of the volute (the pointK),whichcorrespond to thesimilarpoints inFigure15.Here Ipresent thesuccessivewidthsof thevolute intheirradii.

Figure28.Thesuccessivewidthsofthevolute.

005.5________________________________15.95.53211.85.65317.75.84423.66.10529.56.44635.46.85741.37.32847.27.85953.18.421059.09.031165.99.661271.810.29 13 77.710.901483.611.461589.511.941695.412.3017101.312.5318107.212.6119112.112.7520118.013.1721123.913.8722130.814.8523136.716.0924142.617.5325148.518.9926154.420.1727160.320.63______________________________

Table2.

Ascanbeseenfromthetable,thenarrowestplaceofthescrollis11mm,butthebroadestoneis41.26mm.Ofcoursetheseprecisemeasurementsarenotaxiomatic;onecanuseothersizes,leavingthenatureofthesuccessivewidthsofthevolutesimilarwithmydrawing.

Figure29.AStradivari:theviolin,1702.Asalreadyseenintheexampleoftheviolinscroll,foritsconstructionIusethegoldendivision,

whichIdefinedbytherelationbetweentheheightofthescroll(50.83mm)andthedistancebetweentheuppernutoftheneckandthecentreofthevolute(82.25mm).Thesize50.83mmisdeducedbymultiplyingthenumberπ(3.14159...)by1.6180339cm(goldendivision-μ).

Thereby,forthedesignoftheviolinIwillusetwomoduses:thenumberπforrevealingthesizesofthemainpartsofaninstrumentandtheclothoidforthedrawingofitsoutline.AsthemodulorIwillusethegoldendivision,itsderivativesandrelations1/2;2/3;3/4;4/5;3/5;5/8;etc.

Henceforth,Iwilldefinethemainsizesoftheviolinasageometricprogressionofthenumberπ

in the proportional attitude of the golden division. In Figure 29 this progression is defined by thefollowing lengths: AB = 82.25mm; BC = 133.08mm; AC = 215.33mm; CD = 348.41mm; AD =563.74mm.31.4mmistheheightofthebridgeandtheheightoftheribs.

AnoteshouldbetakenthatCD(348.41mm)notthelengthofthebodyoftheinstrument,butthelengthofthemouldwhichhelpstoassembletheribs.ThepointE(theinternalVnotches)dividesCDintotwolengthsinthefollowingratio:

2ED:CE=μ(1.6180339...).

Onecanfindtheselengthsasfollows:

CE=CD:(μ:2+1)=192.6mm;ED=CD:(2μ-1)=155.81mm.

Usually, thelengthofaneckismeasuredfromtheuppernuttotheedgeofthebelly,soIwilldefine it as the lengthAB, solving the right-angled triangleABC,whereAC is the hypotenuse. InStradivari'stimetheneckwasappliedwithaslightbackwardinclination.InthegivenpictureofthemodifiedA.Stradivari'sviolin(Figure30)thisangleCABis7.5º.

AB=AD(133.08mm)-BD(3.5mm-thedistancebetweentheedgeofthebellyandtheblock)=129.58mm,thenAC=AB/cos7.5º=130.7mm

Figure30.TheneckofthemodifiedviolinofA.Stradivari.

THEBODYOFTHEVIOLIN

For the geometric analysis of the violin body I begin with the design of the ff-holes. I havechosenthisstartingpointbecauseitdeterminesthesizeandpositionofthecentrebout,theso-calledwaistoftheviolin.

Aswasshownabove(Figure29),wefindthelocationoftheinternalVnotchesoftheffsonthelineE.Thenthemeasureoftheinstrument(thedistancebetweentheinternalnotchesoftheffsandtheupperedgeofthebelly)is192.6mm+3.5mm=196.1mmwhenthelengthofthebodyis355.41mm(348.41mm+3.5mm+3.5mm).Belowweadducethesummarytableofthelengthandthemeasureoftheinstruments,madebydifferentmastersofItaly(thetable3).

v

____________________________________________________ Master Date Length,mmMeasure,mm ____________________________________________________ A.Stradivari 1686 3541951688358193168835719817003581971705351190170635619617073561961708353.51951708362196171035819717113591951716354.51921718350188nodate354196after1727357.5195after1727356.51951736358198N.Amati16583561951663354.5197nodate3531961678351190A.&J.Amati16283521961629351195Guarneriusdel Jesu "Ysayë"357.51961742350188A.Guarnerius1665353193_______________________________________________

Table3.

THEff-HOLES

Iwill begin the analysis of sizes, configuration and locations of the ffswith drawingswhichweremadebyA.Stradivari(Figure31).Thelinesandarcsofacircle,whichweseeinthedrawing,are only an orientation for carrying the ffs from the paper on to the belly. We don't know howStradivaridrewtheffs,thoughhehasleftthewoodenpatternsofffsforallmodelsofhisinstruments.Andtheyareonlythecopiesofadrawingwhich,ifitonlyexisted,wouldthrowlightonthe'mystery'ofthecreativeprocessnotonlyofStradivari,butalsoofotherluthiersofthepast.

Ishalldemonstratemystudyoftheffsdesignontheseveraldrawings"stepbystep".Ithinkitisof very suitable to primary familiarization with a train of my thought. The further examples,describingboththeffsofA.Stradivariandothermasters,Ishallalreadygiveonlyononedrawing.

Figure31.ReconstructionoftheA.Stradivari'ssketchforthepositionoftheffholesinviolins(the"G"model).

Figure32.Thegeometricreconstructionoftheffholesinviolins(the"G"model).

First,Ihaveelaboratedthediametersoftheeyes(10mmforlowereyesand7.5mmforupperone)andradiusofarc(withthecentreinthelowereye),whichgetsthroughthecentreoftheuppereye and point E (61mm). Hereinafter I have drawn two parallel lines A¹A² and F¹F², which arecoincidenttothelinesofthebridgeandthenarrowestplaceoftheviolin'scentrebouts.ThelinesABand A¹B¹ , which are also parallel to each other and perpendicular to the previous lines, are thetangenttothecentreboutsandinternalsurfacesofthelowereyes.

Withradii38mmand61mmdrawthearcsofacirclecentreE,goingthroughthecentersoftheupperandlowereyesaccordingly.ThesmallercircumferencealsogoesthroughthecentreoftheffsandtoucheswiththelineF¹F².Theratiobetweenbothcircumferencesis:61mm/38mm=1.61,veryclosetothegoldensection.

Sincetheratiobetweenbothcircumferencesismuchclosetothegoldendivision,Itrytoplacetheisoscelesgoldentriangle(theangleatbaseis72º)onmydrawing.Twocornersarerestedinthecentersupperandlowereyes,butthethirdone-onthepointE.Certainly,onecanseethatthecornersof the triangle do not touch the correspondent points exactly, but this may be explained by thediscrepanciesofStradivari'sdrawingatcompasses,whereasIusedthecomputer.

Ihaveadded three lines:HJ,LMandPN.TherectangleGHJKhas thesides in ratio3/2,since103.7mm/69.1mm=1.5andLMNPisasquarewithside38.8mm.

Idrawtheff-holeswiththreeclothoids:a-22;a-29.3;a-39.1.Sincethesizesoftheeyeshavearatio3/4,thesizesoftheclothoidshavethesameratio:

22/29.3=3/4;29.3/39.1=3/4;22/39.1=9/16,i.e.3²/4².

Thepositioningoftheclothoidsisclearlyvisibleonthedrawing.Theclothoidsa-22(theinnerfaceoftheupperconnectingarmofffs)anda-29.3 (theouter faceof the lowerconnectingarmofffs)areinscribedinthecircleoftheeyes;a-39.1beginsfromtheinternalVnotch,touchesthelinesHJandisinscribedinthea-29.3.

Stradivari connected the upper eye to the lower one by means of paper templates, whichreproducedtheffs.Ofteninthecuttinghedidnotfollowtheoutlinecompletely.It is interpretedby

both the acoustic task (when the master works at the ffs, correcting the sound quality) and apeculiarity of the paper template. It consists of three parts: a long body and two small tails. Thetemplatehasaratherbroadcrosspiecebetweentheseparts,butinthecuttingoftheffsacutremainsverynarrowinthisplace.InFigure33Ihaveshownthisdifferencebyunbrokenanddottedlines.

BelowIadducetheanalysisoftheStradivari'sffs,madefromphotographsofhisviolins.Ihavethesameproblemswithopticaldistortion,asbeforewhenanalysing theviolinscroll,but themaingeometricideas,itseemstome,willbemadeobvious.

Figure33.Theff-holes.

Figure34.A.Stradivari:detailofthebellyoftheviolin,1702.

InFigure34Iuseonly twoclothoids:a-25 for theuppereyeanda-37 for the lowerone. It isnoticeablethatthelineofthewaist(F)isatangenttothesmallcircle(r=36.5mm)andcrossesthegreater circle (r¹ = 62.8mm) at the point of the narrowest place of the violin's centre bouts. Thediametersoftheeyesare10mmand6mm.

InFigure35Ianalyzethecentralpartofthe"Emperor"violin.Asinthepreviousviolin,thelineF¹F²isatangenttothesmallcircle(r=38mm)andintersectsthegreatercircle(r¹=62.8mm)atthenarrowestplaceoftheviolin'scentrebouts.Thesmallcircle(r=38mm)getsthroughtheuppereyes,andleftf-hole,butitonlytouchestherightf-hole.Thiscanbeexplainedbyasymmetricalffs,ratherthanopticaldistortion, sinceboth lowereyesare symmetrical.Thegolden triangle restsoneof itsownanglestothepointE,butothertwoslightlydonotcomplywiththecentersofeyes.Idrewtheffswith two clothoids: a-28 and a-34.7.And an though analysis of the patterns with corners will beadducedlater,Ihavetakenthisopportunitytoshowthelocationoftheclothoidsinthecentrebout(C-bout),whichareabsolutelysymmetricalanddescribedbytheclothoidsa-47.7anda-71.7.

Figure35.ThegeometricreconstructionofA.Stradivari'sffsofthe'Emperor'violin.

Figure36.ThegeometricreconstructionofA.Stradivari'sffsoftheviolin.

Figure36showstheanalysisofanotherStradivariviolin.Here,aswesee,thecentreboutsareasymmetrical.The linesofbothbouts sodifferent that Idrew themwithdifferentclothoids,whosesizesareseenonthedrawing.

The geometry of these ffs is completely subordinated to a law of the golden division, forinstance: twocircleswith radii65.38mmand40.4mm(65.38/40.41=1.618...), thegolden trianglePESwithsides65.38mmand40.41mmandthegoldenrectanglePNMVwithsides31mmand50mm.Theffsweredrawnwithtwoclothoids:a-31anda-37.

Figure37.GuarneriusdelJesu:detailofthebellyoftheviolin,1733.

Figure38.ThegeometricreconstructionofA.Stardivari'sffsforaviola.

InFigure37Ihaveanalyzed theviolinofGuarneriusdelJesuby thesamemethod. Iused thegoldenrectanglewithsides31.4mmand50.8mmand thegolden triangle,butascanbeseenfromdrawing,notalloftheirangleshavecompliedwiththecentresoftheeyes.Ihavetriedtodisposetheclothoidson theright f-holeunsuccessfully - theyobviouslydonotcomplywith the linesof the f-hole.Ontheleftf-hole,insteadofaclothoid,Ihavedrawnthecircles(D=17.6mmand22.5mm),whichcomplywiththef-holeenoughwell.

Figure39.ThegeometricreconstructionofA.Stardivari'sffsforacello.

THEVIOLINPATTERN

Thetermviolinpatternreferstheinternalpartofthebodyoftheinstrument,comprisingtheribsandallsixblocks.

Becausethecompositepositionof thecornersof thepattern isalwaysco-coordinatedwith thescroll in the violins of A.Stradivari, in my first drawing (Figure 41), I have shown the wholeinstrument.Additionally this isonlya theoreticalscheme. Its use is for thepurposes of geometricanalysis.

Therearemanyversionsabouttheroleoftheviolinwaist,themajorityoftheseconcerningitspractical functions, allowing a performer to play the violin with a bow comfortably. I hold theopinion,thatthewaistedmusicalinstrumentsmustbeconsideredasacombinationoftworesonatorsunitedtogether.Inalltimesmastersmademusicalinstrumentsbothwithone,andwithtworesonators,anditdidnotmatterwhetheritwaspluckedmusicalinstrumentorabowedone.

BelowIproducethedrawingsofthreepluckedinstrumentswithtworesonators(Figure40).

Figure40.Thepluckedmusicalinstruments.1.TheSouthIndianvina,whichhastworesonatorsmadefromapumpkin.2.Thesitar,whichhastworesonatorsmadefromapumpkin.3.ThewaistedtarofTurkey.

Asarule theresonatorsaremadefromdifferentsizes: the lowerone isgreater,but theupperoneissmaller.Hereby,Iwillconsidertheviolinbodyasawaisteddoubleresonator.Moreover,theupperandlowerboutsareoftwobulbsunitedtogether.

When analyzing the violin moulds of A.Stradivari I will use clothoids of different scalesproceedingfromthesizesofthemouldsthemselves.Thesizesandlocationofthewidestplacesofthelower and upper bouts are defined by both the scales of the clothoids, and by the angle of theirrotation.ForthelowerboutthisangleisKMP,whereMPisthetangent(X)oftheclothoid,butfortheupperboutthisangleisJST,whereSTisalsothetangentoftheclothoid.

The locationof the lowercornersof theviolins (toput itmoreexactly, cornerblocks, ratherthenthebellyitself)isfoundattheintersectionofthreelines:G¹A¹,whichconnectthewidestplaceofthe lower boutwith the eye of the scroll;MD, being the tangent (X) of the clothoid;GH is somestraight line, having some anglewithCD. For the upper corners these lines areG¹A¹, ST and FHaccordingly.

Iwilldrawthesecornerswithsmallclothoids,whichmusttouchtheproperbulbandgetthroughpointsTorP.

Figure41.Theschemeofanalysisofgeometryofaviolin.

ThegeometricreconstructionofNicoloAmati'smould'MB'foraviolin(Figure42).Thelengthofthepattern(CD)is343mm,thewidthoftheupperbout(F¹F¹)is155mm,thewidthofthelowerbout (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).

ThepointM,arootof the lowerbulb,divides the lengthof thepattern into twolengths in theratioof2/μ(1.2360678...),thenMD=189.6mmandCM=153.4mm.

HavingconstructedthelowerbulbIusedtheclothoida-200.Therootoftheclothoidliesatthepoint M, touches the C-bout and is inserted in the lower bout of the mould up to the end block.Moreover,theangleEMP=58º.

Similarly one constructs the opposite side of the bulb. The area around the end block iscompletedbytheclothoida-130sothatitisatangenttothehorizontallineD,andsegmentsofbothclothoidsa-130anda-200arejoinedinsuchawaythatthecurvatureiscontinuousthroughout.

Asitcanbeseenfromthedrawing,thelowerboutwascomposedsymmetrically.Theupperboutisdrawnwiththeclothoida-185.HereIdonotfindanymathematicalregularity

inthepositionofthepointS,butwillonlyindicatethatCS=176mm,andSD=167mm.Theangleoftherotationoftheclothoid,i.e.JST=51.83º.IhavealreadyindicatedthatthisangleliesinthebaseoftheGreatPyramidandisusedwhenbuildingthegoldentriangle.

Ifinishthedesignoftheupperboutwiththeclothoida-96.5Ascanbeseenfromthedrawing,theupperboutisnotsymmetricalatall.Iftheleftclothoidlies

in the point S by its root, the right one is shifted to the right a little and has the smaller angle ofrotation(51º).

Before analyzing the geometric position of the corners, we must obtain the points F and G,whichcorrespondtothewidestplacesoftheupperandlowerboutsaccordingly.IdoitbyasimplemeasurementanddefinethatCF=64mmandGD=67mm.ThelengthFG(212mm)andthelengthofthepattern(343mm)areintheratioofthegoldendivision.

Theline,connectingthewidestplaceofthelowerboutwiththeeyeofthescroll,goesthroughthe lower and upper corners of a violin pattern (G¹A¹). I have only to draw another two lines toindicatetheexactlocationofthesecorners.

DrawthelinefromthepointFthroughtheuppercornerandgetFH,whichisatanangleof54ºtoCD.Rememberthisangleparticipatesinbuildingofthegoldentriangle.Thethirdline,whichisthetangentoftheclothoida-185,passesveryclosetopointT.

Thelowercornerisfoundbyasimilarway,wherethetangentoftheclothoida-200getsthroughthepointPpreciselyandthelineGHmakesanangleof51.83ºwithCD.

Theconfigurationof thecornersandC-bout isdrawnwith theclothoidsa-66anda-44,whichareintheratioof3/2.

IfthelinesTJandPKthroughthecornersdividethelengthoftheviolinpatternintothreeparts,Iget three lengths in thenext ratios:CJ(117.5mm)/JK(90.2mm)=4/3;JK/KD(135.3mm)=2/3.Thecentreofcircle,gettingthroughallcornersliesinpointS;diameter=185mm.

Figure42.ThegeometricreconstructionofNicoloAmati'spattern'MB'foraviolin.

ThegeometricreconstructionofA.Stradivari'spattern"PG"-1689foraviolin(Figure43).Thelengthofthepattern(CD)is348mm,thewidthoftheupperbout(F¹F²)=161mm,thewidthofthe

lowerbout(G¹G²)is200mm,andthewaistis103mm.MoreoverG¹G²/F¹F²=1.24(thatisclosetotheratioof5/4=1.25).


HavingconstructedthelowerbulbIusedtheclothoida-200.Therootoftheclothoidliesinthepoint M, touches the C-bout and is inserted in the lower bout of the mould up to the end block.Moreover,theangleKMP=60º.Rememberthattheangle60ºoccursintheright-angledtrianglewithanglesof90º,60ºand30ºintheratiosof3:2:1.

Similarly one constructs the opposite side of the bulb. The area around the end block iscompletedbytheclothoida-128.5sothatitisatangenttothehorizontallineD,andsegmentsofbothclothoidsa-128.5anda-200arejoinedinsuchawaythatthecurvatureiscontinuousthroughout.

Ascanbeseenfromthedrawing,thelowerboutwascomposedsymmetrically.Theupperboutisdrawnwiththeclothoida-185.CS=176mm(correspondingtotheprevious

mouldofN.Amati)andSD=172mm.Sinceinthispatternthewidestplaceoftheupperboutismore,thanthatofAmati's, theangleoftherotationoftheclothoidisincreased,i.e.theangleJST=54º.Ifinishthedesignoftheupperboutwiththeclothoida-94.3.

Ascanbeseenfromthedrawing,theupperboutissymmetrical.IobtainthepointsFandGbysimplemeasurementanddefinethatCF=66.1mmandGD=75.3

mm.As it can be seen from this construction, Stradivari has enlarged the length of the pattern byincreasingthelowerboutinsize.Thelengthofthepattern(348mm)andthelengthFG(206.6mm)areintheratioof1.68.

Theline,connectingthewidestplaceofthelowerboutwiththeeyeofthescroll,goesthroughtheloweranduppercornersoftheviolinpattern(G¹A¹).

DrawthelinefromthepointFthroughtheuppercornerandgetFH,whichmakesanangleof54ºwithCD.Thethirdline,whichisatangentoftheclothoida-185,goesthroughthepointTexactly.

The lower corner is found by a similar way, where the tangent of the clothoid a-200 goesthroughthepointPexactly,andthelineGHmakesanangleof54ºwithCD.

TheconfigurationofthecornersandC-boutaredrawnwiththeclothoidsa-66anda-44,whichareinratioof3/2.

IfthelinesTJandPKthroughthecornersdividethelengthoftheviolinpatternintothreeparts,Iget three lengths in thenext ratios:CJ(121.3mm)/JK(86.5mm)=7/5;KD(140mm)/JK=μ (goldendivision).Thecentreofcircle,gettingthroughallcornersliesinpointW;diameter=188mm.

Figure43.ThegeometricreconstructionofA.Stradivari'spattern'PG'foraviolin.

Figure44.SuperimposingthedrawingofNicoloAmati'spattern'MB'foraviolin(dashedlines)andthedrawingofA.Stradivari'smould'PG'foraviolin.

ThegeometricreconstructionofA.Stradivari'spattern"SL"-1691foraviolin(Figure45).Thelengthofthepattern(CD)is350mm,thewidthoftheupperbout(F¹F²)=154.5mm,thewidthofthelowerbout(G¹G²)is195.5mm,andthewaistis100mm.MoreoverG¹G²/F¹F²=1.27(thatisclosetotheratioof5/4=1.25).


HavingconstructedthelowerbulbIusedtheclothoida-202.Therootoftheclothoidliesatthe

point M, touches the C-bout and is inserted in the lower bout of the mould up to the end block.Moreover,theangleKMP=58.7º(thatisslightlylessthan60º).

Similarly one constructs the opposite side of the bulb. The area around the end block iscompletedbytheclothoida-128.5sothatitisatangenttothehorizontallineD,andsegmentsofbothclothoidsa-128.5anda-200arejoinedinsuchawaythatthecurvatureiscontinuousthroughout.

Ascanbeseenfromthedrawing,thelowerboutiscomposedsymmetrically.Theupperboutisdrawnwiththeclothoida-185.4.CS=178mm,andSD=172mm.Since in

this pattern the upper bout is a littlemore extended than that of the previous pattern, the angle ofrotationoftheclothoidisdecreased,i.e.theangleJST=51.83º.Ifinishthedesignoftheupperboutwith the clothoid a-100. As it can be seen from the drawing, the upper bout is composedsymmetrically.

IobtainthepointsFandGbysimplemeasurementanddefinethatCF=65mmandGD=73.4mm.Thelengthofthepattern(350mm)andthelengthFG(211.6mm)areintheratioof1.68.


DrawalinefromthepointFthroughtheuppercornerandgetFH,whichmakesanangleof54ºwithCD.Thethirdline,whichisthetangentoftheclothoida-185,goesthroughtheuppercornerofthepatternexactly.

The lower corner is found by a similar way, where the tangent of the clothoid a-202 goesthroughthepointPexactly,andthelineGHmakesanangleof51.83ºwithCD.

TheconfigurationofthecornersandC-boutaredrawnwiththeclothoidsa-66anda-44(whichareinratioof3/2).

IfthelinesTJandPKthroughthecornersdividethelengthoftheviolinpatternintothreeparts,Igetthreelengthsinthefollowingratios:

CJ(118.3mm)/JK(90mm)=4/3;JK/KD(141.7mm)=0.635.

Ifwecompare thispatternwith thepreviousone, it isobvious thatStradivarihasenlarged thelengthof thepatternby increasing thecentrepart.Thecentreofcircle,getting throughallcornersliesinpointS;diameter=188mm.

Figure45.ThegeometricreconstructionofA.Stradivari'spattern"SL"-1691foraviolin.

Figure46.SuperimposingthedrawingofA.Stradivari'spattern 'PG'foraviolin(dashedlines)andthedrawingofA.Stradivari'smould'SL'foraviolin.

The geometric reconstruction of A.Stradivari's pattern "B"-3/6/1692 for a violin (Figure 47).Thelengthofthepattern(CD)is353.5mm,thewidthoftheupperbout(F¹F²)is154.5mm,thewidthofthelowerbout(G¹G²)is194.8mm,andthewaistis102mm.MoreoverG¹G²/F¹F²=1.26(thatisclosetotheratioof5/4=1.25).

ThepointM,arootof the lowerbulb,divides the lengthof thepattern into twolengths in theratioof1.36,thenMD=203.8mmandCM=149.7mm.

Stradivariagainenlargesthepreviousmodelbyincreasingthelowerbout.HavingconstructedthelowerbulbIusedtheclothoida-214.Moreover,theangleKMP=55.6ºmuchlessthan60º,sincetheboutisextendeddownwards.

Similarly one constructs the opposite side of the bulb. The area around the end block iscompletedbytheclothoida-130sothatitisatangenttothehorizontallineD,andsegmentsofbothclothoidsa-130anda-214arejoinedinsuchawaythatthecurvatureiscontinuousthroughout.

Ascanbeseenfromthedrawing,thelowerboutwascomposedalittleasymmetrically.Theupperboutisdrawnwiththeclothoida-184.5.CS=181.3mm,andSD=172.2mm.Sincein

thispattern theupperbout isa littleextended incontrastwith thepreviouspattern, theangleof therotationoftheclothoidisdecreased,i.e.theangleJST=50.5º.Ifinishthedesignoftheupperboutwith the clothoid a-94. As can be seen from the drawing, the upper bout was composedasymmetrically.

I obtain the points F andG by simplemeasurement and define thatCF= 66.2mm andGD=76.5mm.Thelengthofthepattern(353.5mm)andthelengthFG(210.8mm)areintheratioof1.68.


DrawthelinefromthepointFthroughuppercornerandgetFH,whichmakesanangleof54ºwithCD.Thethirdline,whichisthetangentoftheclothoida-194.5,goesthroughtheuppercornerofthepatternexactly.

The lower corner is found by a similar way, where the tangent of the clothoid a-214 goesthroughthepointPexactly,andthelineGHmakesanangleof51.83ºwithCD.

TheconfigurationofthecornersandC-boutaredrawnwiththeclothoidsa-60anda-43(whichareinratioof7/5).

IfthelinesTJandPKthroughthecornersdividethelengthoftheviolinpatternintothreeparts,Igetthreelengthsinthenextratios:CJ(119mm)/JK(89.6mm)=4/3;KD(144.9mm)/JK=μ.

Ifwecomparethispatternwithpreviousone,itisseenthatStradivarihasenlargedthelengthofthepatternbyincreasingthelowerbout.Thecentreofcircle,gettingthroughallcornersliesinpointW;diameter=185mm.

Figure47.ThegeometricreconstructionofA.Stradivari'spattern"B"-3/6/1692foraviolin.

Figure48.SuperimposingthedrawingofA.Stradivari'spattern 'SL'foraviolin(dashedlines)andthedrawingofA.Stradivari'smould"B"-3/6/1692foraviolin.

Thegeometric reconstructionofA.Stradivari'spattern "B"-6/12/1692 foraviolin (Figure49).Thelengthofthepattern(CD)is347.5mm,thewidthoftheupperbout(F¹F²)is154mm,thewidthofthelowerbout(G¹G²)is195mm,andthewaistis102mm.MoreoverG¹G²/F¹F²=1.26(thatisclosetotheratioof5/4=1.25).InthismouldStradivariretainsthegeneralwidthofthepreviousmodel,butmakesitshorter,approachinginsizetothepattern'PG'.


Having constructed the lower bulb I used the clothoid a-207.5.Moreover, the angle KMP isincreased,incontrastwiththepreviouspattern,upto56.7º

Similarly one constructs the opposite side of the bulb. The area around the end block iscompletedbytheclothoida-129sothatitisatangenttothehorizontallineD,andsegmentsofbothclothoidsa-129anda-207.5arejoinedinsuchawaythatthecurvatureiscontinuousthroughout.

Ascanbeseenfromthedrawing,thelowerboutwascomposedalittleasymmetrically.

Theupperboutisdrawnwiththeclothoida-185.CS=176.3mm,andSD=171.8mm.Sinceinthispatterntheupperboutisnotextendedasmuchasinthepreviouspattern,theangleoftherotationoftheclothoidisincreased,i.e.theangleJST=51.83º.Ifinishthedesignoftheupperboutwiththeclothoida-94.8.Ascanbeseenfromthedrawing,theupperboutwascomposedasymmetricallytoo.

I find the points F and G by simple measurement and define that CF = 64.7 mm and GD =73.6mm.Thelengthofthepattern(347.5mm)andthelengthFG(209.8mm)areintheratioof1.66.


DrawthelinefromthepointFthroughtheuppercornerandgetFH,whichmakesanangleof54ºwithCD.Thethirdline,whichisthetangentoftheclothoida-185,goesthroughtheuppercornerofthepattern.

The lower corner is found by a similar way, where the tangent of the clothoid a-207.5goesthroughthepointP,andthelineGHmakesanangleof51.83ºwithCD.

TheconfigurationofthecornersandC-boutaredrawnwiththeclothoidsa-66anda-44(whichareinratioof3/2).Thecentreofcircle,gettingthroughallcornersliesinpointS;diameter=185mm.


CJ(117.1mm)/JK(88.2mm)=4/3;KD(142.7mm)/JK=μ

Figure49.ThegeometricreconstructionofA.Stradivari'spattern"B"-6/12/1692foraviolin.

Figure 50. Superimposing the drawing of A.Stradivari's pattern "B"-3/6/1692 for a violin(dashedlines)andthedrawingofA.Stradivari'smould"B"-6/12/1692foraviolin.

The geometric reconstruction ofA.Stradivari's pattern "S"-1703 for a violin (Figure 51).Thelengthofthepattern(CD)is345mm,thewidthoftheupperbout(F¹F²)is155mm,thewidthofthelowerbout(G¹G²)is195mm,andthewaistis100mm.MoreoverG¹G²/F¹F²=1.26.


ToconstructthelowerbulbIusetheclothoida-199.3.Moreover,theangleKMP,incontrastwiththepreviouspattern,isincreasedupto59º.

Similarly one constructs the opposite side of the bulb. The area around the end block iscompletedbytheclothoida-128.5sothatitisatangenttothehorizontallineD,andsegmentsofbothclothoidsa-128.5anda-199.3arejoinedinsuchawaythatthecurvatureiscontinuousthroughout.

Ascanbeseenfromthedrawing,thelowerboutwascomposedsymmetrically.Theupperboutisdrawnwiththeclothoida-185.CS=175.9mm,andSD=169.1mm.Theangle

oftherotationoftheclothoid,i.e.theangleJST=51.83º.Ifinishthedesignoftheupperboutwiththeclothoida-94.3.

Ascanbeseenfromthedrawing,theupperboutwascomposedlittleasymmetrical.IobtainthepointsFandGbysimplemeasurementanddefinethatCF=62mmandGD=70mm.

Thelengthofthepattern(345mm)andthelengthFG(213.2mm)areintheratioofμ.Theline,connectingthewidestplaceofthelowerboutwiththeeyeofthescroll,goesthrough

theloweranduppercornersofaviolinpattern(G¹A¹).

Drawa line from thepointF through theupper corner andgetFH,whichmakes an angleof51.83ºwithCD.Thethirdline,whichisatangentoftheclothoida-185,getsthroughtheuppercornerofthepattern.

The lower corner is found by a similarway,where the tangent of the clothoid a-199.3 goesthroughthepointP,andthelineGHmakesanangleof51.83ºwithCD.Thereby,forthefirsttimethetriangleFGHisseentobeisoscelesanditcompletelyrepeatsthegeometryoftheGreatPyramid.

Theconfigurationof thecornersandC-bout aredrawnwith theclothoidsa-60anda-42.Thecentreofcircle,gettingthroughallcornersliesinpointS;diameter=185mm.


CJ(119.5mm)/JK(88mm)=1.36;KD(137.5mm)/JK=1.56.

Figure51.ThegeometricreconstructionofA.Stradivari'spattern"S"-1703foraviolin.

Figure 52. Superimposing the drawing of A.Stradivari's pattern "B"-6/12/1692 for a violin(dashedlines)andthedrawingofA.Stradivari'smould"S"-1703foraviolin.

The geometric reconstruction ofA.Stradivari's pattern "P"-1705 for a violin (Figure 53).Thelengthofthepattern(CD)is348mm,thewidthoftheupperbout(F¹F²)is161mm,thewidthofthelowerbout(G¹G²)is200mm,andthewaistis102mm.MoreoverG¹G²/F¹F²=1.24.

ThepointM,arootof the lowerbulb,divides the lengthof thepattern into twolengths in theratioof1.22,thenMD=190mmandCM=158mm.

In the construction of the lower bulb I use the clothoid a-200.Moreover, the angle KMP, incontrastwiththepreviouspattern,isincreasedupto60º.

Similarly one constructs the opposite side of the bulb. The area around the end block iscompletedbytheclothoida-128.4sothatittouchesthehorizontallineD,andsegmentsofclothoidsa-128.4anda-200arejoinedinsuchawaythatthecurvatureiscontinuousthroughout.

Ascanbeseenfromthedrawing,thelowerboutwascomposedsymmetrically.Theupperboutisdrawnwiththeclothoida-182.ThepointSdividesthelengthofthepatternin

halfsoCS=SD=174mm.Theangleoftherotationoftheclothoid,i.e.theangleJST=55º.Ifinishthedesignoftheupperboutwiththeclothoid/b>a-106.

Ascanbeseenfromthedrawing,theupperboutwascomposedalittleasymmetrically.Iobtain thepointsFandGbysimplemeasurementsanddefine thatCF=65mmandGD=73

mm.Thelengthofthepattern(348mm)andthelengthFG(210mm)areintheratioof1.657.Theline,connectingthewidestplaceofthelowerboutwiththeeyeofthescroll,goesthrough

theloweranduppercornersoftheviolinpattern(G¹A¹).DrawthelinefromthepointFthroughtheuppercornerandgetFH,whichmakesanangleof

54ºwithCD.Thethirdline,whichisatangentoftheclothoida-182,goesthroughtheuppercornerofthepattern.

The lower corner is found by a similar way, where the tangent of the clothoid a-200 goesthroughthepointP,andthelineGHmakesanangleof54ºwithCD.Thereby,thetriangleFGHcanbeseenasisoscelesagain,butwithotheranglesthatcorrespondtothegoldentriangle.

TheconfigurationofthecornersandC-boutaredrawnwiththeclothoidsa-66anda-44(whichareinratioof3/2).Thecentreofcircle,gettingthroughallcornersliesinpointW;diameter=190

mm.IfthelinesTJandPKthroughthecornersdividethelengthoftheviolinpatternintothreeparts,I

getthreelengthsinthefollowingratios:

CJ(120mm)/JK(90mm)=1.36;KD(138mm)/JK=1.53.

Figure53.ThegeometricreconstructionofA.Stradivari'spattern"P"-1705foraviolin.

Figure 54. Superimposing the drawing ofA.Stradivari's pattern "S"-1703 for a violin (dashedlines)andthedrawingofA.Stradivari'smould"P"-1705foraviolin.

Thegeometric reconstructionofA.Stradivari'spattern "G"-1715 for aviolin (Figure55).Thelengthofthepattern(CD)is354mm,thewidthoftheupperbout(F¹F²)is161.5mm,thewidthofthelowerbout(G¹G²)is201mm,andthewaistis103mm.MoreoverG¹G²/F¹F²=1.245.

ThepointM,arootof the lowerbulb,divides the lengthof thepattern into twolengths in theratioof1.27,thenMD=198mmandCM=155.9mm.

HavingconstructedthelowerbulbIusedtheclothoida-207.7.Moreover,theangleKMPis59º.Similarly one constructs the opposite side of the bulb. The area around the end block is

completedbytheclothoida-124.5sothatitistangentialtothehorizontallineD,andsegmentsofbothclothoidsa-124.5anda-207.7arejoinedinsuchawaythatthecurvatureiscontinuousthroughout.

Ascanbeseenfromthedrawing,thelowerboutwascomposedsymmetrically.Theupperboutisdrawnwiththeclothoid/b>a-185.ThepointSdividesthelengthofthepattern

inhalfsoCS=SD=177mm.Theangleof therotationof theclothoid, i.e. theangleJST=54º. Ifinishthedesignoftheupperboutwiththeclothoida-124.5.

Ascanbeseenfromthedrawing,theupperboutwascomposedalittleasymmetrically.IobtainthepointsFandGbysimplemeasurementanddefinethatCF=65.6mmandGD=73.2

mm.Thelengthofthepattern(354mm)andthelengthFG(215.5mm)areintheratioof1.64.Theline,connectingthewidestplaceofthelowerboutwiththeeyeofthescroll,goesthrough

theloweranduppercornersoftheviolinpattern(G¹A¹).DrawthelinefromthepointFthroughtheuppercornerandgetFH,whichmakesanangleof

54ºwithCD.Thethirdline,whichisatangentoftheclothoida-185,goesthroughtheuppercornerofthepattern.

The lower corner is found by a similarway,where the tangent of the clothoid a-207.7 goesthroughthepointP,andthelineGHmakesanangleof51.83ºwithCD.

TheconfigurationofthecornersandC-boutaredrawnwiththeclothoidsa-66anda-44(whichareinratioof3/2).Thecentreofcircle,gettingthroughallcornersliesinpointS;diameter=190mm.


CJ(121.4mm)/JK(89.2mm)=1.36;KD(144.3mm)/JK=μ

Figure55.ThegeometricreconstructionofA.Stradivari'spattern"G"-1715foraviolin.

Figure 56. Superimposing the drawing ofA.Stradivari's pattern "P"-1705 for a violin (dashed

lines)andthedrawingofA.Stradivari'smould"G"-1715foraviolin.

ThesummarytableofthemainsizesoftheviolinpatternsofN.AmatiandA.Stradivari.Table4.

Above I introduce the summery table of themain sizes of the violin patterns ofN.Amati andA.Stradivari(thetable4).Theupperrowoflettersrepresentsthemarksmadebythemontheirownmoulds."MB"isthemouldofN.Amati;"B¹"isthemouldofStradivarimadeon3/6/1692;"B²"isthemouldofStradivarimadeon6/12/1692.

WhenIspokeoftheproportionalrelationsbetweenpartsofaviolin(Figure29),Ipointedtothesize of the pattern equal to 48.4 mm. A.Stradivari has this size in his two moulds "PG" and "P".Moreoverthemould"P"wasmadein1705,whenStradivaribeginstocreatehisbestinstruments.

Inthesemouldstheratiosbetweenthewidthofthelowerboutandthewidthoftheupperboutarealsoequalandapproachproportionsof1.2360678=2/μ.Ifweconnectthecentersoftheboutswiththecorners,wewillgetanisoscelestrianglewithanglesatitsbase=54º.Thisisonlyinthepatterns"PG" and "P", but in the rest the angle in the lower bouts is 51.83º. Only one pattern "S" has anisoscelestrianglewithanangleof51.83º.

The proportional relations between the sizes of upper, centre and lower parts of the violinpatterns(CJ,JK,KD)arebasicallyrepeatedfrompatterntopattern.Infourpatterns"PG","B¹","B²"and"G"theratioofthelowerpart(KD)tothecentralone(JK)isthegoldendivision;althoughintherestthisproportionisless,butclosetoμ.

Infivepatterns 'MB","SL","B¹","B²"and"P"theratiooftheupperpart(CJ)tothecentralis4/3,intherestthisproportionis,oriscloseto7/5.

Onlyintwopatterns"MB"and"S"thebase(FG)ofalargetriangle,whosesidesgothroughthecornersoftheviolinpattern,havetheratioofthegoldendivisionwiththegenerallength(CD),intherestthisproportioniseitheralittlemoreorless.

Thelowerboutsofallpatternsirrespectiveoftheirsizesaredrawnwiththeclothoida-200orclose to it, but the upper bouts are drawnwith the clothoida-185 or close to it. The angle of therotation of the lower clothoid is, or is close to 60º. In four patterns "MB", "SL", "B²" and "S" theupperclothoidisrotatedatanangleof51.83º,inthepattern"B¹"thisangleisalittleless,butintherestthisangleis,oriscloseto54º.InthefirstthreepatternsthepointMdividesCDintotwolengthsintheratioof2/μ=1.2360678,intherestthisratioiseitheralittlemoreorless.

***Butnow,usingthemethod,whichIhaveusedwhenanalysingtheviolinsofItalianluthiers,Iwill

buildmyownmodeloftheinstrument,whichwilldifferfrominstrumentsalreadyknownbyus,butwill use the main principles of violin design. It will give me the possibility to show the creativemethod of approach of a luthier in the art of the violin design. Whilst maintaining the cardinalprinciplesofdesign,themasterisfreeenoughinhiscreativeactivitytomakeauniqueandinimitablemusicalinstrumentinbesttraditionsofthegreatmastersofItaly.

Designingtheupperandlowerovalsofviolin(Figure57)IbeginfromtheverticallineCD=348.4mm,whichiscalculatedfromthegeometricprogressionofnumberπintheratioofthegoldendivision.ThepointMisobtainedbydividingCDintotwolengthsMD=192.6mmandCM=155.8mmintheratioof2/μ.

Thetangentoftheclothoida-200ofthelowerboutmakesanangleof60ºwithCD(i.e.theangleKMP=60º)andrestsitsroottopointM.Thewidestplaceofthelowerboutis200mmandremoved

fromtheloweredgeatadistanceof75.3mm(GD).Theupperbout isbuiltwith theclothoida-185,whose tangentmakesanangleof54ºwithCD

(i.e.theangleJST=54º)andrestsitsroottopointS,dividingCDinhalf(i.e.CS=SD=174.2mm).Thewidest place of the upper bout is 160mm and removed from the upper edge at a distance of64.1mm(CF).

Figure57.Designingtheupperandlowerovalsofviolin.

Theratiobetweenthetwowidthsis5/4,i.e.200/160=5/4.Theareasaroundtheendblocksarecompletedbytheclothoidsa-91anda-130accordingly.

HereinafterIconnecttheedgesofwidestplaceofthelowerboutwiththeeyesofthescroll,asIdidearlier,andmarkthecrosspointsoftheselinesandthetangentsoftheclothoidwiththelettersN,L, P andT. Practically I already havemarked the positions of the upper and lower corners of theviolinpatternbythesepoints.Letusconnectthesepointswiththecentresoftheupper(F)andlower(G)boutsaccordingly:theanglesFGHandGFH=54º.

ThelengthFG(209mm)andthelengthofthepattern(348.4mm)areintheratioof3/5.LetusrecallthatinthepasttheproportionsconnectedwiththeFibonacciseries:1,2,3,5,8,13,21,34,55,89andsoon,wereusedoften.Thissetofnumbersisformedbyaddingtwopreviousnumbers,theratiobetweenwhichapproachesthegoldendivision.Soif3/5=0.6and55/34=1.6176,then144/89=1.618.

BeforetryingtodrawtheC-bouts,itisnecessarytoobtainthesizeandpositionofthewaistlinesoftheviolin,whicharedefinedbythesizesandpositionoftheffs.SowegotoFigure58.

In Figure 58 I have shown the terminating stage of the modeling of the violin pattern. ThepositioningofthepointE,throughwhichthelineofthebridgeandinternalVnotchesoftheffsarepassed,isdeterminedbyfindingthepointofbalanceofthefinishedbelly.Withsamelengthofbelly,butwith thedifferent ratiobetween thewidthsof theupperand lowerbouts, thispoint isdisplacedupwardordownwardaccordingtothisratio.Atthelengthofabellyof35.5mmandtheratiobetweenthewidthsoftheboutsof5/4themeasureofinstrument(thedistancebetweentheinternalnotchesoftheffsandtheupperedgeofthebelly)isapproximately196mm.Ifanupperboutisslightlywider,andthereforeheavier, thispoint ispositionedupwardand themeasurecanbe193-194mm.On thecontrary,atanarrowupperbout(orawidelowerboutthatissame)themeasurecangetto198mm.

Ifthelengthofpatternis348.4mm,thePointEmustbeatadistanceof192.6mmfromtheupperend-block. I find this point by dividing the length of pattern into two lengths in the ratio of 2/μ=1.2360678.ThenCE=192.6mmandED=155.8mm.

The following stage concerns the selection of the ffs'model and size. Eachmaster is free tochoosewhat he prefers: to copy the ffs of the greatmasters or to design one's own. I draw yourattention tomyownmodelof the ffs, completelybuiltwith thegoldendivision, themain ideas ofwhich I have demonstrated already when analyzing the ffs of the great Italian masters. On thehorizontal line through the point E I build the golden rectangle, were the smaller side isperpendicular to the horizontal line and 10πmm (31.415mm) in length,while the greater side is50.83mmandbisectedatthepointE.Onthedrawingthisrectangleismarkedbythelettersa,b,candd.

IfweconnectthepointsEandb,thelengthEbcanbecomethebaseofanisoscelestrianglewithanangleatthebaseof72º,whichisidentifiedasagoldentriangle.ThistrianglerestsitsownanglesatthepointE,inthecentreoftheuppereye(thepointb)andinthecentreofthelowereye(thepointe)respectively.

ThelengthEb,asahypotenuseoftheright-angledtriangleaEb,is40.41mm.ThenEe=Ebxμ=65.38mm.

Figure58.Terminatingstageofmodelingoftheviolinpattern.

Draw a circle centre E and radius Eb (r - the small radius), and another circle, centre E andradiusEe (R - the big radius). Let the small circle andCD intersect at f.Draw the horizontal linethrough f. Let this line and big circle intersect at g and h. I define the length of gh, which is thewaistlineofviolinpattern,asfollows:

gh=2√(R²-r²)=2√(65.38mm²-40.41mm²)=102.8mm.

Drawthecirclescentreb(radiusof3.75mmfortheuppereye)andcentree(radius5mmforthelowereye).

ThefurtherworkofbuildingtheffsandtheC-boutwithclothoidsissimilartomyanalysesofthe instruments of the Italianmasters and is easy to read from the drawing. Here I have used theclothoidsa-40 for the lower eyes anda-30 for the upper eyes, but theC-boutwas drawnwith theclothoidsa-71anda-44.

THEVIOLASANDCELLOSPATTERNS

A.Stradivarimadehis firstmouldofaviolacontralto in1672.Therewasonlyone instrumentmadefromthisform.Allthefollowinginstrumentscamefromthemouldof1690madefortheviolacontraltoMedicea.Onthatday,October4th1690,Stradivarimadetwoforms:thefirstmouldwasforthecontralto,andthesecondone-forthetenor.Thegeometricproportionsofviolasarenoticeablydifferfromtheviolins'one.Totracethelogicofthemaster,inchangingtheproportionalrelationsbetweenthepartsofeachinstrument,IwilldosimilarcollationstothoseIdidwithhisviolins.WhencomparingthedrawingoftheviolamouldswithdrawingsofviolinpatternsIwillreducethemtothecomparablesizes.

InFigure60Ihavesuperimposedthedrawingoftheviolamouldof1672withthedrawingoftheviolin pattern "MB".Here one can see that Stradivari has enlarged the volume of the lower bout,leavingadiminishedC-bout.Ifwecomparethispatternwiththoselookedatpreviously,itcanbeseenthatStradivarihasenlargedthelengthofthepatternbyincreasingthelowerbout.

Thegeometric reconstructionofA.Stradivari'spattern foraviolacontralto,1672 (Figure59).Thelengthofthepattern(CD)is403mm,thewidthoftheupperbout(F¹F²)is184mm,thewidthofthelowerbout(G¹G²)is241mm,andthewaistis124mm.MoreoverG¹G²/F¹F²=1.31.ThepointM,arootofthelowerbulb,dividesthelengthofthepatternintotwolengthsintheratioof1.3,thenMD=228mmandCM=175mm.HavingconstructedthelowerbulbIusedtheclothoida-238.5.TherootoftheclothoidliesatthepointM,touchestheC-boutandisinsertedinthelowerboutofthemoulduptotheendblock.Moreover,theangleKMP=60º.

Similarly one constructs the opposite side of the bulb. The area around the end block iscompletedbytheclothoida-148sothatitisatangenttothehorizontallineD,andsegmentsofbothclothoidsa-238.5anda-148arejoinedinsuchawaythatthecurvatureiscontinuousthroughout.Ascanbeseenfromthedrawing,thelowerboutwascomposedalittleasymmetrically.

Theupperboutisdrawnwiththeclothoida-213.6,therootofwhichdoesnotlieonthelineCD.Since in thispattern thewidestplaceof theupperbout to thewaist is less than thatofaviolin, the

angleofrotationoftheclothoidisdecreased,i.e.theangleJUS¹=54º.Ifinishthedesignoftheupperboutwiththeclothoida-108.9.

Ascanbeseenfromthedrawing,theupperboutissymmetrical.IobtainthepointsFandGbysimplemeasurementsanddeterminethatCF=76.7mmandGD=85mm.Thelengthofthepattern(403mm)andthelengthFG(241.3mm)areintheratioof1.67.Thelineconnectingthewidestplaceof the lowerboutwith theeyeof thescrollgoes through the loweranduppercornersof theviolapattern(G¹A¹).DrawthelinefromthepointFthroughtheuppercornerandgetFH,whichmakesanangleof54.5ºwithCD.

Thelowercornerisfoundbyasimilarway,wherethetangentoftheclothoida-238.5getsclosetothepointP,andthelineGHmakesanangleof51ºwithCD.

TheconfigurationofthecornersandC-boutaredrawnwiththeclothoidsa-69.1anda-51.5,andareinratioof1.34.

Figure59.ThegeometricreconstructionofA.Stradivari'spatternforaviolacontralto,1672.

Figure60.SuperimposingthedrawingofAmati'spattern"MB"foraviolin(dashedlines)andthedrawingofA.Stradivari'smouldforaviolacontralto,1672.


CJ(139mm)/JK(96mm)=1.45;KD(168mm)/JK=7/4KD/CJ=6/5.

Themould for a viola contralto (1690)wasmade byA.Stradivari after themould 'PG' for aviolin.So,inFigure69Ihavecomparedtheproportionsofthesemoulds.

Thegeometric reconstructionofA.Stradivari's pattern for theviola contraltoof1690 (Figure61).Thelengthofthepattern(CD)is403mm,thewidthoftheupperbout(F¹F²)is177mm,thewidthofthelowerbout(G¹G²)is233mm,andthewaistis118mm.MoreoverG¹G²/F¹F²=1.32.ThepointM,arootofthelowerbulb,dividesthelengthofthepatternintotwolengthsintheratioof1.37,thenMD=233mmandCM=170mm.

HavingconstructedthelowerbulbIusedtheclothoida-244.TherootoftheclothoidliesatthepointM, touches theC-bout and is inserted in the lower bout of the mould up to the end block.Moreover,theangleKMP=58º.

Similarly one constructs the opposite side of the bulb. The area around the end block iscompletedby theclothoida-145 so that isa tangent to thehorizontal lineD,andsegmentsofbothclothoidsa-244anda-145arejoinedinsuchawaythatthecurvatureiscontinuousthroughout.

Ascanbeseenfromthedrawing,thelowerboutwascomposedalittleasymmetrically.Theupperboutisdrawnwiththeclothoida-184,therootofwhichdoesnotlieonthelineCD.

Since in thispattern thewidestplaceof theupperbout to thewaist is less than thatofaviolin, theangleoftherotationoftheclothoidisdecreased,i.e.theangleFF²S¹=41.3º.Ifinishthedesignoftheupperboutwiththeclothoida-104.4.

Ascanbeseenfromthedrawing,theupperboutissymmetrical.Iobtain thepointsFandGbysimplemeasurementanddeterminethatCF=73mmandGD=

90mm.Thelengthofthepattern(403mm)andthelengthFG(240mm)areintheratioof1.68.Thelineconnectingthewidestplaceofthelowerboutwiththeeyeofthescrollgoesthroughthe

loweranduppercornersoftheviolapattern(G¹A¹).

DrawthelinefromthepointFthroughtheuppercornerandgetFH,whichmakesanangleof54ºwithCD.

The lower corner is found by a similarmethod,where the tangent of the clothoida-244 getsclosetothepointP,andthelineGHmakesanangleof51ºwithCD.

The configuration of the corners and C-bout are drawn with the clothoids a-72.7and a-48.4,whichareinratioof3/2.


CJ(134mm)/JK(98mm)=7/5;KD(171mm)/JK=1.74;KD/CJ=1.28.

Figure61.ThegeometricreconstructionofA.Stradivari'spatternforaviolacontraltoof1690.

Figure 62.Superimposing the drawingof the pattern "PG" for a violin (dashed lines) and thedrawingofA.Stradivari'smouldforaviolacontraltoof1690.

Figure63.Superimposingthedrawingofthepatternforaviolacontraltoof1690(dashedlines)andthedrawingofA.Stradivari'smouldforaviolacontraltoof1672.

ThegeometricreconstructionofA.Stradivari'spatternfor the tenorviolaof1690(Figure64).Thelengthofthepattern(CD)is468mm,thewidthoftheupperbout(F¹F²)is207mm,thewidthofthelowerbout(G¹G²)is257mm,andthewaistis137mm.MoreoverG¹G²/F¹F²=1.24.

Figure64.ThegeometricreconstructionofA.Stradivari'spatternforthetenorviolaof1690.

Figure 65.Superimposing the drawingof the pattern "PG" for a violin (dashed lines) and thedrawingofA.Stradivari'smouldforatenorviolaof1690.

The geometric reconstruction ofA.Stradivari's pattern for the 'Duport' cello (Figure 66). Thelengthofthepattern(CD)is749mm,thewidthoftheupperbout(F¹F²)is340mm,thewidthofthelowerbout(G¹G²)is435mm,andthewaistis220.5mm.MoreoverG¹G²/F¹F²=1.28.


HavingconstructedthelowerbulbIusedtheclothoida-458Moreover,theangleKMPis57.3º.Similarly one constructs the opposite side of the bulb. The area around the end block is

completedbytheclothoida-272.7sothatitisatangentofthehorizontallineD,andsegmentsofbothclothoidsa-272.7anda-458arejoinedinsuchawaythatthecurvatureiscontinuousthroughout.

Ascanbeseenfromthedrawing,thelowerboutwascomposedsymmetrically.Theupperboutisdrawnwiththeclothoida-392.6.ThepointSdividesthelengthofthepattern

inhalfsoCS=SD=374.5mm.Theangleoftherotationoftheclothoid,i.e.theangleJST=54º.Ifinishthedesignoftheupperboutwiththeclothoida-220.

Ascanbeseenfromthedrawing,theupperboutwascomposedalittleasymmetrically.IobtainthepointsFandGbysimplemeasurementanddeterminethatCF=140mmandGD=

161mm.Thelengthofthepattern(749mm)andthelengthFG(448mm)areintheratioof1.67.Thelineconnectingthewidestplaceofthelowerboutwiththeeyeofthescrollgoesthroughthe

loweranduppercornersofaviolinpattern(G¹A¹).DrawthelinefromthepointFthroughtheuppercornerandgetFH,whichmakesanangleof

53.5º with CD. The third line, which is a tangent of the clothoid a-392.6, goes through the uppercornerofthepattern.

The lowercorner is foundbya similarmethod,where the tangentof theclothoida-458 goesthroughthepointP,andthelineGHmakesanangleof49.5ºwithCD.

TheconfigurationofthecornersandC-boutaredrawnwiththeclothoidsa-116anda-85,andareinratioof1.36.

IfthelinesTJandPKthroughthecornersdividethelengthofthecellopatternintothreeparts,Igetthreelengthsinthefollowingratios:

CJ(258mm)/JK(173mm)=3/2;KD(318mm)/JK=1.84.

Figure66.ThegeometricreconstructionofA.Stradivari'spatternforthe'Duport'cello.

THEARCHESOFTHEBELLYANDTHEBACK

Thearchesofthebellyandthebackaresurfaces.Beforegivingtherulesofbuildingthearchesofthebellyandthebackwemustdefinewhatasurfaceis.Thesurfaceisacontinuousextenthavingonly two dimensions (length and breadth, without thickness), whether plane or curved, finite orinfinite.Thepositionofthepointonitisdefinedbytwosurfacecoordinates.

The curved surfaces are subdivided into regular, graphic, topographical, gravitational andothers.

Beforedefiningwhattypeofsurfacesweshallrankthearchesofthebellyandtheback,addresstothehistoryofthisquestion.

Many researchers of the creative activity of Italian luthiers developed the topographicaldiagramsoftheplates'archestostudythem.Thetopographicaldiagramismarkedwithcontourlines,joiningpointsofequalheightofthecurvature.Suchamethodofanalysisoftheviolinarchesisverygoodinmanyevents:itcanbeusedforthestudyofthedeformationsoccurringinthebellyandtheback and it can be used during themaking of new plates as a particular case of checking on thesymmetryofthecurvature.

A.Stradivari himself has left absolutely other method of building of arches. His sixths (onelongitudinalandfivetransversalguides)givetheruleofbuildingforthearchesofthebellyandthebackandtheyareananalyticalinstrumentforthestudyofthedifferentarches.

Proceedingfromthenatureofthesesixths,thesurfaceofabellyorabackcanbedefinedasagraphicsurface,generatedbymovingavariableline(fivetransversalguides)alongafixeddirection(the sixth longitudinal guide) in accordance with the results of calculations, satisfying assessedconditions.Astheconditionsinthisinstancethereareatypicalcurvatureofthelongitudinalguide,the horizontal plane and the outline of the basis of the plate. Five transverse lines, as differentpositions of the variable generator, together with a sixth longitudinal line as the guide, form thelinearframeworkofthesurfaceofthebellyortheback.

ThoughthenatureofthearchesofthedifferentinstrumentsofA.Stradivariisvariable(because

ofparticularitiesinthewood),neverthelessitisgeometricallyanalyzableandcanbeexpressedbyaformula, i.e. the curvature of the arches can be described by the clothoid.Then the generator, andguideofthesesurfaceswillberegularcurvedlinesandthesurfaceofthebellyorthebackwillbearegularcurvilinearsurfacewithavariablegenerator.Thevariabilityofagenerator isassignedbytheoutlineoftheplate,thecurvatureofthelongitudinalguideandtheplaneofbasis.

Theinitialpositionofthegeneratorisdefinedbythegeometriccentreofthebellyortheback.Proceedingfromthespecificsofthebellyandtheback,thesehavedifferentcurvatures.InFigure67Ihavepresentedthetransversesectionoftheviolin'sbelly(a)andtheback(b)ofA.Stradivari.Ascanbeseenfromthedrawing,thedifferentpositionsoftheclothoidonthebellyandonthebackshowthedifferent nature of their curvature. So the top of the belly has the greater radius of curvature, butclosertoedgestheradiusisdecreased.Atthebackisquitethereverse-thesmallerradiusislocatedonthetop,butclosertoedgestheradiusisincreased.

Thelongitudinalcurve(orguide,asdefinedmathematically)of thebelly repeats thenatureofthecurvatureofthegenerator(Figure68,a)i.e.inbothcasestheclothoidisdisposedfromthecentretotheedge.

Thattheguideoftheback(Figure68,b)hasamorecomplexcurvaturethanwhatisclearlyseenfromthedrawing(ofwhichIhaveshownonlyhalf).Ifwedisposetheclothoidinthesamedirection,as with the belly, the clothoid is deviated from the guide far enough from the edge, showinghereunderaratherbigradiusofcurvatureoftheback'sguidebesidetheedgesthatcorrespondtothenature of the generator's curvature. Thereby, the smallest radius of the back's curvature on thelongitudinallineislocatednotatitsgeometriccentre(asthegeneratorhas),butatthecentreoftheupperandlowerbouts.

Figure67.Thetransversesectionoftheviolin'sbelly(a)andtheback(b)ofA.Stradivari.

Figure68.Thelongitudinalsectionoftheviolin'sbelly(a)andtheback(b)ofA.Stradivari.

InFigure69 Ihavedrawn thesixths of curvatureof theviolinbelly,whichareplaced:A)- atcentreof theC-bout;B)-correspondingwith theuppercorners;C)-at themaximumwidthof theupperbout;D)-correspondingwiththelowercorners;E)-atthemaximumwidthofthelowerbout;F)-correspondingwiththeupperhalfofthelongitudinalsection;G)-correspondingwiththelowerhalfofthelongitudinalsection.

Thedistancebetweentheverticallines,dividingthecentralguide(A)intoevenareas,is6mm;theheightofthearchinthecentreis15.5mm;thewidthofthebellyhereis112mm.

Thedistancebetweentheuppercorners(B)is150mm,andtheheightofthearchinthisplaceis14.8mm.

Thewidestplaceoftheupperbout(C)is167mmwhilstthearchhereis12.0mm.Thedistancebetween lowercorners (D) is179mm,and theheightof thearch in thisplace is

15.3mm. The widest place of the lower bout (E) is 207mmwhilst the arch here is 13.0mm. Thelongitudalcurve(FG)issymmetricalandonmydrawingitisdividedinhalf.ThedistancesbetweentheverticallinesintheguidesB,C,D,E,F,andGcorrespondtothepointsoftheirintersectionwiththelinesofthetopographicaldiagram.OnecanseethelocationofthesixthsonthebellyinFigure71,whereforgreaterclarityIhavecombinedthetopographicaldiagramwiththegraphicframework.Thenumbersonthedrawingindicatetheheightsofthearchofthebelly.

The topographical diagram of the belly is shown in Figure 70. The numbers on the diagram

indicatethedistancesbetweenlines.InFigure72Ihavedrawnthesixthsofcurvatureoftheviolinback,whichareplaced:A)-atthe

centreoftheC-bout;B)-correspondingwith theuppercorners;C)-at themaximumwidthof theupperbout;D)-correspondingwiththelowercorners;E)-atthemaximumwidthofthelowerbout;F)-correspondingwiththeupperhalfofthelongitudinalsection;G)-correspondingwiththelowerhalfofthelongitudinalsection.

Thedistancebetweentheverticallines,dividingthecentralguide(A)intoevenareas,is6mm;theheightof thearch in thecentre is14.8mm;thewidthof thebackhere is112mm.Thedistancebetweentheuppercorners(B)is150mm,andtheheightofthearchhereis13.4mm.

Thewidestdistanceoftheupperbout(C)is167mmandtheheightofthearchhereis10.1mm.Thedistancebetween the lowercorners (D) is179mm,and theheightof thearchhere is14.2mm.Thewidestdistanceofthelowerbout(E)is207mmandtheheightofthearchhereis11.0mm.Thelongitudalcurve(FG)issymmetricalandonmydrawingit isdividedinhalf,andIhaveadded thebuttontotheupperhalfoftheback.

The distances between the vertical lines in the guidesB,C,D, E, F, andG correspond to thepointsof their intersectionwiththelinesof thetopographicaldiagram.Onecanseethelocationofthesixths on the belly in Figure 81,where for greater clarity I have combined the topographicaldiagramwiththegraphicframework.Thenumbersonthedrawingindicatetheheightsofthearchofthebelly.

The topographical diagram of the belly is shown in Figure 73. The numbers on the diagramindicate the distances between lines.Hereinafter in Figures from 75 to 88 I show the drawings ofarchesofviolasandcellos.

Figure69.Thesixthsoftheviolinbelly.

Figure70.Thetopographicaldiagramoftheviolinbelly.

Figure71.Combiningthetopographicaldiagramoftheviolinbellywiththesixths.

Figure72.Thesixthsoftheviolinback.

Figure73.Thetopographicaldiagramoftheviolinback.

Figure74.Combiningthetopographicaldiagramoftheviolinbackwiththesixths.

Figure75.Thesixthsoftheviolabelly.

Figure76.Thetopographicaldiagramoftheviolabelly.

Figure77.Combiningthetopographicaldiagramoftheviolabellywiththesixths.

Figure78.Thesixthsoftheviolaback.

Figure79.Thetopographicaldiagramoftheviolaback.

Figure80.Combiningthetopographicaldiagramoftheviolabellywiththesixths.

Figure81.Thefifthsofthecellobelly.

Figure82.Thelongitudinalsectionofthecellobelly.

Figure83.Thetopographicaldiagramofthecellobelly.

Figure84.Combiningthetopographicaldiagramofthecellobellywiththesixths.

Figure85.Thefifthsofthecelloback.

Figure86.Thelongitudinalsectionofthecelloback.

Figure87.Thetopographicaldiagramofthecelloback.

Figure88.Combiningthetopographicaldiagramofthecellobackwiththesixths.

***

The afore-cited analysis of the scrolls, ffs andmoulds of the stringed and bowed instrumentsallowsmetodrawaconclusionaboutthegoal-directedusageofsomeoftheproportionalrelationsandgeometricanglesbygreatmasters.Thereisonlytheneedtosettletheproblemofclothoidusagebythem,incertainpatterns,forthedrawingofthecontouroftheinstrument.

NeitherA.Stradivari,nor fellowmastersused theclothoid in themanner Ihave.Quite simply,theycouldnotdrawwiththeprecisionwecannowadays.

Buttheycouldshapeitfromastringorsomeothersuitablematerial.Thisconclusionisconfirmedbythefact thatavarietyofcurvedlines,madebyStradivariare

still extant them very close to or exactly modeled on this remarkable mathematical curve, the

clothoid.Thesmall inexactnessesareexplainedby thefact thatStradivarididnotstriveforperfectaccuracywhenmaking thedrawing; additionally, the string cannot be as accurate amethod as thatwhichcanbeproducedwithhelpofacomputer.

In thenext chapter Iwill try, as far aspossible, to reproduce themethodofStradivari for thedrawingofallcontouredcurvedlinesoftheviolin,includingthescroll,ffsandthepattern.

Chapterthree

TheReconstructionoftheStardivariusMethodofViolinDesign

In this chapter I will try to show a method of violin design using compasses and a ruler todeterminesizesandproportionsonly;andawayofdrawingcontourlineswiththehelpofstrings(amethodthatcouldhavebeenusedbyoldendaymasters).

Ialreadynotedthatmydesignswereproducedwiththehelpoftwomodules:thenumberπandtheclothoid.

Attheturnofthe16thcentury,whensuchprominentmastersasStradivariandGuarneriuswereworking,mathematicsandgeometryhadreachedastageofratherhighdevelopment,andtherewasdecimalcalculusaroundtheworld,buttheinchwasthefundamentalunitofmeasurement.

Itisimpossibletotalkaboutthesizeofoneinchatthattimewithsufficientaccuracybecauseithas only been since 1959 that the inch has been defined officially as 2.54 centimeters. This unitderivesfromtheoldEnglishunce,orynche,whichinturncamefromtheLatinunituncia,whichwas"one-twelfth"ofaLatinfoot,orpes.TheoldEnglishynchewasdefinedbyKingDavidIofScotlandabout1150asthebreadthofaman'sthumbatthebaseofthenail.DuringthereignofKingEdwardII,intheearly14thcentury,theinchwasdefinedas"threegrainsofbarley,dryandround,placedendtoendlengthwise.Sometimestheinchhasevenbeendefinedasthecombinedlengthof12poppyseeds.

Ifatvarioustimesthegrainsorthethumbwereusedforquicklyfindingthissize,itisimportanttosearchforthetrueoriginoftheinchingeometricproportions.Imeanthenumberπandthegoldenratio:1in.=πμ/2cm=2.5416cm.

Thereby,thenumberπisconnectedwiththeBritishinchandthemodernmetrebymeansofthegoldendivision.Onecansaythattheinch,whichwasusedbytheancientmasters,wasapproximately,asIhaveindicatedinmyformula.

Consequently,theancientconstructorsofviolinsdidnotusethenumberπasamodule,butoneinch instead. Thereby, the main sizes of violin can be defined as a geometric progression of theBritishinchintheproportionalattitudeofthegoldendivision.InFigure29thisprogressioncanbedefinedby the following lengths:AB=82.25mm(2in,μBC=133.08mm(2inμ²);AC= 215.33mm(2inμ³);etc.

IremindthereaderthatasthemodulorIusedthegoldendivisionandnumericallysimpleratios.Concerningthenumericallysimpleratios,theirusageisverysimpleanddoesnotrequireanyspecialexplanations. In Figure 2 I had demonstrated the calibration of lengths in the ratio of the goldendivision.

NowIbegintodesigntheviolinwiththehelpofspirals,usingsimplestways,whichcouldusetheluthiersofthepast.

THESCROLL

Letusbeginwith theBernoulli spiral.Wecan see that becauseof the small sizeof theviolin

scroll, it is unreasonable to use of the method of drawing known to Vitruvius [Marcus VitruviusPollio, an authorof theDearchitectura libri decem (TenBooks onArchitecture)].The degree ofreduction of the arcs of the circumferences (decay) in his method does not help us to solve ourproblems.

For the drawing of the scroll I offer to use the scale compasses (Figure 89), one variety ofwhichwasusedbythemastersforthefrettingoutofguitars,lutes,mandolinsandsoon.

Ifonepairofarmshas,forinstance,alengthof4inches(101.6mm),thentheotherpairmustbeshorterbyapercentagechosenbytheluthierhimself.

Figure89.Thescalecompasses.

Ihavechosenthe15%reductionofradiusandproceedtodrawtheviolinvolute(Figure90).Aspiralisconstructedasaseriesofcirculararcs.Ibeginbydrawingaquadrantofthecircular

arcabcentreOandradiusof16mmwiththespreadof thebigpairofcompasses.ThenI turn thecompassesoverandwiththespreadofthesmallpairofcompassesdrawaquadrantwithacirculararcbccentrex,whichrestsonthelineob.Thenewradiusof13.6mmhasonly85%oftheoriginalradius(16mm).Spreadthebigpairofcompassesanequaldistancetothisradiusanddrawaquadrantwithacirculararccbcentreywiththespreadofthesmallpairofcompasses.

Nextarcdewiththecentreatpointzalsoisaquarterofcircumference.The further process of drawing the volute completely repeats the previous one and is easily

understoodfromthedrawing.Naturallywecanusethecompasseswithanotherpercentagereductionoftheradius(asIdidwhenanalyzingtheviolinscrolls).

The quality of the volute is dependant on the accuracy of the present drawing. If Imademydrawingwiththehelpofacomputerwithahigherdegreeofaccuracy,thanmastersofthepast,whousingonlycompassesandaruler,certainlyallowedforsomeinaccuracy,which,ofcourseinfluencedthefinalresult.

WhenIanalyzetheviolinscrolls,certaininexactnessinthebuildingoftheBernoullispiralbysomeItalianmastersisstriking.Thisinexactnesswasnotsomuchtheeffectofcarelessnessasnon-observanceof themainrulefordesigningtheBernoullispiral:eachnewarcmustbereducedbyaconstantpercentage.

Figure90.Thephaseddrawingofthevolute.

Figure91.ThevoluteofPietroGuarneriI.

Usingthescalecompasses,onecanallowacertaininaccuracy,butinverysmalllimits.Butifwedrawthevolutewiththeusualcompassesandproduceareduction'byeye',thenthespiralalthoughitwillberathergraceful(Figure91),cannotbeidentifiedasagenuineBernoullispiral.Theparametersofthisspiralareasfollowing:arcab-radiusof27.5mm;bc-22.8mm;cd-16mm;de-14.6mm;ef-11.3mm;fg-9.5mm;gh-7.3mm;hi-6.3mm;ij-5.4mm;jk-4.1mm;kl-3.9mm;lm-3.1mm;mn-2.5mm.Thepercentage reductionof the radius fromthegreater radius isas follows:82.8%,70%,91%,77%,84.3%,77.4%,85.3%,87%,75.9%,94%,80.1%,80.1%.Theaveragefactoris82%.

Return toFigure90.On the lasteleventhdrawingofmyconstruction thecentreof thevolute,whichwillserveusforthefurtherconstructionoftheviolinscroll,isalreadyseendistinctly.Ihave

onlytodrawthreeparallel lines: throughthecentreof thevolute,abovethevolute throughpointaandthelowerline,whichgoesthroughthepointaatadistanceof2inches(50.8mm)fromthelinec.Thecenterlineislimitedbytheverticallineatthedistanceof2inμfromthecentreofthevolute.

The next stage of constructing the scroll is concerned with the drawing of the curved lines,whichIhavedonewiththehelpoftheclothoidintheFirstChapter.

To this effect Iuse the springystring,whichcan take the shapeofaclothoidwhenwe roll it.Usingtherolledstringshapedasaloop(Figure92)Icontinuetodrawtheouterfaceofthevolute.Certainly,we can vary not only the size, but also the configuration of the curvature, created by acurledstring,forcingthestringwiththethumbofthelefthandtowardsthenut.

Figure92.Modelingtheexternalcurvatureofthevolute.

Imodeltherearsidesofthepegbox,whichhasanS-form,inonestep,holdingthestringinthewayshowninFigure93.Ascanbeseen,thecurledstringexactlycopiesthesidebarofthescroll.Itwouldnotbeoutofplacetoobservethatthenatureofthecurvaturedependsinfullmeasureonthewayofholdingthestring,onthedistancebetweenhands,onthepressureoftherightthumb,etc.Sotheviolinscrolls,drawninthisway,canbedifferent(withafluentchangeoftheradius),butcannotbeattainedwiththeuseofcompassesalone.

Figure93.Modelingtherearsidesofthepegbox.

Theupperpartofthepegboxisalsomodeledwiththehelpofacurledstring,whichcanbeheldbyamasterinthewayshowninFigure101.EvenA.Stradivari'sscrollshaveadifferentcurvatureinthispartofthepegbox.Asmallerradiusisusedinthecentre,andclosertothevoluteortotheuppernut.So thenatureof the curvature is completelydependentupon themethodofholding the string,whenthethumboftherightor lefthandplaystheleadingrole.Inmyphotograph,for instance, therightthumbforcesthestring,creatingthesmallestradiusofthecurvatureclosertothenut.

Figure94.Modelingtheupperpartofthepegbox.

THEVIOLINPATTERN

Theviolinmoulds,madebyA.Stradivari,haveseveralroundholes,whichhelptoassembletheribs. They correspond with all six blocks. But there are other holes in the moulds, which werechangedintoasemicircularshapewhencuttingthespacesfor theupperandlowerblocks.Insomemoulds such semicircular cuts are placed at the centre of the space (for instance, the model B,6/12/1692),somemouldshave twosemicircularcuts (themodelPG,1689);onehas threecuts (themodel P, 1705). I deduce the reason for these cuts,was that theywere former holes; i.e. auxiliaryholeswhenmodelingthecontourofthepattern.

Whenmodeling the violin scrollwith the help of a springy string, I held the latterwith bothhands.Asavariant,Ihaveofferedtofastenoneendofthestringbyanyknownway.NowIwillshowoneofthepossiblevariantsofsuchfastening.

Inthehole,drilledintheboardofthewillowwoodforafuturepattern,ontheaxisofsymmetrythe violin peg is placed in such away that the small hole for the string is at the surface of board

exactly.Springastringfromthepeg'sholeandrollittodeterminethecurvedshapeofthebout.Thentracethecurvatureofthestringwithapencil.Ascanbeseeninthephotograph,thestringrepeatsthecontouroftheclothoidcompletely(Figure95).

Figure 95. Modeling of the upper bout of the violin pattern from the hole on the axis ofsymmetry.

Itwillbeusefultonoticethatthecorrectangleofrotationofthepeg,holdingthestring,isveryimportantinthismatter.Rotatingthepegtoonesideoranother,Iherebychangethecurvatureofthestringandtheformofthefuturepatterntoo.Inmycaseastringpassesthroughthepegatanangleof54ºtotheaxis.

Weseethatthepositionoftheseholeswasattainedbyexperimentandnotaffixedformula.Onthe Stradivari'smouldswe see the residuum of these holes of different sizes: from deep cuts (themodelB-3/6/1692)tosmallcuts(themodelSL),andalsotheirtotalabsence.Thisisexplainedbythedifferentdistancebetweentheholeandtheedgeofpattern,whereasthedepthofcutsfortheblocksisalways15mm.

On the next photograph (Figure 96) I investigate the possibility of modeling the upper bout,placingthepegintothehole,whichisnotlocatedontheaxisofsymmetry.

Figure 96.Modeling of the upper bout of the violin pattern from the hole not on the axis ofsymmetry.

Ascanbeseen,thisworkswithgreatersuccess,becauseinthiscasethecurvatureofthestringincludes the area of the upper bout, something the clothoid and the previous experiment with thestringdidnotdo.Inbothcasesthestringrepeatsthecontourofthepatternexactly,onlyrightnowitpassesthroughthepegatanangleof60ºtotheaxis.

Theleftsideofthepatternisbuiltaswellastherightone.Someasymmetrybetweentheleftandright sides is discernible because it is practically impossible to repeat this procedure with such aflexiblething,asstring.Thisasymmetrythatdoesnotspoilthegeneralappearanceoftheinstrumentatall,infactitevencontributescertainindividualitytothefinalcontouroftheviolin.

InFigure97Idemonstratethemodelingofthelowerboutoftheviolinpattern.Thisproceduredoes not differ from the previousworkwith the upper bout. The angle atwhich the string passesthroughthepegis62º.

Tofinishthemodelingofboutsattheareaoftheupperandlowerblocksisnotlaboriouswiththeuseofthesamestring(Figures98,99).

Figure97.Modelingofthelowerboutofviolinpattern.

Figure98.Finishingofthemodelingofupperbout.

Figure99.Finishingofthemodelingoflowerbout.

Thenextstageisthedeterminationofthepositionoftheupperandlowercornersofthepattern.Connecttheedgesofthewidestplaceofthelowerboutandthevoluteofthescrollwithastraightline.Thenthelines,passingthroughthecentresoftheboutsatcertainangles(asIdefinedthisintheFirstChapter),intersectthefirstline.Intheintersectionsoftheselinesdrilltheholesforthepegsinsuchawaythattheedgeofthepegtouchesthecrosspoint,andthelinethroughthecentreoftheboutcrossestheholestrictlyatthecentre(Figure100).

Figure100.ThepositionoftheholeforthepegwhenmodelingtheC-bout.

Thepegsare fixed in suchaposition that theirholesarecompliantwith the lines,whichpassthroughthecentresof thebouts.Thestringisplacedinthepegsandpulledthroughthemuptothedesired sizeof thewaist (Figure101).Though in this case the stringdidnot repeat thecontourofStradivari'spattern,suchC-bouts(thesamecurvatureoftheupperandlowercorners,andaroundedcentreofthewaist)arefoundintheviolinsofdifferentmasters,aswiththoseofAndreaAmati.Iwillnamethismethod:modelingwithafreestring.

Figure101.ModelingoftheC-boutoftheviolinpatternwithafreestring.Imodel the Stradivari'sC-bout by anotherway, limiting the excessive concavity of the string

insideofthepatternwithhelpofastop(Figure102).Sincethisstopexertspressureonthestring,notatthecentre,butalittleabove,thecurvatureoftheuppercornerwillbebiggerthanthecurvatureofthelowercorner.Iwillnamethismethod:modelingwitharestrictedstring.

Figure102.ModelingoftheC-boutoftheviolinpatternwitharestrictedstring.

Imodeltheouterfaceofthecornerswiththehelpofthesamepegs,directingthestringalongothercurvedlines,whichmusttouchthebulbsofthecorrespondedbout(Figures103).

Figure103.Modelingtheouterfaceofthecorner.

THEFF-HOLES

InFigures104,105,106and107Ishowfourstagesofmodelingoftheffs.HereIhaveplacedthepegsintotheeyes,butfromthebackoftheboardonlyinordernottooverlaythedrawingofthef-holebytheheadofthepeg.Thestring,whichIuseforthemodelingofthef-hole,isthinnerthaninpreviouscases.

Astothestringswhichhelpedmetodrawtheviolin,Iusedapianostringforthemodelingoflargeobjects:thebulbsoftheboutsandthescroll;aguitarstringfortheC-boutandaviolinstringfortheffs.Inshort,thesmallertheobjectofmodeling,thethinnerthestringused.

Figure104.Modelingoftheupperpartofthef-hole.

Figure105.Modelingofthelowerpartofthef-hole.

Figure106.Modelingoftheinnerpartofthef-hole'sarm.

Figure107.Modelingoftheexternalpartofthef-hole'sarm.

THEARCHESOFTHEBELLYANDTHEBACK

WhenspeakingofthearchesofthebellyandthebackofA.Stradivari'sviolins,wenotedthattodatetwomethodsofconstructionoftheirbuildingsandgeometricanalysisisknown.First,usingthesixthsofthecurvatureandofcoursethetopographicaldiagrams.Bothofthesewaysareverygoodforcopying,butcompletelyunsuitableforprimarymodeling.Evenmymathematicaldescriptionsofthearchesofthebellyandtheback,althoughsuitableforthemodelingofanewviolin,donotanswerthemostimportantquestion:"Howwastheprimarymodelingofthearchesofaviolinproducedbythegreatluthiersof16th-18thcenturies?"

Afteraseriesofexperimentswiththestring,whichhashelpedmetocreatethehypothesisaboutthedesigningoftheoutlineoftheinstrument'sbodywiththehelpofsomeflexiblematerial,andafterfinding the mathematical relationship between a curled string and the clothoid, for a long time Isearchedforanythingsimilarregardingthearches.InReality,neitherAmatinorStradivari,norotherluthiershaveleftusanytracesofthiscreativeprocess.Commonsensetellsusthatthesixthsofthecurvature,madebyStradivari,arenothingmorethanauxiliary instruments,usedbythemasterforthe transferof thecurvatureofsomearchon therealviolinplate. Ihold theview, that therewasacertainobject,identicaltothearchoftheplate,conceivedbythemaster,whichmusthavebeencopiedinwoodbyhimalone.Nodoubthecreatedthisobjecthimself,tomeettherequirementsofeachnewidea.Itisobviousthatthisobject,unlikethestring,hadaconstanthardform,whichallowedhimtouseitsmeasurementeasily.

I do not think thatAmati and Stradivari created the form of the violin arch intuitively, usingmerely the talent of a sculptor and an aesthetic vision of the beautiful. In the first place theywereconcernedwithsound,andso,engineeringproblemsforthemroseabovepureaesthetics.Thus,itispossibletoconjecturethatfirstlytheycreatedsomemodelofafuturearch,butafterwardstheycopieditwith thehelpof thesixths.Mysuggestion is that thismodelwasn'tmadeofwood.Toreach thisconclusionIcanlistseveralmoreorlesscogentarguments:

-theyconsideredthatitisimpossibletodothemodelofwood;-ifitwereeasytomakeofwood,thatwedonotneedthemodel,becauseitispossibletomake

thearchoftheplatedirectly;-thematerialforthemakingofthemodelmustbeflexibleandspringyaswasthestringforthe

designoftheviolinoutline;- the physical force, which changes the form ofmaterial, must be natural and directed to its

wholesurfacetoobtainahigh-qualityconstructionfromanengineeringperspective;- after the completion of the model the material must lose its springy qualities and become

enoughhardasamatterofconveniencetomakeacopyofit.InmyexperimentsI triedagreatdealbut theresultsof thoseattemptswillremainoutsidethis

book.FornowIwillshowonlythefinalresultofmystudies.InthesectionTheArchesoftheBelly

and theBack I listed several typesof curvilinear surfaceswith thegenerator of the variable type:regular,graphic,topographical,gravitationalandothers? I referred toall,except thegravitationalone.Nowitistimetoturnourattentiontoit.

Makeaframebycuttingouttheholeintheformoftheviolinplatewithoutcornersfromsomeboard(Figure108)andglueaknittedfabricaroundtheframe.Bywettingthehorizontallydisposedframewithwater,youwillfinditwillsagundertheweightofitsowngravityexactlyintheshapeoftheviolinarch.Thisdemonstratesthattheviolinplatecanbeseentobearesultofthegravitationalsurface.

Figure108.Theframeformodelingofthearchesoftheplates.

This soft construction can be stiffened with plaster of Paris (gypsum). Of course, if the wetplaster is spread too much, then the fabric will sag too deeply, more than the required 11 mm.Additionally,unequalspreadingoftheplasteronthesurfaceofthefabricwillnotcreatethesmoothsurfaceofanarch,unlessitisspreadequallywithveryfluidconsistency.So,Idilutetheplastertotheconsistencyofmilk,andspreaditonthefabricwiththewideandsoftbrush(Figure109).

Figure109.Modelingofthearches.

Thedifficulty inproperly fulfilling thiswork is akin to thevarnishingof the readyviolin. Inpractice,itisnecessarytocoverthefabricwithabout5-6layersofthegypsummilktogetmoreorlessahardconstruction.Anyincidentalbulgesinthelayerscanchangethenaturesagofthefabricanddisturbthearchofthemodel.Spreadingthefluidgypsumonthesecondandeventhethirdlayermustbeverycareful,sinceitispossibletobreakthestillthingypsumshellandtospoilallthework.Ifanattemptismadetospreadalayerofthickergypsum(forinstancewiththeconsistencyofsourcream)onayetnothardenedshellwithoneortwolayers,thenweriskspoilingtheconfigurationofthearch,becauseitisverydifficulttospreadthethickgypsumonthewholesurfaceevenly.

When the fabric becomes rather stiff and already can not be bent, the gypsum with theconsistencyofsourcreamisspreadinseverallayersinthesameway(Figure110).Ifinishtheworkbyfloodingthemodelwiththegypsumuptotheedges(Figure111).

Figure110.Spreadingthelayerofthickergypsum.

Figure111.Floodingthemodelwiththegypsumuptotheedges.

Weknowthattheconfigurationofthearchofthebellydiffersfromthearchoftheback.Ifwespreadthegypsumabsolutelyevenlyinthefirstlayers,thenwewillgetthearchofthebelly(Figure112).Forobtainingthearchofthebackitisnecessarytospreadthefirstlayerunevenly,i.e.thickerinthecentreandthinnerontheedges.Thegreatersaginthecentre,becauseoftheexcessofgypsum,willgiveusthenecessaryarchoftheback(Figure113).

Figure112.Plastermodeloftheviolinbelly.

Figure113.Plastermodeloftheviolinback.

Ingeneral, thefirst layerofgypsumformsthefuturearch,andtheresilienceofthewetfabricallowsforming thearchsmoothly,without"bignubs"andother roughness.But if thesecond layerwillbespreadunevenly,thentheexcessgypsumwillbendtheweakfinefirstlayermoreandthearchwillbeuneven.

It isknownthatA.Stradivarihadbeendefining theheightofeacharchbefore themakingofaviolinandthesixthsofthecurvatureproveit.Howdoesoneformthearchofthemodelwithagivenheightfortheintendedviolin?Weknowthathissixthsdonotyetshowthecurvatureattheedgesoftheinstrument,becausethiswasdoneafterthepurflingisinlaid.Sobeforethepurflingisinlaidtheheightofthearchinthecentreoftheplatewillbeabout11mmdisregardingthethicknessesoftheedges.Andthereforeitisnecessarytoallowforthesaggingofthefabrictosuchadepthduringthefixingofthefirstlayerofgypsum.Ifthedepthofsaggingfabricisnotenough,thenitisnecessarytospreadasecondlayerofgypsummilk,whenthefirstlayerisnotyetdry.Itisnecessarytorepeattheprocedure until the sagging fabric will meet the intended size. Technically these layers whencompletedwill be considered as the first layer.The secondand thenext layersmustnotbe spreaduntilthepreviouslayerisdry.

Ifweworkwith themodel of the back, thenwe increase the sagging by spreading additionalgypsumonlyalongtheverticalcentralline,moreinthecentreandlessontheedges.

Butifafterspreadingthefirstlayerofgypsumwegettoomuchsaggingofthefabric,thenitisnecessarytocorrectoneoftwothings:eitherreducetheconsistencyofthegypsummilk(andreplacethefabric),orchangethestretchingfabricontheframe.Inthelastcasetwovariantsarepossible:

-replacethefabric(asinthepreviouscase)or-iftheglueisnotyetdry,itispossibletotightenthefabric.Inthelastcase,possibleslantingofthefabricwillinfluencethegeneralqualityofthework,and

so,thestretchingofthefabricduringitsfixingtotheframeisbetter.Onecanmakeacopyofthemodelusingbothways:makingthesixthsofcurvatureanddrawing

thetopographicaldiagram.Itseemstomethatthearchesofthebellyandtheback,builtinsuchaway,withtheirsmoothand

plastic lines of curvature meet the requirements of the distribution of strain in them perfectly.Certainly,intheprocessofworkingwithspecificwoodamastercanchangetheheightofthearchesmildly,butthegeneralnatureofthecurvaturemustbethesameaswehavedeterminedinthemodel.AndA.Stradivarididitlikethis,inthoseviolinswhichhavenotbeenbuiltbyaccordingtohissixthsof curvature absolutely exactly. Thesearching guidesonly helped Stradivari to operate correctly,reserving the right of partial changes for him to adjust according to the acoustic qualities of thewood.

Theframe ismadewithdueregardfor thedesiredwidthof the troughalong theedgesof theplates,sincethiscurvatureattheedgesoftheinstrumentwillbedoneonlyafterthepurflingisinlaid.InmycaseIusedtheoutlineofthefirstinternalcurvedline,shownatFigure70forthebellyandatFigure73fortheback.

Onecanmakethesixthsofthecurvatureofthearchesinthisway:tomakecrosssectionsoftheplastermodelalonglines,intendedforcopying(Figure114),toplacethosesegmentsofthearchonawoodenplate,totraceasidebarofthecurvaturewithapencilandtocutoutthepatternsaccordingtothisdrawing.

Asthiswayisthesimplestwayofcopyingtheplastermodel,thenA.Stradivariprobablyusedit.Ifforsomeviolinformshemadenotonlyfifths,butalso thesixth longitudalpattern, then the lastwasmade beforemaking cross sections of the plastermodel. Common sense tells us thatwe canexpectthatA.Stradivarididnotsavetheseplastersegments.

InFigures115and116Ihavedemonstratedthecoincidenceofthedrawingsofthefifthsof thebellyandthebackwiththeirplastermodels.

Figure114.Thecrosssectionsoftheplastermodel.

Figure115.Superposingthedrawingsofthefifthandthephotosofthecrosssectionsofplastermodeloftheviolinbelly.

Figure116.Superposingthedrawingsofthefifthandthephotosofthecrosssectionsofplastermodeloftheviolinback.

Conclusion

I have finished the geometric analysis of stringed and bowed instruments of the great Italianmastersandhope that thepresentedconsiderationswillhelpothermastersget toknowthecreativeprocess in the design of these instruments better. This, in turn, will inspire them in the quest ofmakingtheirownmodels,whichwillpossesstheparticularitiesofthestyleofthegreatItaliansandstillhavethefeatureofindividuality,whichamodernmasterpossesses.

PracticallyoneachpageofmybookIfoughtwithawishtobegintoprovetheadvantageofmymethodincomparisonwiththeinvestigationsofotherauthors.

The fact that different methods of study of the same instruments yield different results isabsolutely natural. Anyway, all theories, known to the present-day luthiers, reflect the realcharacteristics of the violin geometrywith a greater or smaller degree of approximation and areequallyrightfromthestandpointoftheabstractfeatureofitsproportions,buttheyareincompatiblewithareconstructionofthemethodofworkingoftheold-timeItalianmasters.

Comparingdifferentmethods,inthefirstplacearesearcherturnshisattentiontosuchqualitiesofatheoryasitssystematiccharacter,fullness,accuracyofresultsandversatility(i.e.widerangeofusing).Butoftenitisnotenoughtoconfirmthatthissystemofproportionsoftheinstrumentwastheresultofmethods,whichthemasterhimselfused.Rather,itisasecondary,sideresult.

Howdowefindthemethodswhichwereusedbytheluthier,whenhewasdesigningviolins?Thefactisthattheworkingofthemasterleavesnotonlyfinalresults,butalsodirecttracesofthewaysbywhichitwasmade.Themostobservableofthemisseeninthedifferentsortof'mistakes'theymadealongtheway.

Amistakeitselfcanbeaccidental,but itsnaturedependsupontheintention,whenitwasmade.Let us take, for example, the asymmetry of the upper or lower bout of some violin moulds ofStradivari.What canwe see? The configuration of the left and the right curved lines is the sameabsolutely,buttheyhavenocommonaxisofsymmetry.Itcanhappeneitherwhenusingthepatternofthewholecurvedlinebetweentheblockandthecorner,whichhavebeenplacedbothtimesalittlebitdifferently, orwhenusing the springy string,which although it resembles the configurationof theclothoidinallcases,cannotrepeatitselfasamirrorimageexactly.

SinceStradivari didnot use special patterns for drawing theoutlineof theviolin's body, thennothingelse is left,but tosuppose,heused thespringystring,whichdoesnot retain its formafterhavingbeenused.Theuseof thecompasseswouldgiveanotherkindofdistortion,whichisunlikethatdescribedabove.

We can see the nature of themistakes in designing the curled linewith the compasses in theexample of designing the violin scroll, when a mistake in the determination of the radius of thefollowingarcoftheBernoullispiralchangesthenatureofthecurvedlineitself.

TherewasatimewhenItriedtoanalyzetheviolinscrollbytheclothoid,butitwasunsuccessful.Itispossible,certainly,tofindthescroll,whichcouldbedescribedbytheclothoid,butitwouldbearatheroccasionalevent,notaregularone.

Failure overtook me when using the Archimedes' spiral, which after C.F.Sacconi everybodyconsideredascentralforthedevelopmentthescroll.BelowIadducetwodrawingsoftheArchimedes'spiral:1st-drawnbythehandofC.F.Sacconi,andso,notquiteexact;and2nd-reconstructedwiththehelpofacomputerwiththesufficientdegreeofaccuracy.

Theparticularityof theArchimedes' spiral is that distancebetween thewhorls of the spiral isalikeeverywhere,butsuchacaseexistsextremelyseldomandonlyinsomeviolinscrolls.

Figure117.TheArchimedes'spiral.

If this analysismay seemunconvincing towhomever andhe/shewillwant to correct or evencompletely change themodus operandi, then I will say: "Good luck!" If such new studies will belogicalandmotivatedenough,Iamreadytoconsidertheseforthcomingopinions.

During the research for justification of a certain character of the violin ovals and othergeometricrelationshipsoftheviolinbody,Ifoundagreatsimilaritybetweentheviolincontourandahumanskull.

Bibliography

ABELL,A.M.-ThePartellocollectionofviolins,1909.ALEXANDRE,A.-LesStradivari.Paris,1945.APIAN-BENNEWITZ, P.O. - Die Geige, der Deigenbau und die Bogenverfertigung. Weimar

Voigt,1892.BAGATELLA,A.-Regoleperlaconstruzionedeiviolini,violevioloncellievioloni.Padova,

1914.BACHMANN,A-AnEncyclopediaoftheViolin.NewYork,1966.BOYDEN,D.D.-TheHistoryofViolinPlayingfromitsoriginsto1761anditsRelationshipto

theViolinandViolinMusic,London,1965.COVENTRY,W.B.-NotesontheconstructionoftheViolin.London,1902.DAVIDSON,P.-Theviolin.Glasgow,London,1871.GOODKIND,H.K.-AviolinIconographyofA.Stradivari.NewYork,1972.HAND,L.-Howtomakeafiddle.Chicago,1903.HILL,W.-AntonioStradivari:hislifeandwork(1644-1737).London,1902.JALOVEC,K.-ItalianischeGeigenbauer.Prag,1957.

JALOVEC,K.-BöhmischeGeigenbauer.Prag,1959.JALOVEC,K.-DieSchönstenItalienischenGeigen.Prag,1963.JALOVEC,K.-EnzyklopädiedesGeigenbaues.Prag,1965,2voll.JALOVEC,K.-DeutscheundÖsterreichischeGeigenbauer.Prag,1967.KRESÁK,M.-HusliarskeUmenienaSlovensku.Tatran,1984.MOECKEL,O.-DieKunstdesGeigenbauers.Berlin,1954.NICHOLOSON,J.-Designandplansfortheconstructionandarrangementsofthenewmodel

violin.London,1880.OAKES,W.W.-Constructionoftheviolin.(In:Music,AmonthlyMagazine,Chicago,1897).PANCALDI,C.-ProgressoitalianonellacostruzionedelviolinooperatodaAntonioGilbertini

daParma.Palermo,tip.Maddalena.PELUZZI,E.-Tecnicacostruttivadegliantichiliutaiitaliani.Firenze,1978.PUCCIANTI,A.-AntonioStradivari.Cremona,1959.SACCONI,S.F.-The'secrets'ofStradivari.Cremona,1979.

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