The evolution of air resonance power efficiency in the violin and its ancestors

26 Pages·2015·1.9 MB·English

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The evolution of air resonance
power efficiency in the violin
rspa.royalsocietypublishing.org
and its ancestors
HadiT.Nia1,AnkitaD.Jain1,YumingLiu1,
Research
Mohammad-RezaAlam1,†,RomanBarnas2and
Citethisarticle:NiaHT,JainAD,LiuY,Alam
NicholasC.Makris1
M-R,BarnasR,MakrisNC.2015Theevolution
ofairresonancepowerefficiencyintheviolin
1DepartmentofMechanicalEngineering,MassachusettsInstituteof
anditsancestors.Proc.R.Soc.A471:20140905.
Technology,77MassachusettsAvenue,Cambridge,MA02139,USA
http://dx.doi.org/10.1098/rspa.2014.0905
2ViolinMaking,NorthBennetSt.School,39NorthBennetSt.,
Boston,MA02113,USA
Received:20November2014
Accepted:13January2015 HTN,0000-0003-1970-9901
The fact that acoustic radiation from a violin at
SubjectAreas:
air-cavity resonance is monopolar and can be
acoustics determined by pure volume change is used to help
explainrelatedaspectsofviolindesignevolution.By
Keywords: determining the acoustic conductance of arbitrarily
violinacoustics,musicalacoustics, shaped sound holes, it is found that air flow at
the perimeter rather than the broader sound-hole
violinevolution,Helmholtzresonance,
area dominates acoustic conductance, and coupling
f-hole,soundholeevolution
between compressible air within the violin and its
elastic structure lowers the Helmholtz resonance
Authorforcorrespondence: frequency from that found for a corresponding rigid
NicholasC.Makris instrument by roughly a semitone. As a result of the
former, it is found that as sound-hole geometry of
e-mail:[email protected]
the violin’s ancestors slowly evolved over centuries
from simple circles to complex f-holes, the ratio of
inefficient, acoustically inactive to total sound-hole
area was decimated, roughly doubling air-resonance
powerefficiency.F-holelengththenslowlyincreased
byroughly30%acrosstwocenturiesintherenowned
workshops of Amati, Stradivari and Guarneri,
favouring instruments with higher air-resonance
power, through a corresponding power increase
of roughly 60%. By evolution-rate analysis, these
†Presentaddress:DepartmentofMechanical changes are found to be consistent with mutations
arising within the range of accidental replication
Engineering,UCBerkeley,CA94720,USA.
fluctuations from craftsmanship limitations with
subsequent selection favouring instruments with
higherair-resonancepower.
Electronicsupplementarymaterialisavailable
athttp://dx.doi.org/10.1098/rspa.2014.0905or
viahttp://rspa.royalsocietypublishing.org.
2015TheAuthor(s)PublishedbytheRoyalSociety.Allrightsreserved.
Downloaded from http://rspa.royalsocietypublishing.org/ on October 30, 2015
1. Introduction
2
Acousticradiationfromaviolinatitslowestfrequencyresonanceismonopolar[1]andcanbe
determined by pure volume change [2–4]. We use this to help explain certain aspects of violin ..rsp
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designevolutionthatareimportanttoacousticradiationatitslowestfrequencyresonance.This ...ro
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mostoftheviolin’svolumeisdevotedtohousingtheaircavity,theaircavityhasanimportant ...lish
effectonaviolin’sacousticperformanceatlowfrequenciesbycouplinginteriorcompressibleair ....ing.o
withtheviolin’selasticstructureandairflowtotheexteriorviasoundholes. ..rg
.
Owing to its long-standing prominence in world culture, we find enough archaeological ...Pro
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dataexistfortheviolinanditsancestorstoquantitativelytracedesigntraitsaffectingradiated ..
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..
acousticpowerataircavityresonanceacrossmanycenturiesofpreviouslyunexplainedchange. .S
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Bycombiningarchaeologicaldatawithphysicalanalysis,itisfoundthatassoundholegeometry ...c.A
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inactivetototalsoundholeareawasdecimated,makingairresonancepowerefficiencyroughly ...905
double.Ourfindingsarealsoconsistentwithanincreasingtrendinradiatedairresonancepower
havingoccurredovertheclassicalCremoneseperiodfromroughly1550(theLateRenaissance)to
1750(theLateBaroquePeriod),primarilyduetocorrespondingincreasesinf-holelength.Thisis
basedupontimeseriesoff-holelengthandotherparametersthathaveanatleastornearlyfirst-
order effect on temporal changes in radiated acoustic power at air cavity resonance. The time
seriesareconstructedfrommeasurementsof470classicalCremoneseviolinsmadebythemaster
violin-making families of Amati, Stradivari and Guarneri. By evolution rate analysis, we find
thesechangestobeconsistentwithmutationsarisingwithintherangeofaccidentalreplication
fluctuations from craftsmanship limitations and selection favouring instruments with higher
air-resonancepower,ratherthandrasticpreconceiveddesignchanges.Unsuccessfulnineteenth
centurymutationsaftertheCremoneseperiodknowntobeduetoradicaldesignpreconceptions
arecorrectlyidentifiedbyevolutionrateanalysisasbeinginconsistentwithaccidentalreplication
fluctuationsfromcraftsmanshiplimitationsandarequantitativelyfoundtobelessfitintermsof
airresonancepowerefficiency.
Measurementshaveshownthatacousticradiationfromtheviolinisomni-directionalattheair
cavityresonancefrequency[1].Thesefindingsareconsistentwiththefactthattheviolinradiates
soundasanacousticallycompact,monopolarsource[2–4],wheredimensionsaremuchsmaller
thantheacousticwavelength,atairresonance.Thetotalacousticfieldradiatedfromamonopole
sourcecanbecompletelydeterminedfromtemporalchangesinairvolumeflowfromthesource
[2–4,10,11]. For the violin, the total volume flux is the sum of the air volume flux through the
soundholeandthevolumefluxoftheviolinstructure.Accurateestimationofmonopoleradiation
atairresonancethenonlyrequiresaccurateestimationofvolumeflowchanges[2,3,11,12]rather
than more complicated shape changes of the violin [8,9,13] that do not significantly affect the
totalvolumeflux.Bymodalprinciples,suchothershapechangesmayimpacthigherfrequency
acoustic radiation from the violin. Structural modes describing shape changes at and near air
resonance have been empirically related to measured radiation, where modes not leading to
significant net volume flux, such as torsional modes, have been found to lead to insignificant
radiation [8,13]. Since acoustic radiation from the violin at air resonance is monopolar [1], and
socanbedeterminedfromchangesintotalvolumeflowovertime,arelativelysimpleandclear
mathematicalformulationforthisradiationispossiblebyphysicallyestimatingthetotalvolume
flux resulting from the corresponding violin motions that have been empirically shown [13 ] to
leadtothedominantradiationatairresonance.
Here we isolate the effect of sound hole geometry on acoustic radiation at air resonance by
developingatheoryfortheacousticconductanceofarbitrarilyshapedsoundholes.Weusethis
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to determine limiting case changes in radiated acoustic power over time for the violin and its
3
ancestorsduetosound-holechangealone.Theselimitingcasesareextremelyusefulbecausethey
haveexactsolutionsthatareonlydependentonthesimplegeometricparametersofsoundhole
.rs
shapeandsize,astheyvariedovertime,andarenotdependentoncomplexelasticparametersof ..pa
theviolin.Wethenestimatemonopoleradiationfromtheviolinanditsancestorsatairresonance ...ro
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withinaquarterofasemitoneoraPythagoreancomma[14].Thisindicatesthatrigidanalysismay ...rg
notbesufficientforsomefine-tunedmusicalapplications. ...Pro
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While Helmholtz resonance theory for rigid vessels [3–5] has been experimentally verified ...R
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for simple cavity shapes and elliptical or simple circular sound holes (e.g. [15–18]) as in the ..So
guitar,ithasnotbeenpreviouslyverifiedfortheviolinduetolackofamodelfortheacoustic ...c.A
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theory [3–5] even for rigid vessels. Here it is found both experimentally and theoretically that ..05
whenviolinplatesarerigidlyclamped,theair-cavityresonancefrequencystillfollowstheclassic
rigid body dependence on cavity volume and sound hole conductance of Helmholtz resonator
theory [3–5]. It is also found that when the plate clamps are removed, air-cavity resonance
frequency is reduced by a percentage predicted by elastic volume flux analysis, roughly a
semitone.Discussionsonthedependenceofairresonancefrequency,andconsequentlyacoustic
conductance, on sound hole geometry have typically focused on variations in sound hole area
(e.g.[22–25]).Thefluid-dynamictheorydevelopedhereshowsthattheconductanceofarbitrarily
shapedsoundholesisinfactproportionaltothesoundholeperimeterlengthandnotthearea.
Thisisverifiedbytheoreticalproof,experimentalmeasurementandnumericalcomputation.This
perimeterdependenceisfoundtobeofcriticalimportanceinexplainingthephysicsofair-flow
through f-holes and sound radiation from a violin at air resonance. It is also found to have
significantlyimpactedviolinevolution.
2. Determiningtheacousticconductanceofarbitrarilyshapedsoundholes
SoundholeconductanceC[3,4]isrelatedtoairvolumeflowthroughthesoundholeby
m¨air(t)=C(cid:2)P(t), (2.1)
where(cid:2)P(t)isthepressuredifferenceacrossthehole-bearingwall,formonopolarsoundsources
[2–4] like the violin at the air resonance frequency, and m˙air(t) is mass flow rate through the
sound hole [2], which is the product of air volume flow rate and air density, at time t . The
acousticpressurefieldradiatedfromasoundholeofdimensionsmuchsmallerthantheacoustic
wavelengthisproportionaltothemassflowrate’stemporalderivative[2–4],andconsequently
thetemporalderivativeoftheairvolumeflowrateandconductanceviaequation(2.1).Analytic
solutions for conductance exist for the special cases of circular and elliptical sound holes [3,4],
where conductance equals diameter in the former, but are not available for general sound
holeshapes.
A theoretical method for determining the acoustic conductance of an arbitrarily shaped
soundholeisdescribedhere.Acousticconductanceisdeterminedbysolvingamixedboundary
value problem [2–4,26] for approximately incompressible fluid flow through a sound hole.
This approach is used to theoretically prove that conductance is approximately the product of
soundholeperimeterlengthLandadimensionlessshapefactorα in§3.Itisthennumerically
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implementedin§§4and5toquantitativelyanalysetheevolutionofviolinpowerefficiencyatair
4
resonance.Itisexperimentallyverifiedforvarioussoundholeshapes,includingtheviolinf-hole,
in§§4,5and7.Theapproachdescribedhereisforsoundholedimensionssmallcomparedwith
.rs
theacousticwavelength,asistypicalattheairresonancefrequencyofmanymusicalinstruments, ..pa
includingthoseintheviolin,lute,guitar,harpandharpsichordfamilies. ....roy
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andtheconductanceCis .R
(cid:2)(cid:2) ...S
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the exact solution for the normal velocity un(x,y,z) on the opening from which conductance is ..05
thendetermined[3,4].Thismethodisfoundtobeeffectiveforarbitrarysoundholeshapesand
multiple sound holes. In the limit as the separation between multiple sound holes approaches
infinity,thetotalconductanceequalsthesumoftheconductancesoftheindividualsoundholes.
Whentheseparationbetweensoundholesissmall,thetotalconductanceissmallerthanthesum
oftheconductancesofindividualsoundholes.FortypicalCremoneseviolins,theconductance
theorydevelopedhereshowsthattheinteractionoftwof-holesreducesthetotalconductanceby
roughly7%fromthesumofconductancesofindividualf-holes.Thisleadstoaroughly4%change
inairresonancefrequency,whichisasignificantfractionofasemitone,andaroughly15%change
inradiatedacousticpoweratairresonance.Theconductanceformulationofequation(2.3)isfor
wallthickness,hsh,asymptoticallysmallnearthesoundhole,hsh(cid:2)L,whichisgenerallythecase
fortheviolinanditsancestorsaswellasguitars,lutesandmanyotherinstruments.Theeffectof
finitewallthicknessonCcanbeincludedbyuseofRayleigh’sformulation[3],whichbecomes
negligibleforsufficientlythinwallthicknessatthesoundhole.
Forthespecialcaseofanellipse,bothperimeterlength[28]andRayleigh’sanalyticsolution
fortheconductanceofanellipse[3]aredirectlyproportionaltotheellipse’smajorradius,and
so are proportional to each other for a fixed eccentricity. In the elliptical case, for example,
the eccentricity determines the constant of proportionality, i.e. the shape factor α, between the
conductanceandtheperimeterlength.Withonlytheellipsesolutionavailable,Rayleighobserved
thatsoundholeconductance‘Cvariesasthelineardimension’[3]ofthesoundhole,buthedid
notsupplyageneralproofordescribethephysicalmechanismsleadingtothisdimensionality.
3. Theoreticalproofofthelinearproportionalityofconductanceonsoundhole
perimeterlength
Here the conductance of a sound hole is proved to be linearly proportional to sound hole
perimeter length for L(cid:3)hsh. Total sound hole area S in equation (2.3) can be subdivided into
N elemental areas s, which share a common vertex inside S, such that the total sound hole
j
conductancefromequation(2.3)is
(cid:2)(cid:2)
(cid:2)N
1
C= un(x,y)dS, (3.1)
2
j=1 sj
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(cid:3)
whereS= Nj=1sj,withsjcontainingapiece-wisesmoothboundaryelementljofthesoundhole 5
perimeter.UsingStokestheorem,theintegralinequation(3.1)canbewrittenas
(cid:2)(cid:2) (cid:3)
.rs
sjun(x,y)dS= LjAdx+Bdy, (3.2) .....pa.roy
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wainlohteTnegrghreealLjbbajleonisudstinhnyde(cid:4)ga=utroly0atarcvilatabynlouabuetentphddreaeofirpbyneleecrmiodmnotefnooteurlrjrφsooo(ffxttt,hhhyeae,tzse)oelquehmunaadsetnihaotoansllpe(a3.erA.ce2ia)allorsecjfdaeaaluntccduoer∂oseBrtdo/thi∂naxat−teun∂s(yAxs/,t∂ye,ymz=()xu(cid:4)hn,ay(xs(cid:4)),ayw).iwtheaxk(cid:4), .............lsocietypublish
(cid:2)(cid:2) un(x,y)dS=(cid:2) A(cid:4)dx(cid:4)+(cid:2) A(cid:4)dx(cid:4)+B(cid:4)dy(cid:4), (3.3) ......ing.org
sj lj (cid:2)lj ....Pro
cwohrnereer flLjo=wljs+olu(cid:2)tiloj,n∂[B2(cid:4)9/]∂xw(cid:4)i−th∂A−(cid:4)0/.∂5y<(cid:4)=βu<n0(y.(cid:4)T)haennduA(cid:4)n=(yb(cid:4))+=(aa(/y((cid:4)β)β+f1ro)(cid:4))m(y(cid:4))tβh+e1twhriethe-dBi(cid:4)m=e0nsoiornaanl ......c.R.So
arbitrary constant is a solution to ∂B(cid:4)/∂x(cid:4)−∂A(cid:4)/∂y(cid:4)=un(y(cid:4)), and so ljA(cid:4)dx(cid:4)=blj where a and ....c.A4
barTehceonn,sftraonmts.equations(3.1)and(3.3) .......71:2014
C=1(cid:2)N bl + 1(cid:2)N (cid:2) A(cid:4)dx(cid:4)+B(cid:4)dy(cid:4), (3.4) ....0905
j
2 j=1 2(cid:4)j=1 (cid:2)lj (cid:5)(cid:6) (cid:7)
=0
where the second term vanishes since the total contribution from the edges of s other than l
j j
canceloutfromStokestheorem.Then,theconductanceCofthesoundhole
C=1bL=αL (3.5)
2
isproportionaltothesoundholeperimeterlengthL,bdependsontheshapeofthesoundhole
andα=b/2istheshapefactor.
4. Evolution of sound hole shape from the tenth to the eighteenth centuries
anditseffectonradiatedairresonancepoweroftheviolinanditseuropean
ancestors
Upper and lower limiting cases are determined by exact solutions for the changes in radiated
acousticpowerovertimeduesolelytochangesinthepurelygeometricparameterofsoundhole
shape for the violinand its ancestors (figure1a, electronic supplementarymaterial, §§1 and 2).
These are compared with the radiated power changes over time at air resonance from elastic
volumefluxanalysis(§9andelectronicsupplementarymaterial,§§3and4)(figure1a).Toisolate
theeffectofsoundholegeometryinfigure1a–c,allcaseshaveconstantforcingamplitudeover
time, are normalized to the circular sound hole case, and have all other parameters including
sound-hole area and air cavity volume fixed over time. So, the basic question is, for the same
areaofmaterialcutfromaninstrumenttomakeasoundhole,whatistheisolatedeffectofthe
shapeofthissoundholeontheairresonancefrequencyandtheacousticpowerradiatedatthis
resonancefrequency?
Theupperlimitingcasecorrespondstotheradiatedpowerchangeofaninfiniterigidsound
holebearingwall,wheretheexactanalyticalsolutionfortotalpowerinafrequencyband(cid:2)f is
givenby
(cid:2)
1 |(cid:2)P˜(f)|2
Wwall= d f C2 (4.1)
T (cid:2)f πρaircair
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(a) 140
6
elastic instrument
% radiated power change at air resonance 11864220000000 rinigfiidn iitne swtraulmlent CCC112.7 ........................rspa.royalsocietypublishing.o
0 ..rg
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(b) 111...232500 ce frequency .............Proc.R.Soc.A4
11..1150 ed air resonan ...........71:20140905
elastic instrument 1.05 aliz
rigid instrument m
1.00 nor
(c) 3.0
L)
Cce ()ngth ( 22..62 perimeter length
ne
conductaerimeter l 11..84 conductance
p
1.0
i ii iii iv v vi
century: 10th 12th–13th 13th 15th–16th 16th–17th 16th–18th
Figure1.Acousticair-resonancepowerefficiencygrowsassoundholeshapeevolvesovercenturiesthroughtheviolin’s
Europeanancestorstotheviolin.(a)Changeinradiatedacousticair-resonancepowerforanelasticinstrumentWair-elastic
(equation(4.5)),rigidinstrumentWair-rigid(equation(4.2))andinfiniterigidsoundholebearingwallWwall(equation(4.1))
asafunctionofsoundholeshape,wherepercentagechangeismeasuredfromthecircularsoundholeshape.(b)Air-resonance
frequencyforelasticinstrumentfair-elastic(equation(4.4))andrigidinstrumentfair-rigid(equation(4.3))asafunctionofsound
holeshape,normalizedbyfair-elasticforthecircularopening(i).(c)ConductanceC (equation(2.3))andperimeterlengthLfor
differentsoundholeshapesoffixedsound-holearea,normalizedtobeunityforthecircularopening(i).Shapeoverlapoccurred
betweennearbycenturies.Onlysoundholeshapeischangedandallotherparametersareheldfixedandequaltothoseofthe
1703‘Emiliani’Stradivariviolin[30].Theconductanceofthetwointeractingsoundholesforeachinstrumentisdeterminedfrom
equation(2.3).Datasourcesareprovidedintheelectronicsupplementarymaterial,§5.
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asshownintheelectronicsupplementarymaterial,§1,where(cid:2)P˜(f)isthefrequencyspectrumof
7
thetime-varyingpressuredifference(cid:2)P(t)acrosstheholebearingwall,where(cid:2)P(t)⇔(cid:2)P˜(f)are
aFourierTransformpair[31],Tistheaveragingtime,ρairisairdensity,cairisthesoundspeedin .rs
airandonlyCchangeswithsound-holeshapeinfigure1. ..pa
Thelowerlimitingcasecorrespondstotheradiatedpowerchangeofarigidinstrumentwith ....roy
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abasnoduwniddthhoaleroaunnddtahireHcaevlimtyhowlthzerreesothneanecxea[c3t–5so]lfuretiqounenfocyr itshe total power in the half-power .....lsocie
Wair-rigid=ηrigidC, (4.2) ........typublish
where the exact form of ηrigid is given and shown to be approximately independent of sound ......ing.org
hcoorlreessphoapned,incgonrdiguidctainnscteruamndenctaHvietlymvhoollutzmreesionntahneceel[e3c–t5r]ofnriecqsuuepnpcyleimsentary material, §2. The .....Proc
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fair-rigid=c2aπir (cid:8)VC(cid:9)1/2, (4.3) ........Soc.A4
whereVisaircavityvolume,andonlyCchangeswithsound-holeshapeinfigure1. .......71:2014
Inthecaseofelasticvolumefluxanalysis(§9andelectronicsupplementarymaterial,§§3and ..09
4),derivedfromclassicalCremoneseviolinmeasurements,theairresonancefrequencyfair-elastic ..05
isfoundtobeapproximately
fair-elastic≈κV−0.6(hback)0.1(htop)0.01(ha)0.2C0.5 (4.4)
andthetotalpowerinthehalf-powerbandwidthabouttheairresonancefrequencyisfoundto
beapproximately
Wair-elastic≈βV−0.8(hback)0.6(htop)−0.2(ha)−0.9C1.7, (4.5)
where κ and β are empirically determined constants, hback, htop and ha are, respectively, the
back plate thickness, top plate thickness and mean air-cavity height, which are all assumed
constantovertime,andonlyCchangesovertimeinfigure1a–btoisolatetheeffectofsound-hole
shapechange.
Thepowerchangecurvesinfigure1acanbedistinguishedbytheirdependenciesonsound
hole conductance (figure 1c), where the upper limiting case has power change proportional to
C2 (equation(4.1)),thelowerlimitingcasetoC(equation(4.2))andtheelasticinstrumentcase
toC1.7 (equation(4.5)).TheexpecteddependenciesonsoundholeconductanceareintheC1–C2
range,withelasticanalysisfallingroughlyinthemiddleofthelimitingcasesdeterminedbyexact
analyticsolutions.
When compared to theexact solutions of thelimitingcases, elastic analysis, which involves
more parameters (§9), still yields a similar increasing trend in power change over time due to
geometric changes in sound hole shape alone (figure 1a). The elastic instrument dependence
follows a trend that is roughly the average of the upper and lower limiting cases (figure 1a).
Ifonlysoundholeshapevaried,theseresultsareconsistentwithroughlyadoublingofpower
as sound hole geometry of the violin’s prominent European ancestors slowly evolved over the
centuries from simple circular openings of tenth century medieval fitheles to complex f-holes
thatcharacterizeclassicalsixteenth–eighteenthcenturyCremoneseviolinsofthelateRenaissance
andBaroqueperiod(figure1(i)–(vi)).Thisdoubling(figure1a)isduetoagradualmorphingto
more slender shapes that enables sound hole conductance to increase by roughly 50% through
triplicationofperimeterlengthforthesamesoundholearea(figure1c).
The rigid instrument resonance frequency fair-rigid (equation (4.3)) follows a similar
dependenceonsoundholeshapeasthatofelasticanalysisbutoffsetbyroughly6%(figure1b),
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which will be experimentally confirmed in §§5 and 7. While this may seem to be a small
8
inconsistency, 6% corresponds to roughly a semitone [14], which suggests that idealized rigid
instrumentanalysismaylacktheaccuracyneededforsomefine-tunedmusicalpitchestimates.
.rs
Theresultingairresonancepowerandfrequencydependenciesshowninfigure1a–bindicate ..pa
thatlinearscalingofaviolin,orrelatedinstrument,bypuredilationwillnotleadtoaninstrument ...ro
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resonance frequency, rather than area, as proved in §3. The reason for this dependence can ...rg
be seen by examining the normal velocity field exiting the sound hole for the violin and its ...Pro
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prominent European ancestors (figure 2, §2). Beginning with the Medieval fithele of the tenth ...R
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century,whichcontemporaryimageryshowstohaveasimplecircularsoundhole[33,34],most ..So
acoustic flux is found to be concentrated in a narrow region near the perimeter or outer edge ...c.A
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hole dimension near air resonance frequency. The circular sound hole’s interior then becomes ..05
increasingly inactive as radial distance from the edge increases (figure 2a), following fluid
dynamical principles for flow near wall edges [29], and consequently increasingly inefficient
for acoustic radiation. Our findings from the archaeological record (figure 1(i)–(vi)) indicate
that the ratio of inefficient to total sound-hole area was gradually reduced over the centuries
by increasing aspect ratio and geometric complexity, as exhibited by the introduction of semi-
circular sound holes [34–36] in twelfth–thirteenth century lyras (figure 2b), and then c-holes
[33,36] inthirteenth–sixteenthcenturyMedieval and Renaissance rebecs (figure2c–d). Through
this series of shape changes alone, sound hole perimeter length gradually grew, providing
greater conductance, greater air volume and mass flow rates over time and higher radiated
powerforthesamesoundholeareaattheairresonancefrequency.Theintertwinedevolutionary
trends of decreasing acoustically inactive sound-hole area by increasing sound hole perimeter
length and conductance to increase radiated power efficiency continued with the addition
of taper, circular end nobs and central cusps in vihuela de arcos and viols [33,35,37] of the
fifteenth–seventeenth centuries (figure 2e), and the classical Cremonese f-holes [33–35,38,39 ] of
thesixteenth–eighteenthcenturies(figure2f).
Inthiscontext,aninstructiveexperimentalverificationofthesoundholeconductancetheory
used here is provided in figure 2g, for annular sound holes of varying thickness, where the
dominant mass flow is found to be concentrated near the outer perimeter, which has the most
impactonsoundholeconductance,resonancefrequencyandradiatedpower.Arelatedfinding
is that the purely circular sound holes of classical guitars and the intricate sound hole rosettes
of lutes (figure 2h) and other instruments such as harpsichords, which have complex interior
structureswithinacircularperimeter,havenegligiblydifferentresonancefrequencies(withina
quarter tone) and air resonance powers (within roughly 10%) for fixed outer perimeter length.
Thissupportsassertionsthattheinteriorrosettesoftheseinstrumentfamiliesservedprimarily
decorative purposes [32,40–42] and hence did not evolve to maximize air resonance power
efficiency. The lute in fact became effectively extinct, perhaps partly due to its relatively low
radiatedpower.Thisoccurredastheviolin’sprominencerose,atleastpartiallybecauseitsgreater
radiatedpowerenabledittoprojectsoundmoreeffectivelyasinstrumentensemblesandvenue
sizeshistoricallyincreased.
The gradual nature of sound hole shape changes from the tenth to the sixteenth century in
the violin’s ancestors to the violin is consistent with incremental mutation from generation to
generationofinstruments.Thesteadygrowthofpowerefficiencyacrossthecenturiesisconsistent
withaselectionprocessfavouringinstrumentswithhigherpowerefficiencyatairresonance.
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diameter ratio d/D
Figure2.Soundholeshapeevolutiondrivenbymaximizationofefficientflownearouterperimeter,minimizationofinactive
sound-holeareaandconsequentmaximizationofacousticconductance.Normalairvelocityfieldu (x,y,z)(equation(2.2))
n
through(a)–(f)soundholes(i)–(vi)offigure1and(h)aluterosetteknownasthe‘WarwickFrei’[32]estimatedforaninfinite
rigidsoundholebearingwallatair-resonancefrequenciesbyboundaryelementmethodin§2.(g)Experimentalverificationand
illustrationofthesoundholeconductancetheoryforannularsoundholes.Dashedhorizontallinesindicateequaltemperament
semitonefactors.Theresonancefrequency,conductanceandradiatedpowerofacircularsoundholearerepresentedbyf ,C
0 0
andP ,respectively.Velocitiesin(a)–(f)and(h)arenormalizedbytheaverageair-flowvelocitythroughthecircularsound
0
hole(a).
Downloaded from http://rspa.royalsocietypublishing.org/ on October 30, 2015
5. Time-series analysis over the classical Cremonese period (sixteenth to
10
eighteenthcenturies)
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OvertheclassicalCremoneseperiod,exactsolutionsforupper(∼C2,equation(4.1))andlower ..pa
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forcing. Similar increasing trends are found for all cases (figure 3). So, elastic analysis, which ...lish
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thetemporalvariationsofsixparameters(figures4cand5)thathaveanatleastornearlyfirst- ..R.
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thicknessatthef-holes(figure5e),whichaffectsacousticconductance[3];meanair-cavityheight ..09
(figure 5d), which affects overall stiffnesses and f-hole length (figure 4c), which affects acoustic ..05
conductance.Theempiricallyobservedmodalmotionsthatgeneratetheeffectivelymonopolar
acousticradiationatairresonance[8,13],purelythroughchangesinvolumeflux,aredescribed
byphysicalanalysisinvolvingthesemeasuredparametersin§9.Increasingsoundholelength,
forexample,increasesconductanceandmassflow,butitalsoincreasesradiationdamping.This
lowerstheresonancemaximabutincreasespeakbandwidthsufficientlytoincreasetotalpower
integratedoverthespectralpeak(electronicsupplementarymaterial,§§2and4).Bymakingthe
back plate thicker and denser than the top plate, back plate motion and body volume change
arereduced.Thisleadstolesscoherentcancellationbetweenmassoutflowfromthesoundhole
and violin body contraction, and consequently higher radiated power at air resonance. This is
because at the air resonance frequency the top and back plates move towards each other, air
cavityvolumedecreases,forcingairvolumeoutofthesoundholesothatpositivemassoutflow
fromthesoundholeisthenpartiallycancelled bynegativemassoutflowfromtheviolinbody
contraction,intheobservedomnidirectionalormonopolarradiation(electronicsupplementary
material,§4).Keepingthetopplatelighter,ontheotherhand,enablesittobemoreresponsiveto
directforcingatthebridgeanddrivemoreairmassthroughthesoundhole.Forfixedaircavity
volume,increasingmeanaircavityheightincreasesstiffness.Reducingtopplatethicknessatthe
f-holesreducesairflowresistanceandsoincreasesconductance.
Bycomparisonoffigures3and4a,wefindthatestimatedpowerchangesduetothetemporal
variationsofallsixparameters(figure4a)primarilyfollowtheestimatedpowervariationsdue
totheisolatedtemporaleffectsoff-holelengthvariationalone(figure3).Thiscanalsobeseen
by noting that the total change in radiated power over the Cremonese period estimated from
elastic volume flux analysis with all six parameters is roughly 60±10% (figure 4a), and that
the conductance contribution via C1.7∝L1.7 from f-hole length changes alone of roughly 30%
F
(electronicsupplementarymaterial,equation(S18))leadstoasimilar58±5%powerchangeover
theCremoneseperiod(figure3),withaverysimilartrend.Wefindthattheseresultsareconsistent
withaselectionprocessfavouringinstrumentswithhigheracousticpoweratairresonanceduring
the classical Cremonese period. The increases in estimated power and measured f-hole length
over the classical Cremonese period are found to be relatively steady and gradual beginning
from the Nicolo Amati period until the Guarneri period. During the Guarneri period more
dramaticincreasesinbothoccurred.Byexaminingindividualcontributionsofthesixparameters
(figure6a),itisagainfoundthattemporalchangesinestimatedacousticpoweraredominatedby
temporalchangesinf-holelength.Thenextlargestcontributiontoestimatedpowercomesfrom
clearincreasesinbackplatethicknessduringtheStradivariandGuarneriperiods,whichisstill