PATRIZIO BARBIERI
Giordano
Diameters
C
of
Riccati
on
Strings
and
the
Pipes*
Giordano Riccati of Treviso (1709-1790), though
relativelylittle known to music historians,was a noteworthy
theoristof harmonyand the greatestItalianphysicistin the field of
acoustics. From his correspondencepreserved at the Biblioteca
Comunalein Udinewe learnthatalreadyin 1740he wasworkingon his
Dellecordeovvero
('Of elasticstringsor fibres'),which he
fibreelastiche
eventuallypublishedin 1767.In the prefaceto this book we read:
Thepleasure
of musichasdrawnme to devotea fairamountof thoughtto
thosesolidandfluidstringsof whichmostmusical
instruments
aremade,and
at
various
times
written
on
this
I
having
eightchapters
subject, haverecently
thempagebypageandrevisedthemhereandtheresothatwith
gonethrough
lessembarrassment
to the public.'
theycan[now]be presented
Riccati'sthoughtas a physicistextendswell beyond the scope of a
musician'scompetence,but in chapters5 and 6 the musiciancan find
some very useful informationon the 'solid and fluid strings'of
violinsandorgans.In this articleI shallanalyzefroman
harpsichords,
pointof view the contentsof chapter6, whichis entitled:
organological
'Of the dimensionswhich should be assignedto the stringsof an
instrumentandto organpipes,in orderthattheymayrendersoundsof
equalstrength,and pleasing[to the ear]'?
Accordingto Riccati,the areaS of the cross-sectionof a stringof
lengthL will fallwithinthe followinglimits(of whichthe secondhad
alreadybeen positedby LeonhardEuler):
S = constant
S oc L
Psycho-physicallaws were as yet a thing of the future, so Riccati
calculatedthe 'best' value of S by assumingthat the loudnessof a
vibratingstringis proportionalto its kinetic energy. (Accordingto
moderntheoryit is proportional
to the logarithmof the kineticenergy,
andalsodependson thepitchfrequency.)Theresultssetoutin chapter6
are garnishedwith documentaryevidenceaboutmusicalinstruments
OUNT
*Muchof this articleis drawnfrom my forthcomingAcusticaaccordatura
e
veneto.
ternperamento
nell'llluminismo
20
in the Veneto which scholarshiphas overlooked but which is of unique
historical value.
THE HARPSICHORD
his
Developing
assumptionsmathematically,Riccati found that on a
the
cross-section
S of the strings,'in order to rendersounds
harpsichord,
of equal strength,andpleasing',shouldbe proportionalto the geometric
mean of the two limiting cases shown above. So S oc
and hence
D oc L1/4.
,/L,
To verify this rule Riccati examined a 'harpsichordbuilt by Vito de'
Trasuntiniin 1559', measuringthe stringsfor two Cs three octaves apart
(p. 135). From the lengths given by Riccati one can deduce that they
were for c andc"' of an 8' instrument.(Evidentlyhe did not measurethe
C stringbecause in the lowest octave, as is well known, the curve of the
bridge changes abruptly and quite abandons harmonic proportion
among the string lengths.) For each of the two stringsRiccati measured
the vibrating length in 'inches of the Venetian foot' and, with a
jeweller's balance, the weight in Venetian grani of five feet of string.
More than once in this work Riccati specified that the Venetian pound
which he used as his unit of weight was equal to twelve ounces and
amounted to 12 x 576 grani:this is undoubtedly the 301.3-gram pound
which was at that time the standardof weight among Venetianjewellers
and pharmacists?Unfortunately, Riccati did not specify the materialof
which the strings were made; so in the following calculations I have
used 'Db' and 'Ds' to refer to the diametersas they would be for brass
(8700 kg/m3) and steel (7900 kg/m3 respectively. I use 'W' for the
weight of the sounding length of string:
for c : L=325/6(=949.5mm);W=20 (=0.872g);(Db=0.271,Ds=0.285mm)
for c"': L=
41/4
(=123 mm);W= 6 (=0.262g);(Db=0.149,Ds=0.156mm)
Since S:S"' (=W: W'"=3.3) was not very far from
VL:L'" (= 2.8),
Riccaticoncluded that his rule was adequatelyconfirmed
by practice.If
we assumethat the string lengths would double in each octave, then by
Riccati's rule the diameters should have the ratio 1: 21/4( 1:1.19).
Bearing in mind also that from brassto steel the diameterswould differ
by only some 5%,we can conclude that on Trasuntino'sharpsichordthe
diameters were, approximately:
c=0.28
c"'=0.15 mm
(c'=0.23)
(c"=0.19)
According to Riccati's measurementsL"':L amounted to 1:7.7, or
slightly less than the 'harmonic' ratio 1:8. He explained this as
follows:
21
In all the harpsichordsand spinets that I have examined, when I have compared
the lower strings with the higher ones I have found them to be somewhat
shorter than the ratio of the vibration frequencies would require. From this it
follows that the lower strings,on account of their thickness,are a little less tense
than the higher strings. I believe that the practitionersdo this because the hole
through which the strings are passed [in the process of drawing them]
compresses and strengthensthin strings more than [it does] thick ones, so the
latter cannot bear the tension of a force quite proportional to their [greater]
thickness.There is also the dangerthat the lower stringsmight be broken in the
process of wrapping them around the pin, which flattens them. On the other
hand, when they are protected from these risks and strung to the proper
tension, they last a long time without breaking. The higher strings on the
contrary do not last very well and break easily.5
(This last inconvenience has not been completely eliminated even
today, partly because some harpsichord makers use diameters greater
than 1/4mm for the higher strings.) Riccati also showed (p. 138) that
when the string is plucked, the shorter portion is disturbed more than
the longer one: hence the string is more likely to break at the tuning pin
than at the hitch pin.
Riccati concluded his discussion of harpsichord strings by saying that
his rule should not be considered unalterable:
Experience together with theory has taught practitioners the suitable
thicknesses for the lower and higher strings, and these may be, within certain
discretelimits, done by approximationratherthanexactly. When two stringsof
congruent thicknesses are put on an instrument, there remains another means
- the force of the quills - of equalizing the strength of the sounds. When
tuners play two strings at once, they understandexactly which quill should be
increasedor diminished in force in order that the two lower and higher sounds
may make an equal impression on the ear.6
The diameters which I have inferred from Riccati's measurements
may seem rather small, but the relevant data published by Michael
Thomas in 1971 tend to confirm them! Here are the string-gauge
numbers shown on two late 16th-century Venetian instruments
described by Thomas, and the equivalent values in millimetres as
reckoned by Thomas himself and (in parentheses) by Kenneth
Bakeman:8
1. 'School of Trasuntino' (8' instrument, range C-f'")
c
c'
c"
c"
22
5
0.34
6
0.31
8
0.24
11
0.16
(0.34)
(0.31)
(0.25)
(0.19)
d"'
12
0.14 mm
(0.17) mm
2. 'Trasuntino'(4' instrument,rangeC-c"')
C
c
C'
C1
Ci"
3
7
9
10
12
0.22
0.19
0.14 mm
0.42
0.28
(0.28)
(0.23)
(0.21)
(0.17)mm
(0.42)
of a certain'Franciscus,
Thomasalsofoundon a harpsichord
priestof
Rimini'some old fragmentsof stringwith the followingdiameters:
c : 0.30
c" : 0.14mm
c' : 0.23
He foundthe samediametersin bits of iron or soft steel stringson a
'smallNeapolitaninstrument'(on top of whichsomeonehadlaterput
brassstringsof slightlygreaterdiameter).Accordingto Thomas,'The
conclusionfromthe dataso far mustbe thatstringsat leastas thinas
.0055in. (=0.14 mm)wereusedin Italy'in the 16thand17thcenturies.
Germanmakersused thickerstrings,particularly
in the treble?
I mightmentionthatin chapter1 of Dellecordeovvero
fibreelastiche
Riccatidescribedsome experimentshe had done with a brassstring,
fromwhichit is possibleto deduce'0thatits elasticmoduluswas 8000
kg/mm2:thusthe brassof thatdaywas acoustically'better'thannow,
of unit
giventhattodaythe modulusis some9000kg/mm2.Indications
breakingstress(SB)for differentmetalwirescan alsobe foundin the
samechapter(p. 18),wheretheresultsof measurements
madebyJoseph
Sauveursome decadesearlier" are reported.From these may be
deduced:
steel:SB 10300kg/cm2
brass:SB" 8500kg/cm2
copper:SB 4800kg/cm2
UnfortunatelyRiccatidid not say which metal the stringson his
harpsichordwere made of. (Accordingto his theorythis would not
affectthe choiceof diameters.)I shallnot tryto settlethismatterhere,
but I observethat:
i) In the experimentsdescribedin Riccati'sbook he used,exclusively,
brassstringsof smalldiameter.
ii) If the c"' stringmeasuredby Riccatiwas of brass,it bore some
5900kg/cm2of tension(thatis, somethreesemi-tones'worthlessthan
the breakingstress,accordingto Sauveur'sdatashownabove);if on the
otherhandit wasof steel,thevaluewouldfallto 5350kg/cm2(whichis
nearlysix semitones'worthless thanits breakingstress,a typicalvalue
for a 'slow' string). I have made these calculationsknowing that
Venetianpitcharoundthe middleof the 18thcenturywas430-440 Hz
for a'.12
23
iii) Nowadaysit hasbeenfoundthatin orderto workwithinreasonable
limitsof security,harpsichord
stringsshouldbe stretchedatmostto two
or three semitones'worthbelow their point of rupture.3
iv) At the beginningof chapter6, Riccatiset out the premisethat
'sonorousstringswantto be stretchedto sucha degreethattheywould
breakif theirtensionwereincreased
the
justa littlemore.... Stretching
a
lot
renders
the
of
their
smallest
strings
lively
palpitation
parts,[and]
the soundconsistsmainlyof this [palpitation]'.
Also, as we sawabove,
he remarkedthaton his harpsichord
the high strings'do not lastvery
well andbreakeasily'.
So the balanceof Riccati'sevidencefavoursbrass,at least for the
instrumentas it was in his day. (In whichcase the stringused for the
experimentdescribedin chapter1 would be for c'.)
InAppendix1 I describesomeremarksby Giambattista
Doni (1695which
that
Roman
makers
at
least
someof
1647)
suggest
harpsichord
(or
usedsteelstringsfor the highernotes,
them)at the time of Frescobaldi
and copperfor the very lowest.
THE ORGAN
Riccatiadaptedto 'fluid strings'the rule which he had devisedfor
harpsichord
strings.He beganby showingthata harpsichord
stringof
to L31/12.But
lengthL 'pushestowardthe ear'a massof airproportional
the massof air thatis set directlyinto vibrationby a sonoroustubeof
length L and diameterD is proportionalto the volume of the tube
(oc D2L).Given,then,thatthe airwavesproducedby the pipe andby
string are propagatedat the same velocity, the condition which
determinesthe ratio of their kinetic energies can be reduced to:
D2L oc L31/12; fromwhich(andthisconclusionis emphasized
on p. xiv
of the preface)Riccatiobtained:
D oc L19/24 • L34 (hence S oc L19/12
L3/2)
(Thesamerelationshouldhold for the mouthwidths.)Riccatishowed
that since pipes an octave apartshouldhave, theoretically,the ratio
Li:L2 = 2:1, it follows that (pp. 144-45):
S1:S2
= 23/2 =
(thegeometricalmeanbetween2 and4). Adoptinga geometricdivision
of theoctaveintotwelveequalsemitones,thediametersasonedescends
shouldthusincreaseas follows:
C B Bk A Ab G Gk F E Ek D Dk C B Bk A Ab
L19/24
1.00 1.05 1.10 1.15 1.20 1.26 1.32 1.38 1.44 1.51 1.58 1.65 1.73 1.81 1.90 1.99 2.08
L3/4
1.00 1.04 1.09 1.14 1.19 1.24 1.30 1.35 1.41 1.48 1.54 1.61 1.68 1.76 1.83 1.92 2.00
24
In the firstcasethe doublingof the diameterstakesplaceat aboutthe
minor10th,in the secondcase at the major10th.
Now 1: %is the 'diapason'calculatedby Toipferin 1833andused
by most modernorgan builders.ChristhardMahrenholz,evidently
unawareof Riccati'sstudy,has attributedto G. A. Sorge(1773)the
inventionof the two relationsunderconsideration.4(Sorgeactually
proposedthree kinds of diapason,in which the doubling of the
diameterswouldtakeplaceatthe major9th,minor10thandmajor10th
respectively.)Since theoristsbefore that time had remainedsubtieddownto the traditional
stantially
'pythagorean'
scaling,Mahrenholz
in his lucidanalysisconcludesthata 'new era'in the calculationof the
diapasonbegan with Sorge. This honourshouldbe given ratherto
Riccati,who not only publishedhis conclusionssix yearsbeforeSorge,
but alsoderivedhis calculationsfromobjectiveprinciplesof acoustics
andnot,likeSorgeandTopfer,fromarbitrary
arithmetical
progressions.
found
the
Riccati
as
usual
turned
to practicefor its
rule,
Having
verification:
I measured
the
Marcuzzi,"
verydiligentlywiththe aid of SignorLiberale
worthy organistof this Cathedral[of Treviso],the circumferences,
to the diameters,
of two pipeswhichsounded[two]Fs three
proportional
octavesapart.The organof thischurchis a veryfineinstrument
madeby
Urbanoof Venicein 1420.16
ForanextracheckRiccatigavealso(p. 145)thecircumference
of the
for
the
octave
above
the
lower
F.
From
his
measurements
pipe
(in 'lines
of Venetianfeet')it is clearthatthe lowerpipecorresponded
to the first
key (I indicatethe diameterwith 'D', the circumferencewith 'C'):
F : C1= 140'lines'(=337.5mm);(Di=107.5mm)
f : C2= 84 'lines'(=203 mm);(D2z 65 mm)
f": C4= 29 'lines'(= 70 mm);(D4= 22 mm)
The corresponding
diapasonis very close to 23/4. (Fora perfectmatch
one would have had to obtain D2=64 and D4=22.5.) Mahrenholz
noticedthattheratio1 :23/4( 1: 1.68)approximates
verycloselyto 3: 5
The
(' 1: 1.67),a ratiowhichhe foundin some18th-century
treatises.17
makerof the organdescribedby Riccatiprobablyfollowed such a
procedure.Inanycasewe canaffirmthattheso called'Topferratio'was
alreadyused by Italianorganbuildersin the 15thcentury.
THE VIOLIN
As eachstringon thisinstrumentmustservefor severalnotes,the first
limit(S = constant)is imposedautomatically
in manyinstances.Yet if
we intendthat 'the ear does not judge that [two equalnotes] should
25
belongto one stringratherthanto another',Riccaticoncludedthatthe
crosssectionsof thethreehigherstringsshouldcorrespondto theirpitch
frequencies,thatis, to 3:2, thusyielding:
: E
string:D: A
section:9 : 6
: 4
diameter:3 : 2.45 : 2
(We shallsee presentlywhy the G stringwas not includedhere.)He
now proceededto his usualpracticalverification:
Witha jeweller'sscaleI weighedthreeequallylongsections,1/2 Venetian
knownasthetenor,thecantoandthecantino.
I did
feet,of threeviolinstrings,
notweightheloweststringbecauseit is notof plaingutliketheothers,butis
customarily
overspun
witha thincopperwire."
If we recallthatthe cross-sections
areproportional
to theweights,we
cansee thatthe weightsmeasuredby Riccaticonfirmhis theoryalmost
perfectly.To calculatethe diametersI have assumedthatthe specific
gravityof the gut amountedto 1350kg/m3(testswhichI havemadeon
stringsof differentdiametersand by differentmakershave yielded
valuesrangingbetween 1300and 1400):
D string= 15grani
(D = 1.09mm)
A string= 10grani
(D = 0.89mm)
E string= 6 grani
(D = 0.69mm)
For a perfectfit with Riccati'srule the lastweight shouldhavebeen
62/3grani.Riccati,who was himselfan amateurviolinist,remarkedthat
gut stringswere quiteimperfectandvariable,and anyway'an expert
player'had told him thathis (Riccati's)E stringwas a little too thin.
Riccati'sstringsweresomewhatthickerthanthoseusedby 'baroque'
violiniststoday.LuigiRovighiandEnricoGattihaveinformedme that
they use the followingdiameters:
D
A
E
1.05
0.67
0.52 mm
Rovighi:
Gatti: 1.00 0.70
0.55 mm
We shouldbear in mind, however,that alreadybefore 1743 in the
Veneto, Tartinihad introducedthickerviolin strings.Count GianrinaldoCarliwroteto Tartiniin a letterof August21,1743(fromVenice
to Padua):'Yourecognizedthatit wasnecessaryto thickenthestringsof
theviolinandlengthenthe bow, asyou have[indeed]done,so thatthe
vibrationsmightbe moreregulatedandthe soundcomeoutsweeterand
moresusceptibleto vibrations... .'19DavidBoydenhasconfirmedthat
'violinstringsin Italy,wherea full andstrongtonewasthe ideal,were
regularropescomparedto those used in France'.?o
26
For harpsichordstringswe sometimeshave contemporarygauge
numbers.Verylittlesuchdirectevidenceis availableforviolinstrings,21
but a clue canbe foundinJeromede LaLande'sVoyage
enItalie(1769),
whichdescribesthe 'methodfollowedat Naplesfor the manufacture
of
violinstrings':'theyputtogetheronlytwo strandsof gut(froma 7- or 8month-oldlamb)for the little stringsof the mandolin,three [strands]
for the firststringof theviolin,sevenfor the last'.22
Thiswouldsuggest
a ratio between the diametersof the G and E strings of some
7:33= 1.53. Between the sametwo stringsRiccati'srule (whichof
coursehe didnotconsiderapplicableto the G string)wouldyield 1.84.
Sebastiende Brossardin his early18th-century
treatisesaid
manuscript
of the G string:'if it is merelyof gutit mustbe at leasttwiceas thickas
theD string,butif it is overspunwithsilverit is onlya littlethickerthan
theD string'."Riccati'sG stringwasoverspun,so we mightexpectit to
havebeen 'only a little thicker'than11/2timesthe thicknessof the E
string.Hence the value which we have deducedfrom La Lande's
description(1.53)is quite satisfactory.
In Appendix2 I outlinesome additionaldocumentary
evidencefor
the diametersof 18th-and 19th-century
violin strings.
SUMMARY
Riccati'schapteryields the following organological information:i) the
harpsichordsof Vito Trasuntinohad very thin strings.Their diameters,
starting from a minimum of some 0.15 mm, doubled only after an
interval of at least three octaves down. ii) One of the best Venetian
organ makers of the 15th century (Antonio Dilmani)24scaled the pipe
diameters with a constant ratio equivalent to the modem 'standard
diapason' of Topfer. iii) Mid-18th-century Venetian violinists did
indeed use rather 'robust' strings, for which Riccati furnishes us with
precise diameters - data of unique value for the history of the
violin.
APPENDIX1
GiambattistaDoni, Robert Smith and Carlo Pellegrini
on Harpsichord Strings
Two hitherto overlooked remarks by Giambattista Doni suggest that in
Frescobaldi'sday Roman harpsichordmakers used steel in the higher ranges,
changingover to brassand then to pure copper for the lowest notes. In the first
remark Doni, discussing around 1640 the difficulty of maintaining the pipes
and strings of a claviorgano at the same pitch for the duration of an entire
theatrical production, concluded: '. . . hence [claviorgani] could be quite
useless, if it were not that this inconvenience can be remedied by tuning the
27
entireset of stringsabouta quarter-tone
higherthanthe pipesin orderthatthe
whole instrumentmaycometo its properpitchwhenthe airis warmedby the
torches[whichlightthe theatre];and[indeed]if the higherstringsdidnot also
warmasmuch,eitherbecausetheyaremadeof steelorbecausetheyareso thin,
it wouldbe necessaryto retunemanyof thenotes... .'25Doni indicatedwhich
metalthe loweststringsweremadeof in a letterof August7, 1638,fromRome
to FatherMersenneat Paris,referringto the 'practiceof our harpsichord
makers,who ordinarilyuse, for the lastnotesin the bass,some stringsof red
copper,becausethoseof brassdo nothavea longenoughmeasureto renderthe
requiredtone'?6
In 1759RobertSmithreferredindirectlyto the practiceof changingoverto
copperin thelowestrange:forcertainof hisexperimentshe saidthathe useda
'copperwire,commonlyusedforsomeof thelowestnoteson the harpsichord'.
Fromhis dataone maydeducethatthe diameterof the stringin questionwas
0.563mm,27that is, equal to gauge 11 accordingto Bakeman'stable?sThe
credibilityof thisconclusionis indicatedby the factthatcopperstringsof the
samegaugenumberwereused,forexample,in thelowestrange(GG-BB)of a
describedby JamesTalbotaround1698?9
harpsichord
On theotherhandCarloPellegriniatRomein 1665saidthattheharpsichord
andthat:'The [8'] spinetta
normallyhad45 brassstrings(exoricalcho
confectas),30
is a very notableinstrument,of the sametype as the harpsichord,
havingthe
samenumberof stringsasthelatter;butitssoundis notproducedby brassason
the harpsichord,
butby copperor steel.It is notsonorouslike the harpsichord,
beingof smallerdimensions.Sometimesa redclothis placeduponits stringsin
orderthatit may emit a more submergedsound'?1
APPENDIX2
Plessiard'sRule for Violin Strings
A littlemorethana centuryafterRiccatianotheramateurviolinist,the French
rule for the string
engineerPlessiard,workedout a differentmathematical
His
like
with
was,
Riccati's,
data,
theory
garnished
experimental
dimensions.32
which give us useful informationaboutthe diametersfavouredby French
violinistsaroundthe middleof the last century.
Afterremarkingthatthe firstthreestringsare of plaingut, Plessiardsaid
'thatthe rapidityof executionon the violin, the frequentpassingfrom one
stringto anotherandthe good effectof doublestops[all]requirethatthebow
notencountera greaterresistenceon one stringthanon another,thatis, thatthe
work [required]to drawsoundfromeachof the fourstringsbe the same.To
thisend the ratioof tensionsbetweentwo adjacentstringsshouldbe equalto
the squarerootof the musicalintervalbetweenthem,i.e. V/2, giventhaton
the violinthisintervalis a fifth'?3UnlikeRiccati'srulethisdoes nottakeinto
accountthe intensityof soundof the individualstrings;accordingto Plessiard
the acousticalequilibriumof the instrumentshouldbe regulatedby adjusting
thesoundpostandthebar.Inpassingfromonestringto thenextlarger,thetwo
rulesyield:
28
Plessiard
Riccati
ratioof tensions:
=
0.82
2/3
= 0.67
(2/3)1/2
=
ratioof diameters:
1.355
(3/2)3/4
(3/2)1/2= 1.225
ThusPlessiard's
ruleyieldsa greatervariationin the diametersforthe sakeof a
more uniformdistributionof tensionamongthe four strings.
Plessiardthencalculatedthetensionwhichone setof stringsansweringto his
rulewouldimposeuponviolinsof threedifferentmakers(p. 204).He gavethe
averagespecificweightof thegutwhichhe used(1300kg/m3),thefrequencyof
his a' (435 Hz) andthe soundinglengthof the strings,so I havebeen ableto
calculatethe stringdiameters:
G
D
A
E
violinbyJ.-B. Vuillaume(L= 329.5 mm): 4.50 5.51 6.75 8.26 kg
violin by Bergonzi(L = 326 mm): 4.39 5.38 6.58 8.06 kg
anonymousviolin (L = 333 mm): 4.57 5.61 6.87 8.41 kg
1.20 0.89 0.65 mm)
(diametersof the gut strings: These diametersapproximateto those in use at the time: Plessiardlater
specified(p. 209)that'gutE stringsvariedfrom0.40 to 0.55 grams'permetre,
whichcorresponds
to a diameterof 0.63-0.73 mm.FortheG stringhe saidthat
violin makersusedan E stringoverspunwith 16-gaugesilveredcopperwire,
the weightof whichvaried,accordingto the brand,between0.110and0.139
g/m. Sincethe averagedensityof thiswire was, accordingto Plessiard,8880
kg/m3,its diametermusthave been some 0.13-0.14 mm.
Thesefiguresconfirmthat19th-century
Frenchviolinistshadincreasedthe
thicknesses
of theirstringsso thattheyapproximated
morecloselyto thoseused
in the Veneto in the precedingcentury.Indeed the Frenchstringswere
probablyeventhickerby now, if we areto believewhatFetiswrotein 1856on
the basisof informationwhich he had obtainedfromJ.-B. Vuillaume,the
greatestFrenchviolin makerof the day: 'Twentyyears ago the E string
requireda tensionof 22 pounds[i.e. nearly11kg] to get up to pitch,andthe
otherstringsa littleless;thusthe totalweightwassome80 pounds[morethan
39 kg].Thepitchstandard
hadgoneupa semitonesince1734;instruments
were
equippedwith more robuststrings;and the anglewhich they made on the
bridgewas sharper:hence the necessityof changingthe violins'bar.'34The
datafurnishedin 1806by theAbbotSibirealsosuggestconsiderable
diameters.?
In the followingtableI give the tensionsin poundsas Sibirefurnishedthem,
theirconversioninto kilograms(as done by Plessiard)andthe corresponding
diameters(which I have calculatedby assumingthat L = 33 cm and that
a' = 415-435 Hz):
G
D
A
E
lbs:
13
15
17
19
(total= 64)
6.36
7.34
8.32
9.30
kg:
(total= 31.32)
1.45-1.38
1.03-0.98 0.73-0.70)
(mm:
Fetisalsoreported,on thebasisof informationfromVuillaume,that'Tartini
29
foundasa resultof experimentscarriedout in 1734thatthe tensionof the four
amountedto 63 pounds'?6Butasthisdoesnotspecify
stringson hisinstrument
the kindof pound,the distributionof the tensionamongthe fourstrings,and
their soundinglength,one cannotdrawfromit any secureconclusionsas to
their diameters.
One lastobservation:
we haveseenthatat the timeof Brossardthe gut core
of the G stringhadaboutthe samediameterasthe D string,whereasa century
anda halflaterit hadbeen reducedto the thicknessof the E string.Thismust
have favouredthe adoptionof the overspunD string; indeed Plessiard
remarkedthatif the G stringaloneis overspun,thenthe D string'findsitself
betweentwo stringsthatarethinnerthanit and [hence]soundmorebrilliant.
One couldoverspinit like the G string,andthiswouldrenderthe soundof the
violin more homogenous.Variousisolatedattemptsto carryout this change
havealreadybeenmade'?7Inanycasethistendencywascertainlywitnessedin
the precedingcenturyby BrossardandLaborde?8
Someperformerstodayas
well, forexampleSonyaMonosoff,feel it necessaryto useanoverspunD string
even when they are playinga baroque-styleviolin.
APPENDIX3
Di Colco's Rule for Violin Strings
Serafinmo
Riccatiwas not the firstto workout a rule for violin strings:in 1690another
Venetian,SerafinoDi Colco, had tried to correctthe prevailingcustom
wherebyviolinistschosethe stringsfortheirinstruments
by ear.He postulated
a perfectlyuniformdistributionof tension?9Thus,the diametersshouldvary
still more thanin Plessiard'scase:40
Di Colco
Plessiard
Riccati
ratioof tensions
1
0.82
0.67
ratio of diameters
1.355
1.225
3/2
Di Colco'sproposalappearsto be purelytheoretical,ashe doesnotprovideany
experimentaldata.
Translated
byMarkLindley.
NOTES
'Il piacere della musicami ha invogliato di spenderenon poche meditazioni
intorno alle corde solide, e fluide, colle quali essa compone la maggior partede'
suoi stromenti; ed avendo in varj tempi otto schediasmi distesi sopra questa
materia, li ho nuovamente ripigliati per mano, e ritoccati qua e la, onde con
meno di verecondia possano al pubblico presentarsi' (p. iii).
2 'Delle misure, che debbono
assegnarsialle corde d'uno stromento, ed alle
canne d'organo, acciocche rendano suoni del pari forti, ed aggradevoli'.
3
Horace Doursther, Dictionnaireuniverseldespoidset mesures... (Antwerp,
1840; facsimile ed. Amsterdam, 1965), p. 233. There was also a heavier
Venetian pound of 12 x 768 grani (= 477.05 g). Evidence that Doursther's
301.3-gram estimate applies here can be found in Riccati's article, 'Delle
vibrazioni sonore dei cilindri' (Memoriedi matematica
efisicadellaSocietaiitaliana,
1
30
vol. 1, Verona,1782, pp. 515 and 519),which gives the exact weightsand
dimensionsof two cylinders,one of steel and the other of bronze.
in their
4 Thesedensitiesaregivenby DijldaAbbotandEphraimSegerman
in
the
16th
and
17th
centuries'
58.
xxvii,
'Strings
GSJ
(1974),p.
e le spinetteda me esaminati,posteal paragonele
5 'In tuttii gravicembali,
cordegravicolleacute,le ho trovatealquantopii cortedi quelloche richiedela
proporzionedei tempidellelorovibrazioni.Da ci6 derivala conseguenza,che
le cordegravirispettoalle loro grossezzesono un po meno tese delle corde
acute.Io credo,che i praticicosisi adoperino,perchela trafila,percui si fanno
passarele corde,costipie rendatenacipiuile sottilidelle grosse,dimodoche
queste non possanotollerarela tensione di una forza alle loro grossezze
precisamenteproporzionale.Corrono inoltre pericolo le corde gravi di
mentresi attorcigliano
scavezzarsi,
perattaccarleallostromento,e si rivolgono
intornoal perno,col mezzodel qualesi stirano.Per altrosfuggitii descritti
rischj, e ridotte alla dovuta tensione, si conservanomolto tempo senza
spezzarsi.Le cordeacuteall'opposto,duranopoco, e si romponofacilmente.'
(pp. 137-38).
6 'Non e arbitrario
l'armareunostrumentoconcordedi qualunque
grossezza,
ed armatoche sial'usarforzea capriccioperfarsuonaressecorde.L'esperienza
uniformeallateoricahainsegnatoai praticile convenientigrossezzedellecorde
gravied acute,le qualisurrogandoil prossimoall'esatto,ponno stareentro
certidiscretilimiti.Applicateallo strumentodue cordedi congruagrossezza,
restal'altromezzodellaforzadellepenne,perinteramente
il vigore
pareggiare
dei suoni.Toccandogli accordatorinello stessotempodue corde,comprendonoesattamente
o scemarsidi forza,acciocche
qualepennadebbaaccrescersi,
i duesuonigraveed acutofaccianonell'orecchioegualeimpressione.'(p. 139).
GSJ xxiv,
7 MichaelThomas,'Stringgaugesof old Italianharpsichords'
(1971),pp. 72-3 and 78.
8 KennethBakeman,
'Stringingtechniquesof harpsichordbuilders'GSJ
xxvii, (1974),p. 99.
stringsin
9 Accordingto MichaelThomas(note7, p. 71),Germanclavichord
the treblebecameno smallerthangauge7. Friedemann
in
has,
effect,
Hellwig
confirmedthis ('Stringsand stringing:contemporarydocuments',GSJ xxix,
1976, pp. 91-104). There were exceptions, however. An 18th-century
annotationwhichI havecome acrosson the next-to-lastpageof a
manuscript
volumecontainingtwo worksof AndreaWerckmeister(Paris,Bibliotheque
Nationale,MusRes 1747)prescribesthe followinggaugesfor an unfretted
clavichordof rangeC-c'" (withoutCO):
C DO F
GO c
d'
d" a"-c"'
f
g'
0
1
2
3
4
5
6
7
9-9
8
20-22.
Here
are
the
in
data
the
(the lengths Parisian, weights in
•0 Pp.
Venetianunits):390 'lines'of brassstringweighed7.5 grani(1 ounce= 576
sucha string(thediameterwas0.233mm)was'stretched
firstby a weightof
grani);
two pounds;whentheweightwasincreasedby fourpounds(to six),the string
became13/s'lines'longer,andas muchagainby addinganotherfourpounds.
31
"
Accordingto Sauveur,stringsof steel, brassand copperbroke when
stretchedby weightsamountingto 12000,9000 and5000 times,respectively,
the weight of 40 inches(of the Parisianfoot) of the string.
12 See my 'I1lcoristabolognese,secondoil rilevamentodi V. F. Stancari'
xviii, 1980,p. 25.
L'Organo
13 W. R. ThomasandJ. J. K. Rhodes,'Thestringscalesof Italiankeyboard
GSJ xx, (1967),p. 52; Abbottand Segerman,op.cit.,p. 61.
instruments'
Mahrenholz,Thecalculation
14 Christhard
of organpipescales... (Oxford,
64-66.
1975),pp.
Vallottito Riccati(Udine,
15A letterofJuly 2, 1753,fromFrancescantonio
BibliotecaComunale,MS. 1027)impliesthatMarcuzziwasby thentoo old to
playthe organ- whichin turnsuggeststhatRiccati'saccountwaswrittena
good manyyearsbefore its publication.
16 'Misuraidiligentissimamente
coll'ajutodel Signor LiberaleMarcuzzi
valoroso organistadi questa Cattedralele circonferenzeproporzionaliai
in tripla
diametridi due canne,che suonavanoF fa ut, e si corrispondevano
ottava.L'organodella nominataChiesae un'operaassaiperfettalavoratada
Urbano da Venezia l'anno 1420.' (p. 143). I believe Riccati'shistorical
informationwas inexact.RenatoLunelli,Studie documenti
di storiaorganaria
veneta(Florence,1973),gives the followingaccountof the cathedralorganat
Treviso:in 1436Niccolo d'Allemagnagave the churcha new organ(p. 212);
AntonioDilmani,the sonof the organ-maker
Bernardod'Allemagna,
rebuiltit
ex novoin 1481-83, and carriedout furtherwork between 1493 and 1500
(p. 180);in 1773the instrument,afternumerousrestorations,was rebuiltby
GaetanoCallido(p. 135).Lunellialsosays(pp.227-28) thatUrbanoof Venice
diedbefore1518.It appearsthatRiccatimeasuredpipesforwhichDilmaniwas
most likely responsible.
17 Marenholz,
op.cit.,pp. 41, 66 and69. The samescalingappearsto have
been used by the T6pfer'sRomancontemporary,FrancescoPasquetti,who
probablywould not have known Topfer'streatise;see my 'Rilievi tecnici
xvi,
sull'organo"Franciscus
umbri,
PasquettiRomaefecitA.D.1836"', Quaderni
2 (Terni,March1983),p. 12.
18 'Colle bilancettedell'oro
pesaitre porzioniegualmentelunghepiedi 11/2
Venezianidelle tre cordedel Violino,che si chiamanoil tenore,il cantoe il
cantino.Tralasciai
il pesodellacordapiuigrave;perchequestanone
d'indagare
comel'altredi solaminugia,masuolecircondarsi
conun sottilfilo di rame.'(p.
130).
19GianrinaldoCarli,Delle Opere(Milan,1786),vol. xiv, pp. 338-343.
20 The New Grove,s.v. 'Violin',
p. 828. I imagine Boyden drew this
conclusionfromFran:oisRaguenet'sComparison
between
French
andItalianmusic
(1702);see aproposBoyden'sThehistory
of violinplaying
fromitsoriginsto 1761
(London,1965),p. 203.
21 For some
exceptionalviolin gaugenumberssee note 35.
22
'Methodesuiviea Naplespourla fabrication
descordesde violon.... on
ne met que deux boyeauxensemblepourles petitescordesdes mandolines,
troispourla premierecordede violon,septpourla derniere.'Journal
demusique
32
(Paris,1777; facsimileed. Geneva, 1972),No. 1,
par une societed'amateurs
pp. 11-12.
de boyeau,elle doit estredu moinsle doubleplus
23 'Si elle est simplement
grosseque la 3e, maissi elle est toutefilee d'argentelle n'estquetrespeuplus
grosseque la 3e.' Paris,BibliothequeNationale,MusRes Vm8c 1, fol. 12r.
24 See note 16.
25 ,... dalchesi potrebbono
renderedel tuttoinutili,se nonfossecheancora
a questopareche si possarimediare,con accordaretuttoil sistemadelle corde
un quartodi tuono in circapiu acutodel sistemadelle canne,che cosi poi
che fossel'ariadalletorce,col caloreverrebbeallasuagiustavoce;e
riscaldata,
se le corde acute, o per essere d'acciaio,o per avere mancomaterianon
calasserotanto, allora sarebbenecessarioritoccaremolti tasti, per ridurli
interamenteal suo sesto.' GiovanniBattistaDoni, 'Trattatodella musica
scenica',Opere,ii (Florence,1763),p. 112.Doni diedin 1647andthe 'Trattato'
was writtenbefore 1642.
de nozfaiseursde clavecins;lesquelsmettentd'ordinaire
aux
26 ,. .. pratique
dernieresvoix en bas quelquechordesde cuivrerouge,pourceque celles de
letonn'ontpasla mesureassezlonguepourrendrele ton requis.'Cornelisde
duP. MarinMersenne,
vol. viii (Paris,1963),p. 20.
Waard,ed., Correspondance
Theuseof brassin thebassof Romanharpsichords
atthattimeis alsoconfirmed
by Pier FrancescoValentini:in one of his manuscripts
(c. 1642-45) he gave
instructions
for makinga monochordwith a brassstringsix Romanpalmilong,
and remarked,'qual corda sari della grossezzadei bassi di un cembalo
et novacostitutione
di musica..., p. 23 - Biblioteca
ordinario'(Monocordo
ApostolicaVaticana,MS. Barberinilat. 4430).
27 RobertSmith,Harmonics...(London,1759),pp. 192-93:35.55 inchesof
wire weighed31 grainstroy (1 pound= 7000 grainstroy).
copper
28
Bakeman,note 8, p. 99.
29 Op. cit.,p. 103.
30 CarloPellegrini,Museum
i (Rome,1665),p. 15.
historico-legale
bipartitum,
31 'Spinetta
est instrumentunnotissimum,quodclauicymbalorum
genusest,
totidemchordiscompositum,ac cymbalum,at nonex oricalcho,vt in cymbalo
sit,sedex aere,sei calybe,sonumedit,nontim sonorum,vt cymbalum,cumsit
alteriusfiguraeminoris.& aliquandopannorubeo chordaeoccupantur,ad
effectum,vt submissiussonumedat.' Op. cit.,pp. 26-27.
'Des cordesdu violon',Association
32 Plessiard,
des
franCaise
pourl'avancement
sciences.
deLille,1874,2e Groupe(Sciences
etchimiques),
'Seance
Congres
physiques
du 21 aofit 1874,pp. 192-220. I have not been able to find Plessiard'sfirst
name; he is referredto as 'Ingenieuren chef des Ponts et Chausseesen
retraite'.
33
que la rapidited'executionsur le violon, que le frequentpassage
,...
d'une cordea une autreet le jeu sur doublecorde exigentque l'archetne
rencontrepasplusde resistancesurune cordeque surune autre,c'est-a-dire
que le travailsoit le memepourobtenirle son des quatrecordes.Pourcela le
doit treegala laracinecarrie
rapportdestensionsdedeuxcordescons6cutives
de leurintervallemusicalou / 2, puisquepourle violoncet intervalleest
33
unequinte.'Op.cit.,p. 196.Plessiard
seemsnotto haveknownof Riccati'swork.
He foundhisruleby meansof a formulafortheworkQ of thebow:Q = kmtf,
wherek is a constantand m, t and f representthe string'smass,tensionand
vibrationfrequency.
a sonintonationqu'avecun poids
n'arrivait
3 'I1y a vingtans,la chanterelle
de 22 livres,et les autrescordesun peu moins;la chargeetaitdoncd'environ
80 livres.Le diapasons'etaiteleve d'undemi-tondepuis1734;on montaitles
instruments
en cordesplusfortes,et l'anglequ'ellesfasaientsurle chevaletetait
Fetis,
plus aigu: de 1Ula necessitede rebarrerles violons.' Francois-Joseph
AntoineStradivari
luthiercelbre...(Paris,1856),p. 92. Theprefaceexplainsthat
theworkis basedon investigations
whichVuillaumehadcarriedout (inItalyas
well as France):Vuillaumealso editedthe book. Fetis employedthe french
pound(= 489.5 grams:H. Doursther,note 2, p. 228).
ouleparfaitluthier(Paris,1806),pp.
Sibire,Lachelonomie
3 Sebastien-Andre
112-13.(Theviolinwas tuned'auton de la flute'.)Accordingto Plessiard(op.
cit.,p. 213),Sibire'sworkwas basedupon'notesof the luthierLupot'.Albert
Cohen,'A cacheof 18th-century
strings'(GSJxxxvi,pp. 37-48), describesa
number3 gauge'Chanterellede Violonp. le C. en Gosseque'(i.e. FrancoisJosephGossec)of whichthe diameter,accordingto Cohen,is 0.70 mm. This
matchesthe evidencefrom Sibireand from Riccati.
36 'Tartinia trouve,par des experiencesfaitesen 1734,que la chargedes
quatrecordessur l'instrumentegalait63 livres.'Op. cit.,p. 92.
37 ,... se trouvealorsentredeuxcordesplusmincesqu'elleet qui sonnent
avecplusd'dclat.On pourraitla filercommela quatrieme,quirendraitles sons
duviolonplushomogenes.Diversessaisisolesde cettesubstitution
ontdej' ete
faits.'Op. cit.,p. 197.
38 Brossard,
loc.cit.:'if [theD string]is merelyof gutit mustbe atleasttwice
asthickastheA string,butif it is half-overspun
with silver,asit almostalways
the casenowadays,it is no thickerthantheA string'.RegardingLaborde(Essai
surla musique,
1780),see EduardMelkus,II violino(Florence,1975),p. 27.
da
'Siano
adunViolinole corde,comesi vedenellafigura.
proporzionarsi
39
Iconismo
2. figura2. A.b.c.d. distese, e distirateda pesi vguali E.f.g.h. Se
toccandole,6 suonandolecon l'arco formerannovn Violino benissimo
accordato,sarannobene proporzionate,altrimenticonuerramutarletante
riescadi quintadue,perdue,che appuntotale
volte,sintantochel'accordatura
e l'accordatura
del Violino.' SerafinoDi Colco, 'Letteraprima(Venezia,7
Gennaro1690)',Le vegghie
di MinervanellaAccademia
deFilareti:Per il mesedi
Gennaro1690(Venezia,1690),pp. 32-33. In the sameletterhe describesthe
'Modernaaccordatura
di Clauicimbalo'(1/4-comma
fromEbup
temperament,
to G#).
ratiois the geometricmeanbetweenthose of Di Colco and
40 Plessiard's
Riccati.
34