Instruments of Music TheoryThe Harvard community has made this

article openly available. Please share howthis access benefits you. Your story matters

Citation Rehding, Alexander. 2016. "Instruments of Music Theory." MusicTheory Online 22, no. 4.

Published Version http://mtosmt.org/issues/mto.16.22.4/mto.16.22.4.rehding.html

Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:34391750

Terms of Use This article was downloaded from Harvard University’s DASHrepository, and is made available under the terms and conditionsapplicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA

1

AUTHOR:Rehding,Alexander

TITLE:InstrumentsofMusicTheory

KEYWORDS:HistoryofMusicTheory,CriticalOrganology,SoundStudies,Acoustics,EpistemicThing,Pythagoras,Gaffurius,Vicentino,Cowell,monochord,archicembalo,siren,rhythmicon.ABSTRACT:Thisarticleexploresmusicalinstrumentsasasourceforthehistoricalstudyofmusictheory.ThefigureofPythagorasandhisallegedpenchantforthemonochordoffersawayintothisexplorationofthetheory-bearingdimensionsofinstruments.

Musicianstendtothinkofinstrumentsprimarilyintermsofmusic-making,butinothercontextsinstrumentsare,morebroadly,tools.Inthecontextofscientificexperimentation,specifically,instrumentshelpresearcherscometotermswith“epistemicthings”—objectsunderscrutinythatcarryspecific(butasyetunknown)sourcesofknowledgewithinthem.Aspectsofthisexperimentalpracticecanproductivelybetransferredtothestudyofmusictheoryandareexploredinatwotestcasesfromdifferentperiodsofmusicaltheorizing(andinstrumentbuilding):NicolaVicentino’sarchicembalofrommid-sixteenthcenturyItaly,andHenryCowell’srhythmiconfromtheearly-twentiethcenturyAmerica.

AUTHOR AlexanderRehdingHarvardUniversityDepartmentofMusicCambridgeMA02138arehding@fas.harvard.edu

ACCOMPANYINGFILES:10jpgimages(9Figures)havebeenincorporatedinthistext.2videofilesareaddedtothetextashyperlinks.2soundfiles(tobeembedded)willbeaddedtothefinalversion.

2

InstrumentsofMusicTheory

1.Pythagorasmusicus

[1.1]WithhisemblematicmonochordPythagorasruledmusicandthecosmos.1As

itsnamesuggests,themonochord(monos=single,chordê=string)isaverysimple

instrument,consistingoflittlemorethanastringstretchedoutfromonesideofa

woodenplanktotheothersoitcanfreelyvibrate.Thelengthofthestringcanbe

dividedbymeansofamoveablebridgetoeffectchangesinpitch.Inthecollective

imagination,PythogorasofSamos,themythicalphilosopher,mathematician,and

founderofareligiouscult,whomay(ormaynot)havewalkedtheearthfromc.

570–c.495BCE,hadlongbeenassociatedwiththisinstrumentandwasoftenhailed

asitsinventor.2Thisconnectionbetweenthephilosopherandhismonochordonly

grewstrongerovertime,somuchsothatthelateRomanstatesman,philosopher,

andmusicalthinkerCassiodorus(c.485–c.585CE)awardedtheGreekphilosopher

thehonorific“Pythagorasmusicus.”3Despitehisundeniablystrongaffinitieswith

arithmetic,Pythagoraswasparticularlyassociatedwithmusicthroughoutthe

MiddleAgesandintotheearlymodernperiod,inwaysthatparalleltheastronomer

Ptolemywithhisemblematicplanetaryrulers,orthegeometerEuclidwithhis

compass.1AnearlierversionofthisarticlewaspresentedasthePeterLeHuraylectureatthe50thRMAconferenceinLeedson4September2014.SpecialthanksgotoJosephAuner,BevilConway,EvanMcCarthy,AlexanderNikolaev,StevenRings,DanielWalden,andtheanonymousreviewerswhosecarefulreadingofthisarticlehasgivenmetheopportunitytosharpenandclarifymypoints.2TheiconographysurroundingPythagorasandthemonochordisexploredinBarbaraMünxelhaus,Pythagorasmusicus(Bonn:VerlagfürsystematischeMusikwissenschaft,1976).3Cassiodorus,Variaeepistolae,1.45.4.

3

Fig.1.Mythicalacousticexperimentsonavarietyofinstruments.FranchinusGaffurius,Theoricamusicae(1492).

4

[1.2]Thusitisnottoosurprisingtoseeadetailedfour-panelwoodcutgiving

prideofplacetoamusic-makingPythagorasinFig.1.Thisimageappearsinthe

Theoricamusicae(1492)bytheNorthItalianhumanistFranchinusGaffurius(1451–

1522),oneofthefirstmusictheorybookstoappearinprint.4Whatwesee,

however,takessometimetodecode.IfwedidnotknowthatPythagoras’emblem

wasthemonochord,itwouldbehardtorecognizethemusicalinstrumentinthe

bottomleftpanel.Thesix-stringinstrument,whichPythagorasplayswithtwo

sticks,resemblesmoreazither-typeinstrument,apsalteriumorahammered

dulcimer,thanthetraditionalancientinstrument.Isthisreallystillamonochord?5

Atamoredetailedlevel,thePythagoreanimplicationsoftheimage,however,are

clearbeyondanydoubt:theweightshangingdownontheleftsideoftheinstrument

followaseriesofnumbersthatcanberecognizedasPythagoreanwithoutproblems:

4–6–8–9–12–16.Itseemsthesix-stringmonochordrepresentsthetechnological

andlogicalextensionoftheprinciplethatPythagorasdiscoveredinancienttimes,

accordingtolegend.

[1.3]Weremember,thelegendinquestion—aprimalsceneofmusic—

originatedwiththeGreekphilosopherNicomachusofGerasa(c.60–c.120CE),and

hasPythagorasliftthesecretofsoundserendipitously,ashewalkspastasmithy

andhearstheharmoniousclangingoffourhammersonanvils,weighing6,8,9,and

4FranchinusGaffurius,Theoricamusicae(Milan:FilippoMantegazzaforG.P.daLomazzo,1492),tr.byWalterKurtKreyszig,TheoryofMusic(NewHaven:YaleUniversityPress,1993).5BytheRenaissance,awidevarietyofinstrumentscouldbecalled“monochord,”irrespectiveoftheactualnumberofstrings,includingevenkeyboards.SeetheletterscollectedbyBonnieBlackburnandEdwardLowinsky,eds.,ACorrespondenceofRenaissanceMusicians(Oxford:Clarendon,1991).

5

12unitsofweight.6Thisstory,whichdeterminestheratiosofconsonantintervalsof

theoctave,fifth,andfourthat2:1(=12:6),3:2(=12:8and9:6),and4:3(=12:9and

8:6),wastransmittedthroughouttheMiddleAgesandwellintothemid-sixteenth

century,primarilyinBoethius’sinfluentialvariant,andneednotbefurther

rehearsedhere.7ForallthenumericalbeautyofthePythagoreanratios,weknow

thattheaccountisapocryphal,astheunderlyingphysicsarenotwatertight:the

relationshipbetweentheweightofahammerandthepitchitproducesthatis

suggestedbytheillustrationdoesnothold.Itisnowwell-known,too,thatthestory

inthesmithyneverhappenedinthisway.ItisveryunlikelythatPythagoras—or

anyoneintheancientworld—evercaredtocarryoutthisexperiment.Noristhere

anyrealreasontoassumetheyshouldhavedone:Pythagoreanismisbest

understoodasareligion,notasanempiricalscience.8Infact,itwouldtakeuntilthe

sixteenthcenturyforsomeoneto“fact-check”thisstoryandtocorrectthephysical

basisoftheclaim:VincenzoGalilei(1520–1591),fatherofGalileo,firstconducted

acousticalexperimentstotestthePythagoreanclaimsthathadbeenperpetuated,

unchallenged,formorethanamillennium.Forinstigatingthisgroundbreaking6AndrewBarker,GreekMusicalWritings:Volume2.HarmonicandAcousticTheory(Cambridge:CambridgeUniversityPress,1989),256-8.7SeeBoethius,FundamentalsofMusic,tr.CalvinBower(NewHaven:YaleUniversityPress,1989),17-19(Bk1.10-11).DanielHeller-RoazenhasrecentlytakenupthemedievalPythagorasmyth,inTheFifthHammer:PythagorasandtheDisharmonyoftheWorld(Cambridge,MA:MITPress,2011).Gaffurius’understandingofancienttheorywasclearlychanneledbyBoethius,notearliersources.HeincludesaparaphraseofBoethiusinBk1.8:4-16(Kreyszig,TheoryofMusic,46),includingtheadditionalhammer.8AsAndréBarberasuggests,Pythagoras’standingwassuchthattheratiosnotworkinginpracticemayevenhaveheightenedthemystiquesurroundinghim:“Afterall,theexperimentsdidworkwhenheperformedthem.”Seehis“TheConsonantEleventhandtheExpansionoftheMusicalTetractys:AStudyofAncientPythagoreanism,”JournalofMusicTheory1(1984),200.

6

changeinscientificthoughtandpavingthewayforaturntoempiricism,Galilei

senior,notjunior,hasbeenhailedbysomescholarsasthefirstmodernexperimental

scientist.9

[1.4]Pythagoreans’primaryinterest,asfaraswecanascertainfromthe

historicalrecord,wasnotinmusicassuch,butinthatwhichliesbeyondthe

soundingphenomena.Aristotlewrote,derisively,thatthePythagoreansunderstood

eventheheavensintermsofmusicalscales[harmoniai]andnumbers.10Butthisis

nomereempty,dismissivehyperbole:Pythagoreanswereprimarilyinterestedin

musicinsofarasitwasanexpressionofthenumericalrelationsthatheldthe

cosmostogether.Musicwasfortheminextricablyconnectedtoastronomy,via

arithmeticandgeometry—thesubjectsthatweretobejoinedinthemedieval

quadrivium.Thesoundsandtheexperienceofmusic,aslaterancientcommentators

suchasPtolemynevergottiredofcomplaining,weresecondarytoPythagoreans;

soundingmusiconlygotinthewayoftheperfectionandbeautyofratios.11The

attributionofthemonochordtoPythagoras,asitsinventor,didnotcometothefore

untilthemuchlateraccountsbyDiogenesLaertius(3rdcenturyCE),Gaudentius

9ThereisasizeablebodyofliteraturesurroundingthequestionofwhetherVincenzoGalileishouldcountasanexperimentalscientistornot.SeeStillmanDrake,“RenaissanceMusicandExperimentalScience,”JournaloftheHistoryofIdeas31(1970),483-500,D.P.Walker,StudiesinMusicalScienceintheLateRenaissance(Leiden:Brill,1978),14-33,H.F.Cohen,QuantifyingMusic:TheScienceofMusicastheFirstStageoftheScientificRevolution1580–1650(Dordrecht:Reidel,1984),andClaudePalisca,“ScientificEmpiricisminMusicalThought”inStudiesintheHistoryofItalianMusicandMusicTheory(Oxford:Clarendon,1994).10Aristotle,Metaphysics1.986a.11Ptolemy,HarmonicsBk.1,Ch.5(Barker,GreekMusicalWritingsII:284-6).Tobesure,thisshouldbetakenwithagrainofsalt.Aswitheverythinginthisdiscussion,manyofthepositionsrelyonlatertestimoniesthatareprojectedbackwards.

7

(prob.4thcenturyCE)andBoethius(480-524/5CE).12BythetimeGaffuriuspicksup

thetopic,inthecontextoffifteenth-centuryhumanism,hecanmakeanelegantpun:

Inthisway,Pythagorasdiscoveredthemonochordrule[regula],whichgetsits

name“rule”fromtheobjectnotbecausearuleismadeofwoodorbrassandwe

measuresoundsandmagnitudeswithitbutbecausearuleisacertainfixedand

stableobservationthatleavesthejudgmentinnodoubt.Itissonamedfrom

“ruling”[regendo],asifitruledusinsuchawaythatwecannotfallintoerror.13

Themonochordisalsoknownaskanônorregula(whichmeansruleorrulerin

GreekandLatin),Gaffurius’statement“regulaminvenit”canmeaneither:he“found

therule”orhe“inventedthemonochord.”Bothversionswouldbeequallytrue.

[1.5]Returningtotheinitialimage,whatkindofinstrumentisPythagoras’six-

stringmonochord?Whilethepresenceofmultiplestringswouldnotinitselfhave

beenasstrikingatthetime,wearenowinapositiontodigalittledeeperwith

regardtothenumbersix.WhatappearsunusualinGaffurius’versionisthatthe

numberofstringsonwhichPythagorasplaysisexpanded.Butingoingbeyondthe

numberfouroftheoriginallegend,Gaffuriusweavesaninterestingnewtwistinto

thestory.Byaddingthe4andthe16,heexpandstherangeoftheinstrumentto2

octaves(8:4,16:8),andconsiderablyexpandsthepossibilitiesofPythagorean

intervalsbetweenpairsofstrings,whicharelaidoutinthetablebelow.(Theratios

12SeeDavidCreese,TheMonochordinAncientGreekHarmonicScience(Cambridge:CambridgeUniversityPress,2011),90.13Gaffurius,Theoricamusicae,Bk.1,Ch.8:34-35.(Kreyszig,TheoryofMusic,48.)

8

insquarebracketsmarkcompoundvariantsofthesimpleintervals.)

Octave 8:4 12:6 16:8 [16:4]Fifth 6:4 9:6 12:8 [12:4]Fourth 8:6 12:9 16:12 [16:6]Whole-tone14 9:8 [16:9] [9:4]

[1.6]OnwhatauthoritydidGaffuriusmakethesechanges?Gaffuriusoffersa

disarminglysimpleexplanationinthetextofhistreatisewhyhebelievedthat

Pythagorashadextendedtherangeofhismonochordtosixnumbers.Tounderstand

thisbetter,wehavetogobacktoBoethius’powerfulretellingofthestory,onwhich

Gaffuriusbasedhisowntheorizing.InBoethius’variantofthesmithymyth,

Pythagorasencountersnotfour,butfivehammers.Thisfifthhammerisdissonant,

Boethiusexplains,inwaysthatcannotbereconciledwithPythagoreanprinciples,

andPythagorasquicklydiscardsit.Inthisway,byaddinganewhammertothe

storyandimmediatelydiscardingitagain,Boethiusfurtherunderlinesthenecessity

forperfectnumbersandforeliminatinganyelementthatmightdisturbtheir

harmony.Boethius’srhetoricalflourishseemstohighlight,aboveall,thatexactly

fourhammersareneeded,notmoreorless.ButGaffuriusturnsthisrhetorical

maneuveragainstitselfbymakingexplicitreferencetothe“fifth”hammer.15Onthe

basisofBoethius’influentialvariant,Gaffuriusconcludes:

WemaysupposethatPythagorashimselfentirelychangedtheinconsonantfifth1416:9isanoutlier,asitdescribesnotthewhole-tonebuttheminorseventh,whichIhaveincludedhereforthesakeofcompleteness.Gaffuriusdoesnotcommentonthisscenario.15Tobesure,GaffuriusaddsfurtherlateRomansources,notablyCalcidius’CommentaryonPlato’sTimaeus(c.321CE)andMacrobius’DreamofScipio(early5thcentury).BoththeseauthorsallowformorethantheorthodoxPythagoreanratios,thoughneitheroffersadirectmodelforGaffurius’expansion.

9

hammerintoanotherandaddedasixthabovethefifth,whichhasbeen

establishedinthenumber4,whichwouldsurpassallothersinsmallnessof

weightandalsoasixthhammer,whichwouldexceedalltheothers,thatis,inthe

number16.16

ToparaphraseGaffurius:if,asBoethiusrelates,Pythagorastemporarilyexceeded

thenumberfour,evenonatrialbasis,hemustsurelyhaverealizedthataddingtwo

morehammerswillgreatlyincreasethenumberofpossibleintervals.Gaffurius

blithelyoverlooksanynewcomplicationsthathisadditionsintroduce;heseems

mainlyexcitedabouttheextendedrangeofintervalsthatafifthandsixthhammer

wouldallow.

[1.7]Nolongerdoesthefifthhammerfunctionasamarkerofanouterboundary

thatcannotbeexceeded,asitdidforBoethius,butratheritbecomesthebasisof

furtherexperimentation.Whatismore,Gaffuriusmanagestogivethisoldstoryits

newtwistwithouteversomuchasquestioningBoethius’supremeauthoritybut

ratherbuildingonit:GaffuriuseffectivelyarguesthatifPythagorasheardafifth

hammerandrealizeditwasnotquiteright,hewillhaveappreciatedthatwhathe

neededforgreaterperfectionwasnotonlyafifthbutalsoasixthhammer.“Inthis

manner,”GaffuriussumsuphisexcursionandexpansionofthePythagorasmyth,

“hefoundoutthatwhenthesedifferentweightswerelaidout,allthemusical

consonanceswerecontainedsolelyinthemultipleandinthesuperparticularratios;

andtheheavierhammerssoundedagainstthehigherinacertainmutually

16Gaffurius,Theoricamusicae,Bk.1,Ch.8:20.(Kreyszig,TheoryofMusic,47.)

10

correspondingorder,sothatbothreturnedtothesenseofhearingwithan

establisheddimensionbypresentingoutofthedifferentblowsasingleconsonant

sound.”17InGaffurius’shands,Pythagorasrepresentsnothingsomuchasthespirit

ofperfectibilityintherealmofnumbers.Gaffurius’sPythagoras,inaword,offersan

improvedandperfectedversionofthecosmos.

[1.8]GaffuriusemphasizesthewideapplicabilityofthisuniversalPythagorean

principle.IntheremainingpanelsofFig.1heshowstheGreekphilosopherina

numberofdifferentmusicalsituations,whichallcloselyreflectBoethius’sretelling

ofthemyth.InthebottomrightcornerPythagorasisjoinedbyhisdisciple,the

philosopherandmathematicianPhilolaus,inplayinganumberofpipes.Evengiven

thescantreliablebiographicalinformationwehaveaboutthesetwofigures,it

seemsveryunlikelythatPhilolaus(c.470–c.395BCE)wouldhaveoverlappedwith

Pythagoras,butinthiscontextweprobablydonotneedtolosemuchsleepover

suchdetails.Thelengthsofthepipesareinthesameproportionsashissix-fold

monochord.Theimagesareveryexplicitabouttheratios,allunitscarrynumbers.In

thetoprightcornerofthewoodprinttwoPythagoreanfigurescanbeseenstriking

bellsandtappingglassesofwater,accordingtothesameproportions.ClaudePalisca

haspointedoutthattheseimages,despitethedisplayofmathematicalrigor,are

mostlyphysicallyfalse:infact,theintervalsformedbythebottlesandbellsarenot

governedbytheratiosoftheirdimensionsinthesimplewaysuggestedbythe

numbersinthewoodcut,justastheweightofthehammerswasfoundtobe

unrelatedtotheresultingpitches.Onlythepipesactuallycorrespondtoacoustical

17Ibid.,Bk.1,Ch.8:26.

11

reality.18

[1.9]Finally,themoststrikingpanelofthewoodcutisthetopleftimage.Thisis

probablythefirstpaneltheviewerwilllookat,itthereforecarriesparticular

rhetoricalweight.Weseethefamoussceneintheblacksmith’sworkshop,the

hammerscarrythesamenumbers.Butthepersonlisteningtotheanvilsbeing

struckisnottheancientGreekfigureofPythagoras,butthebiblicalfigureofJubal,

the“ancestorofallwhoplaytheharpandflute,”19andwhoisoftenregardedasthe

inventorofmusicinbiblicalchronology.

[1.10]Thisdisplacementmightseemsurprising,particularlysincethe

accompanyingtext,BookIofTheoricamusicae,isprimarilyconcernedwith

Pythagoras,andonlybringsinJubalfleetinglyattheveryendofthisdiscussion,

almostasanafterthought.Itisnotuntilmuchlaterinthetreatise,inBookV,that

GaffuriuscomesclearaboutJubal’sfoundationalroleinhistext.20Tobesure,there

isnoprecedentfoundintheever-authoritativeBoethius.Butitispossibleto

reconstructthereasonsforthischoice.Thequestionofprimiinventores,orprôtoi

heuretaíastheywerecalledinGreek,the“firstinventors”oftheancientworld,

cametotheforeagainduringthehumanistrevivalduringthefifteenthcenturyasa

centralquestion.Ancientculturaltechniques—suchasfire-making,agriculture,

18SeeClaudePalisca,HumanisminItalianRenaissanceMusicalThought(NewHaven:YaleUniversityPress,1985),229.19Genesis4:21.TheconnectiontothePythagorasinthesmithyisnotamillionmilesaway:Jubal’shalf-brotherTubal-Cainis“theforgerofallinstrumentsofbronzeandiron.”SeeJamesMcKinnon,“JubalvelPythagoras:Quidsitinventormusicae?”inMusicalQuarterly64/1(1978),1-23.20SeeGaffurius,Theoricamusicae,Bk.5,Ch.1:4(Kreyszig,TheoryofMusic,144.)NotethatGaffurius’textofBk1,Ch.8:4(Kreyszig,46),confusingly,assignsthesmithystorytoPythagoras,evenwherethepicturedoesnot.

12

ship-building,bronze-casting,andmusic-making—hadtobeexplainedbycreatinga

firmassociationwithafigure—mythical,divine,orheroic—whocouldcountasits

firstinventor.21Inthecontextofhumanism,however,suchquestionsofprimacy

becameacontestedspacebetweenontheonehandancientsources,thatistosay:

paganmyths,andontheotherbiblicalauthority.ByoptingforJubalastheultimate

inventorofmusic,Gaffuriusmanagedtopassthecrowntobiblicalauthority,even

thoughthereisnothinginthetexttopreparethereaderforthismaneuver.22

[1.11]Butthisdisplacementalsogivesrisetoanotherpossibleconclusion,

whichispossiblymoretroubling.ThewoodcutinGaffurius’Theoricamusicae

visuallyseparatesofftheprimalsceneinthenoticeablydisorderlysmithyfromthe

otherscenes.Jubalislookingon,quiteliterallyoverseeingthesixblacksmithsas

theyhammeraway.Meanwhile,themultiplePythagoraiinFig.1areseentoexplore

themorestrictlymusicalimplicationsofthediscoveryofthemathematicalsecrets

ofsound.Pythagorasisactivelyengaged,playingthebells,akindofglass

harmonica,theflutes,andthesix-stringmonochord.WhereJubalexaminesnoise,

Pythagorasperformsmusic.

[1.12]WeknowthatGaffuriusfollowedBoethius’leadinvaluingthemusicus,the

scholarlymusician,overthemerecantor,whomerelyperformsmusic,bysingingor

21SeeAdolfKleingünther,ΠρῶτοςΕὑρετής:UntersuchungenzurGeschichteeinerFragestellung(Leipzig:Dieterich’scheVerlagsbuchhandlung,1933).SeealsoCreese,Monochord,85.Theconceptof“culturaltechniques”hasrecentlybeenforegroundedbyGermanmediatheoristssurroundingSybilleKrämerandBernhardSiegert.SeeGeoffreyWinthrop-Young,“CulturalTechniques:PreliminaryRemarks,”Theory,Culture&Society30/6(2013),3-19.22GaffuriusisnotthefirsttomakethisconnectiontoJubal.ThistraditioncanbetracedbacktotheRomanhistorianJosephus.SeePalisca,HumanisminRenaissanceMusic,227,andMünxelhaus,Pythagorasmusicus,46-50.

13

playing,withoutreflectingonit.23Indrawingsuchaclearvisualdistinctionbetween

JubalandPythagoras,andbetweenthescienceofsoundontheonehandandthe

practiceofmusicontheother,theimagefromGaffurius’Theoricamusicaeactually

presentsuswithaproblem:themusic-makingPythagoraiofFig.1appearmoreas

cantores,thatis,theywouldseemtofallwrongsideofthescholar/performerdivide.

Fromthisperspective,thezither-likesix-stringmonochordwithwhichwestarted,

andwhichisrathermoreversatilethanthetraditionalone-stringedversion,would

servetounderlinethisimpression.Thistechnologicallyimprovedinstrumentseems

toallowPythagorastoperformmusic,andheseemstorelishhistaskasamusical

virtuoso.HasPythagorasmusicusreallybecomePythagorascantor?Hashebeen

degradedtoamereperformer?Oristhisallamistake?

2.TheMonochordasInstrumentandSystem

[2.1]HowfarcanwetakethisideaofPythagorasasapracticingmusician,against

anyBoethianadmonitions?Weknowthemonochordwasusedasameasuring

deviceinancientGreece,butwasitinfactusedasamusicalinstrument?Theidea

itself,strangeasitseems,isnotwithoutprecedent.Anearly-twentieth-century

studentofclassicalphilology,SigfridWantzloeben,advancedthethesisinhis

Germandoctoraldissertation,DasMonochordalsInstrumentundalsSystem(1911),

thatPythagoraswasactuallyaperformerofthemonochord,whichheregardedasa

23Boethius,Deinstitutionemusica,Bk.1,Ch.34,tr.CalvinM.Bower,FoundationsofMusic(NewHaven:YaleUniversityPress,1989),50-51.

14

musicalinstrument.24

[2.2]Wantzloeben’sExhibitAispreciselythepassagefromNicomachusthat

beganthePythagoreanmyth:Pythagorasrusheshomefromthesmithytotesthis

newlyfoundknowledgeaboutmusiconanumberofdevices:“cymbals[krousin25],

flutes[aulous],panpipes[syringas],monochords,triangularharps[trigona],and

otherslikethem.”26WantzloebennotesthatNicomachusmentionsthemonochord

amongalistofotherinstrumentsthatcouldbeusedinmusicalperformance,and

concludesthatthemonochordmustitselfhavebeenaninstrumentonwhichmusic

wasperformed.27(Likemanyothercommentators,Wantzloebenisrelatively

indifferenttothecomplicatingfactorthatNicomachuswrotethissomesixcenturies

afterPythagoras.)OtherpiecesofevidenceWantzloebenoffersarehighly

questionable;hisinterpretationsoftenhingeondistinctlydubioustranslations.28

24SigfridWantzloeben,DasMonochordalsInstrumentundalsSystem(HalleanderSaale:EhrhardtKarras,1911).Helaterallbutadmitsthathisdocumentaryevidenceisratherthin.25Itisunclearwhatexactlythiswordshouldmeaninthiscontext.IntheirauthoritativedictionaryLiddellandScotttranslatekrousisas“thetappingofearthenvessels,toseewhethertheyringsound.”AndrewBarkerrendersthis,somewhatcautiously,as“beatenpots,”whereasWantzloeben—whotendstothrowinterpretivecautiontothewind—suggestscymbals.Giventhiscontext,Ifollowthosewhouseaslightlyboldertranslationthatforegroundsthemusicalaspects,thoughroomfordoubtremains.26Barker,GreekMusicalWritings,II:258(tr.modified).27Wantzloeben,Monochord,2.28ExibitBisafamouspassagefromAristidesQuintilianus,Demusica(ΠερὶΜουσικῆς)Bk,3,Ch.1:97.4.(Barker,GreekMusicalThoughtII:497):“Attheendofhislife,itissaid,Pythagorasadmonishedhisfriendstoplaythemonochord.”Wantzloeben’sGermantranslationrenderstheunspecificverb“μονοχορδίζειν”(literally“tomonochordize”)tendentiouslyas“toplaythemonochord,”nodoubttounderlinehishypothesisofPythagorasasamusicalperformer.Thebroadertranslation,“usethemonochord,”or“workatthemonochord,”whichcouldalsoincludemeasurementsofintervals,wouldbemuchclosertotheflavoroftheoriginal.(Thismistranslationhasoftenbeenaccepteduncriticallyintheliterature,

15

[2.3]PerhapsthebestevidencethatWantzloebencouldhavemarshaledfrom

ancientGreektextsinsupportofhisboldhypothesisareafewnegativecomments

fromlatercommentators,suchasPtolemy,whoarguesthatthemonochordmakesa

ratherlousymusicalinstrument.ThefactthatPtolemyfeltmovedtoincludethis

passagesuggeststhattherewereindeedsomeeffortstousethemonochordasa

makeshiftmusicalinstrument.29Thereisnoreason,however,toassumethatthis

hadanythingtodowithPythagorasatall.

[2.4]Forallitsobviousproblems,however,thereisakernelofagoodideain

Wantzloeben’shypothesis,whichisthesamethesisthatwejustreadoutof

Gaffurius’sillustration:musictheoristsdowelltotakemusicalinstruments

seriously.Theyareembodimentsoftheoreticalideasaboutmusic.Allmusical

instrumentscarrytheoreticalproperties,anditcanbeinstructivetoconsiderthem

intermsofwhattheycantellusaboutmusiconamoreabstractlevel.Fromthis

angle,Wantzloebengetsonethingexactlyright:“themonochordasaninstrument

andasasystem,”asthetitleofhisthesisrunsinEnglish.Thespecificquestionof

Pythagoras’sroleasaperformerissomethingofaredherring(aseven

Wantzloebenseemstocomeclosetoadmittingoccasionally)thatinfactdistracts

fromamoreimportantpoint.Themaininterestisonthedoublefunctionofthe

instrumentitself.

[2.5]TheideathatIwanttopursuehereisnothingotherthantheliteral

meaningof“instrument”—or,forthatmatter,organon,astheGreekhasit.Atthe

seeAdkinsandMünxelberg.)OthersuchtendentiousinterpretationscanbefoundthroughoutWantzloeben’sthesis.29SeeBarker,GreekMusicalWritingsII:341and497,n.14.

16

heartofourconceptoftheinstrumentistheLatinverbinstruere,toconstructor

equip,justasergon,orwork,isthecognateoftheGreekorganon.30Thatistosay,an

instrumentisadevicethatallowsustoaccomplishsomething,towork;itisan

implementoratool.31Inourcase,musicalinstrumentscanbe,quiteliterally,

instrumentalinshapingthoughtsandideasaboutmusic.

[2.6]Inamusicalcontext,thisleadsustoadoublemeaningoftheterm

“instrument.”Ontheonehand,wehavethetool,orindeedthe“measuringrod”32

thatPythagorassoughtinNicomachus’story—inaword,adevicethatcantellus

somethingabouthowmusicasasystemworks.Inthiscase,thedevicetellsusabout

howsomeofthemostfundamentalmusicalintervalscanbederivedandquantified.

Ontheotherhand,wehavethemusicalinstrumentonwhichcompositionsor

improvisationscanbeperformed.Inthecaseofthemonochord,thisperformative

functionisatbestrudimentary(andatworstfanciful)—whichisexactlythereason

theearlymodernrepresentationsofPythagorasinGaffurius’workpiquedour

interestinitially—buttheprincipleofperformance,ofengaginginmusicalactivities,

isclearlyachiefaspectofmusicalinstruments.Instrumentsarethesiteof

performanceaswellasoftheorizing.

[2.7]ToreturntotheBoethiandistinctionwithwhichweendedtheprevious

section:wecanputasideourworriesaboutthemusicus/cantordivision,wedon’t

needtodecide—itisnotaneither/or.AsWantzloebencorrectlypointedout,the30Andforfriendsofetymology:thearchaicformofergonbeganwithadigamma,*werg-,whichisunmistakablyrelatedtotheGermanicWerkandtheEnglishwork.31Gaffuriusseemsveryconsciousofthisetymology:inBk.1,Ch.4:7(Kreyszig,39)hegoessofarastocalllung,throat,palate,tongue,lips,andteeth“instruments”—where“organs”(eventheLatinizedorgana)wouldseemamoreapparentchoice.32Barker,GreekMusicalWritings,256.

17

monochordservesasbothinstrumentandmusicalsystem.And,asGaffuriusknew,

regulaorkanônmeanssimultaneouslytherulethatgovernsmusicandthe

instrumentfromwhichitisderived.Ourinvestigationofinstrumentswillcheerfully

explorebothaspects,andtheinteractionsbetweenbothsides,whichcanbeattimes

intricate.

3.Music-TheoreticalInstruments

[3.1]Musicologyhasrecentlyrediscovereditsinterestinmusicalinstruments.

EmilyDolanandJohnTreschrevivedthevenerabledisciplineoforganology,the

studyofmusicalinstruments,bycallingfora“NewOrganology.”33Asthemoniker

“New”underlines,themaininterestisnotclassificatoryortaxonomic,whichwas

oneofthechieftasksofthe“Old”organology,butratherexploratoryand

interpretive.Whatexactlyisaninstrument?FollowingBrunoLatour’sclarioncallto

rethinkthesociologyofscience,TreschandDolan,ateamconsistingofahistorian

ofscienceandamusicologist,proposethatinbothmusicandscience,instruments

arenotpassiveortransparent,butthattheypossessagency,takinganactivepartin

socialnetworks.Myambitionhereisalittlelesswide-rangingthantheirs,thoughit

sharessomeoftheircentralconcerns.Iamspecificallyinterestedinexploringwhat

instrumentscantellusaboutthemusicalsysteminwhichtheyoperate,inother

words,theepistemicaspectsofinstruments.Whatcaninstrumentstellusabout

musicalknowledge?Howcanwe“read”aninstrument?Howdoinstrumentshelpus

understandwhatmusicis?Withoutofferingspecificanswersfornow,instruments33JohnTreschandEmilyI.Dolan,“TowardaNewOrganology:InstrumentsofMusicandScience.”Osiris28(2013):278-298.

18

areusefulobjectsforthiskindofinquirybecausetheycanworkontwoepistemic

levels,whichwecouldvariouslycontrastusinganyofthefollowingpairs:hearing

andcounting,sensingandcogitating,empiricismandlogic—or,toreturntoour

earlierdistinction:cantorandmusicus.

[3.2]Sincethe“musical”aspect(inthemodernexperientialsense)ofthe

instrumentismorelikelytobeself-evident,itisperhapsusefultotakeacloserlook

atthe“scientific”senseinwhichIamthinkingofinstrumentshere.Several

influentialfiguresfromthestudyofscienceprovideimportantimpulses,historianof

SimonSchafferandsociologistStevenShapin,whobothfundamentallyre-examined

thecreationandcommunicationofknowledgeduringthescientificrevolution,

highlightingthespecificroleoftheinstrumentwithinscientificexperimentation

duringthatperiod.

[3.3]Inseveralgroundbreakingstudies,focusingonsuchcentralfiguresfrom

theScientificRevolution,asIsaacNewton(1642–1727)andRobertBoyle(1627–

1691),inotherwords,areasthatseemedtobeexhaustivelyknown,Schafferand

Shapindrewattentionnotsomuchtotheresultsofthescientists’work,buttheway

inwhichtheygottheirresults.34FollowingafamouspronouncementbytheFrench

philosopherofscienceGastonBachelard(1884–1962)thatinstrumentsarenothing

34StevenShapinandSimonSchaffer,LeviathanandtheAirPump:Hobbes,Boyle,andtheExperimentalLife(Princeton:PrincetonUniversityPress,1985),andSimonSchaffer,“GlassWorks:Newton’sPrismsandtheUsesofExperiment,”inTheUsesofExperiment:StudiesintheNaturalSciences,ed.DavidGooding,TrevorPinch,andSimonSchaffer(Cambridge:CambridgeUniversityPress,1989),67-104.

19

but“materializedtheories”35(théorêmesréifiées),theystressedthatthe

experimentalset-up,sometimesdowntotheprecisemakeoftheinstruments,

playedamajorroleforthegenerationofscientificknowledge.Theyshowedhowin

theworkofBoyleandNewtonthefundamentalstatusoftheexperimentchanged,

fromademonstrationthatmerelyillustratesascientificphenomenonbutisofno

furtherrelevancetoscientificknowledge,toanintegralpartintheprocessof

discoveryandtheory-making.

[3.4]InthecaseofNewton’sOpticks,theexperimentumcrucis,thecritical

experimentuponwhichtheauthorityofthescientificclaimhinged,wasentirely

dependentonthemakeoftheprism.Thiswasmorecomplicatedthanweimagine,

sinceprismswerewidelyavailable,stuffoffairgroundattractions,butevensmall

flawsintheglasswouldruintheexperimentaloutcome.Newtonhimselfreliedon

high-qualityprismsimportedfromtheNetherlands,butitisnotsurprisingthat

otherresearcherstryingtoreplicateNewton’sexperimentumcrucisfailedintheir

efforts.Understandably,theywerequicktofindfaultwithNewton’shypothesis.

SchaffershowshowmuchworkandeffortNewtonputintoconvincinghis

correspondentsthattheexperimentmustbereplicatedpreciselyandthatthe

physicalpropertiesoftheprismareall-decisiveforthesuccessoftheexperiment,

andhencethescientifictheory.Thesestudieshelpedusherinanewdisciplinary

paradigmfortheHistoryofScience,adisciplinethathadtraditionallybeen

particularlyinterestedinahistoryofdiscoveries,progressivelyunfolding,as

35GastonBachelard,TheNewScientificSpirit,tr.ArthurGoldhammer(Boston:BeaconPress,1984),13.Thisline,inamoreliteraltranslation,functionsasanepigraphinSchaffer’sarticle“GlassWorks.”

20

scientificunderstandingmarchesonthroughtime.Bycontrast,ithadbeenmuch

lessinterestedintheprocessofexperimentationleadinguptothesediscoveries,

andinthematerialconditionsthatmadethempossibleinthefirstplace.

[3.5]Howdoweadaptthisideaforourpurposes?Ofcourse,SchafferandShapin

weretalkingaboutaveryspecificmoment,indeed,aturningpointinthehistoryof

science,theScientificRevolutionintheseventeenthcentury,anditishardlyan

accidentthattheirinterestfocusesonNewtonandBoyle,thatis,figureswhooccupy

acentralpositioninthecanonofthehistoryofscience.Itisimportanttobearin

mindthattheexperimentalmethodoftheScientificRevolutionisspecifictothis

period—infact,thisiswhatmakestheScientificRevolution—anditwouldbewrong

togeneralizebroadly,outsideofthisspecificcontext.Thatsaid,wefarebetterifwe

turnthisthoughtonitshead:werememberthatthePythagoreansshowedno

interestinexperimentalproofofthenumericalhypothesis.Andaswesawearlier,

thefabledstoryofPythagorasinthesmithy—andwithit,theentirephysicalbasis

onwhichthePythagoreanarithmeticparadigmrested—wasquestionedby

VincenzoGalilei,preciselyduringthefirststirringsoftheScientificRevolutionin

Italy.

[3.6]Thissuggeststhatthesignificanceofthemonochordchanged.Itwouldbe

foolhardytoarguethatPythagoreansfromNicomachustoZarlino,thatis,fora

millenniumandahalf,hadused,orthoughtabout,themonochordillegitimately,

andthatVincenzoGalileifinallyputanendtothisfaultyusage.Rather,whatit

suggestsisafundamentalreconfigurationofthetwoquestionsofwhatthe

monochordispurportedtoshowandwhatstatusisassignedtoitsdemonstrative

21

power.Itisperhapsusefultothinkofthisshiftasachangefromanarithmeticway

ofthinkingtoaphysicalone,fromanabstract,numericalwayofthinkingtoan

empiricalone.Thisisanimportantdifference.ItwasthePythagoreanworldview—

withitsfaithinuniversalcorrespondencesanditsliberalassociations—that

assignedmusicitsplacebesideastronomy,geometry,andarithmetic,whichushered

inthemedievalquadrivium.Andfromastrictlyarithmeticpointofview,the

mechanicsoftheinstrumentisirrelevant;thesoundthatthemonochordproduces,

theexperientialdimensionofitseffects,isatbestincidental.Themathematical

distinctionbetweenepideixis(demonstration)andapodeixis(proof)springstomind

here.36ForPythagoreans,themonochordwassimplyadevicewhoseexistencewas

enoughtounderlinetheuniversalvalidityofthemathematicalratiosthatthey

believedtounderlieallworldlyphenomena,fromthesmallesttothelargestscale.In

fact,evenintheancientworldPythagoreanswereregularlytakentotaskforthe

chasmthatopenedupbetweentheirpainstakingcalculationsandthescant

relevancethesepreciseproportionsborewithrespecttomusicalpractice.37

[3.7]OnlyinGalilei’shandsdidthemonochordbecomepartofanexperimental

designinacontextthatcanbeunderstoodfromtheperspectiveofmodernscience.

Ofcourse,thisdidnotchangeovernightthewayinwhichthemonochordwasused.

Cosmologicalmodels,suchasRobertFludd’sfamouscelestialmonochord,from

Utriusquecosmihistoria(1617–1624),whichoutlinedthetraditionalGreatChainof36ForawiderdiscussionoftheepistemologicalstatusofproofsinEuclidianscienceseeRevielNetz,TheShapingofDeductioninGreekMathematics(Cambridge:CambridgeUniversityPress,2003).37SeealsoDavidCreese,“InstrumentsandEmpiricisminAristoxenus’Elementaharmonica,”inAristoxenusofTarentum,ed.C.F.Huffman(NewBrunswick:Transaction,2012),29-63.

22

Being,fromdivinebeingsandangels,viastarsandplanets,allthewaytohumans

andanimals,continuedtofeature,sometimesprominentlyso.38Buttheideaof

correspondencesorresemblancesthatdeterminedsomuchofPythagorean

thinkingallthewaytotheRenaissanceandthatproceededonthebasisofanalogical

orsymbolicthinking,wasonthewayout.Asitbecameharderandhardertothink

togethermusicalintervalsanddivinecelestialorder,newconstellationsbetween

numericalratiosandsoundasanempiricalphenomenon,thatisaperceptually

verifiableobject,emergedaspowerfulandconvincingarguments.Fromthis

perspective,JohannesKepler’sfoundationalastronomicaltreatiseHarmonices

mundi(1619),inwhichthethirdlawofplanetarymotionisproposed—withexplicit

referencetotheMusicoftheSpheres,isperhapsbestunderstoodasafinal

grandiosegaspofthePythagoreanworldview,theviewthatsucceededinholding

thequadriviumtogetherforthebestpartofamillennium.39

4.WhatKindof“Thing”IsAMusic-TheoreticalInstrument?

[4.1]Whatdoesitmeantoconsiderinstrumentsfromtheperspectiveofscientific

inquiry—orperhapsbetter:knowledgeacquisition—aswellasfromtheirmusical

qualities?AsPythagorasputsonhiswhitelabcoat,metaphoricallyspeaking,and

startsexperimentingwithhismonochord,itmakessensetodrawonanothermodel

borrowedfromthehistoryofscience.Hans-JörgRheinbergerdevelopedaconcept

ofthe“epistemicthing”todescribeanimportantaspectoftheprocessofscientific

38RobertFludd,Utriusquecosmihistoria(Oppenheim:JohannTheodordeBry,1617-24).39JohannesKepler,Harmonicesmundi(Linz:GottfriedTampach,1619).

23

experimentation.40Rheinbergeremphasizes“thepowerofmaterialobjects—in

contrasttoideasorconcepts—asdrivingforcesintheprocessofknowledge

acquisition.”41Theepistemicthingisamaterialobject,phenomenon,orprocessthat

arousesourcuriosityandthat,withinanexperimentalsystem,holdsacertain

knowledgethatcanbeuncoveredbythescientist.The“epistemicthing”isneither

identicalwiththemerephenomenonormaterialobject,norwiththescientific

instrumentariumusedtoexamineit;theepistemicthingonlyemergesasthe

conjunctionofboth.WhatmakesRheinberger’sconceptsoproductiveisthatthese

epistemicthingspossessacertainfundamental“fuzziness”or“blurriness”

(Verschwommenheit).Thereisnopre-givenepistemologicalcontentthatthe

researcherextractsfromthethingunderscrutiny;thethingitselfdoesnotexude

knowledge.42Keenlyavoidingtheproblemofdeterminism,Rheinbergerarguesthat

theepistemologicalvalueoftheepistemicthingisonlyconstitutedbyitsplaceand

interactionwithintheexperimentalsystem;itiscontextuallyandhistorically

determined.43

[4.2]Thereisasteadilygrowingbodyofreflectionsontheepistemologyofthe

“thing,”notleastnurturedbyaresurgentinterestinamaterialisticperspective.

40Hans-JörgRheinberger,TowardsaHistoryofEpistemicThings(Stanford:StanfordUP,1997).41Rheinberger,“AreplytoDavidBloor:‘Towardasociologyofepistemicthings’,”PerspectivesonScience13(2005),406.42SeeespeciallyUljanaFeest,“Remembering(Short-term)Memory:OscillationsofanEpistemicThing,”Erkenntnis75(2011),391-411.Feest’sownreadingattributes“blurriness”toconcepts,notobjects.43HasokChanghighlightsthisaspectin“ThePersistenceofEpistemicObjectsThroughScientificChange,”Erkenntnis75(2011),413.

24

Sometimes“thing”and“object”areusedinterchangeably.44ThearthistorianW.J.T.

Mitchell,however,cautions:“objectsarethewaythingsappeartoasubject—thatis,

withaname,anidentity,agestaltorstereotypicaltemplate.…Things,ontheother

hand,[signal]themomentwhentheobjectbecomestheOther,whenthesardinecan

lookback,whenthemuteidolspeaks,whenthesubjectexperiencestheobjectas

uncannyandfeelstheneedforwhatFoucaultcalls‘ametaphysicsoftheobject,or,

moreexactly,ametaphysicsofthatneverobjectifiabledepthfromwhichobjects

riseuptowardoursuperficialknowledge.”45Objects,itisgenerallyagreed,are

characterizedbyacertainpassivity,inthattheyrequireacontemplatingsubject,

whereasThingsarecharacterizedbyanirreducibilitytoobjects,inthattheyhavean

existenceintheirownrightandthatmayimbuethemwithagency,asBrunoLatour

underlined.46AndBillBrownremindsus,inamorepoeticvein,thatthingsare

“encountered”andneverquiteapprehended.47Theepistemologiesunderlyingthese

variousapproachesto“thingness”fallalongacontinuumbetweenidealismand

materialism,withBachelard’s“reifiedtheorems”reachingoverintothematerial

worldfromafirmlyheldpositionattheidealistendofthespectrum,and

Rheinberger’s“epistemicthings”erectingtheirsuperstructurefromamaterialistic

base.48

44See,forinstance,Chang’s“EpistemicObjects,”413-429.45W.J.T.Mitchell,WhatDoPicturesWant?,quotedinJaneBennett,VibrantMatters(Durham,NC:DukeUniversityPress,2010),2.46BrunoLatour,Wehaveneverbeenmodern(Cambridge,MA:HarvardUniversityPress,1993).47BillBrown,“ThingTheory”CriticalInquiry28(2001),1-22.48Forafurtherexplorationoftheepistemicthinginmusicalcontexts,seemy“ThreeMusicTheoryLessons,”JournaloftheRoyalMusicalAssociation141/2(2016),251-282.

25

[4.3]Andinthemusicalrealm?Wecanseehowcertainaspectsofthis

epistemologycaneasilybeadaptedtomusicalpurposes.Justascertainscientific

phenomena,especiallythoseexistingbelowthelevelofperception,mustbebrought

toourattention,examined,andunderstoodwiththeaidofscientificinstruments,so

certainmusicalphenomenacanonlybesubjectedtotheoreticalscrutinywhenthey

manifestthemselvesassounds,producedbymusicalinstruments.Anyonewhohas

taughtanintroductorymusictheoryclassknowsabouttheimportanceofthepiano

inexplicatingmusic-theoreticalconcepts.Or,forasomewhatmorespecialized

example,AnnaGawboyhastracedtheriseoftheWheatstoneConcertinaaccordion

astheinstrumentofchoiceamongtheacousticallyinclinedmembersoftheRoyal

SocietyinVictorianBritain—makingitsurelythemostennobledmusic-theoretical

instrument.49Weshouldbecarefulnottoconstructasimplecause-and-effect

schema.Thedifficultyindecidingbetweenapodeixisandepideixisintherealmof

musicissymptomatichere:doessoundingmusic“prove”or“demonstrate”amusic-

theoreticalproposition?Aswewillsee,differentscenariosrequiredifferent

epistemologicalregimes.Perhapsthebestwayoutofthisdilemma,fornow,isto

thinkoftheinstrumentasfunctioningasafilterthatallowscertainpropositionsto

bemadeinsound,whileinhibitingcertainothers.

[4.4]Whereasthepianoclearlyplaysasuprememusic-theoreticalrole,itis

oftenlesscommoninstrumentsthataretheoreticallymostinteresting.50The

monochordwouldbearepresentativeexamplehere—leavingasidethedogged49AnnaGawboy,“TheWheatstoneConcertinaandSymmetricalArrangementsofTonalSpace,”JournalofMusicTheory53/2(2009),163-190.50ForareconsiderationofthepianoseeRehding,“ThreeMusicTheoryLessons,”264-270.

26

questionofwhetheritshouldcountasamusicalinstrumentornot.Butthereisno

doubtthatitgivesusvaluableinsightsabouthowmusicworks.Themonochord

contains,andmakesreadable,someofthefundamentalsystemicaspectsonwhich

musicdraws.Inproducingsounds,italsoproducesknowledgeaboutmusic.Thisis

theessentialfunctionofamusic-theoreticalinstrument.

[4.4]Inthismodel,inotherwords,wecanimaginemusiciansbecoming

veritablescientistsexperimentingwithinstrumentstouncovermusicalknowledge

andtodemonstrateitsprinciplesinsounds.Wewillexaminetwoshortexamplesof

musicalexperimentalistsfromdifferenthistoricalperiods:thefirstisNicola

Vicentino(1511–1575or76)andhisarchicembalo,whichwasconstructedinthe

1530s,andthesecondistherhythmicon,aninstrumentthatwasdevisedinthe

1930sbytheAmericanexperimentalistcomposerHenryCowell(1897–1965)and

hiscollaborator,theRussianinventorLeonTheremin(1896–1993).Thepointhere

isnottoconstructacoherenthistoryofmusictheoryoranorganologicalsurvey,but

toshowthediversityofpossibilities.Itgoeswithoutsayingthat,inprinciple,every

musicalinstrumentisalsoamusic-theoreticalinstrument.Butitisnotthecasethat

everymusicalinstrumentcarriesveryusefulinsightsaboutthemusicalsystemin

whichitoperates.(Arattleoratriangle,forinstance,willholdlimitedmusic-

theoreticalinformationofinterest.)Themoreinterestingexamples,atleastforour

exploratorypurposes,tendtobethemoreexperimentalones:theyareoftenthose

thatgobeyondaconventionalnotionofwhatmusicisorcanbe,preciselybecause

intestingthelimitstheyshowusmostclearlywhatisatstake.

27

5.TheArchicembalo

[5.1]Ourstoryaboutthearchicembalocentersonastrangebet.51Ataprivate

concertinRomegivenbytheinfluentialbankerBernardoAcciaiuoli-Rucellai,athis

palaceontheTiberinMayof1551,apolyphonicReginaCoeliwasperformed.Inthe

wakeofthisconcert,twomusiciansintheaudiencestartedanargument:Nicola

Vicentino,householdmusicianintheserviceofCardinalIppolitoIId’Este(1509–

1572),CardinalofFerrara,andthepapalsingerVicenteLusitano(d.after1561).

ThesetwowereengagedinadiscussionaboutwhethertheReginacoelibelongedto

thediatonicgenusornot.ItispossiblethatLusitanowasthecomposer,inwhich

casehewouldhaveapersonalstakeinthisdebate,butwedonotknowthiswith

anycertainty.52Asthediscussionquicklygotheated,itwasdecidedthataformal

debatebetweenthetwomusiciansshouldsettlethisquestiononceandforall.The

agreedwagerwastwogoldscudi,ahandsomeamountofmoneyatthetime.Three

judgeswereappointedtoadjudicatethedebate.Intheevent,oneofthejudges,the

NetherlandishmusicianGhiselinDanckerts(1510–1567),wascalledoutoftown

andhadtomisstheactualdebate.InresponsetoarequestbyDanckerts,Vicentino

senthimashortwrittenstatementimmediately,whereasLusitanoapparentlytook

moretimetowriteamuchlongerletteroutlininghisposition.Thisdiscrepancy

struckVicentinoasunfair—especiallyafterLusitanowasdeclaredthewinnerofthe

51ThisstoryhasbeenimmaculatelyreconstructedbyMariaRikaManiates,inhereditionofVicentino’sAncientMusicadaptedtoModernPractice(NewHaven:YaleUniversityPress,1996),xi-lxiii.52Foradetailedandsensitivereconsiderationofthesedocuments,seeGiordanoMastrocola,“VicenteLusitanoentrehistoireethistoriographie:nouvellesperspectives,”inPhilippeCanguilhem,ChantersurlelivredelaRenaissance(Turnhout:Brepols,2013),58-78.

28

debate.Theaffairturnedbittersoon.BothVicentinoandjudgeDanckertscontinued

toholdagrudgeforseveralyearsandpublishedtheirrespectiveversionsofthe

headyeventsof1551,eveninmultipleversions.53Oneofthestickingpointsseemed

tohavebeenadefinitionalissue.AsMariaManiatesobserves,Vicentino’swritten

andnotarizedaccountinsistedonspecifyingthediatonicgenusas“purelydiatonic

music”ormusicadiatonicasemplice,atermthatwasnotincludedintheoriginal

documentssignedbythejudgesandfourwitnesses.54(Vicentino’sdocumentwas

alsosignedbyfourwitnesses,threeofwhichwerethesameastheofficial

document.)Danckertsnoticedthediscrepancyandtookgreatexceptionto

Vicentino’stamperingwiththedocuments.WhywouldVicentinomakethis

apparentlypettydistinction,andwhywoulditmattersomuch?

[5.2]Vicentinotookarathercomplicatedpositiononthequestionof

diatonicism:hearguedthatthechromaticandenharmonicgeneraoftheancients

wereneverabandonedbymusicians,butthattheyhadinsteadbeenfully

internalizedandwerebeingusedunconsciously.55Everytimeasingersangthe

intervaloftheminorthird,orthe“incompositetrihemitone,”56Vicentinoargued,

53Vicentino’ssideoftheeventsisincludedinhistreatiseL’anticamusicaridottaallamodernaprattica(Rome:AntonioBarre,1555),Bk.4,Ch.43.(Maniates,AncientMusic,302-314.)Danckert’ssideexistsinthreeversions,bearingthetitleSopraunadifferentiamusicale(Rome:BibliothecaVallicelliana,MsR56A),nos.15a,15b,and33.SeeManiatesxiv-xv.54ManiatesarguesthatVicentinodidnottamperwiththedocumentandsuspectsthatGhiselinDanckertslikelyfabricatedthisaspectofthestory.Itseemsmoreplausibletomethatthewitnessesdidnotnoticethissmalldifferenceordidnotconsideritsignificantinanyway,whereasforVicentinoitmadeallthedifference.55VicentinoreiterateshisclaimsthatmostpeoplemisunderstandgeneraandmodeinL’anticamusicaBk.3,Chs.15and48(Maniates,AncientMusic,150and203-4.)56ThetermsofVicentino’spositionareoutlinedinL’anticamusicaBk.4,Ch.43(Maniates,AncientMusic,305).

29

theywouldunwittinglyemploythechromaticgenus,andwhentheysangamajor

third,orthe“incompositeditone,”theywereintheenharmonicgenus.Initselfthis

isastrangeclaimthatseemshardtodefend,sincetheseintervalscaneasilybe

constructedwithinthediatonicgenus.Fromthisperspectiveitshouldcomeasno

surprisethatVicentinowaswidelyheldtohavelostthedebate.Butthisisnottosay

thatVicentino’sargumentwascompletelybaseless.Howevercomplicateditmaybe,

itispossibletoreconstructhiscase—aroundhismusic-theoreticalinstrument,the

archicembalo.

Fig.2.ExamplesoftetrachordsinVicentino’sthreegenera(fromL’anticamusica,3.45).Otherconfigurationsarepossible.

[5.3]WecanapproachVicentino’sclaimbyconsideringhisperspectiveon

ancientgenera,andthewayinwhichheimaginedtetrachords,thebasicunitof

ancientGreekmusic.JonathanWildhasrecentlyprovidedalucidaccountof

Vicentino’scomplextheory,whichIwilluseasabasishere.57Fig.2shows

57JonathanWild,“Genus,SpeciesandModeinVicentino’s31-toneCompositionalTheory”inMusicTheoryOnline20/2(2014).<http://www.mtosmt.org/issues/mto.14.20.2/mto.14.20.2.wild.html>

30

diagrammaticallytetrachordsinthethreegeneraandtherelationsbetweenthem.

Thediatonicgenusisquitestraightforward,consistingofonesemitoneandtwo

wholetones.Thechromatictetrachordiscomposedofaminorthirdandtwo

semitones.Theenharmonictetrachordconsistsofamajorthirdandtwodieses(two

microtonalintervals,whichtogethermakeupadiatonicsemitone).Tomarkthese

microtonesVicentinohadtoinventanewnotationalconvention:headdedadot

overthenote,indicatingthatitisraisedbyoneminordiesis.Whilethesegenera

maylookfamiliarfromGreekmusictheory,thedetailofVicentino’sideasputsan

interesting,indeedrevolutionary,twistontheseconcepts.

[5.4]OnemajordifferencefromancientconceptionsisthatVicentino’sgenera

cansmoothlybeconvertedfromoneintoanother.58Vicentinoisquitespecificabout

howthesetransformationswork:thesemitoneofthediatonicgenusistransformed

intotheminorthirdofthechromaticorthemajorthirdoftheenharmonicgenus.

Thismayappearcounterintuitive,ifweexpectthesetransformationstobe

parsimonious—similarityofintervalsizeorshortvoice-leadingdistancesdonot

58KarolBerger’sreadingofVicentinostressesthetransformationalpropertiesofVicentino’stheories.SeeTheoriesofChromaticandEnharmonicMusicinLate16thCenturyItaly(AnnArbor:UMIResearchPress,1976).InL’anticamusicaBk.3,Ch.52(Maniates,AncientMusic,211),Vicentinounderlinesthetransformationalnatureofgenerawiththe(startling)suggestionthatthemusicexamplesinhistreatise,whichareoftenfullyfledgedmadrigalsormotets,beplayedthroughmultipletimes:firstwithoutanyaccidentals,inthediatonicgenus,thenbyaddingthechromaticaccidentals,andfinallybyalsoobservingthedotsthatmarktheenharmonicintervals.Thispointhasoccasionallycausedconfusion,sinceitseemstocontradictmuchofwhatVicentinoargueselsewhere.Fromaperformer’sperspective,however,thisrecommendationisprobablylessparadoxicalthanitmayfirstappear.InWild’sreport,therecentcollaborationbetweenPeterSchubertandJonWildinwhichtheyrecordedVicentino’smicrotonalmusic,operatedalongsimilarlines.SeeWild,“Vicentino’s31-toneCompositionalTheory,”fn.51,ontherecordingandpost-productionprocess.

31

seemtomatterhere.Instead,Vicentino’sruleofthumbistoput“thebigstepinthe

locationofthesmalldiatonicone,andthesmallstepsinplaceofthebigdiatonic

ones.”59

[5.5]Giventhistransformativepotential,itisusefultoapproachVicentino’s

tetrachordsfromtheperspectiveoftheirsmallestconstituents,theminordiesis.It

isthislowestcommondenominatorthatallowsVicentinotomovebetweengenera

smoothlyandeffortlessly.60DuringtheRenaissancethediesiscommonlydescribes

theminuteintervalthatseparatesonetonefromitsenharmonicneighbor.61

Vicentinospecificallydefinesthediesisas“exactlyone-halfoftheminor

semitone,”62—or,expressedinmodernmathematicalterms,√(18:17).Thisinnocent

definitionismoreexplosivethanitmayatfirstappear:thePythagoreantradition

heldthatirrationalnumbers—whichatthattimecouldnotbeexpressed

arithmetically,onlyderivedgeometrically—wereinadmissibleasmusical

intervals.63Eventwodecadeslater,afterthedustoftheRomedebatehadlong

settled,theSpanishmusictheoristFranciscodeSalinas(1513–1590)would

59Vicentino,L’anticamusica,Bk.3,Ch.45(Maniates,198).60Ibid.,Bk.3,Ch.52.(Maniates,210–11.)Thisisperhapsthebiggestdeparturefromancienttheories,whichtendtoproposeseparateratiosforeachgenus,thusforeclosingtheoptionofgeneraconvertingsmoothlyintooneanother.61Vicentino’scontemporaryFranciscodeSalinas,forone,insiststhatadiesiscorrespondstotheratio128:125.Thisratiohederivesfromthedifferencebetweenmajor(16:15)andminor(25:24)semitones.SeeFranciscoSalinas,Demusicalibriseptem(Salamanca:MathiasGastius,1577),Bk.2,Ch.21.62Ibid.Bk.0,Ch.15.(Maniates,18.)63Vicentinocarefullyoutlinestheconsequencesofthis“irrationalratio”fortheenharmonicgenusinBk.3,Ch.50(Maniates,207).PeterPesic’s“HearingtheIrrational:MusicandtheModernConceptionofNumber,”Isis101/3(2010),501-530,explorestheconnectionsbetweenmusicalintervalssuchasVicentino’sdiesisandtheformulationofirrationalnumbersinsixteenth-centurymathematics.

32

condemnVicentinoparticularlyforstrayingfromthepathofrationalnumbers.64

SoundEx.1:Vicentino’squartertones,fromManfredCordes,NicolaVicentinosEnharmonik(2007)

[5.6]ButdespiteoffendingorthodoxPythagoreans,thisdefinitioncertainlyhad

practicaladvantages.AsFig.3shows,Vicentinousedtheconvenientfactthatthis

diesiscorrespondsalmostperfectlytoafifthofawholetone,andsystematizeditby

dividingupthewholetoneintofiveequalmicrotones.(Ademonstrationcanbe

heardonSoundEx.1.)Onthebasisofthisrigoroussubdivisionofthe

wholetone,itispossibletoconceptualizethethreetetrachordsfromthegroundup,

startingwiththesmallestunit,theminordiesis.Eachtetrachordconsistsofthirteen

suchdieses,whicharedifferentlydistributedacrossthesoundingintervals.Going

backtoFig.2,wecanrecapturethediatonictetrachordas3+5+5minordieses,the

64ForaPythagorean,imaginingtheworldincosmicanalogiesonthebasisofperfectproportions,theexistenceofirrationalintervalsmeantthatthecosmoswassomehowoutoforder.(AccordingtoPythagoreanlore,Hippasusdiscoveredirrationalnumbersandwaspunishedbythegodsbydrowning.)SalinasdedicatesawholechapterofDemusica(Bk.3,Ch.27)toanexcoriationofVicentinoandhisarchicembalo.Theirdifferentinterpretationofthediesisisattheheartofthismatter.

Fig.3.Vicentinodivideseachwholetoneintofivemicrotones

33

chromaticas8+2+3,andtheenharmonicas10+2+1.Thewholeoctaveis

subdividedinthissystemintothirty-oneminordieses(5wholetonesand2diatonic

semitones,thatis,5x5+2x3=31).Vicentino’stransformationalconceptionof

generahassomeimportantconsequences:ifthebasicbuildingblockofallthree

tetrachordsistheminordiesis,andwecanmovefreelybetweenthem,thenwecan

onlydistinguishbetweenthegenerabymeansofthecharacteristicintervalsthat

theyemploy.

[5.7]ThesebackgroundconsiderationsputVicentino’sbetagainstLusitanoina

somewhatdifferentlight.ItturnsoutthatVicentinophrasedhisclaimsvery

carefully:everytimeweemployamelodicmajorthird,weareinVicentino’s

enharmonicgenus,andeverytimeweemployaminorthird,weareinhischromatic

genus.Or,putmoresharply:weknowbysoundingthesecharacteristicintervalsthat

wecannotbutbeintherespectivegenus.ItisalsointerestingthatVicentinokeeps

quietinthispublicdebateabouttheotherintervalsofthechromaticand

enharmonicgenera.Theminorchromaticsemitone(2dieses)ismathematically

identicaltothemajorenharmonicdiesis;thetwointervalsonlyderivetheir

differentmeaningsfromthegenerainwhichtheyareemployed;theycannot

thereforedefinethegenus.Thisintervalcanonlysheditsambiguitywhenitis

pairedwiththerespectiveothersmallintervalfromeachgenus:themajorsemitone

ofthechromaticgenus,ortheminordiesisoftheenharmonicgenus.Themajor

semitoneissimilarlyambiguous,asitissharedbetweenthediatonicandchromatic

genera.Theonlyremaininguniqueintervalistheminordiesis,thecontroversial

“irrationalratio”onwhichhissystemisfounded.Needlesstosay,Vicentinodoesnot

34

highlightthediesisinhisbet—though,inanycase,itprobablyhelpsthatthediesis

isnotemployedasamelodicintervalinconventionalsixteenth-centurymusic.

[5.8]Butthishedginggetsustotheheartoftheargument:Vicentino’s

conceptionofwhatmusicis,orshouldbe,wasfundamentallydifferentfromthe

musicthatexisted.(Onecanhardlyblamethesixteenth-centuryRomansin

attendanceatthedebatebetweenLusitanoandVicentinoforbeingmystifiedby

theseideas.)AsWildputsit,itoffersa“tantalizingglimpseofanalternative

pathwayformusicaldevelopment,”drawingonagreatlyextendedmicrotonalpitch

collection.65Vicentino’sscale,withits31-folddivisionoftheoctave,notonlycovers

thethreegenera(albeitinVicentino’sidiosyncraticunderstanding)butitalso

presentsaclosedsystem,spanningtheentireenharmonicsystemoverthirty-one

fifths.Inhisownassessment,hissystemhadtheadvantageofoffering“agreater

abundanceofsteps,consonancesandharmony.”66Hewasconvincedhehadmadea

majordiscovery—nolessthantheperfectdivisionofmusic.

[5.9]Inhissubsequentreflectiononthedebate,Danckertscommentedonthe

extremesecrecyunderwhichVicentinooperated.67Despitethelikelybiasofthis

testimonyfromahostilejuror,thisobservationsoundsfairlyplausible:thecareful

framingofVicentino’sbetsuggeststhathewasconvincedthathisclaimsmust

reflectsomefundamentaltruthaboutmusic,andthathewasunwillingtorevealthe

principle—the31-folddivisionoftheoctave—onwhichthisinsightwasbased.It65Wild,“Vicentino’s31-toneCompositionalTheory,”[1].66Vicentino,L’anticamusicaBk.1,Ch.8(Maniates,AncientMusic,49).67DanckertsspeculatedthatVicentinowashopingforapapalappointment.SeeClaudePalisca,“AClarificationofMusicareservatainJeanTaisnier’sAstrologiae,1559”inStudiesintheHistoryofMusicTheoryandItalianMusic(Oxford:Clarendon,1994),276.

35

seemsthatthissecrecywasamajorcontributingfactortolosingthebet:Lusitano

andtheopposingsidehadnoinsightintotheveryspecific,idiosyncraticmeaningof

histerms.Fromaconventionalmusic-theoreticalperspectiveVicentino’sclaims

simplydonotstanduptoscrutinyandseemclosetononsensical.

[5.10]Meanwhile,Vicentinowashardatworkdevelopingpolyphonicmusicthat

wouldmakeuseofitsfullmicrotonalpotential.AsWildpointsout,hiscompositions

goalongwaytowardclarifyingVicentino’sunderstandingofthethreegenera:

adherencetoonespecificorderofthetetrachord,thatistosay,toonefixedpitch

collection,wasnotimportant.Instead,itwastheintervalsemployedineachvoice

thatmattered:anyminorthirdandsemitonalmovement,majororminor,acrossthe

31-tonegamutconstitutedthechromaticgenus,andmajorthirdandmovementby

eitherofthediesesconstitutedtheenharmonicgenus.68

[5.11]Itisthe31-tonedivisionthatisattheheartofVicentino’sideasabout

music,whichgavehimtheconfidence—theludicrousover-confidence,onemight

say—thatmadehimengageinthefatefulbetwithLusitano.Thematerial

manifestationofthisideawasthearchicembalo,amicrotonalharpsichordthat

Vicentinoprobablyfirstdevelopedinthe1530s,thatis,severalyearsbeforethe

publicdebateandthepublicationofhistheoriesinL’anticamusicaridottaalla

modernaprattica(1555).69Thiswashisexperimentumcrucis.

68Wild,“Vicentino’s31-toneCompositionalTheory,”[32].69Salinas,Demusica,Bk.3,Ch.27,writtenin1571,indicatesthatVicentino’sexperimentswiththearchicembalooccurred“inthelastfortyyears.”

Fig.4.Vicentinocelebrateshisaccomplishmentswithamedal.Therectoshowshisprofile,theversoshowshistwoinstruments,thearchicembaloandthearciorgan.Themedalmarkshimasthe“inventoroftheperfectdivisionofmusic.”(FromMorton&Eden,AuctionCatalog59,November13-14,2012.)

36

[5.12]Asthefifthandfinalbookofhistreatisedetails,thearchicembaloisa

harpsichordwithtwomanuals,eachmanualhasthreeordersofkeys,including

severaldividedones:thelowermanualhas19andtheupper17keystotheoctave.

(Therearesomecomplicatingfactorswiththekeyboardlayout,mostnotablythe

confusingfactthatVicentinoaddsfivemorekeysthannecessaryforhistonal

system,sothathiskeyboardhasthirty-sixkeystotheoctave.Theremainingkeys

arenotpartofthetonalsystem,buttointroducepurerharmoniesandtheyneednot

concernushere.)In1561Vicentinoalsopresentedasimilarorgan,whichhecalled

thearciorgano;themachinationsbehindbothinstrumentsarecomparable.70

Keyboardinstrumentswithdividedkeysthatdistinguishedenharmonically

betweensharpsandflatshadbeenaroundforawhile,butthelengthstowhich

Vicentinowentwithhisinstrumentswereallbutunprecedented.71Histwo

instrumentsmusthaveseemedlikethephilosopher’sstonetoVicentino.Infact,he

70SeeHenryW.Kaufmann,“Vicentino’sArciorgano:AnAnnotatedTranslation”JournalofMusicTheory5(1961),32-53.71SeePatrizioBarbieri,EnharmonicInstrumentsandMusic1470-1900(Rome:Levante,2008).

37

wassoproudofhisaccomplishmentthathehadacoinforged,reproducedinFig.4,

showinghisprofileononesideandhisinstrumentsontheother.Hisinstruments

instantiatedhisideasaboutmusic,makingitpossibletoperformpolyphonicmusic

inanyofthethreegenera—andtoswitchfreelybetweenthem.72Whatismore,

giventhatthebackboneofhismusicalsystemwastheirrationaldiesis,therewasno

wayforhimtoexplainhisprinciplesbyarithmeticmeans—atleastnotaccordingto

thePythagoreanprinciplesatthetime.73Thevalidityofhismusicalsystem,withits

transgressive√(18:17),couldbedemonstratedinsounds,butnotbynumerical

proof.74TheonlywaytounderstandVicentino’smusic,anditsunderlying

principles,wastohearit—andforthat,thearchicembalowascrucial.

[5.13]OnemajorobstaclethatVicentinohadtotackleinordertoputhis

complexmicrotonalsystemintopracticewasthatsingersarenotverygoodat

pickingouttheexactintervalofafifthofatone.Vicentino,whosesurviving

compositionsarevocal,conceivedhismusicverymuchalonginstrumentallines.He

72ThetuningofVicentino’sinstrumentisofteninterpretedas31-equaltemperament,seeforinstance,Barbieri,EnharmonicInstruments,308-324.Vicentinodiscussesatleasttwodifferenttuningsystems,seealsoVolkerRippe,“NicolaVicentinoundseineInstrumente:VersucheinerErklärung”DieMusiktheorie34/4(1981),393-413,ManfredCordes,VicentinosEnharmonik:Musikmit31Tönen(Graz:AkademischeDruck-undVerlagsanstalt,2007),andWild,“Vicentino’s31-toneCompositionalTheory,”[4-9]andfn.16.73ManiatesnotesVicentino’sreluctancetogiveadefinitionofthediesis,whichcausedmuchconfusionamonghisdetractorsandsupporters.See“BottrigariversusSigonio:OnVicentinoandhisAncientMusicAdaptedtoModernPractice,”inMusicalHumanismanditsLegacy,eds.NancyKovaleffBaker,BarbaraRussanoHanning(StuyvesantNY:PendragonPress,1992),99.Thiscoynessisonlytoounderstandable,giventhathisdiesiswreakshavocwithPythagoreancertainties.74Vicentinowasawarethathewaseffectivelyrevivingtheage-olddiscussionbetweentheempiricistAristoxenusandthemathematicalPythagoreans,ashisopeningstatementsonsensevs.reasonmakeclear.SeeL’anticamusica,Bk.0,Ch.1(Maniates,AncientMusic,6).

38

complained:“Ohowimmeasurablyexcellentwouldmusicbeifsingers…could

intoneandsingacompositionasaccuratelyastheorgan!”75Vicentinowasworking

intensivelywithagroupofsingerstotrainthemtointonehismicrotonalintervals

precisely.Hehadswornthesingerstosecrecy,underthreatofsteepfines,lestthey

giveawayhisrevolutionaryideas.Tobesure,thisextremeprecautionraisedseveral

eyebrowsamongVicentino’sdetractors.76

[5.14]Heinsistedthatallmusic,vocalandinstrumental,shouldbebasedonhis

principles.Butgiventhetechnicalcomplexitiesofthismusicalsystem,andthe

difficultysingershadinintoninghisfifthtonescorrectly,thisinstrumentwasclearly

thebackbone,theembodiment,ofVicentino’sideas.77VincenzoGalilei,avoluble

criticofVicentino,recallsthetroublethesingershadwiththisenharmonicmusic:

Ifbymisfortuneoneofthesingerslosthiswaywhilesinging,itwasimpossible

toputhimbackontotherightspot.…Thusthiskindofmusicnecessarily

requiredaninstrumenttoguidethevoicesofthesingersthroughunknown

paths,nottosaythroughprecipitatecliffs.78

75Vicentino,L’anticamusicaBk.3,Ch.52.(Maniates,AncientMusic,302).76SeePalisca,“Musicareservata,”276.77TherecentrecordingofVicentino’smusicbyManfredCordesistelling:allthepartsexceptthesopranoareperformedinstrumentally.Thelonesopranomakesavaliantefforttointonethemicrotonesprecisely;thestruggleofthevoicewiththemusicisanimpressivepartoftheperformance.ContrastthiswiththeexquisiterenditionunderPeterSchubert,whichreliesonthetechnologyofautotunetoproduceaperformancethatexceedsVicentino’swildestdreams.(ExcerptscanbeheardinthesoundexamplesincludedinWild’s“Vicentino’s31-toneCompositionalTheory.)78VincenzoGalilei,Discorsointornoall’usodell’enharmonio,9v.,inFriederRempp,DieKompositionstraktatedesVincenzoGalilei(Cologne:ArnoVolk,1980),166.

39

Apartfromthedisparagingtone,Vicentinomightevenhaveagreed:healso

recommendedthatvocalmusicbealwaysaccompaniedbyinstruments.79

[5.15]ThearchicembalowasineverysenseinstrumentaltoVicentino’s

theory.80Italonecouldproducetherequiredintervalsthatmadethetheorya

musicalreality.81Hisinstrumentwasverymucha“materializedtheory,”in

Bachelard’ssenseasweencounteredabove.Wheremusicalexperiencewasin

conflictwithestablishedscholarlyauthority,itwasclearwhichwayVicentino

wouldturn.Headmitted—insomethingofanunderstatement—thatcertain

theoreticalpositionswerenotinaccordancewithBoethius,theforemostmusical

authorityofVicentino’sage,butcertainlyinagreementwithhisinstrument.82But

thiswasallthatmattered:thearchicembalo,whichallowedlistenerstoexperience

hisideaofmusic,functionedastheultimateepistemologicalauthorityinhismusical

universe.

[5.16]Thesixteenth-centurydebateswirlingaroundVicentinogotstuckonthe

questionofwhetherornothumanvoicescouldbetrainedtosingintervalsassmall

asfifthtones.83Thathistoricaldiscussionisnotquitethesamepointweareraising

here;theissueislessaboutthecapacityofthehumanvoiceandmoreaboutwhat

79Vicentino,L’anticamusica,Bk.4,Ch.42(Maniates,300).80DanielWaldenparticularlyexplorestheconnectionstothevisualarts,andsuggeststhatthearchicembalomaybemodeledontheperspectivalapparatusofDürerandhiscontemporaries.Seehis“DanieleBarbaro,NicolaVicentino,andVitruvianMusicTheoryinSixteenth-CenturyItaly,”inDanieleBarbaro:Vénitien,patricien,humaniste,(Turnhout:Brepols,2016),inpress.81Salinas’outrageatthearchicembalo,whichhecalled“prava”(crookedorperverse),suggeststhathewasalltooawarethatitbroughtirrationalratiosintotheworld.SeeDemusicaBk.3,Ch.27.82Vicentino,L’anticamusica,Bk.1,Ch.15(Maniates,59).83SeeManiates,“BottrigariversusSigonio,”91.

40

suchtrainingshouldbebasedon—itisaquestionof“turtlesallthewaydown”:the

demonstration,theepideixis,ofthefeasibilityofpolyphonicmusicwithina31-tone

systemisultimatelydependentontheexistenceofaninstrumentthatcanproduce

thosetonespreciselyandunambiguouslytotrainthosevoices.Vicentinoworehis

epistemologicalheartonhissleevewhenhespeculatedontheoriginofmusical

intervals:“Itisprobablethatthefirstpersontodiscoverthewaytosingthedistance

ofthestepsofthewholetoneandsemitone…couldnothavedonesowithoutthe

expedientofaninstrument.”84ThesameistrueforthearchicembaloandVicentino’s

challengingmusic.Abetterinvocationoftheepistemicthingishardtoimagine.

[5.17]ToreturntoVicentino’smusicalbetinRomein1551,everything

considered,itisprobablyunderstandablethatthejudgeswerenotconvincedby

Vicentino’sradicalideas.Thesewereexperimentalideasineverysense,andthey

clearlycontradictedBoethius’weightyauthority.Itisalsounderstandablewhy

Vicentinowouldinsistonthe“purelydiatonicgenus,”andwhythisfastidious

distinctionwaslostoneverybodyelse.Hehadveryparticularideasabouthow

musicworkedorshouldwork,ideasthattookseveraldecadestobeseriously

considered,andanothertwocenturiestobecomeworkedoutintheirmusic-

theoreticalimplications.85Eventhoughhefeltitwastooearlytolifttheshroudof

mysterytothepublicthatsurroundedhis“perfectdivisionofmusic,”Vicentinowas

convincedallalongthathewasright.Forhim,theproofwasinthepudding,or

ratherinhisarchicembalo.84Vicentino,L’anticamusica,Bk.1,Ch.12(Maniates,55.)85Barbieridiscussestwocirclesoffifthsbasedonthe31-tonescale,byAmbroseWarren,TheTonometer(1725)andQuiriniusvanBlankenburg,ElementamusicaofniewLicht(1739).SeeEnharmonicInstruments,346-47.

41

6.TheRhythmicon

[6.1]Thesecondcasestudy,HenryCowell’srhythmicon,takesustoBerkeley,

California,ca.1915,whereCowell,thenayoungcompositionstudent,wasworking

withtheethnomusicologistCharlesSeeger(1886–1979).Seegerintroducedhimto

polyrhythms,butCowellbecameincreasinglydisillusionedbytheinabilityofhuman

musicianstoexecutethesecomplexrhythmsprecisely.Hewrote:

Itishighlyprobablethataninstrumentcouldbedevisedwhichwould

mechanicallyproducearhythmicratio,butwhichwouldbecontrolledbyhand

andwouldthereforenotbeover-mechanical.Forexample,supposewecould

haveakeyboardonwhich,whenCwasstruck,arhythmofeightwouldbe

sounded;whenDwasstruck,arhythmofnine;whenEwasstruck,arhythmof

ten.”86

Cowell’sopeningtag“itishighlyprobable”shouldbereadastongue-in-cheek,since

heknewquitewellthattheunderlyingmechanismlongexisted.Afterall,earlierin

thesametext,hadCowellexplained,enigmaticallybutperfectlyaccurately:

Thereisawell-knownacousticalinstrumentwhichproducesasoundbrokenby

silences.Whenthesilencesbetweenthesoundoccurnottoorapidly,theresultis

arhythm.Whenthebreaksbetweenthesoundarespeeded,however,they86HenryCowell,NewMusicalResources,ed.DavidNicholls(Cambridge:CambridgeUniversityPress,1996),65-66.

42

produceanewpitchinthemselves,whichisregulatedbytherapidityofthe

successivesilencesbetweenthesounds.87

Thewell-knownacousticalinstrumenthewasthinkingofwasthemechanicalsiren,

whichhadbeenaroundforahundredyears.

[6.2]Letusbrieflyrewindfromtwentieth-centuryAmericanexperimentalismto

nineteenth-centuryFrenchengineering.In1819CharlesCagniarddelaTour

(1777–1859)presentedhislatestinvention,whichhecalledthesirène.88Itwas

originallynotthewarningsignalofmodernlifethatweallknowtoday,butrather

anexperimenttotestthetheoryofsoundgeneration.Atthetime,followingErnst

ChladniandThomasYoung’simportantworkonacoustics,itwasagreedthatall

musicalsoundshadtofollowthemodelofwindorstringinstruments,whichsetup

astandingwaveofregularoscillationsofpressurechangesinpositiveandnegative

directionsaroundaneutralzeropoint(whichcanberepresentedgraphicallyby

sinosoidsofvaryingdegreesofcomplexity).89Thesiren,bycontrast,produced

soundinaradicallydifferentway,whichcanbest,ifanachronistically,bedescribed

as“digital”:thesirencreatedaseriesofalternatingonandoffimpulses,justas

Cowelldescribed.

87Ibid.,51.88CharlesCagniarddelaTour,“Surlasirène,nouvellemachined’acoustiquedestinéeàmesurerlesvibrationsdel’airquicontientleson,”Annalesdechimieetdephysique12(1819),167-171.89SeeespeciallyThomasYoung,“TheoryofSoundandLight,”PhilosophicalTransactionoftheRoyalSociety90(1800),106-128,andErnstFriedrichChladni,DieAkustik(Leipzig:BreitkopfundHärtel,1802).SeealsoRobertBeyer,200YearsofAcoustics(NewYork:Springer,1999),1-25.

43

[6.3]Fig.5,takenfromapopularsciencearticleinHarper’sNewMonthly

Magazine(1872),showshowthesirenworks.90Asthecross-sectionshows,airis

blownthroughabellowsatthebottomofthedevice.Theairpassesthroughametal

discwithdiagonalholes,whichsetsanothermetaldiscontopinrotation.This

rotatingdiscalsohasholesinregularintervals.Everytimetheairpassesthrough

theholesanairpuffwillbereleased.(Twocountersatthetopkeeptrackofthe

numberofrotations.)Whentheairpulsesareslow,wewillhearthemasaseriesof

regularpulsations,asteadyrhythm.Butwhentherotationspeedpasses20

impulsespersecond,inotherwords:20Hz,ourearwillconvertthesefastrhythms

intoapitch,whichrisesasthefrequencyincreases.(Cagniard’ssireninactioncan

beheardonVideo1.)That’sthefamiliarnoiseofthesiren.Theprinciplewas

nothingshortofrevolutionary:thesirendemonstratedthatthetwomusical

parameters,rhythmandpitch,areinfactnotseparatedimensionsatall,butthey

existonacontinuum.90Anon.“TheSirenofScience;OrtheModeofNumberingSonorousVibrations.”Harper’s270(1874),844-849.

Fig.5.Apopularintroductiontothemechanismofthesiren,fromHarper’sNewMonthlyMagazine(1872).

44

Video1.DemonstrationofCaignard’ssiren,fromtheNationalMuseumofAmericanHistory,SmithsonianInstitution,WashingtonDC.<https://www.youtube.com/watch?v=Rs7CC4pdJeM>

[6.4]ThescientificworldwasaghastwhenCagniard’ssirenbegantowailand

scream,buthehadmadehispoint.91Thesirenwastheperceptualproofthatpitched

soundscouldbecreatedoutofrhythmicpulsations,dependingonwhetherthe

patternsthatcreatedthemwerepresentedaboveorbelowtheauditorythreshold

around20Hz.ForCaignard,thisphenomenonwasameresideeffect,whichhedid

notpursueanyfurther.Buttheideaofarhythm-pitchcontinuumgenuinelyexcited

musicians.

[6.5]Itwaslefttootherthinkers,scientificandmusical,toexploretheimplications

formusic,ofwhichtheOhm-Seebeckdebateisthebest-knownscientificdispute.92

Butinthemusicalworlditisworthdrawingattentiontothelittle-knownfigureof

FriedrichWilhelmOpelt(1794–1863),astateofficial,astronomer,andhobbymusic

theorist.Opelterectedawholeconceptionofhowmusicworksonthebasisofthe

91SeealsoPhilipvonHilgers,“Sirenen:LösungendesKlangesvomKörper,”PhilosophiaScientiae7/1(2003),85-114.92SeeStephenTurner,“TheOhm-SeebeckDispute,HermannvonHelmholtz,andtheOriginsofPhysiologicalAcoustics,”BritishJournalfortheHistoryofScience10(1977),1-24.

Fig.6.Theratio3:2correspondstotheintervalofthefifth.Opelt’smultiphonicsirenshowshowthecompoundrhythms,translatedintospatialsequencesofdots,asshownontheright,producetwo(ormore)sounds.(ThecirculardiagramincludedinFig.7showsthispatterninitssecondringfromthecenter.)

45

mechanismofthesiren,whichcausedall-too-briefexcitementinthe1830s.93Opelt

madethestartlingclaimthatthesirenprovidedthemechanismtoexplainall

aspectsofmusic,bywhichhemeantrhythm,pitch,andharmony.Buildingonthe

knowledgethatmultiplefrequenciescouldbesoundedsimultaneouslyononedisc,

Opeltexperimentedwithmultiphonicsirensandexploredtheeffectsofmerging

twoseparatefrequenciesintoone.Takethesimplehemiolicrhythmtwo-against-

three,asshowninFig.6.WeknowfromPythagorasthatthisratiocorrespondsto

theintervalofthefifth.Thiscompoundrhythmcanbeimprintedonasirendisc.

Whenrotatingatsufficientspeed,the[1/8–1/16–1/16–1/8figure]<insertnote

valuesinprint>rhythmwillturnintotheintervalofthefifth.Itmakesnodifference

whetherthefrequenciesarekeptseparateonthediscorcombined.

Video2:AdemonstrationofOpelt’ssiren,fromFlorenceScienceandTechnologyFoundation(FirST,FirenzeScienziaeTechnica)<https://www.youtube.com/watch?v=9OHfQLtMWWc>[6.6]Thesameisobviouslytrueforotherratiosaswell.Opeltproducedadisc

withmultipledifferentfrequencieshappeningatthesametime.Thismultiphonic

sirencanbeheardonVideo2.ThecircularfigureincludedinFig.7showsOpelt’s

diagramforboresintheratio4:5:6:8.ThefourcirclesmarkedG,T,Q,O(for

Grundton,Terz,Quinte,Oktave)describetheseparateregularfrequencies,the

outermostcircleoutlinestheirregularcompoundrhythmofallfourputtogether.

Wealsoseethiscompoundrhythm,translatedintospatialdistancesandlaidouton

93Opelt’stheory,UeberdieNaturderMusik(Leipzig:HermannundLangbein,1834),isallbutforgottennowadays.ErnstRobel,DieSirenen(Berlin:Gaertner,1891-1900),5-12,considersOpeltfoundational,anddiscusseshiscontributionindepth.HansUlrichHumpertundHerbertEimertacknowledgeOpeltatvariouspointsintheirLexikonderelektronischenMusik(Regensburg:GustavBosse,1973).

46

astraightline,inthemiddleportionofFig.7.Asthesirendemonstrates,this

complexrhythm4:5:6:8isthesameasamajortriadinclosepositionwithitsoctave

doubled.94

94Technically,alltheholesof4coincidewiththoseof8,sotheloweroctaveisnotsoundedseparately,asOpeltrecognizedinhistext.Itseems,though,thathedidnotrealizethiscomplicationuntilafterheproducedthedisc.

Fig.7.Evenchordscanbecapturedascompoundrhythms.Theratio4:5:6:8,firstmarkedbyarrowsalongtheharmonicseries,thentranslatedintoasequenceofholes,correspondingtothecompoundrhythmofourcomplexratio,andfinallyprojectedontoaschematizedsirendisc.

47

[6.7]Backinearly-twentieth-centuryCalifornia,alltheseideasstoodbehind

Cowell’scallforanewmusicalinstrumentthatwouldcouplespecificrhythmic

pulsationswithspecificpitches.Around1930Cowellfoundacongenialcollaborator

forthisprojectintheinventorandRussianémigréLeonTheremin,whohadmadea

nameforhimselfwithhiseponymouselectricalinstrument,andwhocreatedthe

rhythmicononthebasisofCowell’sideas.DepictedinFig.8,apparentlythe

rhythmiconwasoriginallyconceivedaslittlemorethana“highlysophisticated

metronome,”95anditisnowsometimeshailedasthefirst-everbeatmachine—both

thesecharacterizationsmisssomeofthecentralfeaturesoftheinstrument,

especiallyasregardspairingrhythmswithpitches.Earlyreviews,itistrue,suggest

thattheoriginalinstrument,demonstratedattheNewSchoolinNewYorkon

January19,1932,hadverylittletoofferinthewayofpitchortimbre,andvarious

commentatorswereparticularlyconcernedaboutimprovingthepitchdimension.96

Butitsconceptionwasclearlywasasamusicalinstrument:Cowellandhisfriend

NicolasSlonimsky(1894–1995)composedanumberofpiecesforit,including

Cowell’sownConcertoforRhythmiconandOrchestra.97Ahistoricrecordingofthe

rhythmicon,fromtheholdingsofhisfriendJosephSchillinger(1895–1943)that

nowsurvivesintheSmithsonianInstitution,isincludedinSoundExample2.

95WilliamLichtenwanger,TheMusicofHenryCowell(NewYork:InstituteforStudiesinAmericanMusic,1986),132.96SeeRitaMead,HenryCowell’sNewMusic1925-1936(AnnArbor:UMIResearchPress,1978),188-90.SeealsoRogerNicholls,AmericanExperimentalMusic1890-1940(Cambridge:CambridgeUniversityPress,1990),140-41.97AreconstructionofCowell’sRhythmicanawaspremieredin1971atStanfordUniversity.SeeLelandSmith,“HenryCowell’sRhythmicana,”AnuarioInteramericanodeInvestigaciones9(1973),134-147.

48

SoundEx.2:Demonstrationoftherhythmicon,fromtheSmithsonianInstitution

[6.8]Likethesiren,themechanismoftherhythmiconisbasedontwometal

discswithvaryingnumbersofholesarrangedinregularintervals.98AsSlonimsky

describes,by“manipulatingarheostatwitharudimentarycrank,theperformer

automaticallyproducedprecisesynchronizationoftheharmonicseries,thenumber98Asmallnumberofrhythmiconswerebuilt.Oneinstrument,atStanford,wasdiscarded(tobesure,withCowell’spermission).SlonimskysoldhisinstrumenttoJosephSchillinger,whichendedupattheSmithsonian.AfurtherinstrumenthassurfacedinMoscow—doubtlessbuiltbyThereminafterhisreturntotheSovietUnion.Itcanbeheardonhttps://www.youtube.com/watch?v=HkodVcuPVAo<pleaseembedfile,ifpossible,includedindocuments>.ItispossiblethatthisisthesamerhythmiconthatJoelSachslocatedinBudapest,seehisHenryCowell:AManMadeofMusic(Oxford:OxfordUniversityPress,2012),539,n.110.ThesoundqualityisnotablyimprovedfromtheearlierinstrumentoftheSchillingerrecording.

Fig.8:HenryCowell’sfriendJosephSchillingerbehindtherhythmicon.(1932),photonowheldatStanfordUniversity

49

ofbeatspertimeunitbeingequaltothepositionintheseries.…Theresultwasan

arithmeticallyaccuratesynchronyscoreof32differenttimepulses.”99

[6.9]WecanseehowCowell’srhythmiconpursueseffectivelythesame

phenomenonasOpelt’smusicalsirenahundredyearspreviously,withone

importantdifference:therhythmiconhadakeyboardattachedtothemechanism,

whichmadeitplayableasamusicalinstrument,withfixedpitchesandrhythms.

CharlesIves(1874–1954),whofinancedthedevelopmentoftherhythmicon,

expressedreliefthatthedevicewas“nearertoaninstrumentthanamachine.”100

Thepresenceofthekeyboardnodoubthadalotdowithitsstatusasamusical

instrument.101Thekeyboardsuggestedvisuallyandhapticallythatthesounds

wouldqualifyasmusical.

99Slonimsky,MusicSince1900(4thedn,NewYork:Scribner’s,1971),1495.100SeeSlonimsky,PerfectPitch(Oxford:OxfordUniversityPress,1988),151.Similarly,Mead,NewMusic,190,citeslettersinwhichCowellstellsIvesthattherhythmiconhasbeenacceptedasa“realartisticinstrument,”“withalmostwildacclaim,”“asopeningupafieldforbothmusicandinvestigation.”101TrevorPinchandFrankTroccoconvincinglyarguethatthepresenceofakeyboard—clearlysignifying“musicalinstrument”—gavetheMoogtheedgeoverotherkindsofsynthesizer.SeeAnalogDays:TheInventionandImpactoftheMoogSynthesizer(Cambridge,MA:HarvardUniversityPress,2004).

50

[6.10]Asacomposer,Cowellwasclearlyexcitedbythiscorrelationbetweenhis

musicalintervalsandrhythmicpulsations.Heusedthisprincipleasthebasisof

youthfulQuartetRomantic(1915–17).102Thiscomposition,fortwoflutesandtwo

violins,worksontwolevelsatthesametime:thefourinstrumentsplayfreelyatonal

102ThequartetwaswrittenaroundthesametimeasNewMusicalResources,thoughnotpublisheduntilmuchlater.AdetailedanalysisisfoundininNicholls,AmericanExperimentalMusic,140-148.

Fig.9.(a)TheopeningofHenryCowell’sQuartetRomantic(1915)codifiesamajorharmonyinfourvoicesinthepulsatingrhythmsofeachpart.

Fig9.(b)InhisownanalysisofQuartetRomanticCowelldecodestherhythmicstructureofhismusicforusintohigher-orderharmonies.

51

melodiesatfixedrhythms,buttherhythmsthemselvescontainencodedpitch

information,sothattheproportionsbetweentheserhythmicpulsationsoutlinean

underlyingtonalstructure.Togivejustoneexample,inFig.9a,thefirstmeasure

juxtaposes6,5,4,and2pulsations,ratiosthatcorrespondtoamajortriadovera

rootinthebass.TheserhythmsactuallyencryptafullytonalBach-stylechoralein

fourparts,asecretmeta-compositionthatremainsunheardbyhumanears,along

thelinesindicatedinFig.9b.Itistrue,thispiecedoesnotsoundremotelylikeaBach

chorale.ButwecouldimagineetchingthecompoundrhythmsontoanOpeltsiren

andspeedingthemup.Playedatsufficienttempo,theproportionsofthissecret

musicwouldbecomeaudibleas(admittedly,extremelyshort-lived)harmonies.

Conclusions

[7.1]Turningtothebiggerquestions,whatdoestheexaminationofmusical

instrumentsasdeeplyengagedintheprocessofmusicaltheorizingdoforus?What

goodisthis“material”turnwithintheory?Let’stakeawiderlookatthecultural

historyofmusictheory.Oneofthekeyissuesthatthefieldhasbattledwithfora

whileisitsrelationshipwithspecificmusicalrepertoires.Attemptstorelate

theoristsfromthepasttothecomposersoftheirowntime,tomaphistorical

theoriesontocontemporaneouscomposers,tendtoberelativelyfrustrating,quite

simplybecauseourinterestsdonotnecessarilyalign:thequestionsthatwe,inthe

twenty-firstcentury,askfrommusicarenotnecessarilythesamethatthefigures

52

fromthepastwereinterestedindiscussing.103Ahistoricallymoresympatheticand

musicallymoresensitiveapproachisafundamentalnecessity.

[7.2]Byincorporatingmusic-theoreticalinstruments,asIsuggestedearlier,we

turnthetheorist’sstudyintosomethinglikeascientist’slaboratory—orperhaps,

dependingontheperiodwearediscussing,somethingakintonaturalphilosophers

andWunderkammern.Inoperatingwithsounds,withthematerialsthatmakeup

whatweusuallymeanbymusic,thesemusic-theoreticalinstrumentsarelocated

somewherebetweencompositionalpractice,theoreticalspeculation,and

experientiallistening.Itisnocoincidencethatthetwoexamplesdiscussedhere,

fromthemid-sixteenthandtheearly-twentiethcenturies,aretakenfromperiodsof

intenseexperimentation.Butthisisnottosaythatothers,lessintellectuallyfluid

periods,wouldnotbeopentothiskindofinquiry.Anynumberofotherpossible

scenariossuggestthemselves—andtheyarenotrestrictedtotheWest:theChinese

12lü十二律ortheArabicoudعودareprimeexamplesofmusic-theoretical

instrumentsinothercultures.

[7.3]Ultimately,theissueofhowamusic-theoreticalinstrumentworksisalways

amatterofthespecificquestionsweaskfromthemusicandfromthetheory.A

preciseanswerchangesfromhistoricalperiodtohistoricalperiod—or,ifyouwill,

fromoneFoucauldianepistemetothenext—andalsofrominstrumentto

instrument.Certainrecurringfactorsandissues,however,arenoticeable.Firstofall,

instrumentstendtocometotheforewheneverhumanperformersreachtheir

103SeeCristleCollinsJudd,“Thedialogueofpastandpresent:Approachestohistoricalmusictheory,”Intégral14/15(2000/01),56-63.

53

limit—pitchaccuracyinVicentino’scaseorrhythmicaccuracyinCowell’scase.The

instrument,asanobjectlocatedoutsidethehumansphere,canmakegreaterclaims

toobjectivity(ineverysenseconceivable)andisthereforeapreferred

argumentativestategy.104Attimes,thisnecessitytoholdhumaninterventionatbay,

orremoveitaltogether,iscounterbalancedbyaconcernattheotherendofthe

spectrum:ifnohumaninputisrequired,isitstillmusicthatwehear?Thisconcern

isvoicedfromtimetotime,andthespecterofthedehumanizedmachine,fedby

essentialhumanisticanxieties,invariablyloomslargebehindthoseconcerns.This

suggeststhatthedesiredepistemicobjectivityofmusic-theoreticalinstruments

operatesinaforcefieldthatisdeterminedbyacarefullycalibratedbalancebetween

humanagencyontheonehand,whichexpressesitselfin(soulful)music-making,

andthedehumanized,soullessmachineontheother.

[7.4]Second,allthesemusic-theoreticalinstrumentsrelyinsomesenseona

scientific—or,morebroadly,numerical—conceptionofsound.Thisseems

fundamentallyalientomuchoftheworkthatwearefamiliarwithincontemporary

musicologicalandmusic-theoreticalthought.ThefigureofPythagorasismorethan

emblematichere:thenumericalwayofthinkingaboutsound,whichplayssucha

smallpartintoday’smusicaldiscourse,isfundamentaltovirtuallyallmusic-

theoreticalinstruments.Sincethisfundamentalconceptionfeelssounfamiliartous,

wedowelltolearntoappreciatethesubtletiesbetweendifferentconceptions.Not

allnumber-basedapproachestosoundarePythagorean,andaswesawabove,there

areimportantdistinctionsbetweenanabstractmathematicalwayof104SeealsoPeterGallisonandLorraineDaston’sclassicstudy,“TheImageofObjectivity,”Representations40(1992),81-128.

54

conceptualizingsoundandphysical-acousticalones—eventhoughfromourmodern

perspective,thesimilaritiesbetweenthemseembyfartooutweighthedifferences.

[7.5]Andthird,wedowelltoreconsidertherelationshipbetweenmusic-

theoreticalinstrumentsandthemusictheorytheyoccasion.Ourstartingpoint,

Pythagorasplayingthemonochord,providesagoodexamplehere.ClassicistDavid

Creese,aftercarefullyreviewingtheancientsourcesmentioningPythagorasandthe

monochord,concludedthattherewasnocontemporaneousevidencethat

Pythagorasactuallyusedthemonochord.Henotedthatthefirstdocumenttomake

theconnectionbetweenthetwowastheverypassagefromNicomachuswe

encounteredinitially.Nicomachus,itshouldberemembered,livedsomesix

centuriesafterPythagoras;sohistestimonywasatbestbasedonhearsaypassed

downthegenerations.Nicomachus’claimhasbeenrepeateduncriticallythroughout

thecenturies,andovertimePythagorasevenmorphedintotheinventorofthe

instrument.Temptingasitis,weshouldnotassumethemonochordwasaround

whenthemathatthebasisofmusicalrelationswasfirstworkedout.Itbecamean

objectofscientificexperimentationmuchlaterthanthemathematicalrelationsit

proved.Creeseconcludesforcefully:“[D]oingmathematicalharmonicswithoutthe

monochordwasnotonlypossible,but…thereisnocredibleancientevidenceto

suggestthattherewasanyotherwaytostudythesubjectbeforethelatefourth

century.105Hepointsoutthatthe“mirage”thatPythagorasoperatedwith,oreven

inventedthemonochord,hasbeenconsistentlyfedbytheassumptionthatthe

105Creese,Monochord,92.Onthebasisoftheavailableevidence,CreesetracesthemonochordtotheEuclidiantreatiseSectiocanonis(Κατατομὴκανόνος)datingfromthe3rdcenturyBCE.Pythagorasisnotassociatedwiththisinstrumentinthistreatise.

55

discoveryofharmonicratiosandtheinventionofthemonochordmusthavebeen

concurrent.Themonochordseemstoservenootherpurposethantomeasureand

demonstratetheratiosofmusicalintervals,whichencouragedauthorsfromlate

antiquitytothemostrecentpasttoassumeaconcurrencyorevencausalrelation.

[7.6]Inaword,Pythagorasdidnotplaythemonochord.Wecanputthismythto

restonceandforall.LooseningthefirmtiesbetweenPythagorasandhis

monochord,betweentheoristandinstrument,willalsoallowustorevisittwoparts

ofourearliermethodologicalobservations:Wedefinedthemusic-theoretical

instrument,ontheonehand,intermsofBachelard’s“materializedtheory,”andon

theother,intermsofRheinberger’s“epistemicthing.”Infact,thetwomodelscover

oppositeendsatthespectrumofpossibilities:whileBachelardconsidersthe

instrumenttobethereifiedembodimentofatheory,Rheinbergerconceptualizes

theinstrumentasamaterialobjectthatallowscertaintheoreticalpropositionsto

issueforth.Putdifferently,RheinbergerisMarxtoBachelard’sHegel.Yet,thisdoes

notmeaninthisexplorationofmusic-theoreticalinstrumentsthatwemustmakea

firmcommitmenttoonepositionortheother.Itisnotnecessarythatthemachine

givesriseto,oremergesfromatheory,butratheritcanembodyorexemplifyit.In

thetwoexamplesatplaywesawtwodifferentwaysinwhichtheinstrument

featured:inthecaseofVicentinoandthearchicembalo,theinstrumentwasthe

linchpinthatheldmusicaltheoriesandpracticetogetherinasituationwherethe

paradigmofunassistedvocalmusicnecessarilyfailed.InthecaseofHenryCowell

andtherhythmicon,bycontrast,wesawhowtheinstrumentexplicitlypiggy-backed

onmechanicaldevicesthathadbeenaroundforacenturyandtransformedsome

56

preexistingtheoriesofmusicintoapracticalapplicationthatcouldthenbeusedin

composition.Alloftheseinstrumentsarelinkedbythefactthattheyputtothetesta

specificideaabouthowmusicalsoundworks—howitcanorshouldwork.Inthis

way,theorizing,composing,andhearingcometogether,inoftensurprisingways.

Othercasesarepossibleinwhichmusicalinstrumentsfulfillthefunctionof

“epistemicthings”andprovideuswithmusicalknowledge—ifweonlyknowhowto

makeuseofthem:eitherbyperformingexperimentsonthem,ormusic,orboth.