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23 Pages·2016·0.49 MB·English

by  Stefano Isola

NISSUNAUMANAINVESTIGAZIONE SI PUO DIMANDARE VERASCIENZIA
S’ESSA NON PASSA PER LE MATEMATICHE DIMOSTRAZIONI
LEONARDO DAVINCI
vol. 4 no. 3-4 2016
Mathematics and Mechanics
of
Complex Systems
STEFANO ISOLA
“MATHEMATICS” AND “PHYSICS”
IN THE SCIENCE OF HARMONICS
msp
MATHEMATICSANDMECHANICSOFCOMPLEXSYSTEMS
Vol.4,No.3-4,2016
M ∩ M
dx.doi.org/10.2140/memocs.2016.4.213
“MATHEMATICS” AND “PHYSICS”
IN THE SCIENCE OF HARMONICS
STEFANO ISOLA
Someaspectsoftherolethatthescienceofharmonicshasplayedinthehistory
of science are discussed in light of Russo’s investigation of the history of the
conceptsof“mathematics”and“physics”.
1. Theramblingrouteoftheancientscientificmethod
In several places in Russo’s writings on the history of science, one can find en-
lightening discussionsabout themeanings ofthe conceptsof “physics” and“math-
ematics”, along with the particular notions of truth involved in them; see, e.g.,
[58,Chapter 6.6;60, Chapter15; 56;57]. Bothtermsderivefromthe Greek: the
originalmeaningoftheformerwastheinvestigationofeverythingthatlives,grows
or,more generally,comesintoexistence,whereas thelatterreferredtoallthatis
studied,thusderivingitsmeaningnotfromitscontentbutfromitsmethod. Inthe
Hellenisticperiod,theterm“physics”continuedtobeusedtoindicatethatsector
ofphilosophythataddressednature(theothersectorsbeingethicsandlogic),thus
corresponding to what came to be called “natural philosophy” in modern times.
On theother hand, theterm “mathematics”was usedto indicateall the disciplines
(includinggeometry,arithmetic,harmonics,astronomy,optics,mechanics,hydro-
statics, pneumatics, geodesy and mathematical geography) that shared the same
methodofinvestigation,basedontheconstructionoftheoriesbywhich“theorems”
areproved,leaningonexplicitlystatedinitialassumptions. Itsmeaningthuscorre-
spondedtowhatwecall“exactsciences”andreferstoaunitarybodyofscientific
disciplinesalientothemoderndistinctionbetweenphysicalandmathematicalsci-
ences.
Inantiquity,howthescopeofmathematicscontrastedwiththatofphysicswasa
topicof muchdebate. Accordingto somekeytestimonialsreportedand discussed
in[56]—inparticularthatduetoGeminus(andreportedbySimplicius)inthefirst
CommunicatedbyRaffaeleEsposito.
MSC2010: 01A20.
Keywords: harmonictheory,mathematics,physics.
213
214 STEFANOISOLA
centuryB.C.1 and,muchlater,thatofThomasAquinas2—the“physicist”would
be able to grasp the “substance” of reality using philosophical categories, whereas
a characteristic feature of the work of the astronomer, that is, the “mathemati-
cian”, is its incapability to assert absolute truths, in that he is able to rigorously
deduce/construct a number of consequences from previously stated hypotheses
whoseultimatevalidityremainshoweveroutofcontrol.
Tobetterunderstandthisdiscrepancy, let’sstepbackagainto highlight another
important methodological difference between natural philosophy and exact science
inantiquity,inthattheformeroperatesonasinglelevelofdiscourse,wheredata
fromexperienceandthoughtsareorganizedsoastoproduce“directly”arational
account of the perceptions themselves.3 In particular, natural philosophy starts
“fromthethingswhicharemoreknowableandobvioustousandproceedstowards
thosewhichareclearerandmoreknowablebynature”[4,p.184a],thusrevealing
the alleged genuine, mind-independent nature of things. This is also reflected in
the useof language. As reported bythe fifth-century Alexandrianscholar Ammo-
nius,“...Aristotleteacheswhatthethingsprincipallyandimmediatelysignifiedby
sounds[e.g.,namesandverbs]are,andthesearethoughts. Throughtheseasmeans
1“Thephysicistwillproveeachfactbyconsiderationsofessenceorsubstance,offorce,ofits
beingbetterthatthingsshouldbeastheyare,orofcomingintobeingandchange;theastronomer
willprovethembythepropertiesoffiguresormagnitudes,orbytheamountofmovementandthe
timethatisappropriatetoit.Again,thephysicistwillinmanycasesreachthecausebylookingto
creativeforce;buttheastronomer,whenheprovesfactsfromexternalconditions,isnotqualifiedto
judgeofthecause...sometimesheinventsbywayofhypothesis,andstatescertainexpedientsbythe
assumptionofwhichthephenomenawillbesaved.Forexample,whydothesun,themoon,andthe
planetsappeartomoveirregularly?Wemayanswerthat,ifweassumethattheirorbitsareeccentric
circlesorthatthestarsdescribeanepicycle,theirapparentirregularitywillbesaved;anditwillbe
necessarytogofurtherandexamineinhowmanydifferentwaysitispossibleforthesephenomena
tobebroughtabout...”[34,p.276].
2“Reasonmaybeemployedintwowaystoestablishapoint:firstly,forthepurposeoffurnishing
sufficientproofofsomeprinciple,asinnaturalscience,wheresufficientproofcanbebroughttoshow
thatthemovementoftheheavensisalwaysofuniformvelocity.Reasonisemployedinanotherway,
notasfurnishingasufficientproofofaprinciple,butasconfirminganalreadyestablishedprinciple,
byshowingthecongruityofitsresults,asthetheoryofeccentricsandepicyclesisconsideredas
establishedinastronomy,becausetherebythesensibleappearancesoftheheavenlymovementscan
beexplained;not,however,asifthisproofweresufficient,forasmuchassomeothertheorymight
explainthem”[1,pp.63–64].
3IncriticizingProtagoras’sstatementthatmanisthemeasureofallthings,Aristotlesays,“Wesay
thatknowledgeandsense-perceptionarethemeasureofthingsbecauseourrecognitionofsomething
isduetothem”[41,p.184].Tohim,therefore,sense-perceptionandknowledgearethefacultiesthat
furnishallourunderstandingofthingsandthusexhaustedallpossiblemeaningsoftheexpression
“criterionoftruth”. InHellenisticpractice,however,othermeaningsofthisexpressionwereput
forward(ofwhichProtagoras’sdictumcouldbeconsideredalikelyprecursor)includingtheStoics’
infallibleactofcognitionbasedonkatalepticimpressions(self-certifyingactsofsense-perceptions)
aswellasthehypothetico-deductivemethodofexactsciences.
“MATHEMATICS”AND“PHYSICS”INTHESCIENCEOFHARMONICS 215
wesignifythings;anditisnotnecessarytoconsideranythingelseasintermediate
betweenthethoughtandthething,astheStoicsdo,whoassumewhattheyname
tobethemeaning[lekton]”[61,p.77].
Thus,atvariancewiththeAristotelianpointofview,theearlyStoicsconsidered
itnecessarytodistinguishbetweenthepronouncedsoundandthemeaningofwhat
ispronouncedasanintermediatelinkbetweenathoughtandasound. Thesame
kind of epistemological attitude characterized the exact sciences—which flour-
ished in the same period of the early Stoic school—with their specific effort to
overcometheillusionofbeingabletobuildintellectualschemesbaseddirectlyon
perceptiblerealityand theelaborationofabstractlanguagescapable ofdescribing
notonlyaspectsofthesensibleworldbutalsootherdesignablerealities;see[58],in
particularChapter 6. Theexistenceof adouble levelof discourseseems therefore
anessentialfeatureofexactsciences,inthattheirassertionsdonotdirectlyconcern
the things of the natural world but rather theoretical entities which are obtained
by a procedure of “pruning” which allows one to focus on certain aspects of the
phainomena—that is, what appears to the senses and calls for an explanation—
andtoignorethoseconsideredunessential. Inbrief,themethodologicalmarkof
exactsciencesconsistsintheconstructionofsimplifiedmodelsofaspectsofreality
which,startingfromsuitablebut“unjustified”hypotheses,operateontheirinternal
entities in a logically rigorous way and then move back to the real world. Note
that,byitsverynature,everyhypothesisissomehow“false”,sonothingprevents
different models based on different hypotheses of being capable of “saving” the
samephenomena. Inaddition,whiletheassertionsobtainedatthetheoreticallevel
are“objective”anduniversallyvalid,thecorrespondenceruleswhichtransformthe
entitiesinvolvedintherealworldandtheclaimsaboutthemintotheoreticalentities
and theoretical statements are instead historically determined. For example, in
Hellenisticscientifictheoriesdealingwithphenomenarelatedtothesenseofsight,
devices such as ruler and compass, designed to assist in the construction of the
straightlineandthecircle,aswellasinthemeasurementoftheirparts,incorporate
thecorrespondencerulesrelatingtheoreticalstatementsofgeometryoropticsto
concreteobjects. Intheories ofacoustic-musicalphenomena, thisrolewas played
insteadbythecanon(seebelow). Inbothcases,the“concreteobjects”—drawings
withrulerandcompassandpitchesproducedbyapluckedstring,respectively—
arenotroughnaturaldata: rather,theyaretheresultofasomewhatrefinedhuman
activity,whichinturnisrootedinthehistoricalandculturalcontext.
Although rarely acknowledged, the scientific method, as a cultural product of
earlierHellenistictimes,underwentarapiddeclineinthecontextofamoregeneral
culturalcollapsethatoccurredduringthesecondcenturyB.C.4 Notwithstanding
4Particularly dramatic were the years 146–145 B.C., with the sharp hardening of the Roman
policyintheMediterraneanthathadamongitsconsequencesthereductionofMacedoniatoaRoman
216 STEFANOISOLA
thelossofamajorpartofancientknowledge,thememoryofHellenisticscience
survived thanks to a series of geographically localized revival periods.5 On the
other hand, a peculiar feature of these revivals was the insertion of individual
contents, recovered from ancient science or derived from it, into foreign overar-
ching systems of thought which provided their main motivating framework.6 In
particular, “in the Age of Galileo”, Russo says, “the exact science preserved the
unity that distinguished the Greek models, from which it drew the terminology,
but the ancient method was rarely understood. Not that the explanation reported
by Simplicus had been forgotten, but few, as Stevin, used the freedom of choice
ofthehypothesestobuildmodels;muchmorefrequentlytherelativearbitrariness
oftheinitialassumptionsappeared(asithadappearedtoSimpliciusandThomas
Aquinas)asaparticularity(aswellasanoddity)ofthemethodofthe‘mathemati-
cian’,whichdetermineditsinferioritywithrespecttophilosophersandtheologians,
whoknewhowtodistinguish‘truth’from‘falsehood”’[56,p.37]. Theideathatthe
hypothetico-deductivemethodwasmostlyalimit thatpreventedapproachingthe
absolutetruthpeakedwithNewton. InthewellknownGeneralScholiumaddedto
thePrincipiain1713,hewrites,“ButhithertoIhavenotbeenabletodiscoverthe
causeofthosepropertiesofgravityfromphenomena,andIframenohypotheses
[hypothesesnonfingo]. For whatever isnot deduced from the phenomena, is to be
calledanhypothesis;andhypotheses,whethermetaphysicalorphysical,whether
of occult qualities or mechanical, have no place in experimental philosophy. In
thisphilosophyparticularpropositionsare inferred fromthephenomena,andafter-
wardsrenderedgeneralbyinduction”[45,p.392].
It is worth stressing that the term “phenomena” is used here with a meaning
which differs considerably from the ancient one, in that it refers to something
which lies beyond our perception.7 Likewise, the term “hypothesis” was given
thenewmeaning—stillinuse—ofastatementlyingatthebeginningofourinter-
pretationoftheexternalworldbutwaitingtobecorroboratedorrefutedassoonas
the“facts”areknownwithsufficientdetail. Thus,ineverygenuinesearchforthe
province, the razing of Carthage and Corinth and the heavy political interference in Egypt with
persecutionandexterminationoftheGreekintellectualclass[59,Chapter5].
5Thefirstofthemwastheresumptionofscientificstudiesinimperialtimes,whosemainprotago-
nistswereHeron,PtolemyandGalen.Thenextonesoccurredinthesixth-centuryByzantineworld,
theninthemedievalIslamicworld(eighthtoninthcenturies)andfinallyinWesternEurope,from
the“twelfth-centuryRenaissance”untiltheRenaissanceparexcellenceofearlymoderntimes[58,
Chapter11].
6WeshalldiscussbelowanexamplewhichillustratesthisfactinconnectionwithPtolemy’swork.
7Thinkoftheabsolutemotionsofmaterialbodieswithrespecttotheimmovablespacewhich,
coexistingwithAristarchanheliocentrism,cannotcorrespondtoanyobservabledatum(seethedis-
cussiongivenin[58,Chapter11.7]).Inaletterof1698,Newtonaffirmed,“Iaminclinedtobelieve
somegenerallawsoftheCreatorprevailedwithrespecttotheagreeableorunpleasingaffectionsof
alloursenses”[46,LetterXXIX].
“MATHEMATICS”AND“PHYSICS”INTHESCIENCEOFHARMONICS 217
truth, hypothesescannot be anythingbut a hindrance.8 As itis wellknown,New-
tonianismwaspresentedintheEuropeancontinentasthephilosophyofprogress.
ThemostfamousofhissupporterswasVoltaire,whointheprefaceoftheFrench
translationofthePrincipiadismissedas“foolish”thefollowersofvorticesformed
by the “thin matter” of Descartes and Leibniz and affirmed that only a follower
of Newton could be truly called a “physicist”. Indeed, according to Russo, the
spread of Newtonian mechanics has brought with it the way of reasoning on the
basisofwhich“theexactsciencegotbrokenintotwostumps: ‘mathematics’and
‘physics’. Bothoftheminheritedfromtheancient‘mathematics’thequantitative
approachandseveraltechnicalresults,andfromtheancient‘physics’(thatisfrom
naturalphilosophy) theidea ofproducing statementswhichare absolutely‘true’.
The essential difference was lying in the nature of such truth. While the truth of
theassumptions of‘mathematics’(calledpostulates) wasconsideredimmediately
evident,theassumptionsof‘physics’(calledprinciples)wereregardedtrueinas-
much as they are ‘proven by the phenomena’...It is plain that these differences
were strictly connectedto the diverse nature attributed to the entities studied bythe
twodisciplines: the‘mathematical’entities,althoughusabletodescribeconcrete
objects, wereconsideredabstract, whereasthe ‘physical’entities wereconsidered
asconcreteastheobjectstheywerereferringto”[56,pp.42–43]. Inbothcases,the
“truth”ofascientifictheory(e.g.,atheoryoftheplanetarymotions)doesnotliein
its capabilityto “savethe phenomena”(e.g., to determinewith someaccuracy the
observablepositionofaplanetatanytime)butbecomessomethingthatonecan
“prove”by means ofits owninstruments, inthe same wayin whichone can prove
a statement on the entities internal to the theory itself. If so, a scientific theory
would cease to be a theoretical model, instead becoming a system of statements
settodescribethetruenatureoftherealworld.9
8Asd’Alembertwrote,“itisnotatallbyvagueandarbitraryhypothesesthatwecanhopeto
knownature;itisbythoughtfulstudyofphenomena,bythecomparisonswemakeamongthem,
bytheartofreducing,asmuchasthatmaybepossible,alargenumberofphenomenatoasingle
onethatcanberegardedastheirprinciple”[23, p.22], andalittlefurther, “letusconcludethat
thesingletruemethodofphilosophizingasphysicalscientistsconsistseitherintheapplicationof
mathematicalanalysistoexperiments,orinobservationalone,enlightenedbythespiritofmethod,
aidedsometimesbyconjectureswhentheycanfurnishsomeinsights,butrigidlydissociatedfrom
anyarbitraryhypotheses”[23,p.25].Asanaside,thissemantictransformationmayhaveplayeda
roleinclaiminga“historicmission”tohumanknowledge,fromthenaivetyofthemythtowardsthe
finalenlightenment,passingthroughanincreasingcontrolofthesourcesoferrorwhichallowsone
toprogressivelyovercomeall“falsehypotheses”(thatis,“prejudices”):akindofsecularizedversion
ofthemedievalmillenarianism,ofwhich,amongothers,Newtonwasanardentsupporter.
9SuchaprescientificpositionisproudlymaintainedbyVoltaireintheentry“System”of[65,
p.224], whichstartsbystating, “Weunderstandbysystemasupposition; forifasystemcanbe
proved, itisnolongerasystem, butatruth. Inthemeantime, ledbyhabit, wesaythecelestial
system,althoughweunderstandbyittherealpositionofthestars”.Bytheway,andnotsurprisingly,
218 STEFANOISOLA
Followingthisreshapingofthescientificenterprise,somedisciplineshavebeen
counted on one side and others on the other; still others have been somehow in-
ternally divided or eventually disappeared, as we shall see below in a particular
example. Referring to the already cited writings of Russo for a discussion of
the splitting between “mathematics” and “physics” in nineteenth and twentieth
centuries,letusjustremarkthatthefinalfailureoftheeffortstowardsamethod-
ological reunification of the exact science, of which Poincaré was a prominent
exponent, and the prevailing of powerful trends towards specialization and frag-
mentation ofthe scientificdisciplines, ifon oneside hasled some to wonder what
mysteryliesbehindthe“unreasonableeffectiveness”ofmathematicsinproviding
accuratedescriptionsofthephenomena[67],ontheothersidepromptedoneofthe
greatestcontemporarymathematicianstoacknowledgeinthistrendaseverecrisis
ofscienceitself: “Inthemiddleofthetwentiethcenturyitwasattemptedtodivide
physicsand mathematics. The consequences turnedoutto becatastrophic. Whole
generationsofmathematiciansgrewupwithoutknowinghalfoftheirscienceand,
ofcourse,intotalignoranceofanyothersciences. Theyfirstbeganteachingtheir
ugly scholastic pseudomathematics to their students, then to schoolchildren (for-
gettingHardy’swarningthatugly mathematicshasnopermanent placeunderthe
Sun)”[7];seealso[57]forafurtherdiscussion.
2. Acoustic-musicalphenomena
“Ho detto che la nostra scienza o arte musicale fu dettata dalla
matematica. Dovevadirecostruita. Essascienzanonnacquedalla
natura,...ma ebbe origine ed ha il suo fondamento in quello che
ègiustamentechiamatosecondanatura,machealtrettantoatorto
quanto facilmente e spesso è confuso e scambiato...colla natura
medesima, voglio dire nell’assuefazione. Le antiche assuefazioni
de’greci...furonol’origineeilfondamentodellascienzamusicale
da’grecideterminata,fabbricataeanoine’librienell’usotraman-
data, dalla qual greca scienza vien per comun consenso e confes-
sionelanostraeuropea”(G.Leopardi,Zibaldone[40,3125–3126]).
Todaymusicaltheoryismainly“thestudyofthestructureofmusic”,whereas
originallyit waspartofmathematics. Whathappenedin themeantime? In order
to getan idea, itis necessaryto go backagain to the rambling routeof the ancient
scientificmethodthroughthesubsequenthistory. Resumingwhatwassaidinthe
previoussectioninaconcisealbeitvagueway,wecansaythatthegeneralobjects
of Greek science were not so much the “laws” of the natural world viewed as
thisentryproceedsbystrengtheningtheideaofanecessaryprogressionoftheknowledgebydenying
thatAristarchusintroducedheliocentrism.
“MATHEMATICS”AND“PHYSICS”INTHESCIENCEOFHARMONICS 219
an entity independent from the man who observes it but rather those indubitable
epistemological data provided by the phainomena resulting from the interaction
betweensubjectandobjectthroughactiveperception. Inparticular,themodelsof
Hellenistic exact science were primarily suited for that purpose: the creation of
theoreticalentities asintermediateutterancesbetween therealobjects andabstract
truths has the effect of making that interaction available to conscious manipula-
tion. This gets a peculiar meaning within the context of music theory which, as
such, establishes sound, the material aspect of music, as something which can
be knowingly investigated in connection with human experience. Although mu-
sic—perhapsthemostunfathomableexpressionofpsychicactivity—mightnot
seem properly suited to scientific analysis, the investigation of acoustic-musical
phenomenanonethelessprovidesanexamplewheretheepistemologicalopposition
sketchedintheprevioussectionoccurredwithastrikingcharacterwithinthesame
domain,asweshallnowbrieflyoutline.
ItisratherwellknownthatPythagoreanmusictheory—asapartoftheirpro-
gram of liberation of the soul by means of the intellectual perception of propor-
tions in all things—starts from the recognition that the harmonic intervals can
be expressed as simple numerical ratios. The following “Pythagorean principle”
hasbeenviewedasthefirst“naturallaw”expressedintermsofnumericalentities
(see, e.g., [11]): if two sounding bodies, such as stretched strings or sounding
pipes, havelengths whichare insimple proportions,and allother aspectsare kept
fixed, together they will produce musical intervals which are judged by the ear
tobeinharmoniousagreement,or“consonant”. Conversely,allintervalsthatthe
earacceptsasconsonantcanberepresentedasratiosofnumbersfromthetetrad
1,2,3,4.10 The harmonic system of Philolaus (see, e.g., [18; 19]), for example,
isastructureofintervalsexternallylimitedbytheoctave(diapason),whoseratio
is2:1,andinternallyarticulatedbyintervalsoffifths(diapente),withratio3:2,
and fourths (diatessaron), with ratio 4 : 3. If we want to find four quantities—
for instance the lengths of the strings of a four-string lyre—that, taken in pairs,
reproduce these ratios, then we can choose a unit of measure so that the longest
stringis12units,theintermediateones9and8andtheshortest6. Itisclearthat
the systemof reciprocal ratios, andtherefore the wholeharmonic structure, does
notchangeifthestringshavelengths12,9,8and6meters,centimeters,stadiums,
etc. Finally,observingthat(3:2):(4:3)=9:8,theintervalofatone,equivalent
to thedifference betweena fifthand afourth, isrepresented bythe ratio9:8. The
octaveisthus“harmonically”dividedintotwofourthsspacedbyatone.11
10The question of which observations lay behind the detection of these ratios and when this
happenedishardtoanswer[13].
11Notethat6·12=8·9,i.e.,thefournumbersareingeometricalproportion. Moreover,8=
2:(1+ 1 )and9=(6+12):2;namely,8and9aretheharmonicmeanandthearithmeticmean,
6 12
220 STEFANOISOLA
LetuspointoutthatinthetransitionfromtheHellenictotheHellenisticperiod
mathematicsbecomesanexactscience,inthesensespecifiedabove,notonlyby
distinguishingtheoreticalentitiesfromconcrete objectsbut alsofrompureabstrac-
tions in the “platonic” sense. While discussing the subjects for the education of
the“Guardians”oftheRepublic,PlatoletsSocratesconceivethat“astheeyesare
designedtolookupatthestars,soaretheearstohearharmonicmotions”,therefore
agreeing that astronomyand music theory aresistersciences, as the Pythagoreans
said[50,530d]. Ontheotherhand,thosescholarsarejudgedinadequatetoreach
the “true knowledge” beyond the sensible world in that “their method exactly cor-
responds to that of the astronomer; for the numbers they seek are those found
in these heard concords, but they do not ascend to generalized problems and the
considerationwhichnumbersareinherentlyconcordantandwhichnotandwhyin
eachcase”[50,531c]. Clearly,thejustmentioned“ascension”aboveexperience
does not need to be embedded in a theory. Rather, it would rely on “evidences”
perse.
In a different direction, music theory, or at least that part of it dealing with
tuning systems, was set to become a scientific discipline by putting together the
arithmetictheoryofproportionsandtherecognitionoftheproportionalitybetween
the pitch of the sounds and the speed of the vibrations that produce them,12 a
conceptual step that according to some sources had been made in the circle of
Archytas in about 400 B.C. [13; 36]. The “experimental device” enabling the
establishmentofacorrespondencebetweenconcordsandnumericalratioswasthe
canon (kanon harmonikos), an instrument that in its simplest form is made of a
single string stretched between two bridges fixed on a rigid base and equipped
with another movable bridge by which one may divide the string into two parts,
yieldingsoundsof variablepitch. One canfurtherimagine arowfixed atitsbase
on which the positions of the movable bridge corresponding to the notes can be
marked. The name of the entire device is then a metonym for the line segment
thatrepresentsitasatheoreticalentity. ThetheoryoutlinedintheSectiocanonis,
attributed to Euclid, deals precisely with the harmonic divisions of this segment,
i.e.,withthosedivisionscorrespondingtomusicalintervalsjudgedtobeconsonant
[25];seealso[26]. Inthiswork,farawayfromanymysticalefflorescenceabout
themusicofthecosmos,aschemeofdivisionoftheoctavebymeansofthetheory
of proportions contained in the Elements is proposed with the aim of producing
patterns of consonant intervals adoptable in practice, e.g., when tuning musical
respectively,oftheextremes6and12.Theexclusionofthegeometricmeanin“Pythagorean”music
theoryisjustifiedbyanimpossibilityresultduetoArchytas(seebelow).
12Therecognitionofthenatureofsoundasvibrationofair,withalternationofrarefactionand
compression,canalreadybefoundinAristotle’sProblemata[3]aswellasinthePeripateticDe
audibilibus[5],whereastheideaofasoundwaveisattestedatleastasearlyasintheStoa.
“MATHEMATICS”AND“PHYSICS”INTHESCIENCEOFHARMONICS 221
instruments. Alongthispath,thebranchofGreekmusictheoryreferredtoasthe
scienceofharmonicsenteredtheunitarybodyofHellenisticmathematics,along
withastronomy,arithmetic,geometry,optics,topography,pneumatics,mechanics
andotherdisciplines[58,Chapter3].
In the short introduction of the Sectio canonis, the author establishes a corre-
spondence between musical intervals and numerical ratios and states the main
hypothesis underlying the model: consonantintervalscorrespondtomultipleor
epimoric ratios.13 The rationale of this postulate relies on the observation that,
as consonant intervals produce a perception of unity or tonal fusion between the
notes,theymustcorrespondtonumberswhicharegivena“singlename”inrela-
tion to one another.14 On the other hand, this postulate is clearly false not only
becauseit includesamongtheconcords alsointervalsconsidereddissonantby the
Pythagoreanprinciplestatedabove,suchasthetone9:8ortheratio5:4(natural
majorthird),butalsobecauseitcountsasdissonanttheintervalcomposedbyan
octave plus a fourth, represented by the ratio 8 : 3, unanimously recognized as
consonant bythe music theorists ofantiquity (exactlyas an octave plus afifth, that
is,3:1). However,thisisnotaprobleminitself,forallhypothesesaresomehow
“false”: what matters is that the theory based on them is consistent and suited to
savethephenomenawhichitaimstomodel. Theintroductionisthenfollowedby
twentypropositions: thefirstnine,ofpure“numbertheory”,provideadeductive
constructionofthePhilolausharmonicsystemsketchedabove,whereastheremain-
ingonesform thepartproperlyrelevantto tuning systems. Of particularinterest
isthethirdproposition,whichstatesthatneitheronenormoremeanproportionals
canbeinsertedwithinanepimoricinterval. Inparticular,itisnotpossibletodivide
theoctaveintoequalpartsthatformarationalrelationshipwiththeoctaveitself.15
The consequences ofthis simple result have been the subjectof a controversy
whichhaslastedforovertwomillennia,atthebasisofwhichthereisthedistinction
13Greekarithmeticclassifiedratiosintothreebasictypes,whichreducedtolowesttermscorre-
spondington:1(multiple),(n+1):n (epimoricorsuperparticular)or(n+m):n,n>m >1
(epimericorsuperpartient). Notethatthefirsttwoareinaone-to-onecorrespondence: p :q is
multipleifandonlyif p:(p−q)isepimoricsothatq isthegreatestcommondivisorofboth p
and p−q.Inparticular,theoctave2:1,participatingtobothconsonantclasses,isthe“consonance
oftheconsonances”.
14Theinterpretationofthisseeminglyarbitrarycorrespondenceiscontroversial. Accordingto
somescholars[11],the“singlename”hastobeascribedtothefactthat,unliketheepimericratios,
multipleandepimoricratioswereindicatedwithaone-wordname,likeepitritos,“thirdinaddition”,
for4:3.Accordingtoadifferentinterpretation[26](basedon[54,§I.5]),the“singlename”isnota
linguisticunitybutanumericalone,correspondingtothegreatestcommon“part”whichcomposes
thenotesinbothmultipleandepimoricratios(seefootnote13).
15Norwoulditbepossibletodivideinthiswaythefifth,thefourthorthewholetone.Thisseems
tobethefirstimpossibilityresultsurelyascribabletoanauthorasBoethius[12,§III.11]reportsa
proofofitgivenbyArchytas.