Let’s Calculate Bach: Applying Information Theory and Statistics to Numbers in Music PDF
Preview Let’s Calculate Bach: Applying Information Theory and Statistics to Numbers in Music
Quantitative Methods in the Humanities and Social Sciences Alan Shepherd Let’s Calculate Bach Applying Information Theory and Statistics to Numbers in Music Quantitative Methods in the Humanities and Social Sciences SeriesEditors ThomasDeFanti,UniversityofCaliforniaSanDiego,LaJolla,CA,USA AnthonyGrafton,PrincetonUniversity,Princeton,NJ,USA ThomasE.Levy,UniversityofCaliforniaSanDiego,LaJolla,CA,USA LevManovich,TheGraduateCenter,CUNY,NewYork,NY,USA AlynRockwood,KAUST,Boulder,CO,USA Moreinformationaboutthisseriesathttp://www.springer.com/series/11748 Alan Shepherd Let’s Calculate Bach Applying Information Theory and Statistics to Numbers in Music AlanShepherd Dierdorf,Germany ISSN2199-0956 ISSN2199-0964 (electronic) QuantitativeMethodsintheHumanitiesandSocialSciences ISBN978-3-030-63768-2 ISBN978-3-030-63769-9 (eBook) https://doi.org/10.1007/978-3-030-63769-9 MathematicsSubjectClassification:94A15,62P99,65C05 ©SpringerNatureSwitzerlandAG2021 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Illustration:©bytheauthor. InGermanthenoteB iscalledBandBiscalledH,hencethenameBACHgivesthemelodyshown. Thenotevaluesinquarternotesare2,1,3,8whichisBACHinthenumberalphabetwithA=1, B=2,etc. Calculemus. Lasstunsrechnen. Letuscalculate. GottfriedWilhelmLeibniz(1646–1716) Foreword There is always one statement in a book that the author regrets almost as soon as it is published. In my case, it appears on page 140 of Bach’s Numbers and reads: “Theprobabilityofsixtermsbetween272and524fallingrandomlyintoaperfect double2:1proportionisminimal.Bachmusthaveplannedit”.Combiningcompo- sitionalintentionwithprobability,itwasaprovocativeinvitationtostatisticians.My regretwasshort-lived,though,asitquicklyledtoafruitfulcollaborationwithAlan Shepherdandtothisbook. In October 2016, I received a short email from Alan inquiring about the 2017 BachNetworkDialogueMeeting,whichincludedthepassingcomment:“Iamvery interestedinyourresearchonBach’snumbersandhaveyourtwobooks.Imayeven beabletocontributesomethoughts”.AsBach’sNumbers:CompositionalProportion andSignificancehadonlyjustbeenpublished,Irealisedthismustbeaseriousreader. Twomonthslater,Alansentanabstractofthetopichewashopingtopresentatthe DialogueMeeting:“Thispaperappliessomemathematicsandinformationtheoryto mapoutthepossibilitiesofencodinginformationinmusicalscoresandputsthisin perspectivewithsomecommonbiasesandfallaciestowhichmosthumansareprone whenevaluatingprobabilitiessuchashowlikelyorimprobableanobservationis”. WesetupaFaceTimecall. Overtheyears,myresearchintothenumericalaspectsofBach’scompositional processhashadamixedreception,withequalmeasuresofoutrightdismissal,enthusi- asticacceptanceandlukewarmscepticism.Criticalscepticismwasmydefaultsetting asamusicalanalyst.Myaimwastodiscoverwhatevidence,ifany,therewasbehind thepopularcrazeofinterpretingnumbersinthestructuresofBach’sscores.Todo thissystematically,IneededtoknowwhichnumbermethodsBachcouldhaveknown andwhichcouldhavebeenappliedtomusicalcomposition.Myresearchturnedup theoriginsofnumberalphabets,whichBachwouldhaveknown,andanexplanation of how and when the fashion for gematria became associated with Bach’s music (Tatlow 1991). Next I had toestablish aclear analytical method that would, as far aspossible,uncoverevidenceofBach’sworkingmethod.Ratherthanusingestab- lishedanalyticalmethods,IchosetousetoolsandconceptsthatBachhimselfwould recognise,astheypromisedtotakemeclosertoBach’sthoughtprocesses.Iscoured treatisesthatheownedorcouldhavereadforevidenceofhowhemighthaveordered ix x Foreword hiscompositionsnumerically,ifatall.Iusedpencil,paperandsimplearithmeticasI lookedatthestructuresofhisscores,intheirearly,revisedandfinalforms.Thistook methroughanexplorationofwhetherornothecouldhaveusedthegoldensection ortheFibonacciseries(Tatlow2006),tothediscoveryin2004ofwhatseemedtobe anextraordinaryone-offuseoftheproportions1:2inseverallayerswithintheSix SolosforViolin(BWV1001–1006).LatercametheunforgettablemomentwhenI realised that it was only in the final revisions that the bar totals were whole round numbersdividedbyclearlarge-scalelayersofproportion.Untilthatpoint,Ihadstood safely on the side of the sceptics, but when I tentatively presented the first results in December 2006 a colleague challenged me to cross the line. The first iteration of the theory of proportional parallelism was published soon after (Tatlow 2007). Like it or not, I found myself presenting new evidence to suggest that Bach had deliberatelyusednumberstocreateproportionalorderinhiscompositions.Tatlow (2007)heldmetoatheorywithathree-parttestthatIcontinuedtorefineasIcrit- icised,wroteandrewrotewhatwouldbecomeBach’sNumbers(Tatlow2015).As part of the test, I asked a colleague in 2010 to run a statistical exploration of my results for the Six Solos for Violin to see the probability that these patterns were intentionalonBach’spart.Heconcludedthatthe“nullhypothesis”gavenoreason tobelieve thatmyhypothesis wastrue.Thisrancounter toagutinstinctbasedon bothmygrowingunderstandingofhowBach’scontemporariesthoughtaboutmusic andthenumerousexamplesinBach’smusicwithlayersoflarge-scaledoubleand sometimestriple1:2or1:1proportion.Wasthenegativeresultbecausemycolleague hadnotfactoredinsufficientdata?Wasitthatstatisticaltechniqueswereunsuitable forhistoricalstudies?WhenAlanandIhadourfirstconversation6yearslaterthese considerationswereuppermostinmymind. InBach’sNumbers,Iwantedtopresenttheevidencetransparentlysothatscholars from all disciplines, with negative or positive expectations, could read and weigh up the evidence, test the theory for themselves, accept what rang true and reject andreplacewhatdidnot.InPartI—Foundations—Ipresentthestrandsofhistorical evidenceoncompositionalprocess,paralleltechniquesandbeliefsaboutproportions. InPartII—Demonstrations—Igiveevidenceofthecompositionalhistorywithtables of the final totals and their layers of 1:1 and 1:2 proportions for each of Bach’s publishedworks.Sometablesaremoreconvincingthanothers,andIlikesomeof themmorethanothers.Itseemedafairerandmorehonesttestofthetheorytoinclude thedemonstrationsthatIfoundlessconvincing.Itisclearfromthevarietyofresults that Bach had principles of ordering, but that there was no rigid code to which he adhered. I discovered that Bach’s works were layered with numerous proportioned units nesting within each other, which I called parallel proportions. Combined with the historicalevidence,itwastheingenuityoftheselayersthatpersuadedmetheyhad been planned. A single 1:1 or 1:2 proportion formed by the number of bars, the numberofmovementsorworksorbytheirsymmetricalarrangementcouldeasilybe coincidental.Adouble1:1or1:2proportion,formedbythenumberofbars,andthe number of movements or works, and their symmetrical arrangement seemed more intentional,particularlywhensubdividedintoafurther1:1or1:2proportion,andeven Foreword xi moresowhenthesamethreeelementscombinedtoformatriple1:1or1:2proportion. Thecaseforintentionalitybecameevenstrongerwhentheseproportionswerefurther combinedwithdoubleortriple1:1or1:2proportionsatdifferentstructurallevels, especiallywhenthesetwowereoftenlayeredwithtwoormorelayersofdoubleor triple1:1or1:2proportion. In the structure of Bach’s Six Solos for Violin, for example, the largest-scale double 1:1 proportion is between the 6 Solos:6 Sonatas each with 2400:2400 bars (Tatlow2015,133–158).Thenextlarge-scaledouble2:1proportioniswithintheSix Solos,where4:2Solosareproportionedas1600:800bars.Thisisparalleltoanother large-scaledouble2:1proportionwithintheSixViolinSonatas,with4:2Sonatasin 1600:800bars. Atasmallerscale,single1:1and1:2proportionsarefoundinthefourmovements oftheGminorSonata(BWV1001),with136:136barsand136:272bars(including repeats).Thisisparalleltoyetanothersmaller-scalesingle2:1proportioninthefour movementsoftheBminorPartita(BWV1002),both272:136barsand544:272bars withrepeats.Astheproportionsareformedwithunitsof136bars,thereareseveral additionallayersofproportionbetweentheGminorSonataandtheBminorPartita. These results were supported by evidence of compositional adjustments that Bach made as the work evolved. Combined with the many good theological and aestheticreasonsthatBachandhiscontemporarieshadforcreatingperfectlyexecuted symmetry,1:1and1:2proportions,itfeltacceptableinthisinstancetoclaiminten- tionality. In 2016, Alan Shepherd set out with an open mind to test my hypoth- esis,initiallydevelopingtheProportionalParallelExplorerprogramtofindsimple proportionsandovertimerefiningittoisolatesimultaneouslayeringofproportions withsymmetricallayouts.Ofinteresttousbothweretoseehowfarstatisticaland computer-basedmethodscouldshedlightonwhetherBachintendedtheresultsIhad discovered,whetheritwas“analyticalcoincidenceorBach’sdesign”,asIaskedin 2007(Tatlow2007). It is fairly alarming to have one’s work scrutinised in minute detail. It is also a great honour. When Alan Shepherd and I first spoke in 2016, we had no idea of whatwouldresult.Forme,ithasmeantgainingacolleagueandfriend,buryingany regretsaboutthatsentence(see10.4)andrealisinganewthatgoodold-fashionedgut instinct still has a place in scholarship. I am grateful to Alan for his patience and humility,anddelightedthatwiththisbookhehassucceededinpushingtheboundaries of musicological methods, to narrow the gap at the intersection of compositional intentionalityandnumericalconstruction. Danderyd,Sweden RuthTatlow July2020
The list of books you might like
