Detectio of Historical Period i Symbolic Music Text Michele Della Vetura Abstract Despite the various studies o computer-aided musical aalysis, there have bee relatively few attempts at tryig to locate, by meas of aalysis, a give melody i a certai stylistic period (Baroque, Classical, Romatic or Cotemporary). The mai problem is that a compositioal style of a certai historic period is difficult to formalize. This study presets a model of aalysis based o the theories of Warre Weaver ad Claude Elwood Shao, able to progressively explore the symbolic level of a melody, idetifyig the historic period o the basis of the iformatio that it carries. The cocept of iformatio has already bee used for several years ow i liguistic aalysis ad it has also bee applied to musical laguage. This approach was dealt with o the melodic level, omittig cocepts like toality, modulatio ad moreover rhythm. The efficiecy of the model was verified by aalyzig a series of melodies by differet authors ad from differet times (tryig to rage through the differet compositioal techiques by meas of a uique aalysis methodology) emphasizig both the strog poits ad the weak poits of the approach. Idex Terms Iformatio theory, musical surface, historical aalysis, segmetatio, etropy. sigle idetified fragmet to a previously compiled list of rules [3]: rules that may go well for a certai musical piece ad ot for some other musical piece depedig also o the compositioal style that chages i accordace with the historical time period [4]. The latter is oe of the aspects that are rather difficultly take ito cosideratio i computer-aided musical aalysis, because of the lack of well-defied parameters that would allow its formalizatio. TABLE I: AN EXAMPLE OF THE MAIN MUSICAL FUNCTION FOR MELODIC ANALYSIS Musical Fuctio Traspositio Iversio Mathematical Operator Tx(h) = h + x I i (h) = I (h I) = 2i h Musical Segmet I. INTRODUCTION Simulatig the huma itellect with the help of iformatio ad computer systems is a research area that cotiues to egage scholars i the field of musical aalysis: melodic, rhythmic ad harmoic aalysis. I all these cases we are dealig with aalyses that allow, a posteriori, a formalizatio of the aalysis tools, i.e. is the developmet of the istrumets required i order to describe i mathematical terms, the musical fuctios extesively exploited by composers (see Table I) [1]: fuctios as the traspositio (the process of movig a sequece of otes up or dow i pitch by a umber of semitoes (which represet the traspositio degree x) while keepig costat the relative distace betwee its composig otes), the iversio (the process through which a melody is trasformed by ivertig the directio of its sigle itervals i relatio to the iitial ote, so that every ascedig iterval becomes descedet ad vice versa.) or the retrogradatio (the process that cosists i represetig the melody backwards, from the last to the first ote) of a melodic or rhythmic cell [2]. These musical fuctios are used i the segmetatio algorithms of a musical piece i order to idetify the sigificat melodic ad rhythmic cells ad i may cases the fial choice of the segmet is performed by comparig every Mauscript received May 2, 2013; revised August 11, 2013. Della Vetura Michele Vetura is with the Music Academy Studio Musica Departmet of Techology, 31100 Treviso, Italy (e-mail: Michele.dellavetura@ti.it). DOI: 10.7763/IJEEEE.2014.V4.297 32 Retrogradatio R(h) h R(wh) = hr(w) This article will preset a model of melodic aalysis able to explore the symbolic level of the musical text, idetifyig the historical period o the basis of the iformatio that it carries. This paper is orgaized as follows. Sectio II describes the characteristics of the historical periods. Sectio III describes the iformatio theory. Sectio IV describes the aalysis of the musical message. Sectio V shows some experimetal tests that illustrate the effectiveess of the proposed method. Fially, coclusios are draw i Sectio V. II. CHARACTERISTICS OF HISTORICAL TIME PERIOD The style of a certai epoch, of a school, of a composer, of a musical work does ot emerge accidetally, they are ot casual products ad maifestatios of artistic will. O the cotrary, they are based o the laws of becomig, developmet ad decay of a orgaic ature. Music is a orgaism, a multitude of orgaisms that by meas of their variable relatioships ad iterdepedecies make up a etirety [5]. The evolutio of music may be see as a sufficietly cotiuous, uiterrupted, pheomeo, urtured by listeig to music from the past ad motivated by a desire to improve it [5]. Toality, for istace, is a laguage the maturatio of which was very slow ad did ot cease to develop for several
ceturies, before reachig its breakig poit i the early 900. This evolutio was achieved by virtue of a cotiuous culture ad experiece, eriched by meas of writte ad oral commuicatio forms. May we hope to fid models that iclude the style ad the rules typical to every epoch? A fudametal fact emerges from a macroscopic aalysis of the musical text of pieces belogig to differet historical time periods ad differet composers: A progressive icrease, i the various historical periods, of the diversity of itervals betwee various souds that make up the melody (see Fig. 1) [6]: there is a gradual trasitio from the itervals of secod, third ad fourth (a iterval that becomes a fifth if iverted) of the Baroque period to wider itervals, as the sixth, the seveth ad the ith (alog with a icreased use of dissoace) i the Romatic period up to eve wider itervals i the Cotemporary period. I the Romatic period, melody also displays some other characteristics (still o a melodic level) as [6]: the presece of chromaticism; the (expressive) use of ote repeats. (a) Baroque (J.S. Bach, ivetio BWV 772). (b) Romatic (F. Chopi, Nocture op. 9. 2). (c) Cotemporary (B. Bettielli, Prelude for piao). Fig. 1. Examples of melodies belogig to the time periods. Fig. 1 Esempi di melodie apparteeti rispettiv amete al periodo Barocco (J. S. Bach.), Romatico (.) e Cotemporaeo (.). A aalysis of this kid requires well-defied musical kowledge. The average listeer (this expressio is used to mea a perso who lacks specific kowledge of musical grammar) maages evertheless to comprehed, by simply listeig to various musical fragmets, more or less ample ad belogig to the same epoch or to differet epochs, a stylistic differece ad though ot maagig to idetify the related historical time period, he is able to perform a cataloguig based o persoal schemata (remidig, i this regard, that oe's past experieces ad ow culture may ifluece perceptio) [7]. I the listeer's perceptio it is ot oly the presece of certai types of itervals, but also the way i which these itervals follow oe aother that defie this cataloguig. I fact, it is the structure of the musical text that supplies precise iformatio to the listeer i this regard [8]. By meas of the quatificatio of the iformatio cotaied i the musical message it will be possible to idetify the historical time period ad, therefore, defie potetial rules of aalysis. III. INFORMATION THEORY The aalysis based o iformatio theory sees music as a liear process supported by sytax of its ow [9]. Nevertheless we are dealig with a sytax formulated ot o the basis of grammar rules, but o the basis of the probabilities of occurrece of every sigle elemet of the musical message i relatio their precedig elemet [10]. The fact that music is equated to a message trasmitted by a seder to a receiver, or that iformatio is dealt with may give the impressio of a aalytical approach of a hermeeutic kid, iterested i the meaigs of commuicatio [11]. I fact, here, the iterest goes exclusively to the maer i which the expectatios of the receiver are stimulated, satisfied or disappoited. From the defiitio of message beig a chai of discotiuous uits of meaig follows that the musical uits of meaig correspod to the miimal evets of a compositio: usually isolated otes, chords. Ay evet of such a chai demads a previsio about the evet that will follow: there is iformatio trasmissio whe the previsio is disregarded, there is o iformatio trasmissio whe it is cofirmed. Besides, the fact that the evets of a compositio are orgaized modularly brigs forward the possibility to calculate usig a formula, or to express with a idex the total amout of iformatio trasmitted by a certai musical segmet [12]. I a commuicatio that occurs by meas of a give alphabet of symbols, iformatio is associated to every sigle trasmitted symbol. Hece, the iformatio may be defied as the reductio of ucertaity that might a priori have ested i the trasmitted symbol [10]. The wider the message rage that the source ca trasmit is (ad the greater the ucertaity of the receiver with respect to the possible message), the larger the quatity of trasmitted iformatio ad alog with it its measure: the etropy. I the iformatio theory [8], etropy is a positive value as opposed to its origial egative couterpart i physics. Mathematically, the measure of the cotet of a piece of iformatio (I) is obtaied with Shao's formula: ' p I log 2 (1) p where p is the probability of the message to be trasmitted, p' is coected to the use that the observer makes of the message ad correspods to the probability for the cotet of the iformatio expected by the latter to be fulfilled after the trasmissio of the message. For every symbol (of a message) that we trasmit we have a certai quatity of iformatio associated to that symbol. I most practical applicatios of iformatio theory a choice must be made amog the messages of a set, every sigle oe havig its ow probability of beig trasmitted. Shao gave a defiitio of the etropy of such a set, idetifyig it with the iformatio cotet that the choice of oe of the messages will trasmit. If every sigle message has the probability pi of beig trasmitted, the etropy is obtaied 33
as the sum of all the set of fuctios p i log 2 p i, every sigle oe related to a message, that is: H ( X ) E I ( ) I( ) P( ) P( ) log (2) i 1 i 1 2 P( ) The term etropy, borrowed from thermodyamics, desigates therefore the average iformatio cotet of a message. I the light of what has bee said so far, the musical message may be defied as a sequece of sigals orgaized accordig to a code. 1 B. Quatitative Aalysis The data idetified by meas of aalytical aalysis (alphabet ad matrix of trasitios) allow the quatificatio of the iformatio cotaied i the musical message calculatig its etropy value o the basis of the formula specified i paragraph 3: H ( X ) E I ( ) I( ) P( ) P( ) log (3) i 1 i 1 2 P( ) TABLE II: EXAMPLE OF AN ALPHABET RELATED TO THE MUSICAL SEGMENT OF FIG. 1A 1 IV. ANALYSIS OF THE MUSICAL MESSAGE Give a melodic segmet, the idetificatio of the historical time period may be performed by meas of two distict aalyses which are cosecutive ad complemet each other: Aalytical aalysis: through the idetificatio of the itervals ad the probabilities for a certai type of iterval to resolve to aother type of iterval; Quatitative aalysis: through the idetificatio of the iformatio cotaied i the melodic segmet. A. Aalytical Aalysis To calculate the etropy ad therefore quatify iformatio, oe must refer to a specific alphabet: i the case of musical laguage, the classificatio of the various melodic itervals as symbols of the alphabet was cosidered. The classificatio of a iterval cosists i the deomiatio (geeric idicatio) ad i the qualificatio (specific idicatio) [13]. The deomiatio correspods to the umber of degrees that the iterval icludes, calculated from the low oe to the high oe; it may be of a 2 d, a 3 rd, 4 th, 5 th, ad so o.; the qualificatio is deduced from the umber of toes ad semi-toes that the iterval cotais; it may be: perfect (G), major (M), mior (m), augmeted (A), dimiished (d), more tha augmeted (A+), more tha dimiished (d+), exceedig (E), deficiet (df). For every sigle musical piece, or part of it, a table represetig its ow alphabet is compiled, cosiderig also for every iterval its ascedig or descedig evolutio (Table II). Determiig a alphabet is ot eough to calculate the etropy of a musical segmet: the way i which the itervals follow oe aother withi the musical piece must be take ito cosideratio. To do so the Markov process (or Markov stochastic process) is used: the choice was made to acquire the probability of trasitio that determies the passage from oe state of the system to the ext uiquely from the immediately precedig state (Table III). Please ote the way i which the cofiguratio of the statistic values i the table is a visual idicator of the probability of the itervals alphabet of the musical piece ad, therefore, of the redudacy of the melodic articulatios. The first colum idicates the deomiatio of the iterval (classificatio), the secod colum displays the ascedig or descedig movemet, ad the third colum presets the umber of semi-toes that make up the iterval (qualificatio) while the fourth colum shows the umber of semi-toes of that specific type idetified i the musical piece. TABLE III: EXAMPLE OF A FINAL MATRIX OF TRANSITIONS RELATED TO THE MUSICAL SEGMENT OF FIG. 1A The historical time period is idetified o the basis of the etropy value of the musical segmet (that may be represeted by a phrase or by a etire compositio). This kid of aalysis allows the presetatio of the results i statistic tables (which are easily covertible i diagrams so as to icrease their comprehesibility) [11] with a immediate idetificatio of the result, ulike aalytical aalysis where the result may be idetified oly by meas of a iterpretatio 34
of the fial matrix of trasitios (see Table III): a iterpretatio that presupposes specific musical kowledge. V. THE RESULTS OBTAINED The model of aalysis proposed i this article was verified by realizig a algorithm the structure of which takes i cosideratio each ad every sigle aspect described above: The algorithm does ot provide ay limitatio with respect to the dimesios of the table represetig the alphabet ad the matrix of trasitios that will be automatically dimesioed i every sigle aalysis o the basis of the characteristics of the aalyzed musical piece. This allows coferrig geerality to the algorithm ad specificity to every sigle aalysis (Stregth), due to the statistics o the type of itervals beig preset: every formal expectatio with respect to the itervals expected to be sigificat might have limited the aalysis to musical pieces of a certai time period. For a superior precisio of the result the choice was made to submit to aalysis ot etire musical pieces but oly parts of every oe of these: i particular, a stadard legth segmet made up of 40 musical otes startig from the begiig of the musical piece was take ito cosideratio. This choice was made for two reasos: 1) the begiig of every musical piece cotais the pre-thematic elemets, i.e. the elemets that will characterize the etire musical piece; 2) the loger the segmet, the larger the umber of itervals cosidered i the aalysis, therefore more probabilities for a certai situatio to occur, that is for a certai iterval to resolve to aother iterval: this reduces the degree of ucertaity ad thus dimiishes the value of the iformatio (see paragraph 3). For the defiitio of etropy described i the precedig paragraphs we deduce that there will ot be values above zero, but, o the cotrary, very small values, therefore it is importat to evaluate the umber of decimals that may ifluece, albeit slightly, the iterpretatio of the segmets (Weakess): the bigger the umber of decimals ad the higher the accuracy of the aalysis. TABLE IV: TABLE SUMMARIZING SOME ELEMENTS OF MUSICAL PIECES ANALYZED WITH THE INDICATION OF THE RELATED ENTROPY VALUE (COLUMN 3) Author Title Etropy J.S. Bach Allemada 0,000000000013002 D. Scarlatti Soata L413 K9 0,000000000000191 F. Couperi Preludio 0,000000000001608 A. Vivaldi Primavera 0,000000000000098 F. Chopi Mazurka op. 69. 2 0,000001872379605 F. Schubert Mometo Musicale. 1 0,000001872379605 F. Schubert Valzer op. 50. 11 0,000000002506347 F. Medelssho Romaza op. 53. 2 0,000000001836063 R. Schuma Arabesque op. 18 0,000000898651929...... A. Schoberg Klavierstuke op. 19. 1 0,00317076873789 G. Petrassi Petite Piece (1950) 0,015682496448 B. Bettielli Preludio (1984) 0,425593113228291 Baroque Romatic Cotemporary Musical pieces by differet authors ad of differet epochs were subject to aalysis. Table IV illustrates some of the represetative results of the aalysis performed, divided ito historical time periods. As it ca easily be see, for every historical time period, the Etropy values of the segmets are placed iside specific itervals which are clearly differet from oe aother (Stregth): this allows us to obtai idexes (equal to the mamum values) i order to delimit every sigle historical time period. The graphic represetatio (Fig. 2) of these results may help uderstad their meaig. Fig. 2. Graphic represetatio of the iformatio coveyed by the various segmets aalyzed (Table IV). The scale used for the diagram is the logarithmic scale that allows the represetatio of both very small values ad very large values. I spite of the fact that the results achieved are satisfactory, the artistic part of the author, that may lead to a result differig from the expected oe, still remais to be cosidered (Weakess): for istace there are musical pieces composed by Bach i the Baroque period that display certai characteristics which are typical of the Classic style ad sometimes of the Romatic style; Beethove with his Great Fugue (op. 133) tries to surpass the cocept of Toality usig, therefore, itervals which are typical to the cotemporary period. VI. CONCLUSION The tools preseted i this article, developed o the basis of specifically musical objectives, represet a meas of support for the melodic aalysis withi the field of automatic segmetatio of a musical piece: we are dealig with a a priori aalysis that allows the idetificatio of the body of rules to be used at a later stage, i the field of melodic segmetatio of the musical piece. The extesio of this methodology to rhythmic aalysis of a musical piece might help acquire greater accuracy i idetifyig the historical time period. The high degree of complety of musical pheomea imposes certai forms of achievemet that must be adequate ad that, for completeess' sake, must cope with the problems uder a sufficietly large umber of agles. Thus, eve from a theoretical - musical poit of view, the possibility to itegrate differet approaches appears as a precursory way of iterestig developmets. Ad it is really thaks to the ew techiques of artificial itelligece that such forms of itegratio ad verificatio of the results become achievable. 35
REFERENCES [1] L. Camilleri, Musical Grammars ad Computer Aalysis, Florece: Leo Olschki Editore, 1984. [2] M. D. Vetura, L improta digitale del compositore, Roma: GDE Edizioi Musicali, 2010 [3] O. Lartillot-Nakamura, Fodemets d u système d aalyse musicale computatioelle suivat ue modélisatio cogitiviste de l écoute, Doctoral Thesis, Uiversity of Paris, 2004. [4] N. Chomsky, Sytactic structure, Mouto: The Hague, 1957. [5] I. Bet ad W. Drabki, Music Aalysis, Turi: EDT, 1990. [6] D. de la Motte, Il cotrapputo. U libro da leggere e da studiare, Uiversal Music MGB, 1991. [7] F. A. Carmeci, Processi cogitivi ell ascolto della musica, Psychofeia, vol. 7, o. 11, 2004 [8] M. D. Vetura, Aalysis of algorithms implemetatio for melodical operators i symbolical tectual segmetatio ad coected evaluatio of musical etropy, i Proc. 1st Models ad Methods i Applied Scieces, Drobeta Turu Severi, 2011, pp. 66-73. [9] W. Weaver ad C. Shao, The Mathematical Theory of Iformatio, Urbaa: Illiois Press, 1964. [10] E. Ageleri, Iformatio, Meaig ad Uiversalit, UTET, Turi, 2000. [11] Moles ad Abraham, Teorie de l iformatio et Perceptio esthetique, Paris : Flammario Editeur, 1958. [12] F. Lerdhal ad R. Jackedoff, A Grammatical Parallel betwee Music ad Laguage, New York: Pleum Press, 1982. [13] B. Coltro, Lezioi di Armoia Complemetare, Zaibo, 1979. Michele Della Vetura defied his professioal traiig, sice high school, withi the framework of two distict areas of study: music ad mathematics. He embarked o a course of study of the Piaoforte uder the guidace of Doato Cuzzato, cotiuig the with Fracesco Becivega uder the guidace of whom he brilliatly graduated from the A. Steffai Coservatory of Music of Castelfraco Veeto i 1993. Cocurretly to the music studies he graduated i techology disciplies with the highest hoors ad distictio, obtaiig a scholarship, defedig the thesis etitled Study o the implemetatio of algorithms for the melodic operators i the symbolic text segmetatio ad cojoit evaluatio of musical etropy. His dual formatio, i Iformatio Techology ad music, drives him ito carryig out research activities o the relatio betwee Music ad Mathematics. The developmet of educatio-related techologies draw him to focus his attetio o the iovatios of iformatio techology associated to musical programmig laguages ad to atted a Post-Graduate Master's Degree o E-Learig (E-Learig: methods, techiques ad applicatios) at the Uiversity Tor Vergata of Rome, graduatig from it with the highest marks with the thesis "Learig ad ew techologies". His research activity cotiues withi the framework of computer-aided musical aalysis, publishig articles ad holdig atioal ad iteratioal cofereces ad semiars o Music ad ew techologies. He combies his research ad IT cosultat activity with the positio of Music iformatics Professor at Music Academies ad Coservatories ad the positio of Musical Techologies Teacher i High Schools specializig i Music. 36