A quantitative approach to evolution of music and philosophy

A quan titativ e approac h to ev olution of m usic and philosoph y Vilson Vieira 1 , Renato F abbri 1 , Gonzalo T ra vieso 1 , Osv aldo N. Oliv eira Jr. 1 , Luciano da F on toura Costa 1 1 Instituto de F ´ ısica de S˜ ao Carlos, Univ ersidade de S˜ ao P aulo (IFSC/USP) E-mail: vilsonvieira@usp.br , fabbri@usp.br , gonzalo@ifsc.usp.br , chu@ifsc.usp.br , ldfcosta@gmail.com Abstract. The dev elopmen t of new statistical and computational metho ds is increasingly making it p ossible to bridge the gap betw een hard sciences and humanities. In this study , w e propose an approac h based on a quan titativ e ev aluation of attributes of ob jects in fields of h umanities, from which concepts suc h as dialectics and opp osition are formally defined mathematically . As case studies, we analyzed the temp oral ev olution of classical m usic and philosoph y by obtaining data for 8 features c haracterizing the corresponding fields for 7 well-kno wn comp osers and philosophers, whic h w ere treated with m ultiv ariate statistics and pattern recognition metho ds. A b ootstrap metho d w as applied to av oid statistical bias caused b y the small sample data set, with whic h hundreds of artificial comp osers and philosophers were generated, influenced by the 7 names originally chosen. Up on defining indices for opp osition, sk ewness and coun ter-dialectics, we confirmed the intuitiv e analysis of historians in that classical music evolv ed according to a master-apprentice tradition, while in philosophy c hanges w ere driven by opposition. Though these case studies were mean t only to sho w the p ossibility of treating phenomena in humanities quan titativ ely , including a quantitativ e measure of concepts such as dialectics and opposition the results are encouraging for further application of the approac h presented here to many other areas, since it is en tirely generic. P A CS num b ers: 89.75.Fb,05.65.+b Keywor ds: Arts, music, musicology , philosophy , pattern recognition, statistics A quantitative appr o ach to evolution of music and philosophy 2 1. In tro duction Philosoph y and natural sciences were b orn together in the 6th cen tury B.C. within Greek civilization [1]. Logic started with Aristotle, geometry with Thales and Euclid, arithmetic with Diophan tus, while others created algebra, astronom y and p olitics, with all fields b eing part of a common kno wledge space [2]. What we know to da y as mo dern philosoph y and science w ere not originally separate, and arts w ere also presen t almost univ ersally in this space, not only as cultural expression. The harmonic prop erties of m usic were of interest to philosophers suc h as Pythagoras. Aristotle with his P o etics discussed a dramatic theory for theatre. In this m ultidisciplinary space, music w as addressed with mathematics and science with philosophy . Quan titative methods were applied to explain h umanities, while the scientific method had its origin in philosophy . The segregation of this space into philosophy , science and arts w as inevitable in the scien tific rev olution, thus leading to its division in to individual, indep enden t areas. Although sp ecific, these areas are gro wing in complexity and their domains o verlap. Metho ds from sp ecific areas are not sufficien t to deal with their complexit y . In this study , w e cross the in terdisciplinary b orders and apply a quan titative metho d to philosoph y and m usic, in an attempt to understand how humanities ev olve. While philosoph y and music ha v e their history constantly analyzed and discussed by critics, the nature of their works — text do cuments and m usic scores — are difficult to analyze and interpretation is alw ays sub jectiv e. Here we prop ose a generic approac h to analyze features from any field in a quan titativ e manner. The approach, described in the next Section, was then applied to c haracterize comp osers of classical music and philosophers. The data collected from scores assigned to these comp osers and philosophers were treated with statistical and pattern recognition metho ds, with the conclusions drawn b eing then compared to the literature based on critics of m usic and philosophy . 1.1. The Appr o ach The approac h devised to study the evolution of music and philosophy is completely generic, and can be applied to any sub ject. It consists of the following steps: (i) Given a sub ject (or area) to be studied, ob jects b elonging to this area are identified. Here, philosophers and comp osers were chosen arbitrarily for the fields of philosophy and classical music, resp ectively . In other applications, the choice could b e ob jective; for example, in a study of metrop olitan areas the choice of the ob jects (cities) would ob ey a very precise criterion (p opulation). (ii) A set of attributes is established, whic h are used to characterize the ob jects. In our case, a few attributes w ere c hosen arbitrarily based on well-kno wn c haracteristics in the fields of philosophy and music. In order to make the analysis by humans feasible, the n um b er of attributes had to be lo w. But in other types of w ork, the set could b e chosen based on ob jective criteria and the num b er of attributes could b e large. A quantitative appr o ach to evolution of music and philosophy 3 (iii) F or a quantitativ e analysis, scores are assigned for each of the attributes to eac h of the ob jects. In this study , three of the authors assigned scores based on their kno wledge of the fields under study . This can b e generalized, with scores assigned with ob jective criteria. Moreov er, schemes ma y be applied to chec k the qualit y of the assignmen t, for example using the Kappa index [3] to v erify agreemen t among the p eople who assigned the scores. (iv) The data generated from the steps ab ov e may b e sparse and in small amoun t, therefore unsuitable for the application of statistical methods. In order to ov ercome this limitation, w e in tro duced a step to verify the robustness of the analysis, which consists in generating “artificial data” via a b o otstrap metho d [4]. It is w orth noting that in other applications, this creation of artificial data may not b e needed. F or instance, w e could ha ve tak en data for a m uch larger num b er of philosophers and comp osers. Ho wev er, in this pap er this w ould hamp er the task of man ual score assignmen t, and interpretation of the final results would b e muc h more difficult (with so man y names to be analyzed). (v) Because the study is comparative, metrics must be found to identify similarities (or dissimilarities) among the ob jects, and pro jection metho ds are applied to visualize the data. There are several p ossibilities for this task, and here w e used Pearson correlation and principal comp onent analysis [5] to analyze the distances among ob jects. It serv ed to establish a time line, with whic h the ev olution of philosoph y and music could b e studied. F urthermore, with this time line it w as p ossible to analyze the data in terms of concepts such as dialectics and opp osition, whic h were obtained in a quan titative manner. With regard to the sp ecific metho dology in the presen t w ork, w e first iden tified prominen t m usic composers and philosophers along history , and established a set of main m usical and philosophical features. Grades w ere then assigned to each of the composers and philosophers for all features. The assignment of scores was not arbitrary , for they w ere based on research ab out tec hniques and st yles used by comp osers and philosophers. The grades reveal a tendency index on characteristics of comp osers or philosophers. F or example, to say that Bac h is more contrapun tist than the other composers corresp onds to assigning a higher grade — e.g. 8.0 or 9.0 — to the Barro que master and smaller grades to others. W e chose a reduced set of philosophers and comp osers for the sak e of simplicit y and clarit y . Then, to a void statistical bias o wing to the small num b er of samples, w e developed a b o otstrap metho d [6] with which a larger data set of 1000 new artificial c omp osers and philosophers were generated, b eing directly influenced by the original sample and representing the contemporaries of the philosophers and comp osers c hosen. The scores assigned to the c haracteristics of eac h comp oser — and philosopher — define a state vector in its feature space. This quan tification of comp osers and philosophers allo wed the application of sound quan titative concepts and metho ds from m ultiv ariate statistics [7, 8, 9] and pattern recognition [10, 5]. Correlations b etw een A quantitative appr o ach to evolution of music and philosophy 4 these characteristic v ectors w ere identified and principal component analysis (PCA) [5] w as applied to represent the m usic — and philosoph y — history as a planar space where dev elopment may b e follow ed as v ectorial mo vemen ts. On this planar space, concepts lik e dialectics, innov ation and opp osition, originally non-quantitativ e, can b e mo deled as mathematical relations betw een individual states. It is imp ortan t to note that application of statistical analysis to music is not recen t. In m usicology , statistical metho ds hav e b een used to iden tify many musical c haracteristics. Simonton [11, 12] used time-series analysis to measure the creative pro ductivit y of comp osers based on their m usic and p opularity . Kozb elt [13, 14] also analyzed pro ductivit y , but based on the measure of p erformance time of the comp ositions and inv estigated the relation betw een pro ductivit y and v ersatilit y . More recen t works [15, 16] used mac hine-learning algorithms to recognize m usical styles of selected comp ositions. In con trast to the studies ab o ve, we are not in terested in applying statistical analysis to music but on c haracterizing comp osers by iden tification of scores based on their styles. On the other hand, automatic information retriev al — that has been applied to music — is not common in philosoph y . The metho d prop osed here is a wa y to analyze b oth fields indep enden tly of the nature of their works — i.e. music pieces and textual documents — but based on a w ell-formed opinion of review ers or critics on these fields and their analysis of temp oral ev olution. 2. Mathematical Description Eac h comp oser or philosopher and their characteristics (i.e. opp osition and skewness) are defined for eac h pair of subsequent comp osers or philosophers along time. Therefore, the choice of comp osers and philosophers is crucial for the time-ev olution analysis. A sequence S of C m usic comp osers and P philosophers was chosen based on their relev ance in eac h p erio d of classical m usic and w estern philosophy history , resp ectively . The set of C measurements defined a C -dimensional space, henceforth referred to as the music al sp ac e . Likewise, w e defined a P -dimensional philosophic al sp ac e based on P measurements. The vector ~ v i for each comp oser or philosopher i defines a corresp onding c omp oser state in the music space or philosopher state in the philosophy space. F or the set of C comp osers and P philosophers, we defined the same relations summarized in T able 1. Some details ab out these relations are worth noting from Figure 1 for comp osers, and similar remarks apply to philosophers. Giv en a set of C comp osers as a time-sequence S , the aver age state at time i is defined as ~ a i . The opp osite state is defined as the “coun terp oin t” of a music state ~ v i , considering its av erage state: ev erything running along the opp osite direction of ~ v i is understo o d as opposition. In other w ords, an y displacemen t from ~ v i along the direction ~ r i is a c ontr ary move , and any displacemen t from ~ v i along the direction − ~ r i is an emphasis move . Given a comp oser state ~ v i and its opp osite state ~ r i , the opp osition ve ctor ~ D i is defined. A quantitative appr o ach to evolution of music and philosophy 5 T able 1. Description of mathematical relations for each comp oser or philosopher i , j and k , giv en a set of C composers or P philosophers as a time-sequence S . The opp osition index W i,j quan tifies how muc h a comp oser or philosopher state i is opposed to j . The skewness index i,j quan tifies the exten t to which the new comp oser or philosopher state j departs from the corresp onding opp osition state. And the c ounter- diale ctics index quantifies ho w muc h a state k could b e considered a synthesis b etw een the thesis state i and the antithesis state j . Av erage state ~ a i = 1 i P i k =1 ~ v k . Opp osite state ~ r i = ~ v i + 2( ~ a i − ~ v i ) Opp osition v ector ~ D i = ~ r i − ~ v i Comp oser or philosopher state mo ve ~ M i,j = ~ v j − ~ v i Opp osition index W i,j = h ~ M i,j , ~ D i i || ~ D i || 2 Sk ewness index s i,j = q | ~ v i − ~ v j | 2 | ~ a i − ~ v i | 2 − [( ~ v i − ~ v j ) . ( ~ a i − ~ v i )] 2 | ~ a i − ~ v i | 2 Coun ter-dialectics index d i → k = |h ~ v j − ~ v i , ~ v k i + 1 2 h ~ v i − ~ v j , ~ v i + ~ v j i| | ~ v j − ~ v i | F or the time-sequence S relations b et ween pairs of comp osers can b e defined, as follo ws. The mo ve b etw een tw o successiv e c omp oser states at time i and j corresp onds to the ~ M i,j v ector extending from ~ v i to ~ v j . Given the comp oser state we can quan tify the intensit y of opp osition by the pro jection of ~ M i,j along the opp osition vector ~ D i , normalized, yielding the opp osition index W i,j . With the same comp oser state, the skewness index s i,j is the distance b et ween ~ v j and the line L i defined by the v ector ~ D i , thus quan tifying the exten t into whic h the new comp oser state departs from the corresp onding opposition state. A relationship b et ween a triple of successive comp osers can also b e defined. T aking i , j and k as the thesis , antithesis and synthesis states, the c ounter-diale ctics index d i → k w as defined by the distance b etw een the comp oser state ~ v k and the middle line M L i,j defined by the thesis and an tithesis, as shown in Figure 2. In higher dimensional musical or philosophical spaces, the middle-h yp erplane defined by the p oints whic h are at equal distances to b oth ~ v i and ~ v j should b e used instead of the middle line M L i,j . The prop osed equation for counter-dialectics scales to h yp erplanes. The counter-dialectics index is used here, instead of a dialectics index, to A quantitative appr o ach to evolution of music and philosophy 6 Figure 1. Gr aphic al r epr esentation of the me asur es derive d fr om a comp oser state mo v e . Figure 2. Gr aphic al r epr esentation of the quantific ation of diale ctics. main tain compatibilit y with the use of a distance from p oint to line as adopted for the definition of sk ewness. A quantitative appr o ach to evolution of music and philosophy 7 3. Characteristics T o create the music and philosophy spaces we derived eight v ariables corresp onding to distinct c haracteristics commonly found in music comp ositions and w orks b y philosophers. While the selected characteristics cannot b e considered a summary of all the relev an t features in music and philosoph y history , they are initial indicators that rev eal differences in st yle of comp osers and philosophers. W e emphasize that the fo cus of this work is not on the specific c haracteristics used or the scores assigned, whic h can b e disputed, but on the tec hniques for a quantitativ e analysis. 3.1. Music al Char acteristics These characteristics are related to basic elemen ts of m usic — melo dy , harmon y , rh ythm, tim bre, form and tessitura [17] — in addition to non-musical issues lik e historical ev ents that influenced comp ositions, such as the imp ortance of the Ch urch. The eight charac- teristics are listed below: Sacr e d - Se cular ( S - S c ): the sacred or religious m usic is comp osed through religious influence or used for its purp oses. Masses , motets and hymns, dedicated to the Christian liturgy , are w ell-kno wn examples [18]. Secular m usic has no or minimal relation with religion and includes p opular songs lik e Italian madrigals and German lie ds [17]. Short dur ation - L ong dur ation ( D s - D l ): compositions quan tified as short duration ha v e a few min utes of execution. Long duration comp ositions hav e at least 20 min utes of execution. The same criterion w as adopted b y Kozbelt [13, 14] in his analysis of time execution. Harmony - Counterp oint ( H - C ): harmon y regards the v ertical combination of notes, while coun terp oin t fo cuses on horizontal com binations [17]. V o c al - Instrumental ( V - I ): comp ositions using just vocals (e.g. c antata ) or exclusiv ely instruments (e.g. sonata ). Note the use of vocals ov er instruments on Sacred comp ositions [18]. Non-discursive - Discursive ( D n - D ): comp ositions based or not on v erbal discourse, lik e programmatic music or Baro que rhetoric, where the composer w ants to “tell a story” in voking images to the listeners mind [17]. Its contrary part is known as absolute music where m usic was aimed to b e appreciated simply by what it is. Motivic Stability - Motivic V ariety ( M s - M v ): motivic pieces present equilibrium b etw een rep etition, reuse and v ariation of melo dic motiv es. Bac h is noticeable for his variation of motiv es, contrasting with the constan tly inv entiv e use of new materials b y Mozart [19]. R hythmic Simplicity - R hythmic Complexity ( R s - R c ): presence or not of p olyrh ythms, the use of indep enden t rh ythms at the same time — also kno wn as rhythmic c ounterp oint [17] — a characteristic frequen tly found in Roman ticism and the w orks of 20th-cen tury composers like Stra vinsky . A quantitative appr o ach to evolution of music and philosophy 8 Harmonic Stability - Harmonic V ariety ( H s - H v ): rate of tonality change along a piece or its stabilit y . After the highly p olyphonic developmen t in Renaissance, W eb ern regarded Beetho ven as the comp oser who returned to the maxim um exploration of harmonic v ariety [19]. 3.2. Char acteristics in Philosophy W e deriv ed eight v ariables corresp onding to some of the most recurrent philosophical issues [1, 2, 20]. Eac h of these v ariables, which defined an axis in the philosophy space, are briefly describ ed in the follo wing. R ationalism - Empiricism ( R-E ): the rationalists claim that the h uman acquain tance of knowledge/concepts is significan tly independent of sense exp erience. Empiricists understand sensory exp erience as the main w ay to gain knowledge. Empiricism is in the origin of the scien tific method where kno wledge m ust b e based on sensible observ ation of the world instead of faith or intuition. F requently , rationalists understand the world as affected b y intrinsic prop erties of the h uman brain, in con trast to the empiricist approach where the world w ould imprint itself on to our minds (the principle of “tabula r asa” ). F or rationalists, deduction is the sup erior metho d for in vestigation and privileges the reason instead of exp erience as the source of true kno wledge. Historically , Descartes, Espinoza and Leibniz introduced the rationalism in mo dern philosoph y . Essenc e - Existenc e ( E-E ): An existence-based understanding of the w orld has its basis on the fact that things are as an existen t unit. In Existen tialism, the life of an individual is determined by its self, that constitutes its essence, and not a predefined essence that defines what is to b e human. Essence fo cuses on a substance (e.g. in tellectual) that precedes existence itself. F or essentialists, an y sp ecific entit y has essen tial c haracteristics necessary for its function and identit y . F or example, to ha v e t wo legs and the ability to run is not an essential c haracteristic defining humans, b ecause other animals ha ve the same c haracteristics. Plato w as one of the first essen tialists while Nietzsc he and Kierk egaard w ere fundamen tal to Existentialism in the 19th cen tury . Monism - Dualism ( M-D ): Dualism requires the division of the human p erson in to t w o or more domains, suc h as matter and soul. Monism is based on a unique “category of b eing”. Plato is a recognized dualist while his disciple, Aristotle, w as a notable monist. The o c entrism - A nthr op o c entrism ( T-A ): In theo cen trism, Go d is the most imp ortan t thing in the universe. The anthropo centric view has man as prev alent, with Nietzsc he b eing its main representativ e. It is imp ortan t to distinguish theo centrism from deism, where the figure of Go d is considered as an abstract entit y , as for Espinoza, but do es not play a central role in the univ erse. Holism - R e ductionism ( H-R ): Reductionism attempts to explain the w orld in terms of simple comp onen ts and their emerging prop erties. Holists fo cus on the fact A quantitative appr o ach to evolution of music and philosophy 9 that the whole is more than its constitutiv e parts. De ductionism - Phenomenolo gy ( D-P ): Phenomenology relies on systematic reflection of consciousness and what happ ens in conscious acts. Deductionism is based on deriving conclusions from axiomatic systems. Determinism - F r e e Wil l ( D-F ): F ree will assumes that humans mak e c hoices, whic h are not predetermined. Determinism understands that ev ery even t is fatidic, e.g. p erfectly determined by prior states. Natur alism - Me chanism ( N-M ): Metho dological naturalism is the thinking basis of mo dern science, i.e. h yp otheses must b e argued and tested in terms of natural la ws. Mechanism attempts to build explanation using logic-mathematical pro cesses. 3.3. Bo otstr ap metho d for sampling T o eliminate the bias intrinsic in a small sample group, we used a b o otstrap metho d for generating artificial c omp osers and philosophers contemporaries of those seven chosen. The b o otstrap routine generated new scores ~ r , whic h are not totally random b ecause they follow a probability distribution that mo dels the original n = 7 scores, giv en by p ( ~ r ) = P n i =1 e d i 2 σ 2 where d i is the distance b et w een a random score ~ r and the original score c hart. F or eac h step a v alue p ( ~ r ) is generated and compared with a random normalized v alue, as in the Mon te Carlo [21] method to choose a set of samples. These samples sim ulate new randomized comp osers and philosophers score charts — while preserving the historical influence of the main 7 original names in each field. Higher v alues of p ( ~ r ) imply a stronger influence of the original scores o ver ~ r . F or the analysis w e used 1000 b o otstrap samples obtained b y the b o otstrap pro cess together with the original scores, taking σ = 1 . 1. Other v alues for σ w ere used yielding distributions with b o otstrap samples that did not affect the music or philosophy space substan tially . 4. Results and Discussion Memorable comp osers w ere chosen as k ey representativ es of classical m usic, whic h w e b eliev e had impact on their con temp oraries, thus creating a concise parallel with m usic history . The chronological sequence is presen ted in T able 2 with eac h comp oser related to his historical p erio d. The same was done for philosophy where a set of seven philosophers w ere c hosen spanning the p erio d from Classical Greece until contemporary times, and ordered chronologically as: Plato, Aristotle, Descartes, Espinoza, Kant, Nietzsche and Deleuze, as sho wn in T able 3. The quan tification of the eigh t c haracteristics for music and philosoph y was p erformed join tly b y three of the authors of this article, based on research of history of m usic and western philosophy . The scores sho wn in T ables 4 and 5 for philosophers and comp osers, resp ectiv ely , were n umerical v alues b etw een 1 and 9. V alues closer to 1 reveal the comp oser or philosopher tended to the first elemen t of eac h c haracteristic pair, and vice-v ersa. A quantitative appr o ach to evolution of music and philosophy 10 T able 2. Sequence of music comp osers ordered chronologically with the p erio d each represen ts. Comp oser Mo vemen t Mon teverdi Renaissance Bac h Baro que Mozart Classical Beetho ven Classical → Roman tic Brahms Roman tic Stra vinsky 20th-cen tury Sto c khausen Con temp orary T able 3. Sequence of philosophers ordered chronologically with the p erio d each represen ts. Philosopher Era Plato Ancien t Aristotle Ancien t Descartes 17th-cen tury Espinoza 17th-cen tury Kan t 18th-cen tury Nietzsc he 19th century Deleuze 20th-cen tury T able 4. Quan tification of eight music characteristics for each of the seven comp osers. Comp osers S - S c D s - D l H - C V - I D n - D M s - M v R s - R c H s - H v Mon tev erdi 3.0 8.0 5.0 3.0 7.0 5.0 3.0 7.0 Bac h 2.0 6.0 9.0 2.0 8.0 2.0 1.0 5.0 Mozart 6.0 4.0 1.0 6.0 6.0 7.0 2.0 2.0 Beetho v en 7.0 8.0 2.5 8.0 5.0 4.0 4.0 7.0 Brahms 6.0 6.0 4.0 7.0 4.5 6.5 5.0 7.0 Stra vinsky 8.0 7.0 6.0 7.0 8.0 5.0 8.0 5.0 Sto c khausen 7.0 4.0 8.0 7.0 5.0 8.0 9.0 6.0 This data set defines an 8-dimensional space for music or philosoph y where each dimension corresp onds to a characteristic that applies to all 7 composers or philosophers. Suc h small data sets are not adequate for statistical analysis, whic h could b e biased. This is the reason wh y w e used the b o otstrap metho d for sampling describ ed in section 3.3. F or the extended data set after applying the bo otstrap metho d, P earson correlation co efficien ts betw een the eigh t characteristics chosen are giv en in T able 6 for comp osers A quantitative appr o ach to evolution of music and philosophy 11 T able 5. Quantification of eight philosophy c haracteristics for eac h of the sev en philosophers. Philosophers R-E E-E M-D T-A H-R D-P D-F N-M Plato 3.0 3.5 9.0 5.0 4.5 3.5 5.0 4.5 Aristotle 8.0 7.5 7.0 5.5 7.5 8.0 2.5 2.5 Descartes 1.5 2.5 9.0 6.5 7.0 2.5 7.5 7.5 Espinoza 8.0 2.0 1.0 5.0 2.0 3.0 1.0 1.0 Kan t 7.0 2.5 8.5 6.5 4.5 3.5 7.5 5.0 Nietzsc he 7.5 9.0 1.0 9.0 5.0 8.0 1.0 1.5 Deleuze 5.5 7.5 1.0 8.0 2.5 5.5 5.0 6.0 and in T able 7 for philosophers. The co efficient was 0.69 for the pairs S - S c (Sacred or Secular) and V - I (V o cal or Instrumen tal), which indicates that sacred music tends to b e more v o cal than instrumen tal. The co efficient 0.56 for the pairs S - S c and R s - R c (Rh ythmic Simplicity or Complexit y) also shows that this genre do es not commonly use p olyrh ythms. A negativ e co efficien t of -0.33 for the pair V - I and D n - D (Non-discursive or Discursive) indicated that comp osers who used just voices on their comp ositions also preferred programmatic music techniques suc h as baroque rhetoric. Strong correlations w ere also observ ed for philosophers. F or instance, the P earson co efficien t of − 0 . 46 for R-E and N-M suggests that rationalists tend to be also mechanists. An ev en stronger correlation of 0 . 74, now p ositiv e, is observ ed betw een E-E and D-P , i.e. existentialists also tend to b e phenomenologists, as could be exp ected. Other imp ortan t correlations app eared b etw een D-F and N-M (co efficient = 0 . 61) and b et ween M-D and D-F , which seems to b e directly implied b y religious bac kground. T able 6. Pearson correlation coefficients b et w een the eight m usical characteristics. The en tries with absolute v alues ≥ 0.30 hav e b een highlighted. - S - S c D s - D l H - C V - I D n - D M s - M v R s - R c H s - H v S - S c - -0.2 -0.06 0.69 -0.18 0.19 0.56 -0.16 D s - D l - - -0.14 -0.13 0.2 -0.48 -0.2 0.37 H - C - - - -0.23 0.26 0.05 0.46 0.03 V - I - - - - -0.33 0.17 0.42 -0.06 D n - D - - - - - -0.3 0.02 -0.22 M s - M v - - - - - - 0.26 -0.15 R s - R c - - - - - - - -0.02 H s - H v - - - - - - - - The data w ere analyzed with principal comp onent analysis (PCA), from whic h it w as found that sev eral of the characteristics con tributed to the v ariability of the data, A quantitative appr o ach to evolution of music and philosophy 12 T able 7. Pearson correlation co efficients betw een the eight philosophical c haracteristics. The entries with absolute v alues ≥ 0.35 hav e b een highlighted. - R-E E-E M-D T-A H-R D-P D-F N-M R-E - 0.37 -0.23 0.15 0.1 0.46 -0.27 -0.46 E-E - - -0.53 0.19 0.15 0.74 -0.61 -0.3 M-D - - - -0.43 0.41 -0.3 0.35 0.01 T-A - - - - -0.21 0.06 0.19 0.26 H-R - - - - - 0.32 -0.22 -0.25 D-P - - - - - - -0.63 -0.47 D-F - - - - - - - 0.61 N-M - - - - - - - - as shown in T ables B1 and B2 in the Supp orting Information. Figures 3 and 4 displa y a 2-dimensional space with the first t wo main axes. The arro ws follo w the time sequence along with the sev en comp osers and philosophers. Eac h of these arrows corresp onds to a vectorial mo v e from one comp oser or philosopher state to another. F or clarity , just the lines of the arro ws are preserv ed. The b o otstrap samples define clusters around the original comp osers and philosophers. In subsidiary exp eriments, we verified that the results from the b o otstrap metho d were robust. This w as p erformed by applying 1000 p erturbations of the original scores b y adding to each score the v alues -2, -1, 0, 1 or 2 with uniform probability . In other words, we tested if scoring errors could b e sufficien t to cause relev an t effects on the PCA pro jections. Interestingly , the v alues of av erage and standard deviation for b oth original and p erturb ed p ositions listed in T ables B3 and B4 in the Supp orting Information show relatively small c hanges. It is therefore reasonable to assume that small errors in the scores assigned had no significant effect on the ov erall analysis. Bac h is found far from the rest of the comp osers, which suggests his key role ac knowledged b y other great comp osers lik e Beetho v en and W eb ern [19]: “In fact Bac h comp osed ev erything, concerned himself with everything that giv es foo d for thought!”. The greatest subsequent change takes place from Bac h to Mozart, reflecting a substantial difference in st yle. There is a strong relationship b et w een Beethov en and Brahms, supp orting the belief b y the virtuosi Hans von B ¨ ulo w [22] when he stated that the 1 st Symphon y of Brahms was, in realit y , the 10 th Symphony of Be ethoven , app ointing Brahms as the true successor of Beetho v en. Stra vinsky is near Beetho v en and Brahms, presumably due to his heterogeneit y [17, 18]. Beethov en is also near Mozart, who deeply influenced Beethov en, mainly in his early works. F or W eb ern, Beethov en was the unique classicist who really came close to the coherence found in the pieces of the Burgundian Sc ho ol: “Not even in Haydn and Mozart do we see these tw o forms as clearly as in Beethov en. The p erio d and the eigh t-bar sentence are at their purest in Beethov en; in his predecessors we find only traces of them” [19]. It could explain A quantitative appr o ach to evolution of music and philosophy 13 − 4 − 3 − 2 − 1 0 1 2 3 4 − 4 − 3 − 2 − 1 0 1 2 3 4 B o o t s t r a p s a mp l e s O r i g i na l s a m pl e s 0 1 Monteverdi 2 Bach 3 Mozart 4 Beethoven 5 Brahms 6 Stravinsky 7 Stockhausen Figure 3. 2-dimensional pr oje cte d music sp ac e. the mov e of Beethov en in direction of the Renaissance Mon teverdi. Sto ckhausen is a deviating p oint when compared with the others, which could b e more even so had w e considered v anguard c haracteristics — e.g. timbre exploration by using electronic devices [18] — not shared b y his precursors. The opp osition and skewness indices for each of the six mov es among comp osers states in T able 8 indicate that the mov emen ts were driven b y small opp osition and strong skewness. In other w ords, most mov ements seem to seek inno v ation rather than opp osition. F urthermore, the counter-dialectics indices in T able 9 are smaller than for the philosophers, as will b e discussed later on. As for philosoph y , Figure 4 shows an opp osite mov ement from Plato to Aristotle, whic h confirmed the antagonistic view of Aristotle when compared with Plato, ev en though Aristotle was his disciple [1]. Opp osition is presen t along all the mo ves among philosophers states. This oscillatory pattern mak es it p ossible to iden tify tw o well defined groups. The first con tained Aristotle, Espinoza, Nietzsc he and Deleuze, in the left side of the graph. The other con tained Plato, Descartes and Kan t,in the right side. This division is consistent with the points of view shared b y eac h group mem b er. Another t wo groups are identified in the y-axis, separating all the philosophers from A quantitative appr o ach to evolution of music and philosophy 14 − 4 − 3 − 2 − 1 0 1 2 3 4 − 4 − 3 − 2 − 1 0 1 2 3 4 Bo o t s t r a p s a m p l e s Or i g i n a l s a m p l e s 1 Plato 2 Aristotle 3 Descartes 4 Espinoza 5 Kant 6 Nietzche 7 Deleuze Figure 4. 2-dimensional pr oje cte d philosophy sp ac e. T able 8. Opp osition and skewness indices for each of the six comp osers states mov es Musical Mov e W i,j s i,j Mon teverdi → Bac h 1.0 0. Bac h → Mozart 1.0196 1.9042 Mozart → Beetho v en 0.4991 2.8665 Beetho ven → Brahms 0.2669 1.7495 Brahms → Stra vinsky 0.4582 2.6844 Stra vinsky → Stockhausen 0.2516 3.1348 Nietzsc he and Deleuze, who are represented b y the most distan t p oin ts. When opp osition and sk ewness indices in T able 10 are considered, all the mov es among philosophers states tend to take place according to a w ell-defined, intense opp osition from the a verage state. This was already noticeable in the PCA analysis. An in teresting relationship is the minor opp osition and strong skewness b etw een Nietzsche and Deleuze, suggesting the return to Nietzsche as noted in the works of Deleuze while considering the v anguard c haracteristics of the 20th century philosopher [20]. Espinoza A quantitative appr o ach to evolution of music and philosophy 15 T able 9. Counter-dialectics index for eac h of the five subsequent pairs of mo ves among comp osers states for the 8 comp onents. Musical T riple d i → k Mon teverdi → Bac h → Mozart 2.0586 Bac h → Mozart → Beetho ven 1.2020 Mozart → Beetho v en → Brahms 1.0769 Beetho ven → Brahms → Stra vinsky 0.2518 Brahms → Stra vinsky → Sto ckhausen 0.2549 T able 10. Opp osition and skewness indices for each of the six philosophers states mo v e s. Philosophical Mov e W i,j s i,j Plato → Aristotle 1.0 0 Aristotle → Descartes 0.8740 1.1205 Descartes → Espinoza 0.9137 2.3856 Espinoza → Kan t 0.6014 1.6842 Kan t → Nietzsc he 1.1102 2.9716 Nietzsc he → Deleuze 0.3584 2.4890 T able 11. Coun ter-dialectics index for each of the five subsequen t pairs of philosophers states mo v es. Philosophical T riple d i → k Plato → Aristotle → Descartes 3.0198 Aristotle → Descartes → Espinoza 1.8916 Descartes → Espinoza → Kant 1.1536 Espinoza → Kan t → Nietzsc he 1.1530 Kan t → Nietzsc he → Deleuze 0.2705 tended tow ard Nietzsche, and their similarity was admitted by Nietzsche, by naming Espinoza his direct precursor [1]. While similar to Espinoza, Nietzsche presen ted strong opp osition to Kan t, consisten t with Nietzsc he b eing the strongest ob jector to Kant ideas [1]. Also surprising w as the rather small skewness among most of the mo ves among philosophers states, whic h would b e driv en almost exclusively b y opp osition to the curren t philosopher state. The results for dialectics in T able 11 show a progressiv ely stronger dialectics among subsequen t pairs of mo ves in philosophers states. The analysis ab ov e indicates distinct c haracteristics for the evolution of classical m usic and philosophy . Philosophers seem to ha ve developed their ideas driven by opp osition ( W i,j ), as sho wn in T able 10, while comp osers tend to b e more influenced A quantitative appr o ach to evolution of music and philosophy 16 b y their predecessors according to the dialectics measuremen ts (1 /d i → k ). In general, the mo vemen ts among comp osers had minor opposition, thus reflecting the master- appren tice tradition. There is then a crucial difference in the memory tr e atment along the developmen t of philosoph y and m usic: using the same tec hniques, w e v erified that a philosopher was influenced b y opp osition of ideas from his direct predecessor, while comp osers were influenced by their tw o predecessors. W e can argue that philosophy exhibits a memory-1 state, while m usic presents memory-2 , with memory-N having N of past generations that influenced a philosopher or comp oser. F urthermore, Figure 5 sho ws a constant decrease in the coun ter-dialectics index, whic h means a constan t return to the origins for the developmen t of m usic based on the searc h for unit y . Using the w ords of W eb ern, the searc h for the “comprehensibility” but alw ays influenced b y their old masters. 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 Com p o se r s c o u n t e r -d i a l e c t i c s P h i l o s o p h e r s c o u n t e r -d i a l e c t i c s Plato - Aristotle - Descartes Monteverdi - Bach - Mozart Aristotle - Descartes - Espinoza Bach - Mozart - Beethoven Descartes - Espinoza - Kant Mozart - Beethoven - Brahms Espinoza - Kant - Nietzshe Beethoven - Brahms - Stravinsky Kant - Nietzshe - Deleuze Brahms - Stravinsky - Stockhausen Figure 5. Comp arison b etwe en c omp osers and philosophers c ounter-diale ctics indic es The comparison b etw een comp osers and philosophers is complemen ted with the W ards hierarc hical clustering [23], whic h clusters the original scores taking into account their distance. The generated dendrogram in Figure 6 shows comp osers according to their similarit y , while the corresp onding dendrogram for philosophers is sho wn in Figure 7. These dendrograms are consisten t with the previous observ ations. F or example, Beetho ven and Brahms are close, reflecting their heritage. Stravinsky and Sto ckhausen A quantitative appr o ach to evolution of music and philosophy 17 form another cluster, while Mozart app ears on its o wn, like Bac h and Montev erdi. Both relations were also presen t in the planar space in Figure 3. Be e t h o v e n Br a h m s S t r a v i n sk y S t o c k h a u se n M o za r t M o n t e v e r d i Ba c h 2 3 4 5 6 D i st a n c e Figure 6. War ds hier ar chic al clustering of the seven c omp osers. 5. Concluding Remarks An approac h has b een prop osed whic h allows for a quan titative assessment of features from an y given sub ject or area. F or classical mu sic and philosoph y inv estigated here, a time line could b e established based on the c haracteristics of comp osers and philosophers. The quan titative analysis inv olv ed the collection of data for these c haracteristics, extended with a bo otstrap metho d to yield robustness to the statistical analysis, and the use of m ultiv ariate statistics. Also imp ortant was the establishment of indices for opposition, skewness and counter-dialectics. Though the emphasis of our w ork has b een on the metho d of analysis, the in terpretation of the data w as already sufficien t to draw conclusions that confirm intuition and sub jectiv e analyses of classical m usic and philosoph y . F or instance, the results p oin ted to the developmen t in m usic following a dichotom y: while composers aim at inno v ation, creating their o wn st yles, their tec hnique is based on the w ork of their predecessors, in a master-apprentice tradition. Indeed, in the history of m usic, comp osers dev elop ed their o wn st yles with a con tin uous searc h for coherence or unit y . In the words of Anton W eb ern [19], “[...] ev er since music has b een written most great artists hav e striv en to make this unit y ever clearer. Everything that has happ ened aims at this [...]”. A frequent inheritance of st yle can b e iden tified from one comp oser to another as a gradual developmen t from its predecessor, contrasting with the necessit y A quantitative appr o ach to evolution of music and philosophy 18 Plato Aristotle Nietzsche Espinoza Deleuze Descartes Kant Figure 7. War ds hier ar chic al clustering of the seven philosophers c onsidering al l the eight fe atur es. for innov ation. Quoting Lo velock: “[...] b y exp eriment that progress is p ossible; it is the man with the forw ard-lo oking t yp e of mind [...] who forces man out of the rut of ‘what was go o d enough for my father is go o d enough for me’.” [18]. In contrast, our quantitativ e analysis confirmed that philosophy app ears to exhibit a w ell-defined trend in innov ation: unlike m usic, the quest for difference seems to drive philosophical c hanges as expressed b y Gilles Deleuze [24]. According to F erdinand de Saussurre’s principle [25], concepts (words) tend to b e differen t in the sense of meaning distinct things. The paradigm of difference is particularly imp ortant b ecause it is related to the o wn dynamics of philosophical ev olution. Again emphasizing that our aim was to prop ose a generic, quan titative metho d, we highligh t some limitations of the sp ecific analysis made here for music and philosoph y . F or the scores and c hoice of main c haracteristics in m usic and philosoph y w ere largely arbitrary and could be disputed. Nevertheless, the p erturbation analysis performed in this w ork suggests that the effect of non-systematic errors in assigning the scores do es not seem to b e critical and has little ov erall impact on the conclusions dra wn. Most imp ortantly , the formal quantitativ e metho dology described here ma y b e com bined with other metho ds, including information retriev al and natural language pro cessing, to in vestigate other issues in h umanities. F or example, still connected with the present w ork, it can b e adapted to the inv estigation of musical and philosophical sc ho ols, individual pieces (e.g. music suites or b o oks), or ev en the con tributions from the same comp osers or philosophers along distinct p erio ds of time. Ob viously , this metho dology can also b e applied to other areas suc h as p o etry , cinema and science. A quantitative appr o ach to evolution of music and philosophy 19 6. Ackno wledgmen ts Luciano da F. Costa thanks CNPq (308231/03-1) and F APESP (05/00587-5) for sp onsorship. Gonzalo T ravieso thanks CNPq (308118/2010-3) for sp onsorship. Vilson Vieira is grateful to CAPES. App endix A. A Brief Explanation of Principal Comp onent Analy sis (PCA) PCA is a dimensionality reduction pro cedure p erformed through rotation of axes. It op erates b y concen trating disp ersion/v ariance along the first new axes, whic h are referred to as the principal comp onents. The technique consists in finding the eigen vectors and eigenv alues of the cov ariance matrix of the corresp onding random v ectors (i.e. the v ectors asso ciated with eac h philosophical state). The eigenv alues corresp ond to the v ariances of the new v ariables. When multiplied b y the original feature matrix, the eigen vectors yield the new random v ariables which are fully uncorrelated. F or a more extensiv e explanation of PCA, please refer to [5] and references therein. App endix B. Supp orting Information T ables B1 and B2 show the normalized w eigh ts of the contributions of each original prop ert y on the eight axes for comp osers and philosophers. Most of the c haracteristics con tribute almost equally in defining the axes. T able B1. Percen tages of the con tributions from eac h musical c haracteristic on the eigh t new main axes. Musical C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 Charac. S - S c 19.78 4.04 10.38 10.60 17.55 36.60 4.41 0.63 D s - D l 13.63 9.21 19.17 3.55 3.13 1.65 25.55 24.05 H - C 1.44 26.62 8.26 13.97 21.71 7.76 13.98 12.20 V - I 18.35 12.82 9.29 8.02 9.37 40.95 2.12 2.03 D n - D 6.31 10.73 15.48 26.29 4.04 1.86 25.29 2.35 M s - M v 16.94 13.28 15.03 4.84 32.25 1.70 2.62 4.37 R s - R c 14.13 3.26 15.58 13.80 7.48 1.88 1.36 35.99 H s - H v 9.38 20.00 6.75 18.88 4.45 7.56 24.62 18.36 References [1] Bertrand Russel. A History of Western Philosophy . Simon and Sch uster T ouchstone, 1967. [2] D. Papineau. Philosophy . Oxford Universit y Press, 2009. A quantitative appr o ach to evolution of music and philosophy 20 T able B2. Percen tages of the con tributions from eac h philosophical characteristic to the four new main axes. Philos. C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 Charac. R-E 13.35 1.81 13.11 31.12 15.06 10.99 8.20 3.56 E-E 18.13 7.88 2.10 14.68 11.41 5.31 7.98 33.50 M-D 10.07 24.59 10.71 2.91 0.69 19.39 20.95 2.91 T-A 1.08 24.18 21.51 0.68 23.72 6.31 7,27 2.06 H-R 5.90 21.49 22.29 14.90 6.35 18.81 8.69 2.05 D-P 18.99 1.29 6.85 8.41 9.67 20.50 11.47 24.93 D-F 14.57 13.67 8.37 18.93 21.12 9.06 12.77 14.98 N-M 17.86 5.06 15.03 8.33 11.96 9.59 22.62 15.97 T able B3. Averages and standard deviations for each comp oser and for the 8 eigen v alues. Comp osers µ ∆ σ ∆ Mon teverdi 3.7347 0.8503 Bac h 5.3561 0.9379 Mozart 4.4319 0.8911 Beetho ven 3.4987 0.7851 Brahms 3.0449 0.6996 Stra vinsky 3.6339 0.7960 Sto c khausen 4.2143 0.9029 Eigen v alues µ ∆ σ ∆ λ 1 -0.1759 0.0045 λ 2 -0.0638 0.0026 λ 3 -0.0411 0.0021 λ 4 -0.0144 0.0019 λ 5 0.0578 0.0021 λ 6 0.0736 0.0023 λ 7 0.0080 0.0027 λ 8 0.0835 0.0030 A quantitative appr o ach to evolution of music and philosophy 21 T able B4. Av erage and standard deviation for each philosopher and for the 8 eigen v alues. Philosophers µ ∆ σ ∆ Plato 3.3263 0.7673 Aristotle 4.0896 0.8930 Descartes 4.3081 0.9225 Espinoza 4.9709 0.9131 Kan t 3.2845 0.7749 Nietzsc he 5.3195 0.9797 Deleuze 4.0990 0.8970 Eigen v alues µ ∆ σ ∆ λ 1 -0.2618 0.0068 λ 2 -0.0976 0.0035 λ 3 0.0154 0.0025 λ 4 0.0212 0.0024 λ 5 0.0697 0.0026 λ 6 0.0807 0.0030 λ 7 0.0877 0.0032 λ 8 0.0846 0.0036 [3] J. Cohen et al. A co efficien t of agreement for nominal scales. Educ ational and psycholo gic al me asur ement , 20(1):37–46, 1960. 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