SENSE AND INTUITION IN MUSIC (ARGUMENTS ON BACH AND MOZART)

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1 SENSE AND INTUITION IN MUSIC (ARGUMENTS ON BACH AND MOZART) CARMEN CHELARU George Enescu University of Arts Iași, Romania ABSTRACT Analyzing in detail the musical structure could be helpful, but not enough to understand the profound sense of the music. Mathematics and arithmetic involve intuition and imagination, but have to be entirely precise. In the mean time, music is more original while it beaks the rules and principles. The composers are creating exceptions all the time. In The Art of Fugue, Bach reveals a magnificent mathematic thinking and musical intuition, at the same time. Mozart s opera The Marriage of is the result of both musical inspiration and mathematical intuition. KEY WORDS mathematics, music, Bach, Mozart, intuition, sense, construction, form, theme. 1. INTRODUCTION According to the remark generally attributed to Mozart, "It is easy to write notes but hard to write rests!" The mankind has always been concerned about the two spheres of the art, music including: construction, form, structure, semiotics, and specific language on the one hand; the meaning, the inexpressible, the esoteric, the inimitable in the art on the other. For a long time (even nowadays there are many people who resume art to this) it seemed that analyzing the artistic structure in detail could be enough to understand and define the work of art. No doubt, mathematics includes the aesthetic beauty the same as the art does [1]. Take, for instance, some geometric forms as the triangle, the square, the rectangle, the circle, the pyramid, the pentagram and the hexagram etc. All these are so beautiful that they have been adopted as mystic, philosophic and artistic symbols. More than that, they have been applied to the science researches, to the building planning, to the artistic forms. The same is the aesthetics and programming, semiotics and games etc.[2] Let's take for example the Egyptian Triangle. It is a right triangle, where the sides are in the ratio of the integers 3:4:5. After ancient Egypt, it was Pythagoras who enclosed this beautiful triangle in his theorem. Then, the Italian Fibonacci wrote about the Golden Ratio and his numbers series [3] related to the Egyptian triangle too. But the beauty of the mathematics does not end here. Those who work with numbers often say that Arithmetic means pure aesthetic pleasure. Even if it is so, we have to admit a substantial difference: arithmetic must be absolutely accurate, while the art is even more impressive, more original and more beautiful as it deviates from pre-established rules and principles. We choose to

2 prove this here by two illustrious examples: Bach's Art of Fugue and Mozart's Marriage of. 2. BACH: THE MATHEMATICS OF ART THE ART OF MATHEMATICS The supreme proof of the above title is we dare say The Art of Fugue. The theme is the essence of the masterpiece. Kapelmeister Bach has given to this theme the properties of an embryo that will generate the complex musical body. Fig. 1 Bach, Art of Fugue. Matrix Theme Its features are as following: Simplicity condition and guarantee for the entire musical processing Perfect tonal stability (tonic dominant tonic) Structure: Thesis (D minor arpeggio, first 4 notes) Antithesis (next 4 notes) Synthesis (last 4 notes). Rhythm: progressive diminution Thematic transformations are multiple and complex. We include some of them bellow. Rhythmic ornamentation: Rhythmic transformation (from binary pulse to ternary pulse): Melodic inversion: Ternary rhythm and melodic inversion: Inversion, diminution, ornamentation: Complete (Character) variation: Permutation: etc. Fig. 2 Bach, Art of Fugue. Theme transformations Formal Aspects In the Art of Fugue, each fugue plan involves as much imagination as strict construction. We may find here both symmetry and asymmetry. Let s take for instance, Fugue X, two subjects, 120 bars: Subject I (S 1 ):

I Subject II (S 2 ): Fig. 3 Bach, Art of Fugue. Fugue X. Subjects. The Fugue form includes three distinct elements: S 1, S 2 and counterpoint conducting passages [C].

3 I Subject II (S 2 ): Fig. 3 Bach, Art of Fugue. Fugue X. Subjects. The Fugue form includes three distinct elements: S 1, S 2 and counterpoint conducting passages [C]. We represented the general form of the Fugue X like this: GRi GR II III IV Fig. 4 Bach, Art of Fugue. Fugue X. General plan. Short lines = S 1 ; long lines = S 2 ; grey rectangles = C. Golden Ratio = GR (bar 75); Golden Ratio Inversion = GRi (bar 44) The Golden Ratio of the number 120 is: 120 x = 74.16; Bach puts it on bar 75! There, he brings simultaneously two versions of S 2 and one version of S 1 : Fig. 5 Bach, Art of Fugue. Fugue X. Bars The Golden Ratio Invertion of the number 120 is: 120 x = 45.84; Bach puts it on bar 44! There, he brings simultaneously S 1 and S 2. Further more, the combinations of two elements [S 1 and S 2 ], sung one by one or simultaneously, by the four voices are as many permutations among the components of an arithmetic set. In the first half of the Fugue, the element C is shorter; in the second half it is almost twice the number of bars. This generates proportionality and balance. And the speculations could continue! 3. MOZART SENSE AND SENSIBILITY IN MUSIC For more concision, let s take the opera Le nozze di. Everybody is acquainted with the author s well-known opinion regarding the primacy of the melody upon the poetry in the musical drama. Actually, for him it was Music the main character in the so-called dramma per musica. That means not only melody, but each part of the musical language, including sense, expression, internal dynamics etc. The macro- and the micro-structure of this opera brilliant result of his collaboration with Lorenzo da Ponte, music the same as theatre is the proof of perfection. We represented bellow the general plan of the four acts, and the characters [Fig. 6]. Let s consider each act as an arithmetic crowd of elements, ignoring the recitatives [4]. Here there are a few observations:

4 a. First of all we note a perfect balance between the solo and the parts in each act, and in all four acts together. Each Act except the first one begins by solo moments, ending by. On the other hand, Act I includes arias, s, terzet and less ensemble moments. Act II the final is longer. Act III the final ensemble is much longer, including orchestra intermezzi (ballet). Finally, the Act IV presents a balance between the five arias and Finale ensemble. b. Mozart gives each character a specific musical weight: 3 arias (2 act I; 1 act IV), 1, sextet. He is treated as the main character, even his stage presence is not too long. The Act I begins and ends with him singing. Susanna 2 arias (act II; act IV), 5 s (acts I, II, III), 2 terzets [5], sextet. According to this, she is the real main character of the opera. Countess 2 arias (acts II, III), 2 s (act II, III), terzet (act II). Count Almaviva aria (act III), 2 s (II, III) terzets, sextet. [6] etc. c. The distribution of the arias and s is proportionate by the moment of the drama and by the importance of each character. Except the two important arias in the first act, the main characters have not two arias during the same act. d. There is a musical balance between the main and the secondary characters in each act: Act I and Susanna are more present. The others are passing through. Act II Susanna remains. Cherubino, Countess, Count are more visible. Act III Countess, Count and ensemble in the first plan. Act IV First phase: secondary characters Barbarina, Marcellina, Basilio. Second phase:, Susanna. Third (last) phase: Tutti. Act I Act II Act III Act IV Barbarina Countess Marcellina Count Overture Bartolo Cherubino sextet Basilio Susanna Countess Cherubino Susanna terzet terzet chorus Intermezzo Fig. 6 Mozart, The Marriage of. General plan of the drama

5 4. SUMMARIES After the above references regarding the mathematics in Bach s and Mozart s music, the questions are: What s these all about? What s the use? Do we need all this analysis to feel the music for real? We consider the answers both Yes and No! It is obvious for us that Kapelmeister Bach played in the Art of Fugue. We are convinced that he knew mathematics more than he seemed to know. He played for his own pleasure and satisfaction and he also challenged his contemporaries, but most of all the next generations. Art of Fugue is as much construction as intuition. We consider that the intuition (called here inspiration) begins and ends the composers action. Bach needed intuition to conceive the magnificent Matrix Theme. He also needed intuition to choose the appropriate variant of the theme transformations. Finally, he acted by intuition when he has built the entire monument. This is the reason we shall never know his final intention regarding the real form of this work. If he would act as a pure mathematician, we could nowadays provide the final solution, conceive a program, use a computer to end it! Mozart we dare say involves much more intuition in his working process [7]. Actually, his intuition included not only the unique and inexhaustible melodic inspiration, but also his options towards the almost mathematical planning of his works, from the shortest to the monumental ones. Le nozze di involves the same intuition as rational and precise construction. And this is not due to Da Ponte only. We tried to prove that above the text, the poetry, the drama structure brilliant created by the Italian writer we find here a musical architecture, a formal aesthetic, a musical profile of the characters all these due to Mozart himself. 6. REFERENCES [1] George D. Birkhoff, Aesthetic Measure, Cambridge, Harvard University Press, Comments by C.A. Garabedian. [2] Victor Ernest Masek, Artă și Matematică (Art and Mathematics), Ed. Politică, București [3] In the 12th century, Leonardo Fibonacci discovered a simple numerical series that is the foundation for an incredible mathematical relationship. Starting with 0 and 1, each new number in the series is simply the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,... The ratio of each successive pair of numbers in the series is Florica T. Câmpan, Povestiri cu proporții și simetrii (Proportions and Symmetry Stories), Ed. Albatros, București 1985, pp [4] Even they are music, the recitatives (secco or acompagnato) represent the story, the theatrical part of the libretto. [5] For economy, we prefer this form to vocal trio. [6] The Count and the Countess are mostly equally represented in the musical part of the drama. [7] This is why we consider the Peter Shaffer s drama Amadeus, a so happily exposal of the Mozart genius.

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