The perception of the contacts between the two domains, with their positions, can only enrich the act of creation. A decline in this act is present when the inspiration is missing, when fantasy and lyricism are incomplete and when there intervene (out of snobbism) a search for new laws in composition, other than the ones specific to the art, whose final goal is the esthetic category of the beautiful.
	Musical composition – an ineffable act between fantasy and arithmetical and geometrical rigor by Dr.Liana Alexandra Biographical materials concerning Dr. Alexandra are available online: http://romania-on-line.net/whoswho/AlexandraLiana.htm http://romania-on-line.net/general/duointermedia.htm Résumé This article is reproduced by permission of the author who retains full copyrights. Motto: “We can reduce all to numbers, including Beethoven’s music. But we do not hear numbers, we hear music”. Pierre Schaeffer Chapter 1 Link between music and mathematical sciences – always present and proved since Antiquity      The topic of the relation between music (the art of sounds) and exact sciences is more and more complex in the light of the achievement of 20th century music. Today, the new computer technique and the electronic devices are intensely present both in the composition act and in the interpretation and the understanding of the works of art. Until reaching this level, music had always relations with the sciences, which were present at an organic level. The perception of the contacts between the two domains, with their positions, can only enrich the act of creation. A decline in this act is present when the inspiration is missing, when fantasy and lyricism are incomplete and when there intervene (out of snobbism) a search for new laws in composition, other than the ones specific to the art, whose final goal is the esthetic category of the beautiful.

The music-mathematical sciences relationship was present early in Antiquity and was emphasized by a large number of Greek authors: Pythagoras of Samos (centuries 4-5 BC), Aristoxenos of Tarentum (4th century BC), Plato (centuries 4-5 BC), Aristotle (4th century BC), Vitruvius Pollio (1st century BC). During the Middle Ages, the art of sounds was considered an important subject in education, next to the other sciences of the Quadrivium: arithmetic, geometry, music and astronomy. Severinus Boethius (475-524 AC), Leonardo Fibonacci (centuries 12-13), Gioseffo Zarlino (1517-1590), René Descartes (Cartesius) (1596-1650), Jean Philippe Rameau (1683-1764), Herman con Helmholz (1821-1892) are some of the significant authors for Middle Age, Renaissance and Modern times, who conducted research on the everpresent connection between music and mathematics, as it was known from philosophy and from artistic output. Gottfried Wilhelm Leibniz (1646-1716) wrote: “music is an exercise of covert arithmetic and the one who devotes himself to it is not aware of the fact that he is handling numbers.”

From the above-mentioned authors I kept the following assertions, which I considered the most significant: a) From Pythagoras – the Pythagorean concept of acoustics and the theory of figured numbers (triangular numbers, square and rectangle numbers, applicable in modal structures) b) From Plato – his discoveries of “the golden ratio” and on the heterophonic mode (this acoustic mode is constructed on the proportions of the numbers 2, 3, 4, 9, 8, 27 and represents the harmony of the universe) c) From Aristotle – the philosophical implications on the relation music-mathematics (through the classification he proposes inside the sciences: theoretical sciences, practical sciences and poetical sciences). Music, together with poetry and architecture is in the poetical sciences group. d) Aristoxenos of Tarentum (considered the greatest musician of Antiquity) was also interested in the relation between the art of sounds and mathematics in his studies on acoustics, researching micro-intervals. e) From Vitruvius Pollio I took the idea of symmetry of form, his theory of the tetrachords (diatonic, chromatic and enharmonic tetrachords), the relationship between theatrical acoustical ceramic and the tetrachords systems. f) Leonardo Fibonacci left mankind an interesting arithmetic equivalent of the “golden ratio” in the law of organic growth and the law of Fibonacci series. g) Gioseffo Zarlino, important theoretician from the Renaissance, was preoccupied by the acoustic non-tempered standards and by the definition of the major and minor scales. h) Acoustical studies were made by René Descartes (Cartesius) too, who looked on music aesthetics in concordance with acoustics and music psychology. i) Jean Philippe Rameau, an important theoretician, was considered the founder of classical harmony concepts j) The relationship between music and mathematical sciences was studied by Hermann von Helmholz too, who focused on the theory of physiology and acoustics of music; in his opinion the musical “harmony“ (the consonances) creates continuous excitations and the dissonances discontinue excitations.       The evolution of this strong connection between music and science and its particularity according to historical periods is the result of continuous research, which in the 20th century developed into several aspects connected to the investigation of the artistic conception act, to the understanding of the works of art, to modern analytic methods. In the 20th century there were some artistic styles which emphasized this connection (perhaps, some times, not in favor of the lyric expression of the artistic message): 12 tone music, serial music, atonal music, stochastic music, random music, minimal music, repetitive music etc. Musical instruments developed very much, according to the technical and scientific evolution, next to the traditional sources using electronic devices and computers. From the aesthetics point of view, semiotic and semantic theories determined the development of quite an industry of explanations of the role of graphic sign and of the sound signal, explanations that are interesting to some extent, but which cannot get inside the profound mechanism of artistic creation and cannot establish precise rules of how to compose. The relationship between music and the exact sciences must not have a forced effect, coming from outside human creativity. In this line, I express some reserve towards those styles and those composers which have no talent to create a lyric expression and which transforms the art of sounds into a master craft of mixing uninteresting frequencies and rhythmic impulses, without coming to something expressive and clearly constructed. I see expressiveness and the national character of the music as being two essential and eternal coordinates of any work of art, regardless the historic moment when it was created. In my opinion, the presence of rigor in the construction of music is capable on generating beautiful music even in this century, when technology competes with the sentiment; so the rigor of construction is not a characteristic of the ugly and anti-human music only. In fact, from another angle, my interest is directed toward a plea for the presence of consonance in music, this being not an old-fashioned concept, but an essential way of expressing the harmony, the beautiful, the light, the most beautiful arithmetic relations which human intelligence can produce, of expressing the human soul and spirit.

Chapter 2 The magical squares and their presence in music A. Definition of the magical square       We call “magical square” a square of numbers where there are n2 numbers, aligned consecutively or not, so the sum of the numbers placed on each of the two diagonal lines is equal to the sum of the numbers on each column or row. This constant sum is named ”the magical number” of the square. B. History of the magic square       Astrologists of Antiquity, for instance the Chinese, in the 7th century BC, then the Arabs, were building talismans to which they were giving magical powers. These squares were popular in Europe in the Renaissance – for instance Albrecht Dürer, in his painting The Melancholy, painted a magic square with the magic number 34. In the same period (14th century) the Greek mathematician, Manuel Moscopoulos wrote about the magical squares, which he named arithmetic squares (“tetragonon arithmon”). He was first to present a general method of building a square of odd squares and double even squares. The issue of magical squares, seen as arithmetic fun, became a delight during those times, being of interest to great mathematicians like Euler or like Benjamin Franklin. Magical squares remain an attractive domain until our day, without the magical attribute, of course. C. Rules of making magical squares       The odd magical squares may be made following the method of Bachet de Méziriac, who published in 1612 the book “Nice and delightful problems which may be solved by numbers”, or following the method of Philippe de la Hire (1700). D. Applications in music of the odd magical squares       Those arithmetic squares can be used in music, their numeral equivalence creating very interesting modal systems, frequently used by composers, during the centuries. If we connect an interval to each figure, in ascendant order, starting with the semitone, we can trace the following parallelism:      In this chapter, I presented a demonstration of modes applicable to the arithmetic squares of the 3,5,7,9 order and to the square named “hypermagical square”, from which results constant rules of forming chords: 1) The reading of the figures in rows gets the intervallic sum of the 2 diagonal lines. 2) The reading of the figures in columns gets the result of the substraction of the 2 diagonal lines 3) Both the chords in rows and in lines are disposed symmetrically 4) There is a symmetry also in the plan of the modal structures 5) Each arithmetic square presents 2 intervallic constants of the 2 diagonal lines, by which the whole modal system may be built, which will have as symmetrical axes the numeric correspondence given by its “magical sum.” 6) The system may be repeated indefinitely and is applicable also to arithmetic squares made of random figures Making a numeric equivalence in the plan of durations too, there will often appear rhythmic imitation structures, both when reading rows and columns.

Example: The square of the 5th order used for rhythmic structures, with the basic value of the 16th note. Horizontal reading (rows) Vertical reading (columns) Chapter 3 The relationship between musical forms and geometric forms “In art everything is made up after three fundamental geometrical forms: sphere, cube, and cylinder. You have to learn to draw those very simple forms, and afterwards you can draw whatever you want.” Paul Cézanne       The music-mathematical sciences relationship has always been present and proved since Antiquity not only in acoustics but also in the field of musical architecture. Making musical forms often has to do with geometrical forms. It was not incidentally said that music is an architecture which evolves in time. This relation may be seen and proved from various angles, which seems to have been fundamental during the ages. A. The relation between the five solids of Plato and musical form       The five convex regular solids (three-dimensional geometrical shapes) are: the tetrahedron, the hexahedron (the cube), the octahedron, the dodecahedron and the icosahedron. What can be the link between the shape of a musical construction (form) and those Plato solids? It exists organically, can be proven, and one can show the close connection between space geometrical thinking and musical creative thinking. It is a known fact that in the art of sounds there are few fundamental construction schemes, which have generated over the centuries various specific forms and genres like A (single-part form) AB (binary form) ABA (the ternary form) AAB or ABB (bar-form) ABACABA (rondo-form) If we associate the above-mentioned regular tetrahedrons with the reading of a mathematic Hamiltonian circuit, we will have succeeded in discovering the presence of musical construction schemes used in all styles. The basic circuit imagined by Hamilton on a regular polyhedron is the one which forms a closed circuit, along the edges, passing only once through each peak. 1) If this circuit is made along the edges of a hexahedron (cube), having the coordinates A, B and C, the road to be taken is ABACABA, which is the classical rondo-form. 2) Following the same principle, the relationship between a tetrahedron and musical architecture generates the scheme ABA or AAB (bar-form) 3) Reading by the same rules, an octahedron develops the following musical pattern: ABBBA or ABCBA. 4) The relationship between musical form and the dodecahedron (considered the measure of the Universe) is a unit: A. 5) The icosahedron, consisting of 20 equilateral triangles will suggest a mono-part form also, replicable to infinity: A. Concluding, we may observe that: a) the octahedron and the hexahedron are 2 Plato solids which generate musical forms of the same family (ABCBA or ABACABA) b) the icosahedron and the dodecahedron give the same musical pattern (A) c) the tetrahedron has a unique form (AAB or ABB) B. The relation of symmetry present in sounds architecture       Symmetry is an essential coordinate in composition and a rigorous geometric concept. Both in geometry and music there are several types of symmetry: 1) Bilateral symmetry (plane symmetry) – which exists in music in palindrome canons in which the melody which imitates is the recurrence of the first melody. Many composers, among them Joseph Haydn, Johann Sebastian Bach, Ludwig van Beethoven, Paul Hindemith, Arnold Schönberg, Alban Berg etc., used this kind of schemes for obtaining effects in counterpoint. 2) Translational symmetry. This can be of two kinds: a) rhythmical translation symmetry b) cylindrical translation symmetry Both types are used in music, in various composition processes. For instance, rhythmical translational symmetry (of infinite ratio) can be frequently found in all repetitive music, and cylindrical translational symmetry (of finite ratio) is present in the typology model – sequence –cadenza. 3) Plane rotating symmetry. This can be of two kinds: c) rotating cyclic symmetry (rotation without reflection) d) rotating dihedral symmetry (with reflection) The rotating cyclic symmetry has equivalence in musical composition in the articulation of the architecture, a characteristic example being the exposition of the fugue. The musical equivalent of the dihedral rotation may be considered the palindrome type structures, transposed n times. 4) Rotating symmetry in space. This can be of three kinds: e) symmetry made of a rotation in plane and an orthogonal translation in plane f) symmetry made of a rotation in plane accompanied by dilatation g) symmetry made of a rotation, a translation and a dilatation All these kinds of rotating symmetries are frequently used in the musical creation process. The Theme and variations form clearly belongs to the typology of rotating symmetry with translation and dilatation in space because the periodic replay of an initial theme asks for a cyclic re-evaluation, and its more and more ornamented shape suggests a dilatation of this symmetry. C. Structural rhythm – dynamic symmetry – logarithmic spiral       In musical scores, rhythm is present both as a succession of equal or different values and as a succession of structures, which ends uniformly or asymmetrical in the fields of an architectural configuration. Musical forms in their articulation, both at the micro-structural and macro-structural levels, present somewhat interior symmetries, which by their continuous projection during the piece suggests the existence of a logarithmical spiral. Structural cadenzas in musical language may be grouped on binary or ternary periodicity bases, which are present both at the level of primary configurations and at the level of large surfaces. The archetype of construction model-sequence-cadenza is also a translation of the logarithmical spiral from geometry to the art of sounds. The structural binary and ternary rhythms present in music, generated some typologies of architectural organic articulation such as the verse-rhythm relationship established early in Antiquity two large groups of rhythms: dissyllabic and trisyllabic.

Therefore, the structural binary rhythms may have their roots in the antic rhythms of spondees () and pyrrhic (), and the structural ternary rhythms in the antic rhythms of the dactyls (), anapests () and amphibrach (). Also, the rhythms molos () and tribrachic () may be found in music, but their uniformity converge more towards static cadenzas. The rhythm anapest () presents 2 short values and a third long value. This sequence is identical to the architectonic form AAB which I mentioned before regarding the relationship between Platonic solids and musical architecture and regarding the typology model-sequence-cadenza. Cornel Ailincăi, in his book “Introduction to the grammar of the visual language” concludes: „If we think of rhythm in the most general meaning, its existence requires a continuous, uninterrupted movement of extension (the Greek word rhythmos comes from reo – to flow), but at the same time requires a division of the infinite extension in periods to be repeated by a rule. Therefore, rhythm is different from ordinary movement by a succession of phases whose discontinuity ensures the reconstruction of the movement”. Chapter 4 The relation between the mosaic and musical architecture       Geometrically, the mosaic is built up of pre-defined elements, like squares, rhombs or triangles. Inside the mosaic one can make original shapes, using various combinations of the above mentioned elements. Mosaic has been frequently used since Antiquity in the decorative arts, using its fundamental elements which made an homogeneous field paved with squares, equilateral triangles and hexagons. The Romans, the Persians, the Chinese, and the Japanese were masters of ornamentation, and they made all kinds of mosaic surfaces, using repeated elements.       The mosaic may be used in music architecture, at the micro-structural level, as well as at the macro-structural level. It can be found, for instance, in the technique of collage where music entities apparently dissimilar may give, together, a nice ensemble impression. We can take Gustav Mahler’s symphonies as a characteristic example of this. Also, we can easily disclose mosaic in the juxtaposing of micro-forms, differently repeated in two dimensions. For this, the typical example is the passacaglia form. Here we meet the phenomenon of figure – background, where the perception discovers two simultaneous aspects of the same image. One is the background, the rhythm and the melody of the passacaglia, which remains unchanged, which forms the homogeneous pavement, and the other is the figure, built up of different polyphonic variations, laid over the background.       The ostinato form (which may have found its equivalent in mosaic decoration) is frequently found in music. Among the famous examples there are the 32nd C minor Beethoven variations, the 4th Hindemith string quartet op. 32 (last movement), the 3rd Bartok quartet (first movement), J. S. Bach’s Crucifixus from the D minor Mass, D. Buxtehude’s E minor Ciaccona for solo organ, M. Reger’s Introduction, Passacaglia and Fugue for 2 pianos op. 96, J. Brahms’ Variations on a Haydn theme op 56a (last movement) and his 4th Symphony (last movement), I. Strawinski’s Psalms symphony, A. Webern’s Passacaglia op. 1 for orchestra, A. Honegger’s Pacific 231 etc. Chapter 5 A. The notation of the “parlando rubato” sections used in my own compositions       Around 1977-78, when I elaborated Incantations I and II, inspired by manuscripts of Filothei Sin Agăi Jipei, which I analyzed, focusing on their rhythmic and temporal aspect, I was more and more preoccupied with finding a more rigorous way of writing what we call “parlando rubato”. It may seem strange to want to write exactly a “parlando rubato” music, but I wanted to create a type of music consisting of rhythmical configurations at the micro-structural level, to be used by each of the interpreters, so that each concert variant is as close to my psychological tempo and, especially, to the refined waves I imagined in the creational process. I used the first attempts of this kind in the cycle Incantations, and after I introduced that writing with consistency in all slow tempo music that followed (3rd, 4th, 5th and 6th Symphonies, the Concerto for flute, viola and chamber orchestra, opera The Snow Queen, ballet The Little Mermaid, the Concerto for strings etc.), this writing is nowadays a constant method of imagining rhythmic rows for various texts or polyphonic layers (some technique may be applied also in homophonic music).       Starting from the well-known Fibonacci series (1, 2, 3, 5, 8, 13, 21 etc.), including its translations (like 1, 3, 4, 7, 11, 18, 29, 47 or 0, 2, 2, 4, 6, 10, 16, 26, 42, 60 etc. or 0, 3, 3, 6, 9, 15, 24, 39 etc. or 1, 4, 5, 9, 14, 23, 37) I started to imagine various non-demoting series applied to a time unit (for example the quarter note) and to its equal divisions in 2, 3, 4, 5, 6. Those rhythmic structures resulting from the use of non-demoting, above-mentioned partitions present some constant characteristics: 1. Each configuration has a certain phrasing, both at the micro-structural and macro-structural levels, given by the sum of the terms used in those series 2. The usage of simultaneous series (Fibonacci’s series translations) may give the same pattern on the vertical plane (regardless of the subdivision of the main unit), if sum of the terms is the same Example: the superposing of the series 3, 2, 3, 1, 3, 2, 3, 1 … and 1, 3, 1, 4, 1, 3, 14, … 3+2+3+1=9 1+3+1+4=9 3. It becomes clear that by applying Fibonacci series to 5, 6, 7 etc., equal impulses subdivisions of the main value, the result is rhythmic structures with refined waving, which are not always very strict (especially if they are used in combinations, like quintuplet and sextuplet); but if they are written in this technique, they may become a pattern to which each interpretation is close, with no major difference from the initial structure had in mind by the composer. 4. The Fibonacci series is the most pure type of the two-beat accumulational series, but the infinite reproduction of the initial series is the Pascal triangle, the richest diagram in algebric and geometric properties. This diagram includes figurative numbers series – triangular, tetrahedric, pentagonal etc., often present in the rhythmic configuration of musical pieces. B. The notation of some rhythmic structures based on the accumulational series of triangular numbers.       The series of the triangular numbers is 1, 3, 6, 10, 15, 21, 36, 45 etc. From their various combinations (taken in row or by one, two or several leaps), by their application to various basic time units or to subdivisions of these units, by their use in various partitions of non-demoting series, one can creat endless rhythmic combinations. These rhythmic structures always have the same invariable, which is the sum of the terms used in the numeric loops gives always the periodicity of the rhythmic figures at the micro-structural or macro-structural level. This phenomenon may be the explanation of the frequent assertion of the musicologists that “micro-structure generates macro-structure.” Example: The periodicity is of 20 impulses (1+3+6+10=20) Example of notation of rhythmic structures based on Fibonacci series numbers: Liana Alexandra – 3rd Symphony, Editura muzicala, 1985, 2nd movement, page 69. Chapter 6 Epilogue       The complex relations between the art of sounds and mathematic sciences which I tried to investigate in this study and to demonstrate in musical examples, with which creative thinking operates frequently, may define, to a certain extent, the rational aspect of artistic creation. But in addition to these investigations and demonstrations there must always be inspiration, the ineffable aspect of human creativity, which constantly provides connexions and original ways of expressing the beautiful in art. Together with the intellect, inspiration is that which gives aesthetic value to a work of art. When inspiration is not present, one may produce abstract schemes, inventive charts, and scholarly written explanations, but these totally lacking that warmth specific to any creative act.       In order to emphasize these assertions, I cited in the final part of my study some words of wisdom written by great individuals in the history of mankind, which stress the aesthetic dimension of the work of art, without which no final product of human fantasy, harmoniously knitted with brainpower, may exist. These words were written by Democritus, Heracles, Simylus, Plato, Pascal, Michelangelo, Shakespeare, J. J. Rousseau, Nicolae Iorga, Barbu Ştefănescu Delavrancea, Tudor Vianu, Arthur Honegger, Anton Webern, George Enescu, Abraham Moles, P. A. Michelis and Confucius. For this résumé I have chosen fragments from the ideas of Arthur Honegger, George Enescu and Confucius.

Arthur Honegger: “Writing music is like putting a ladder without fixing it. With no scaffolding, a building in construction may stay in place only by a miracle, a miracle of the internal logic, of an inner sense of proportion. I am in the same time the architect and the spectator of my works; I work and I analyze my work…” George Enescu: “It is true that music is related with mathematics. But the great composers were no mathematicians; or, if you like better, they were, but in an unconscious way. Bach, with his genius, sensed the superior connection between the fragments of his works. His pieces may disclose mathematical ratio and proportions, but Bach himself has not created them by logical, deductive thinking. The composer is a mathematician, or more precisely, the mathematical spirit dominates him like the profound intelligence”. Confucius: “If you want to know if a country is well governed, you have only to listen to its music.”

Bibliography Ailincăi, Cornel. Introducere în gramatica limbajului vizual (Introduction to the grammar of visual language). Editura Dacia, Cluj-Napoca, 1982. Berry, Wallace. Form in music. Prentice Hall, New Jersey, 1966. Banu, Ion. Platon heraclitul (Plato Heraclitus). Editura Ştiinţifică şi Enciclopedică, Bucureşti, 1972. Birkoff, George David. Aesthetic measure. Cambridge, Massachusets, 1933. Cîmpan, Florica T. Povestiri cu proporţii şi simetrii (Stories with proportions and symmetries). Editura Albatros, 1985. Descartes. Colecţia texte filosofice (A Collection of Philosofical texts). Editura de Stat pentru Literatură şi Ştiinţă, Bucureşti, 1952. Dumitru, Anton. Teoria logicii (The theory of logic). Editura Academiei, 1973. Domoread, A. P. Jocuri şi probleme distractive de matematică (Games and funny maths problems). Biblioteca de ştiinţe matematice din România, Editura Didactică şi Pedagogică, Bucureşti, 1969. Eco, Umberto. Opera deschisă (The open oeuvre). Editura pentru literatură universală, Bucureşti, 1969. Giuleanu, Victor. Principii fundamentale în teoria muzicii (Fundamental principles of music theory). Editura muzicală, Bucureşti, 1981. Giuleanu, Victor. Tratat de teoria muzicii (Music theory treatise). Editura muzicală, Bucureşti, 1986. Giuleanu, Victor. Ritmul muzical (Musical rhythm). Editura muzicală, Bucureşti, 1968, 1969. Ghica, Mattila. Estetica şi teoria artei (Aesthetics and the theory of art). Editura Ştiinţifică şi Enciclopedică, Bucureşti, 1981. Georgescu, Corneliu Dan. Semnale de bucium (Bucium signals). Editura muzicală, Bucureşti, 1987. Gardner, Martin. Amuzamente matematice (Mathematical amusements). Editura Ştiinţifică, Bucureşti, 1988. Helmholz, Hermann von. Despre semzaţiile de ton (On the sensations of the tone). Braunschweig, 1863. Joja, Athanasie. Istoria gândirii antice (History of the thinking in Antiquity). Editura Ştiinţifică şi Enciclopedică, Bucureşti, 1982. XXX. Maxime şi cugetări (Expressions of wisdom). Editura tineretului, Bucureşti, 1968. Matei, Dumitru. Originile artei (The origins of art). Editura Meridiane, Bucureşti, 1968. Messiaen, Olivier. Technique de mon langage musical (The technique of my musical language). Paris, Leduc, 1944. Michelis, P. A. Estetica arhitecturii (The aesthetics of architecture). Editura Meridiane, Bucureşti, 1982. Moutsopoulos, Evanghelos. La musique dans l’oeuvre de Platon (Music in Plato’s writings). Paris, 1959. Moles, Abraham. Creaţia artistică şi mecanismele spiritului (The artistic creation and the mechanisms of the spirit). Estetică, informare, programare. Editura ştiinţifică, Bucureşti, 1972. Moles, Abraham. Artă şi ordinator (Art and computer). Editura Meridiane, Bucureşti, 1970. Platon. Opere (Oeuvres). Editura ştiinţifică şi enciclopedică, Bucureşti, 1976. Pohonţu, Eugen. Iniţiere în artele plastice (Initiation in visual arts). Editura Albatros, Bucureşti, 1980. Radian, H. R. Cartea proporţiilor (The book of proportions). Editura Meridiane, Bucureşti, 1981. Roman, Tiberiu. Simetria (The symmetry). Editura tehnică, Bucureşti, 1963. Read, H. Originile formei în artă (The origins of the form in art). Editura Univers, Bucureşti, 1971. Stancovici, Virgil. Logica limbajelor (The logic of the languages). Editura Ştiinţifică, Bucureşti, 1972. Schillinger, Joseph. The mathematic basis of the arts. Philosophical Library, New York, 1948. Vianu, Tudor. Estetica (The aesthetics). Editura Minerva, Bucureşti, 1968. Toduţă, Sigismund. Formele muzicale ale barocului în operele lui Johann Sebastian Bach (The forms of the baroque music in Johann Sebastian Bach’s music), vol. I-III. Editura muzicală, Bucureşti, 1969, 1973, 1978. Vitruvio Pollio. Despre arhitectură (On architecture). Editura Academiei, Bucureşti, 1964. Webern, Anton. Calea spre muzica nouă (The way towards new music). Editura muzicală, Bucureşti, 1988. ---------- NB: Titles of the books of Romanian authors were translated for the understanding of English-speaking public. The books were printed in Romanian.

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