composer, pianist, music theorist

Costère Calculator


This project was developed by Dr. Marcus Alessi Bittencourt and demonstrated in 2007 at the Brazilian Symposium of Computer Music (Simpósio Brasileiro de Computação Musical - SBCM-07) and at the XVII Congress of the Brazilian National Association of Graduate Research in Music (XVII Congresso da ANPPOM), both in São Paulo, Brazil. It presents a computational model in PHP of the analytical music theories by Edmond Costère (Costère, 1954), combined to Set-Theory (Forte, 1973 and Rahn, 1980). This model was implemented in the form of an analytical calculator available to the community via the internet as an HTML page. Such a tool allows the fast access to the analytical diagnostics generated by the computational model by means of an HTML page containing a concise form easily viewable and printable.

On Costère's Theories

The music theories of the french musicologist Edmond Costère, pseudonym of the famous magistrate of the French Supreme Court Edouard Coester (1905-2001), are for the most part unkown to the academic public but have never ceased to intrigue and fascinate those who know them, both positively and negatively. The general goal of Costère's work consists in the tentative of formulating a general music theory based on supposedly indisputable physical and universal grounds which could serve to explain the music craftmanship of all ages and cultures, thus demonstrating the non-existence of a real break between the music of the past and that of the present. Despite the multiplicity of possible musical styles, Costère observed the existence of something objective, acoustical and physical and not cultural and subjective, capable of generating relationships of polarization, of attraction between pitches, a principle always active in any sonority. Costère formalized this idea in the form of his Law of Universal Attraction, which establishes that the shortest distances between two points represent the main gravitational-flow passages for the forces of sound attraction. From this principle, Costère developed an extensive system of classifications and analytical diagnostics of all possible harmonic combinations using the western 12-tone equal temperament, reducing the number of these to 351 types by means of relationships of transpositional equivalence. Refutable or not, Costère's books promote an acute reflexion on musical craftsmanship, specially the 20th century one, and are presented with a fascinating fluidity and logic, with a prose of a passionate and delicious character. In Brazil, the interest in Costère's theories has its origins in the figure of the composer Willy Corrêa de Oliveira and his circle of pupils and former pupils. For an understanding of Costère's theories, Costère's own writings (Costère, 1954 and 1962) are obligatory stops, as are the excellent works by Marisa Ramires (Ramires, 2001) and Brian Ellard (Ellard, 1973).

On Set-Theory

Musical Set-Theory does not need a bigger introduction here, given its large influence and use in the contemporary academic world (Forte, 1973) (Rahn, 1980) (Straus, 1990) (Oliveira, 1998) (Perle, 1968). It is perhaps surprising here to see its use together with Costère's principles, but it is a fact that both theories share the concern to see similarities and equivalences between the sonorities with the goal of creating typologies and taxonomies of harmonic combinations. This is the basis to a process of reducing the number of possible harmonic combinations to the smallest possible number of basic types. Brian Ellard (Ellard, 1973) had already compared in his doctoral dissertation several contemporary theories such as the ones by Costère, Forte, Perle, Messiaen (Messiaen, 1944), Hindemith (Hindemith, 1945), Slonimsky (Slonimsky, 1947) and Hanson (Hanson, 1960), pointing at their similarities and differences in an extremely lucid and constructive way.

The Project of the Analytical Calculator

The study of the combinatorial properties and the equivalence relationships of pitch sets, mapping their attraction flows and their invariance potentials under operations of transposition and inversion is a very important task for compositional and analytical work. Nonetheless, such methodology requires a certain amount of calculations which are, although not too complex from the point of view of mathematicians, certainly boring and cumbersome from a musician's standpoint. The simple consultation to the “Table Synoptique des 351 Échelonnements” and the “Tableaux Analytiques des Échelonnements” of the book “Lois et Styles” by Costère (Costère, 1954) or to the tables found in the works of Straus (Straus, 1990) or Forte (Forte, 1973) does not provide a convenient access to the study of pitch-class sets. It is with this realization in mind that this calculator was created: to provide quick and clear access to the interpretations and diagnostics done from the analysis of musical data submitted by a user. The project was implemented in PHP so that the calculator would be absolutely multi-platform and accessible online at the internet to all the music community, without the need to install special software, except for the usual web browser normally present in any modern computer.


COSTÈRE, Edmond. Lois et Styles des Harmonies Musicales. Paris: Presses Universitaires de France, 1954.
COSTÈRE, Edmond. Mort ou Transfigurations de l'Harmonie. Paris: Presses Universitaires de France, 1962.
ELLARD, Brian. Edmond Costère's Lois et Styles des Harmonies Musicales, an English translation and commentary. Rochester: University of Rochester PHD thesis, Eastman School of Music, 1973.
FORTE, Allen. The Structure of Atonal Music. New Haven: Yale University Press, 1973.
HANSON, Howard. Harmonic Materials of Modern Music. New York: Appleton, Century and Crofts, 1960.
HINDEMITH, Paul. The Craft of Musical Composition. New York: Associated Music Publishers, 1945.
MESSIAEN, Olivier. Technique de mon language musical. Paris: Leduc, 1944.
OLIVEIRA, João Pedro Paiva. Teoria Analítica da Música do Século XX. Lisboa: Gulbenkian, 1998.
PERLE, George. Serial Composition and Atonality. Califórnia: University of California Press, 1968.
RAHN, John. Basic Atonal Theory. New York: Schirmer Books, 1980.
RAMIRES, Marisa. A Teoria de Costère, uma perspectiva em análise musical. São Paulo: Embraform, 2001.
SLONIMSKY, Nicolas. Thesaurus of Scales and Melodic Patterns. New York: Coleman-Ross Co., Inc., 1947.
STRAUS, Joseph Nathan. Introduction to Post-Tonal Theory. New Jersey; Prentice Hall, 1990.

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