The early Aristotelian mechanics found its parallel conception of motion in the melodies of cantus firmus. The distinction between the potential to the actual of this motion would become in the centuries to come the foundation of impetus. Before the distinction between force and impetus, Medieval Music prevailed; movement was seen as a global driving mechanism in the modes of Gregorian Chants.

Impetus is later equated with acceleration as an essential component of force, and with the ascendance of Homophony, the potential of motion is fully transformed into action.

The Euclid’s geometry and porism (theorema) were based on local transformations oriented towards figural results as seen in the melodic contours of the Gregorian Chants’ styles.

It is worth to notice that during the Middle Ages, music was quite properly grouped with arithmetic, geometry and astronomy in the concept of the Quadrivium. Music rather than being simply an art form, it was allied with the three major sciences of those ages.

It is interesting to see that in the development of Greek geometry from Euclid s elements (which represents the type of geometry that was to predominate during the entire period from Antiquity to modern times), the methodology and system of rules remained apparent the same. Rene Discates’s Discourse on Method and Rules for the Direction of Mind to find Truth in the Sciences [1637] marks the beginning of a new concept of analytical geometry and the modern period of the history of mathematics.

Descates and Fermat were to replace the points in a plane by pairs of numbers and curves by equations. Thus, the study of the properties of curves was to be replaced by that of the algebraic properties of the corresponding equations.

In this way, geometry was reduced to algebra as we see homophony reduced to equated laws of properties of harmonic intervals.

The motion of transformation of coordinates from geometry to algebra had its purpose in the way to the discovery of properties of alignment and concurrence which can be verified in the counter-point development of Renaissance Music.

The early geometry figure of the Quadratix of the Greeks as represented by the intersection of the radius of a circle which rotates about the center, became in music, the spiral relation of harmonic functions known as the Circle of Fifths in the Baroque Era.

The development of transformation of spherical triangles and the formulation of the Treatise of Trigonometry to the application of algebra to integral calculus, allowed the possibility of quantified spatial relation of the Circle of Fifths, the quantization of the Major and Minor Triads, and the establishment of the Equal Temperament System, as well as the rise of Polyphony in Baroque Music.

The development of transformations into analytic geometry and the possibility to show how the movement and the symmetries of figures were related with the problem of changing the axes of coordinates been specified in terms of calculus, found its application in the counter-point symmetry of the music of Johann more than any other composer of the Baroque Period.

Transformation in spherical trigonometry would last almost two centuries up to the end of the Classical Period in music.

Transformation from the sixteenth to the eighteenth century never achieved the status of a Universal Method for geometry, however its change from usage, or implicit application to consequent use, and conceptualization, became what (to be known) under the term Thematization corresponding to 19th century musical harmony.

In the 18th century, transformations were applied by means of equations. Geometry appears only in the beginning of the process, with formulation of the problem, and at the end, as a translation of the result of applying algebraic transformations.

The parallel of the above description can be more evidently verified in the music of the Classical Period, where rules of tonal music are in the beginning of the process of formulation, and at the end, as a translation by means of recapitulation. The translation of the result applied by rules of transformation as seen in developmental sections by means of modulations and transposed themes, becomes the subject of the known geometric properties of the Sonata Form in music.

Throughout the 18th century, algebra was seen in itself as a system of forms which generate their own exclusive contents, a type of randomness in which all transformations are subordinated to an algebraic system corresponding to the Sonata Form in music.

In the 19th century, algebra enters into the domain of intrinsic necessities in which the observer does not sense the constraint of geometry (or musical form), instead, feels free to construct the transformations that seem useful. Geometry becomes nothing more than a system of forms modeled by the observer after he/she links its contents to other objects.

The process of the interaction of the observer with geometrical forms, constitutes the transformation of an endogeneous construction of the observer. The observer senses his emotions in the harmonic depth of music out of the necessity to dimensionalize the object (musical content) in his/her own.

The effort to adjust algebra to space and space to algebra developed in the composition of movements and axes of rotation in conjunction with curved lines, becomes the factor for the use of determinants and the use of imaginary numbers. Sense endogeneous factors no longer elaborate structure consisting of figures, they integrate all realizable constructions within total systems.

This new process and abstractive phase within the observer, represented the apogee of Romantic Music, where integrated structures are realized by means of chromaticism and all possible modulations within the rotational tonal system.

In the 20th century mathematics expands its notion of systems with the discovery of new dimensions of spaces. Integrated structures become highly coordinated by the use of symmetry as one of the determinant factors within systems. Whether the new forms of geometry are created or discovered, real or surreal, has been one of its most unrevealed mystery.

In 20th century mathematics, negative and positive numbers become equated on the formulation of matrixes on the compositional systems of symmetry; and 20th century music becomes ruled by a system of matrixes known as Dodecaphony, in which negative and positive numbers are translated into the notion of mirror intervals by the use of inversions and retrograded motions ruled by matrixes composed of 12 tone rows. Later on, referred as Serial Music.

At this point, I made an adjacent turn, and headed back in the direction of my apartment, cutting through a wide field of grasses. As I entered my bedroom, I sat quietly on my chair, and facing the window, I focused in the outside dark, until images of my dream started being projected in the inner space of the frame. The ideas from Dunne’s An Experiment with Time and Piaget’s Psychogenesis and the History of Science, began to ripple the images in the inner space, until they were, calmly solved in a strange dark tunnel.

I could still sense my consciousness moving away from the boundaries of the world line of the actual physical time... when suddenly, I stumbled on a realization within the inner frame of the tunnel, before the outer beyond:

THE END

© 2010 Nuclear Music Journal