Much has been said and published recently on chord geometry. This paper intends to provide scientific proof that the essence of harmonic progression is non-geometric. As an example of harmonic progression I will take Beethoven's theme of the slow movement from Piano Sonata op. 13. In a larger scope, this is the same as saying that visualization is not a replacement for theorizing on phenomena as they are. Even within mathematics, some new categories, such as function, were introduced first strictly as geometric (Leibniz and Bernoulli at the end of the 17th century), but then rose above visualizations into the abstractions that do not require references to geometry (Dirichlet, Euler, Frege). In today's calculus, the second order derivation and integration operate within purely topological system. in this regard. In this regard, music theory fell behind. Pythagorean numerology was taken for the face value and striped of all its metaphorical and theological aspects. Hence, the current condition with reliance on geometry of basic shapes and calculation with positive whole numbers. Specific derivation of the line notation from the simplified Euclidian ideas of two-dimensional space may have led theorists to speculations of a simplified geometric character, with the abscissa denoting temporal and ordinate spacial dimensions. Other systems of notation, such as neumatic or numeric, do not offer themselves to such speculations. Musical syntax can be related to the visual domain only through the power of a metaphor. It is undeniable that music moves forward and that there is a sense of harmonic progression. These events are happening not in visual space but in the specific musical space. It is not even aural or sounding; the core structures and processes of this space are inaudible, or in terms of Rameau, sous-entendu. Indeed, there is no visual equivalent of dominantness, subsominantness, tonicity or leading-tonness. Boleslav Yavorsky, perhaps sensing this deficiency, offered the non-geometric physical categories, such as force, gravitation, balance, momentum, ictus and predictus. Similar developments can be found in Hugo Riemann's discussion of rhythm (dynamische Schattierung). At this juncture, one may only ascertain that harmonic progression exerts centripetal and centrifugal forces, namely the force of constitution of tonality and the opposing force of modulation, gathering and dispercing. These forces are directional; one can point at the goal of each of these two. However, mapping of this directionality involves rough approximation. Its product is a topological and not topographic map. Its object is only partially available for verbalization and visual schematisation. Music making is thus always rooted in intuition and the subconscious cognitive processes.

Bartoli, Jean-Pierre. L'harmonie classique et romantique (1750-1900). Minerve, Paris, 2001.

Khannanov, Ildar. “Towards a Topology of Music.” The Routledge Companion to the Sound of Space. Philosophy & Politics: Virtual Spaces. Oxfordshire: Routledge, 2024.

Riemann, Hugo. System der musikalischen Rhythmik und Metrik. Breitkopf und Härtel Verlag: Leipzig, 1903.

Tymoczko, Dmitri. A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. New York and London: Oxford University Press, 2011.

Yavorsky, Boleslav. The Design of Musical Speech. Moscow: Kompozitor, 2022