The Abstract Seed of the Inverse System
At the core of the system lies the numerical relation 1–3–5, defined as the Abstract Seed. It is not a historicized tonal triad, but a generative structure capable of producing simultaneous coherence between melodic development, intervallic stability, and imitation.
Polyphony is not added; it emerges from the structure.
The Law of Unique Pairs
In an eight-voice canon, there exist nine possible ordered pairs of the elements 1, 3, 5. To construct a canonical cycle without parallel fifths or octaves, any repetition of a pair must be avoided.
Out of 560 theoretical permutations, only 24 cycles are valid. This result shows how apparent complexity turns into constrained selection.
The Inverse System, in fact, eliminates the error before it becomes sound.
The Three Matrices
The Sound Matrix (MS), the Clef Matrix (MK), and the Duration Matrix (MT) operate jointly. Horizontal construction is calculated to satisfy vertical constraints. Harmony is implicit in the generation.
Download the paper
The complete model is presented in the preprint available on Zenodo:
Download the full paper in English (PDF).
Original version: Composing Music with the Pythagorean Table.
Download the Italian version of the paper (PDF).
Inverse 1–3–5 System and Pythagorean Table.
Official DOI: https://doi.org/10.5281/zenodo.18497758
Conclusion
The Inverse 1–3–5 System demonstrates that complex polyphony can be generated in a controlled, verifiable, and reproducible way. The Pythagorean Table thus becomes a sound matrix capable of producing high-density polyphonic architectures without resorting to empirical corrections.
The Inverse 1–3–5 System integrates mathematics and music theory.
Discover the other structural models developed in the ItalianOpera research.