Abstract
This contribution presents the formal core of the so-called “Inverse System 1–3–5,” a deterministic model for generating infinite canons built upon three symbolic pillars. The system does not begin from a vertically superimposed score, but from a horizontal line, according to a logic akin to choirbook writing.
The problem addressed is the following: given a reduced alphabet {1, 3, 5}, is it possible to construct a cyclic seed which, once superimposed at canonical distance d = 1, never produces within the same vertical alignment the co-occurrence of the pairs 11 and 55? The verification is conducted through exhaustive enumeration of all possible cases.
The result shows that, under these constraints, the maximum compatible cyclic seed length is 8. No seed of length 9 satisfies the vertical condition, despite the existence of 216 theoretically admissible cyclic structures with nine distinct bigrams.
Theoretical Premise: the Abstract Seed and Horizontal Writing
The Inverse System 1–3–5 originates from the idea that the structural framework of a canon can be conceived as an autonomous cyclic line, independent of vertical score representation. The alphabet is reduced to three symbols: 1, 3, and 5. They represent structural pillars, not yet concrete pitches, but relational functions.
The canon is conceived as infinite. After the entrance of the last voice, all parts continue without interruption. Consequently, the sequence must be treated as cyclic: the pair formed by the last and the first element is structurally meaningful.
Two fundamental constraints are introduced.
The first is a horizontal constraint: within a cycle, bigrams (consecutive pairs) must not repeat. If a pair is repeated, it will eventually be superimposed upon itself through canonical translation.
The second is a vertical musical constraint: within the same alignment, the pairs 11 and 55 must not appear simultaneously. In the adopted model, such a configuration generates structurally incompatible parallelisms.
The canonical distance is fixed at d = 1. Each new voice enters shifted by one position relative to the previous one. This distance constitutes the structural backbone of the system and is not altered in the present verification.
Objective of the Script
The Python script developed for this verification has a precise aim: to enumerate all cyclic seeds of length L over the alphabet {1,3,5} that satisfy the uniqueness constraint on bigrams, and to verify for each whether, in the canon at distance d = 1, there exists at least one vertical slice containing simultaneously 11 and 55.
For L = 9, the number of theoretically possible structures with nine distinct bigrams is finite and computable. The objective is to determine whether at least one of them satisfies the vertical constraint, or whether the conflict is structural.
The code does not generate music; it verifies a combinatorial property of the abstract system.
General Structure of the Code
The core of the analysis consists of three fundamental functions.
The first constructs the list of cyclic bigrams of a sequence. Given a list of symbols, all consecutive pairs are generated, including the pair formed by the last and the first element. This step is essential, since the canon is infinite and therefore cyclic.
The second function verifies that all such bigrams are distinct. If the cardinality of the bigram set coincides with the length of the sequence, the horizontal constraint is satisfied.
The third function simulates canonical superposition. For each position i of the cycle, the bigrams of the different voices are collected by shifting the sequence by k positions (with step d = 1). This produces a vertical “slice,” that is, the set of simultaneous pairs at that instant.
Within each slice, the simultaneous presence of the pairs (1,1) and (5,5) is checked. If both are present, the seed is discarded.
Enumeration of the sequences is performed via backtracking with pruning. The sequence is built symbol by symbol; whenever a previously used bigram is generated, that branch is immediately abandoned. In this way only potentially valid structures are explored.
Exhaustive Verification for L = 9
For L = 9, the number of cycles with nine distinct bigrams is 216. This means that there exist 216 cyclic sequences using all nine possible bigrams of the alphabet {1,3,5}.
The script examines each of these 216 structures and checks the vertical constraint. The result is unequivocal: none of them passes the test.
In all 216 cases, at least one vertical slice contains simultaneously 11 and 55. The conflict is not accidental, but structural.
It follows that the maximum cyclic seed length compatible with the adopted constraints is 8.
The verification is reproducible through automated testing (pytest) and does not rely on random sampling. It constitutes a complete exploration of the combinatorial space for L = 9.
Counting Cyclic Classes
Since the canon is infinite, rotations of the same cycle are equivalent. The sequence 1–3–5 is structurally indistinguishable from 3–5–1 or 5–1–3. For this reason, counting is also performed modulo rotation.
The number of valid cyclic classes (distinct up to rotation) for L = 2…8 is 93. For L = 9 the number is zero.
These values constitute the repertory of structural frameworks compatible with the vertical constraint in the Inverse System 1–3–5 at unit canonical distance.
Expected and Effective Results
Prior to computational verification, it was reasonable to hypothesize that simultaneous use of all nine bigrams might force the alignment of 11 and 55 within the same slice. Exhaustive enumeration confirms this intuition.
Expected result: possible structural collapse at L = 9.
Effective result: out of 216 possible cases, 216 violate the vertical constraint. No exception.
The limit at 8 is not a convention, but a property of the system under the adopted constraints.
Perspectives
The distance d = 1 represents the structural backbone. A future extension may densify the structure by introducing intermediate symbols (for example 1(2)3(4)5), without altering the underlying logic.
The Inverse System 1–3–5 thus constitutes an open formal core: a minimal deterministic engine from which contrapuntal realizations, rhythmic expansions, and concrete tonal mappings may be derived.
The complete code and verification tests are publicly available and allow full reproduction of the results presented here.
The Inverse 1–3–5 System is verifiable and reproducible.
The archived and citable version of the software is available on Zenodo (DOI), while the source code and verification scripts are accessible in the development repository on GitHub. The computational tools demonstrate the structural limit of the cyclic canon (maximum 8 voices with d = 1).