The Abstract Seed and Generative Seeds

The Abstract Seed: Three Numbers That Are Not Three Notes

We begin with a minimal sequence:

1 – 3 – 5

These numbers do not indicate a fixed melody (such as C–E–G). Rather, they indicate a relation: each step is a third above or below, and the sequence describes a form of motion. At this stage, it does not matter whether the third is major or minor, because what matters is the structural distance between one event and the next.

For this reason, we call it an abstract seed. It may take many melodic forms, yet it preserves the same underlying framework. It functions like a mould: change the material and the colour, but the shape remains recognisable.

This abstraction has a practical advantage: it allows us to treat polyphony as a problem of coherence between movements before treating it as a matter of choosing attractive notes.

Three Readings of the Same Structure: Horizontal, Vertical, and Diagonal

The seed 1–3–5 becomes particularly interesting because the same object can be read in three different directions.

1) Horizontal reading (melody).
Read from left to right, 1–3–5 produces a temporal succession: a melodic line built in thirds.

2) Vertical reading (chord).
If we place 1, 3, and 5 in a column, we no longer have succession but simultaneity: root, third, and fifth. In other words, the triad.

3) Diagonal reading (canon).
If we repeat the sequence while shifting it by one position (what we would today call a staggered entry), the same structure begins to function as imitation. One voice enters after another, preserving the same profile.

We can visualise what happens when the same sequence is read diagonally as a canon:

Voice 1:  1   3   5
Voice 2:      1   3   5
Voice 3:          1   3   5
----------------------------
Vertical:       Triad

Each voice sings the same sequence, but enters with a delay. If we observe a vertical column (for example, the central one), we find 1, 3, and 5 sounding simultaneously. The melodic structure, simply shifted in time, automatically generates harmony.

This is the decisive theoretical point: the triad is not added from the outside. It is already implicit in the shape of the seed. The diagonal projection produces verticality.

This is the key idea. We are not gluing melody and harmony together; we are using a single structure that, depending on the direction of reading, produces melody, chord, and canon.

Why the Numbers Cannot Be Random

As long as the number of voices is small, almost everything appears to work. Problems arise when density increases, because voices overlap and, if the structure is weak, errors inevitably emerge.

Let us take a deliberately simple example. Consider a seed of four symbols:

1 – 1 – 3 – 5

This sequence is perfectly usable: each voice repeats the same pattern, and staggered entries produce a canon that can continue indefinitely.

But if we introduce repetitions that appear harmless within a single line, such as:

1 – 1 – 3 – 1 – 1

the pair 1→1 appears both at the beginning and at the end. When the voices overlap, two different voices will eventually perform the same transition at the same moment. The result is not a subjective “ugliness,” but a structural parallelism (parallel unisons or octaves), traditionally considered an error.

The problem, therefore, is not the number 1. The problem is the repetition of the same transition.

The Practical Rule: Control the Pairs, Not the Notes

To avoid forbidden parallelisms, it is not enough to control which notes appear. We must control the movements, that is, the consecutive pairs.

With the symbols {1, 3, 5}, there are nine possible pairs:

1→1, 1→3, 1→5
3→1, 3→3, 3→5
5→1, 5→3, 5→5

If a seed generates the same pair twice (for example 1→1 in two different positions), then in the infinite canon that pair will eventually appear in overlap between two voices. When this happens, the inevitable effect is that two voices move in parallel in the same way.

This is the simple (and decisive) rule: to build a robust seed, the internal pairs must be all different up to the density of voices we wish to achieve.

Put bluntly: if pairs are repeated, the error is not possible, but inevitable.

The Limit Case: Eight Voices and Rare Seeds

When aiming for a very dense canon (for example, eight real voices), the space of possibilities narrows drastically. We must construct a chain of eight symbols generating seven consecutive pairs, all distinct, and capable of closing cyclically without recreating previously used pairs.

One sequence satisfying this condition is:

1 – 1 – 3 – 1 – 5 – 3 – 3 – 5

The consecutive pairs are all distinct, and the final connection (5→1) does not repeat any previous transition. This makes the cycle suitable for an infinite canon without collisions.

A historical insight emerges here: although theoretical combinations are numerous, truly safe ones are very few. A composer working by trial and error may eventually find a solution, but statistically navigates through a vast number of sequences that will eventually collapse and produce structural faults.

This is why we speak of generative seeds: not an infinite number of good possibilities, but a finite set of robust, enumerable, reusable cycles.

Valid Seeds Are Finite and Cataloguable

By systematically applying the control of pairs (and, when necessary, additional constraints to avoid parallel fifths), the set of valid seeds for infinite canons between two and eight voices reduces to a finite number. In other words, a complete catalogue of safe cycles exists for writing canons.

This changes the working method. Instead of composing and correcting, one begins from structures that cannot collapse for combinatorial reasons. Of course, the musical work remains: rhythm, durations, dissonances, style, articulation. But the structural framework does not fracture while building the upper levels.

Those who wish to consult the full formalisation, including the census of cycles, the proof of validity criteria, and examples of constructing any type of canon, may refer to the scientific paper with permanent DOI.