Musical zygotes are three fundamental intervallic structures called alpha, epsilon, and sigma. The term “zygote” here refers to a minimal cell of completeness: a reduced set of notes capable of concentrating within itself a particularly rich relational field.
In the σεα (sea) system, these three zygotes are arranged in increasing order of size and complexity: alpha is the minimal three-note form, epsilon is the intermediate four-note form, and sigma is the largest five-note form.
α Alpha chord
The alpha chord is the smallest form in the system. With three notes, it generates the three fundamental diatonic classes—second/seventh, third/sixth, and fourth/fifth—concentrated in their most compact form. For this reason, it represents the minimal level of diatonic completeness.
ε Epsilon chord
The epsilon chord is the intermediate form of the system. With four notes, it generates all six possible diatonic distances within the heptatonic system—that is, second, third, fourth, fifth, sixth, and seventh—each present only once. It thus constitutes a broader form of diatonic completeness compared to alpha.
σ Sigma chord
The sigma chord is the most extended form of the three musical zygotes. With five notes, it generates all the distances of the spatial interval vector within the octave, except for the tritone alone. Moreover, each interval appears only once, with no redundancies. In this sense, sigma represents the highest degree of completeness achieved with the minimal use of sonic material.
Structural uniqueness
The alpha chord, epsilon chord, and sigma chord are, each in their own vector domain, unique structural types. Considering also their respective mirror chord as a reflected variant of the same structure, they in fact represent the only solution capable of fully satisfying the vector form required by the reference system.
The alpha chord is defined by the diatonic class vector: [1 1 1]
The epsilon chord is defined by the diatonic interval vector: [1 1 1 1 1 1] and, consequently, also by the diatonic class vector: [2 2 2]
The sigma chord is instead defined by the spatial interval vector:
[1 1 1 1 1 0 1 1 1 1 1]
In this sense, these are not three open families of equivalent chords, but three privileged structures that respond exclusively to a precise condition of completeness. The mirror chord, when present, therefore does not constitute a second independent type, but the reflected form of the same structural solution.
Structure and connection with the Golomb ruler
The structure of musical zygotes can be understood starting from a principle similar to that of the Golomb ruler: a minimal arrangement of points in which all distances between pairs are different. Transposed into music, this principle leads to the search for sets of notes capable of producing the maximum number of distinct intervals with the minimum number of sounds.
From a combinatorial point of view, the number of internal intervals grows with the number of notes: with a single note, no interval is generated; with two notes, one interval; with three notes, three intervals; with four notes, six; with five notes, ten. The problem is not only to increase the number of relationships, but to find structures in which these relationships are all different from each other, avoiding redundancies.
Musical zygotes arise precisely from this search. The alpha chord, epsilon chord, and sigma chord represent three progressively broader solutions of the same combinatorial principle, applied to three different intervallic domains: the fundamental diatonic classes, the diatonic intervals, and the intervals of the spatial interval vector.
From this perspective, alpha can be considered a perfect structure in the reduced domain of the three fundamental diatonic classes: three notes generate three relationships, and each of the three classes appears only once.
Epsilon, instead, realizes a perfect structure in the domain of generic diatonic intervals. Its form 0 1 4 6 in fact generates all six possible distances only once, from 1 to 6, and coincides with the perfect Golomb ruler of order 4.
Sigma brings the same principle into the twelve-tone chromatic system. Its form 0 1 4 9 11, together with its mirror chord 0 2 7 10 11, generates ten all-different distances. Since perfect Golomb rulers with five points do not exist, sigma is not perfect in the strict mathematical sense, but corresponds to an optimal Golomb ruler of order 5: it achieves the greatest possible economy for five notes, covering ten different distances within a span of eleven semitones.
In this sense, musical zygotes can be read as three musical realizations of the Golomb ruler principle: alpha and epsilon as perfect structures in their respective domains, sigma as an optimal structure in the chromatic spatial domain.
Relationship with all-interval chords
The search for structures capable of containing the maximum number of different intervals in a reduced number of notes is already present in twentieth-century music theory. An important reference is that of all-interval tetrachords, that is, tetrachords containing each of the six chromatic interval classes exactly once.
The zygote chords are in dialogue with this tradition, but follow a different logic. They do not seek completeness of the standard chromatic interval classes, but instead build a nested family of structures based on three distinct domains: diatonic classes, generic diatonic intervals, and chromatic distances.
In this sense, the alpha chord, epsilon chord, and sigma chord should not be understood as substitutes for categories already present in music theory, but as a specific family built from formally verifiable intervallic properties.
Related entries
Alpha chord
Epsilon chord
Sigma chord
Diatonic class vector
Diatonic interval vector
Spatial interval vector
Catalog of spatial interval vectors in the chromatic system