Musical zygotes are three fundamental intervallic structures called alpha, epsilon and sigma. The term “zygote” indicates, in this context, 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 breadth and complexity: alpha is the minimal three-note form, epsilon the intermediate four-note form, sigma the most extensive five-note form.
α Alpha Chord
The alpha chord is the smallest form of the system. With three notes it generates the three fundamental diatonic classes — second/seventh, third/sixth and fourth/fifth — concentrated in their most synthetic 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 — second, third, fourth, fifth, sixth and seventh — each present exactly once. It therefore constitutes a form of diatonic completeness broader than that of alpha.
σ Sigma Chord
The sigma chord is the most extensive of the three musical zygotes. With five notes it generates all the distances of the spatial interval vector within the octave, with the sole exception of the tritone. Moreover, each interval appears exactly once, without redundancy. In this sense, sigma represents the maximum degree of completeness obtained with the minimum use of sonic material.
Structural Uniqueness
The alpha chord, the epsilon chord and the sigma chord are, each within their own vector domain, unique structural types. Considering also the respective mirror chord as a reflected variant of the same structure, they represent indeed 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 the musical zygotes can be understood starting from a principle akin to that of the Golomb ruler — that is, a minimal arrangement of points in which all distances between pairs are different. Transposed to a musical context, 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 standpoint, the number of internal intervals grows with the number of notes: with a single note no interval is generated, with two notes one is generated, with three notes three, with four notes six, with five notes ten. The problem therefore consists not only in increasing the number of relationships, but in finding structures in which such relationships are all different from one another, avoiding redundancies.
The musical zygotes arise precisely from this search. The alpha chord, the epsilon chord and the sigma chord represent three progressively broader solutions to the same combinatorial principle, applied to three distinct intervallic domains: the fundamental diatonic classes, the generic diatonic intervals and the distances of the spatial interval vector.
In 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 exactly once.
Epsilon instead realises a perfect structure in the domain of generic diatonic intervals. Its form 0 1 4 6 indeed generates all six possible distances exactly once, from 1 to 6, and coincides with the perfect Golomb ruler of order 4.
Sigma carries 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 no perfect Golomb rulers with five marks exist, sigma is not perfect in the strict mathematical sense, but corresponds to an optimal Golomb ruler of order 5: it realises the maximum economy possible for five notes, covering ten different distances within a span of eleven semitones.
In this sense, the musical zygotes can be read as three musical realisations of the Golomb ruler principle: alpha and epsilon as perfect structures in their respective domains, sigma as an optimal structure in the spatial chromatic 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 — tetrachords that contain 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 the 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, the epsilon chord and the sigma chord should not be understood as substitutes for categories already present in music theory, but as a specific family constructed from formally verifiable intervallic properties.
Related Entries
Alpha Chord
Epsilon Chord
Sigma Chord
Diatonic Class Vector
Diatonic Interval Vector
Spatial Interval Vector
Catalogue of Spatial Interval Vectors in the Chromatic System