Proliferating series by Jean Barraqué: a study and classification in mathematical terms

Barraqué’s proliferating series give an interesting turn on the concept of classic serialism by creating a new invariant when it comes to constructing the series: rather than the intervals between consecutive notes, what remains unaltered during the construction of the proliferations of the given base series is the permutation of the notes which happens between two consecutive series, that is to say, the transformation of the order of the notes in the series. This presents new possibilities for composers interested in the serial method, given the fact that the variety of intervals obtained by this method is far greater than that of classic serialism. In this manuscript, we will study some unexplored possibilities that the proliferating series offer from a mathematical point of view, which will allow composers to gain much more familiarity with them and potentially result in the creation of pieces that take serialism to the next level.

💡 Research Summary

The paper provides a rigorous mathematical treatment of Jean Barraqué’s “proliferating series,” a technique that expands the traditional twelve‑tone serial method by focusing on the permutation that maps one series onto another rather than on the intervallic relationships between successive notes. The authors begin by defining the basic construction: two twelve‑tone rows are chosen, and the permutation π that transforms the first row into the second is identified. By repeatedly applying π to each newly generated row, a chain of rows—called proliferations—is produced. Because π belongs to the symmetric group S₁₂, its repeated application eventually returns to the original row; the number of distinct rows before this return is the order of π. The order is computed as the least common multiple of the lengths of the disjoint cycles in the cycle decomposition of π, a standard result from group theory.

The paper then examines how the four classical serial transformations—Prime (P), Retrograde (R), Inversion (I), and Retrograde‑Inversion (RI)—affect the structure of π and consequently the possible number of proliferations. For each transformation the authors derive explicit formulas for the cycle structure:

• Inversion (I): The mapping x ↦ −x + t (where t is a transposition) always yields only 1‑ or 2‑cycles. Consequently the order of the resulting proliferating permutation (PP) is at most 2, offering no new material beyond the original row and its inversion.

• Prime (P): The mapping x ↦ x + t produces cycles all of the same length k = n / gcd(n, t) (with n = 12 for the chromatic case). The order of the PP is therefore k, but the proliferated rows are merely transpositions of the original row, again providing no novel intervallic content.

• Retrograde (R) and Retrograde‑Inversion (RI) generate more intricate cycle structures. In the RI case, the mapping can be expressed as x ↦ −x_rev + t, leading to cycle lengths that depend on both t and n. When t is odd, the order can reach the full n (i.e., 12), whereas even t often collapses the order to a divisor of n/2. This yields a substantially richer set of rows, because the permutation may contain cycles of varying lengths, producing intervallic relationships that are unattainable through ordinary serial operations.

The authors also generalise the method to any modulus n, allowing microtonal divisions (e.g., 24‑tone quarter‑tone, 36‑tone third‑tone) and to other musical parameters such as rhythm, dynamics, or timbre. By numbering the elements of any ordered set modulo n and applying the same permutation logic, the same cycle‑decomposition analysis holds, opening the door to “total serialism” across arbitrary pitch or parameter spaces.

A key practical contribution is the discussion of the limits on the number of proliferations when the two generating rows are not independent. If the rows are arbitrary, any permutation in Sₙ can be used, so the maximal order is bounded only by the orders of elements of Sₙ (i.e., by the divisors of n!). However, when the second row is derived from the first by one of the four classical transformations, the set of admissible permutations shrinks dramatically. The paper systematically enumerates the possible orders for each transformation, showing that only certain cycle structures can arise. This knowledge enables composers to design an initial row and a transposition value t that guarantee a desired number of distinct proliferations, effectively giving them control over the size of the material pool before the process cycles back.

To aid composers, the authors provide a Python script (Appendix A) that takes a row and a chosen transformation, computes the associated permutation, displays its cycle decomposition, and reports the order. This tool allows rapid experimentation with different rows, transpositions, and microtonal divisions, making the theoretical results immediately applicable in compositional practice.

In conclusion, the paper demonstrates that Barraqué’s proliferating series can be fully understood through the lens of permutation group theory. By analysing the cycle structure of the underlying permutation, one can predict exactly how many distinct rows will be generated, what intervallic content those rows will contain, and how to manipulate the process to obtain desired musical outcomes. The extension to arbitrary modulus n and to non‑pitch parameters further broadens the technique’s relevance, offering contemporary composers a mathematically grounded yet creatively expansive method for generating serial material beyond the constraints of traditional twelve‑tone serialism.