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

Isabel Tardón, Pablo Martín-Santamaría

TL;DR

The paper formalizes Barraqué's proliferating series by treating the generation of related dodecaphonic series as a group-theoretic proliferation driven by a permutation between two generators. It analyzes the structural consequences of four transformations—Primes, Inversions, Retrogrades, and Retrograde Inversions—and extends the framework to microtonal divisions via modulo $n$, introducing generalized transpositions (GT). The key contributions include precise order and cycle-structure results for each transformation (with detailed subcases for RI and R), a classification scheme for RI-based proliferations, and a practical computational toolkit implemented in Python to explore and verify proliferations. The work provides composers with a rigorous, architecture-driven approach to composing with proliferating material, highlighting when proliferation yields genuinely new material (notably RI and R) versus when it recovers traditional serialism (I and P). Overall, the study bridges mathematical structure and musical practice, offering both theoretical insight and actionable methods for extending serial techniques into richer, higher-variability material.

Abstract

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.

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

TL;DR

The paper formalizes Barraqué's proliferating series by treating the generation of related dodecaphonic series as a group-theoretic proliferation driven by a permutation between two generators. It analyzes the structural consequences of four transformations—Primes, Inversions, Retrogrades, and Retrograde Inversions—and extends the framework to microtonal divisions via modulo , introducing generalized transpositions (GT). The key contributions include precise order and cycle-structure results for each transformation (with detailed subcases for RI and R), a classification scheme for RI-based proliferations, and a practical computational toolkit implemented in Python to explore and verify proliferations. The work provides composers with a rigorous, architecture-driven approach to composing with proliferating material, highlighting when proliferation yields genuinely new material (notably RI and R) versus when it recovers traditional serialism (I and P). Overall, the study bridges mathematical structure and musical practice, offering both theoretical insight and actionable methods for extending serial techniques into richer, higher-variability material.

Abstract

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.
Paper Structure (21 sections, 7 equations, 8 figures)

This paper contains 21 sections, 7 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Methodology
  3. Construction of the proliferating series
  4. Series constructed through the transformations of traditional serialism
  5. Generalization of proliferating series: micro-tonal scales
  6. Possible number of proliferations for a series of $n$ notes
  7. Study of proliferations using inversion (I)
  8. Study of proliferations using primes (P)
  9. Study of proliferations using retrograde inversion (RI)
  10. When $n$ is even and $t$ is odd
  11. When $n$ is even and $t$ is even
  12. When $n$ is odd and $t$ is zero
  13. When $n$ is odd and $t$ is not zero
  14. Study of proliferations using retrogradation (R)
  15. When $t$ is zero
  16. ...and 6 more sections

Figures (8)

  • Figure 1: Mapping the series from Webern Symphony Op. 21 and Webern String Quartet Op. 28 in order to obtain a permutation which generates proliferations
  • Figure 2: Proliferations of the series from "...Au Dela du Hasard", as used by Jean Barraqué
  • Figure 3: Comparison of the proliferations of A using RI with $t=2$ and B using IR with $t=5$
  • Figure 4: Inductive process of the construction of a cycle when $n$ is even and $t$ odd
  • Figure 5: Inductive process of the construction of a cycle when $n$ is even and $t$ is also even
  • ...and 3 more figures