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.

翻译：巴拉凯的增殖序列通过引入一种新的不变性来构建序列，从而对经典序列主义概念进行了有趣的转变：在给定基础序列的增殖构造过程中保持不变的不是连续音符之间的音程，而是两个连续序列之间发生的音符排列，即序列中音符顺序的变换。鉴于这种方法获得的音程多样性远超经典序列主义，这为关注序列方法的作曲家提供了新的可能性。在本手稿中，我们将从数学角度研究增殖序列所提供的一些尚未探索的可能性，这将使作曲家能更深入地理解其特性，并可能催生将序列主义推向新高度的音乐作品。