We introduce the notion of multi-patterns, a combinatorial abstraction of polyphonic musical phrases. The interest of this approach in encoding musical phrases lies in the fact that it becomes possible to compose multi-patterns in order to produce new ones. This composition is parameterized by a monoid structure on the scale degrees. This embeds the set of the musical phrases into an algebraic framework since the set of the multi-patterns is endowed with the structure of an operad. Operads are algebraic structures offering a formalization and an abstraction of the notion of operators and their compositions. Seeing musical phrases as operators allows us to perform computations on phrases and admits applications in generative music. Indeed, given a set of initial multi-patterns, we propose various algorithms to randomly generate a new and longer phrase emulating the style suggested by the inputted multi-patterns. The designed algorithms use types of grammars working with operads and colored operads, known as bud generating systems.

翻译：我们引入多重模式（multi-patterns）的概念，即复调音乐乐句的组合抽象。这种乐句编码方法的价值在于，它使得通过组合多重模式生成新乐句成为可能。该组合过程由音阶度数上的幺半群结构进行参数化，从而将乐句集合嵌入代数框架——多重模式集合被赋予Operad结构。Operad作为一种代数结构，为算子及其组合的概念提供了形式化抽象。将乐句视为算子，既可对乐句实施运算操作，又可在生成音乐领域获得应用：给定初始多重模式集合，我们提出多种算法，通过随机生成模仿输入多重模式风格的新长乐句。所设计算法采用基于Operad与染色Operad的语法类型（即芽生成系统）进行运作。