The fundamental results in mathematical music theory due to Clough and Myerson, Cardinality equals Variety (CV) and Structure yields Multiplicity (SM), correspond to statements about Christoffel words. These results may be stated and proved in a purely word-theoretic environment by means of the concept of a partitioned factor. In this setting, the generalized circle of fifths of Clough and Myerson is replaced by the lexicographic order of word theory, and their notion of the generic description of a musical line is replaced by that of a partitioned factor of a circular Christoffel word. Sturmian words are the infinite counterparts to Christoffel words, characterized as aperiodic but of minimal complexity, i.e. words such that for all $n\in \mathbb{N}$ there are n+1 factors of length n. Berthé showed that the factors of a given length have at most three frequencies (probabilities). Drawing on previous work, we extend to results on factors under a fixed partitioning (decompositions of factors of length n into concatenations of words whose lengths are given by a composition of n into k parts). Any factor of a Sturmian word thus partitioned into k components belongs to one of k+1 types (varieties). We show how to compute the frequencies of the varieties. These results recapture CV and SM in Sturmian words, and we suggest musical interpretations, parallel to those in the finitary case.

由于Clough和Myerson的缘故，数学音乐理论的基本结果是：基数等于变异（CV）和结构产生多重性（SM），与有关Christoffel单词的陈述相对应。这些结果可以在纯字理论环境中通过分配因子的概念来陈述和证明。在这种情况下，Clough和Myerson的五分之二的广义圆被单词理论的词典顺序取代，而他们对音乐线的一般性描述的概念被圆的Christoffel单词的分隔因子所取代。Sturmian单词是Christoffel单词的无限对等词，其特征是非周期性但复杂性极低，即对于所有人而言，$ñ\in \mathbb{ñ}$有Ñ长度的1个因素Ñ。Berthé表明，给定长度的因子最多具有三个频率（概率）。借助先前的工作，我们扩展到固定分区下的因子结果（长度为n的因子分解为单词的串联，其长度由n分为k个部分组成）。这样被划分成k个成分的Sturmian单词的任何因子都属于k +1种类型（变体）。我们展示了如何计算品种的频率。这些结果重新获得了Sturmian单词中的CV和SM，并且我们建议音乐解释与最终案例中的解释相似。