AI 中文总结
本文为广义调性实践建立数学框架，以群和torsor形式化调式音阶，构建Grothendieck纤维层级，推广Rohrmeier生成语法，定义和声实践范畴，源自作者Op.26-31作曲实践。
AI 中文摘要
我们为广义调性实践发展数学原理。我们从调式和音阶的代数理论开始，分别形式化为群和循环群上的 torsor。我们识别出三类音乐上显著的调式同构——平移、旋转和变形。在此基础上，我们将和声结构的连续层次组织成 Grothendieck 纤维范畴的层级，由音阶类型（推广自然音阶）、调式（推广自然调式）、轨道覆盖（推广其由三度叠置和弦的覆盖）、变形族（推广半音变化，如调式混合和副属和弦）以及和声功能分配（推广主功能、下属功能和属功能的分配）构建。为所得结构配备生成语法——推广 Rohrmeier 的调性和声生成语法——产生和声实践的正式定义：和声词汇连同和声功能分配以及容纳层级和声组织的生成语法。我们证明和声实践构成一个范畴，其态射构成从一个和声实践到另一个的结构继承与丰富。该理论源于作者自身的作曲实践，最初通过作品 Op. 26--31 系列发展而来。
英文摘要
We develop mathematical principles for a generalized tonal practice. We begin with an algebraic theory of modes and scales, formalized as groups and torsors over cyclic groups, respectively. We identify three musically salient classes of mode isomorphism---translations, rotations, and deformations. Upon this basis we organize successive layers of harmonic structure into a hierarchy of Grothendieck fibrations, built from scale types (generalizing the diatonic scale), modes (generalizing the diatonic modes), orbit covers (generalizing their covering by tertian triads), deformation families (generalizing chromatic alteration, such as modal mixture and secondary chords), and harmonic-function assignments (generalizing the assignment of tonic, subdominant, and dominant function). Equipping the resulting structure with a generative syntax---generalizing Rohrmeier's generative grammar of tonal harmony---yields a formal definition of a harmonic practice: a harmonic vocabulary together with an assignment of harmonic functions and a generative syntax accommodating hierarchical harmonic organization. We show that harmonic practices form a category, with morphisms constituting an inheritance and enrichment of structure from one harmonic practice to another. The theory originates in the author's own compositional practice, developed initially through the series of compositions Op. 26--31.
Comments111 pages, 20 figures
