发表机构
Florida State University（佛罗里达州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究具有无限解的无限卡农，基于离散素因数分解和连续对数法两种构造描述最大自相似旋律线结构，其声部垂直音程由同态给出，逆行倒影等同于移调，展示了结构并规划了未来工作。
AI 中文摘要
无限卡农是一系列具有无限解的卡农。具体而言，每个卡农基于一条旋律线，可在任意数量的声部、任意速度比例（有理或无理）下组合，各声部可正向或逆行倒影进行，同时保持和声一致性。本文基于两种不同构造描述这些最大自相似旋律线的结构：一是离散素因数分解法，在所有有理比例下产生自相似性；二是连续对数法，将其扩展到无理比例。在这两种情况下，速度比例为\(\lambda_i / \lambda_j\)时声部间的垂直音程由同态\(\phi(\lambda_i / \lambda_j)\)给出，它是与时间无关的常数。此外，在这些构造下逆行倒影等同于移调，可实现各种形式的表格卡农。通过对几种不同无限卡农的示意性实现展示了这些结构。未来工作包括更完整的数学处理、音乐应用以及一个允许用户探索这些作品无限多种实现的交互式程序。
英文摘要
Infinite Canons is an ongoing series of canons with infinite solutions. More specifically, each canon is based on a melodic line that can be combined in any number of voices, at any tempo ratios (rational or irrational), and with each voice moving either forward or in retrograde inversion, while maintaining harmonic consistency. This paper describes the structure of these maximally self-similar melodic lines based on two different constructions: 1) a discrete prime-factorization approach yielding self-similarity under all rational ratios, and 2) a continuous logarithmic approach extending this to irrational ratios. In both cases the vertical interval between voices in a tempo ratio of $λ_i / λ_j$ is given by a homomorphism $ϕ(λ_i / λ_j)$, which is a constant independent of time. Furthermore, under these constructions retrograde inversion collapses to transposition, allowing for all manner of table canons. These structures are demonstrated with suggestive realizations of several different infinite canons. Future work includes a more complete mathematical treatment, musical applications, and an interactive program that allows users to explore an unlimited number of realizations of these pieces.
Comments19 pages, 15 figures. Versions of this material were presented at the Joint Mathematics Meetings (2011), Nancarrow in the 21st Century (2012), and the joint AMS/SEM/SMT meeting (2012). A more developed treatment, with musical applications, an interactive program, and greater mathematical detail, is in preparation
