发表机构
National Taiwan Normal University（国立台湾师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为数字低音实现建立显式数学模型，将约束表示为谓词，证明固定声部下可行性与最小成本实现为多项式时间问题，并通过实例展示合法性、优化及贪婪选择的失败。
AI 中文摘要
数字低音实现可以被描述为一系列选择，这些选择既受每个和声音响内部的约束，也受连续和声音响之间的约束。本文给出了一个受限的、考试风格的四声部实现问题的显式数学模型。音高拼写、音域、和弦成员资格、重复、省略、间距、交叉、重叠、旋律进行、连续纯音程以及选定的解决要求均被表示为谓词。我们区分硬约束和可选的偏好成本。四个标记的音符在视觉上表示为四边形的顶点，在计算上表示为一个有序的声部配置状态。合法的进行变为通过分层图的路径。我们证明，对于固定数量的声部、显式有限的音符域和固定的局部规则，可行性和最小成本实现是多项式时间问题。对于固定音域、固定音符字母表和相邻事件规则，图操作的数量与事件数量呈线性关系。两个、四个和八拍的实例说明了合法性、优化以及贪婪选择的失败。该结果涉及所述的形式模型；并非声称每个音乐判断都被局部谓词所捕获。
英文摘要
Figured-bass realization can be described as a sequence of choices constrained both within each sonority and between successive sonorities. This paper gives an explicit mathematical model of a restricted, examination-style four-part realization problem. Pitch spelling, range, chord membership, doubling, omission, spacing, crossing, overlap, melodic motion, consecutive perfect intervals, and selected resolution requirements are expressed as predicates. We distinguish hard constraints from optional preference costs. Four labeled notes are represented visually as the vertices of a quadrilateral and computationally as one ordered voicing state. Legal progressions become paths through a layered graph. We prove that feasibility and minimum-cost realization are polynomial-time problems for a fixed number of voices with explicit finite note domains and fixed local rules. For fixed ranges, a fixed note alphabet, and adjacent-event rules, the number of graph operations is linear in the number of events. Worked two-, four-, and eight-beat examples illustrate legality, optimization, and the failure of a greedy choice. The result concerns the stated formal model; it is not a claim that every musical judgment is captured by local predicates.
