发表机构
University of New Hampshire（新罕布什尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究等程音律中半音三音和弦循环，证明之字形轮廓缺失源于拓扑荷守恒，仅在特定条件下出现，并给出精确普查与机器验证。
AI 中文摘要
在十二音等程音律中，九个音级的循环可以这样排列：其九个连续的三音窗口恰好各一次地实现每个包含半音的三音和弦类型——这些类型在移调（但不包括倒置）下被视为等价。此类循环大量存在，并可推广到 n 音等程音律，其中窗口实现 n-3 个包含半音的类型。在十二音等程音律中所有 1764 个此类循环里，半音三音和弦 012 总是以音阶形式出现，即作为直接的半音进行，而从不以破碎的“之字形”轮廓（如 1-0-2）出现。为什么之字形缺失了？我们证明，之字形呈现表现为底层音程流网络的拓扑荷守恒：它出现当且仅当 3 整除 n 且 n 既不是 6 也不是 12。正向障碍是一个 Z/3 取值的守恒律，通过权重函数论证对所有 n 证明；两个例外音律——其中之一是常见的西方调律——由有限证书排除；而其他每个 3 的倍数的存在性则来自显式构造。我们还量化了稀有性，并给出了远超直接搜索范围的精确普查。该分类在 Lean 4 中双向机器验证。本文面向两类读者。音乐部分仅需音级集合词汇；数学推导自足且初等，但完全严谨。
英文摘要
A cycle of nine pitch classes in twelve-tone equal temperament can be arranged so that its nine consecutive three-note windows realize, exactly once each, every trichord type containing a semitone -- types taken up to transposition but not inversion. Such cycles exist in abundance, and generalize to n-tone equal temperament, where the windows realize the n-3 semitone-containing types. In every one of the 1764 such cycles in twelve-tone equal temperament, the chromatic trichord 012 appears scalarly, as a direct chromatic run, and never in a broken "zigzag" contour such as 1-0-2. Why is the zigzag missing? We show that the zigzag presentation behaves as a conserved topological charge of an underlying interval-flow network: it occurs if and only if 3 divides n and n is neither 6 nor 12. The forward obstruction is a Z/3-valued conservation law, proved for all n by a weight-function argument; the two exceptional temperaments -- one of them the common Western tuning -- are excluded by finite certificates; and existence for every other multiple of 3 follows from an explicit construction. We also quantify rarity, and give exact censuses well beyond the reach of direct search. The classification is machine verified in Lean 4 in both directions. The paper is written for two audiences. Musical sections require no more than pitch-class set vocabulary; the mathematical development is self-contained and elementary, yet fully rigorous.
Comments29 pages, 4 figures
