发表机构
Abstract Math Institute（抽象数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出ARS框架，用k阶马尔可夫模型结合约束MAP问题重建符号拉格序列，经合成与真实音频实验验证了方法可行性，为音乐片段重建提供了概率建模方案。
AI 中文摘要
重建受损音乐片段是一个逆问题：观测序列仅包含部分信息，而拉格（raga）编码的约束可限制允许的补全方案。本文为此建立了数学框架，提出人工罗塞塔石碑（Artificial Rosetta Stone, ARS）。我们区分了常被混淆的三个主张：符号序列可通过概率方法重建、序列可与显式语法一致、历史表演可被验证，仅支持前两个主张。我们通过有限字母表和约束系统对拉格建模，使用k阶马尔可夫模型计算旋律概率，对称狄利克雷先验可得到易处理的后验分布。我们将缺失音符重建表述为约束MAP问题：对于固定长度序列和有限阶约束，优化问题可采用精确动态规划求解，最坏情况时间复杂度为$O(TN^{k+1})$；推导了参数数量为$N^k(N-1)$，在显式混合假设下证明了浓度界，并分析了估计误差传播。可复现的合成实验使用6个拉格启发的字母表、$k \u2208 \{1,2,3\}$阶、最高50%的掩蔽率，这是概念验证而非历史重建。真实音频可行性试点从42段Yaman片段中通过自动音高提取、分割和量化得到30个可用序列，但因缺乏记录来源且依赖自动转录，并非经专家验证的档案重建，相关主张受限于所述条件，并非印度斯坦音乐的普遍属性。代码见此https URL。
英文摘要
Reconstructing a damaged musical fragment is an inverse problem: the observed sequence contains partial information, while a raga encodes constraints limiting allowable completions. This paper formalizes a mathematical framework for this, proposing the Artificial Rosetta Stone (ARS). We separate three claims often conflated: a symbolic sequence can be reconstructed probabilistically; a sequence can be consistent with an explicit grammar; and a historical performance can be authenticated. We only support the first two. We model a raga via a finite alphabet and constraint system, using an order-k Markov model for melodic probabilities. A symmetric Dirichlet prior yields a tractable posterior. We pose missing-note reconstruction as a constrained MAP problem. For fixed-length sequences and finite-order constraints, optimization admits an exact dynamic-programming solution with worst-case time complexity $O(TN^{k+1})$. We derive the parameter count $N^k(N - 1)$, prove a concentration bound under explicit mixing assumptions, and analyze estimation error propagation. A reproducible synthetic experiment uses six raga-inspired alphabets, orders $k \in \{1, 2, 3\}$, and masking rates up to 50%. This is a proof of concept, not historical reconstruction. A real-audio feasibility pilot evaluates 30 usable sequences from 42 Yaman clips via automated pitch extraction, segmentation, and quantization. Lacking documented provenance and relying on automated transcription, this is not expert-validated archival reconstruction. Claims are tied to stated conditions, not universal properties of Hindustani music. Code: https://github.com/mathacker23/ArtificialRosettaStone.
Comments37 pages,5 tables,11 Graphical Representations
