提出更简单的不对称公式，解释音乐音程的和谐度。
An Asymmetric Formula for Interval Consonance and its Relation to Harmonic Coincidence
- 用分子加分母素因数权重和定义新音程和谐度公式。
- 在标准数据上表现与欧拉公式相当，且能生成连续谐波间隔的和谐值。
- 公式暗示听觉感知中谐波识别的两阶段机制，适合音乐理论研究者。
欧拉1739年提出的音程和谐度公式G(p/q) = 1 + Ω^(p) + Ω^(q)，通过加权素因数指数计算音程不协和度。本文提出更简洁的不对称公式f(p/q) = p + Ω^(q)，区分分子与分母作用，在标准协和性数据上表现相近。我们进一步证明：在谐波为整数索引且截断至固定层级的模型下，欧拉公式等价于加权谐波重合计数，权重w(n) = Ω^(n)，从而连接到伽利略1638年的脉冲重合模型。该公式自然导出一个互质整数三角形T(n,k) = n + Ω^(k)，其右对角线给出超部分音（连续谐波）音程的两阶段不协和度。公式可解释为谐波上下文与分部识别的两阶段过程，作为听觉感知的假设性解释。
原文摘要 · Abstract (English)
Euler's Gradus Suavitatis (1739) assigns a dissonance value to a musical interval p/q by the formula G(p/q) = 1 + Ω^(p) + Ω^(q), where Ω^(n) = \sum_i e_i(p_i - 1) sums the weighted prime exponents of n. We propose the simpler asymmetric formula f(p/q) = p + Ω^(q), which treats numerator and denominator differently and performs comparably on standard consonance data. We also show that, under a model in which harmonics are integer-indexed and counted uniformly up to a fixed truncation level, Gradus is equivalent to a weighted harmonic coincidence count with weights w(n) = Ω^(n), connecting it to Galileo's earlier pulse-coincidence model (1638). The formula naturally generates a coprime integer triangle T(n,k) = n + Ω^(k), whose rightmost diagonal gives the two-stage dissonance of the superparticular (consecutive-harmonic) intervals. The formula f admits a simple two-stage interpretation in terms of harmonic context and partial recognition, which we offer as a speculative perceptual hypothesis.
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