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.

翻译：欧拉在1739年提出的《愉悦度》中，通过公式G(p/q)=1+Ω^(p)+Ω^(q)为音乐音程p/q赋予不协和值，其中Ω^(n)=∑_i e_i(p_i - 1)对n的加权质数指数求和。我们提出更简单的不对称公式f(p/q)=p+Ω^(q)，该公式对分子和分母进行不同处理，并在标准协和数据上表现相当。我们还证明，在谐波按整数索引且均匀计数至固定截断水平的模型下，《愉悦度》等价于一个权重为w(n)=Ω^(n)的加权谐波重合计数，从而将其与伽利略更早的脉冲重合模型（1638年）联系起来。该公式自然生成一个互质整数三角形T(n,k)=n+Ω^(k)，其最右对角线给出了超特比（连续谐波）音程的两阶段不协和值。该公式f允许在谐波背景和部分识别方面进行简单的两阶段解释，我们将其作为推测性的感知假说提出。