Equal temperament, in which semitones are tuned in the irrational ratio of $2^{1/12} : 1$, is best seen as a serviceable compromise, sacrificing purity for flexibility. Just intonation, in which intervals given by products of powers of $2$, $3$, and $5$, is more natural, but of limited flexibility. We propose a new scheme in which ratios of Gaussian integers form the basis of an abstract tonal system. The tritone, so problematic in just temperament, given ambiguously by the ratios $\tfrac{45}{32}$, $\tfrac{64}{45}$, $\tfrac{36}{25}$, $\tfrac{25}{18}$, none satisfactory, is in our scheme represented by the complex ratio $1 + \rm{i} : 1$. The major and minor whole tones, given by intervals of $\tfrac{9}{8}$ and $\tfrac{10}{9}$, can each be factorized into products of complex semitones, giving us a major complex semitone $\tfrac{3}{4}(1 + \rm{i})$ and a minor complex semitone $\tfrac{1}{3}(3 + \rm{i})$. The perfect third, given by the interval $\tfrac{5}{4}$, factorizes into the product of a complex whole tone $\tfrac{1}{2}(1 + 2\rm{i})$ and its complex conjugate. Augmented with these supplementary tones, the resulting scheme of complex intervals based on products of powers of Gaussian primes leads very naturally to the construction of a complete system of major and minor scales in all keys.

翻译：十二平均律将半音调律为无理数比 $2^{1/12} : 1$，可视为一种可用的折中方案，以牺牲纯正性换取灵活性。纯律以质因子 $2$、$3$、$5$ 的幂乘积构成音程，虽更自然但灵活性有限。我们提出一种新体系，以高斯整数之比作为抽象音调系统的基础。在纯律中极具争议的三全音，曾被含糊地表示为 $\tfrac{45}{32}$、$\tfrac{64}{45}$、$\tfrac{36}{25}$、$\tfrac{25}{18}$ 等不理想比例，而在本体系中则由复比 $1 + \rm{i} : 1$ 表示。由 $\tfrac{9}{8}$ 和 $\tfrac{10}{9}$ 音程构成的大全音与小全音可分别分解为复半音之积：大复半音 $\tfrac{3}{4}(1 + \rm{i})$ 与小复半音 $\tfrac{1}{3}(3 + \rm{i})$。由 $\tfrac{5}{4}$ 音程表示的纯三度可分解为复全音 $\tfrac{1}{2}(1 + 2\rm{i})$ 与其复共轭之积。通过引入这些辅助音，基于高斯素数幂乘积的复音程体系自然构建出完整的大小调音阶系统，覆盖所有调性。