论文标题
复杂音调的数学基础
Mathematical Foundations of Complex Tonality
论文作者
论文摘要
相等的气质，其中半音符以$ 2^{1/12}的非理性比率调整为1 $，最好被视为可维修的折衷，牺牲了纯度以提高灵活性。只是语调，其中$ 2 $，$ 3 $和$ 5 $的功率产品的间隔更为自然，但灵活性有限。我们提出了一个新方案，其中高斯整数比例构成了抽象音调系统的基础。 tritone在气质上如此有问题，被比率$ \ tfrac {45} {32} $，$ \ tfrac {64} {45} $，$ \ tfrac {36} {36} {25} {25} {25} $，$ \ tfrac {25} \ rm {i}：1 $。由$ \ tfrac {9} {8} $和$ \ tfrac {10} {9} $给出的主要和次要整体音调，可以将每个音调分为复杂的半音产品，从$ \ tfrac {1} {3}（3 + \ rm {i}）$。完美的第三，由间隔$ \ tfrac {5} {4} $给出，将复杂的整体音调$ \ tfrac {1} {1} {2}（1 + 2 \ rm {i}）$分配给了分解。随着这些补充色调的增强，基于高斯素数低幂的产物的复杂间隔的结果方案导致在所有钥匙中建立一个完整的主要和次要尺度的系统。
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 are 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 low powers of Gaussian primes leads to the construction of a complete system of major and minor scales in all keys.
