重构12音律系统，让五度音程完美契合泛音结构。
Elastic overtones: an equal temperament 12 tone music system with "perfect" fifths
- 基于电子音乐特性，放松泛音为整数倍的限制。
- 实现12音律中五度谐波与基频三倍频完全重合。
- 兼具纯律的和谐感与十二平均律的转调灵活性。
12半音音阶的可转调性受限于数学事实：$2 \times 2^{7/12} \neq 3$，即五度音的第二泛音无法精确匹配基频的第三泛音。这源于西方音乐的整数泛音结构，以及八度音程频率比为2的物理基础，该基础源自一维振动体的声学特性。在电子音乐时代，可放宽上述假设，构建一种新音乐系统：保留标准音乐体系的所有结构性特征，但允许泛音非整数倍于基频，且八度不再严格为2倍频率。由此可构造一个可转调的12半音系统，使五度音的第二泛音恰好与基频的第三泛音对齐。该系统在近似上恢复了纯律的谐和性，同时保持了12平均律的转调能力。
原文摘要 · Abstract (English)
The impossibility of a transposable 12 semitone tuning of the octave arises from the mathematical fact that $2 \times 2^{7/12} \neq 3$ i.e., the second harmonic of the fifth can not exactly match the third harmonic of the fundamental. This in turn, stems from the whole number harmonic structure of western music, and the subsequent fundamental character of the octave interval as multiples of 2 in frequency, a property inherited by our music system from the physics of instruments with vibrating elements being to a good approximation one dimensional. In the current era of electronic music, one can relax the above assumptions to construct an analogous music system where all the structural properties of the standard music system are preserved, but where harmonics are not whole number multiples of the fundamental frequency, and the octave is no longer a factor of 2 in frequency. This now allows to construct a transposable 12 semitone music system where the second harmonic of the fifth exactly matches the third harmonic of the fundamental. The enhanced harmonic qualities of this system recover to a good approximation the musical qualities of Just Intonation, whilst retaining by construction all the versatility and modulating ability of 12TET.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。