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.

翻译：八度音程无法实现可移调的十二半音调音，源于一个数学事实：$2 \times 2^{7/12} \neq 3$，即五度的第二泛音无法与基音的第三泛音精确匹配。这又源于西方音乐的整体数泛音结构，以及随之而来的八度音程在频率上为2的倍数的基本特性——这一特性由我们的音乐体系继承自振动元件近似一维的乐器物理原理。在当今电子音乐时代，我们可以放宽上述假设，构建一个类似的音乐体系，其中标准音乐体系的所有结构特性得以保留，但泛音不再是基频的整体数倍数，且八度音程在频率上也不再是2的倍数。这使得构建一种可移调的十二半音音乐体系成为可能，其中五度的第二泛音能够精确匹配基音的第三泛音。该体系增强的谐波特性较好地恢复了纯律的音乐品质，同时通过构造保留了12TET的全部灵活性与转调能力。