Incommensurability of the Octave Interval with any other Interval
Wilmer T. Bartholomew · The Journal of the Acoustical Society of America · 1930
Although it is common knowledge that the well-known tempered scale of twelve semitones is imperfect, relatively few musicians understand the necessary cause of that imperfection, which lies in the mathematical relations of the ratios involved. In any method of tuning, the ear demands that at least the octaves must be true. It is demonstrated that if this is so, no other true interval, if successively stepped off from a starting tone, can ever arrive at an exact octave of the starting tone. The mathematics, though elementary and obvious to the scientific musician, is important since it underlies all attempts at tempering scales.