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English: A mathematical definition solves a mathematical problem. In Equal Temperament, the derivation of the Twelfth Root of Two creates equal sounding intervals and a closed circle of Fourths and Fifths allowing movement from key to key throughout the musical range of frequencies (Modulation).

Interest and acceptance, which is never final, arose from considerations of Harmonic Progression, perhaps most significantly starting in Europe in the early-mid 17th Century though it had been considered much earlier in China.

It is a useful, practical way of making music, so long as we are not trying to prove the almighty infinite through music. And its fun.

The musical problem it creates is a slight artificiality which no longer matches precisely, and only approximates, the natural vibrations of vibrating objects as in the Harmonic Series.

Much of the Musical Treatment of Intervals in Music Theory involves the techniques and approaches that have been effective in creating a viable and interesting artform out of this kind of insolvable problem. Art does not solve problems as math does, it explores them