How to calculate the ratio equal temperament with your calculator, or even by hand!

If you try to understand what is equal temperament (ET) and go to Wikipedia, the article says which we divide the octave into 12 equal parts (so far so good and easy to understand) and with a ratio equal to the 12th root of 2 (12√2 ≈ 1.05946). The problem is how can we use a household calculator or the app on our phone to calculate the 12th root of 2, let alone in the age without a calculator that Chu-Tsaiyu in 1584 was the first person who calculated the ratio of ET in China. Chu only wrote the answer/result only but we can only guess his process. 

Back to square 1, we divide the octave with 12 equal ratios (r), the distance between two semitones. When 12 semitone distances sum up, we arrive at the octave of the original tone. We know that an octave is the original frequency double itself.


Right now hands on your calculator. Step 1, calculate the square root of 2. Step 2, calculate the square root of the answer of step 1. Step 3, calculate the cube root of the answer of step 2. There you go the answer.

If you really want to calculate by pen and paper, these are the links of the method of the long division algorithm. 

This teaches you how to calculate the square root:

https://www.youtube.com/watch?v=HBdVVFqTrUU

And cube root:

https://www.youtube.com/watch?v=y0qWHMmCY4E













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