In 'the false piano (2)' the number 554,365 Hz is mentioned for the major third upon 440 Hz. The calculation of this number is as follows:
In the Well Temperament the fifth is calculated by multiplication with \(2^{\frac{7}{12}}\). We pile 4 fifths upon 440 Hz and go back again 2 octaves. This means: multiply four times with \(2^{\frac{7}{12}}\) and after that divide twice by 2:
\(440\times \left ( 2^{\frac{7}{12}} \right )^{4}\div 4\)
This equals 554,365 (rounded off).
In the Pythagorean temperament the major third is calculated on the basis of the pure fifth: \(440\times \left (\frac{3}{2} \right )^{4}\div 4\). This number is even higher: 556,875.
Furthermore \(\left (\frac{3}{2} \right )^{4}\div 4= \frac{81}{64}\). The number of the pure major third is \(\frac{5}{4}= \frac{80}{64}\). The ratio between these numbers is called syntonic comma or didymean comma.
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