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calculation major third

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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