In the ‘chord of nature’ described, the frequencies of the partials are

where is partial number 1, the lowest position in the ‘chord of nature’. Thus the frequency of partial is given by

This conformity to a harmonic series is sufficiently accurate for the purposes of temperament theory applied to most tensioned musical strings. The frequency ratio defining the size of an octave that it would be necessary to tune between the strings, in order to produce exact consonance between the coinciding partials, is thus

All ratios will be expressed as the ratio of the upper note frequency to that of the lower note. The frequency ratio defining the size of the fifth that it is necessary to tune between the strings, in order to produce exact consonance between the coinciding lower partials in the ‘chords of nature’, will similarly be

On inversion, this becomes a fourth, whose frequency ratio is

The inversion is made taking down an octave the upper note, thus the ratio of the new lower note (the upper one taken down an octave) to the new upper note (which was originally the lower note), becomes

and the ratio of the new upper to the new lower note, will be

In the tuning sequence described in the picture, when the starting note C has a frequency of , the sequence to G sharp includes 4 rising fifths and 4 falling fourths (inverted rising fifths). In frequency ratios this becomes

for the ratio between C and G sharp.

The ratio between C and E flat, in the second section of the sequence, is

So the frequency ratio of G sharp to E flat is

If this interval had been a perfect fourth, this ratio would be

but is larger, than this, because

The interval between a perfect fourth and our interval G sharp to E flat, expressed as a frequency ratio, is therefore

where 1.013643 is the frequency ratio of the Pythagorean comma.