What would be the best musical scale? That's a difficult question. For the equitempered scale all the steps are of equal size but the ratios of the frequencies and therefore the harmonics are slightly off. The image below shows a plot of the harmonic versus the number of a note. The scale is based on A440. From the harmonics one can see that some notes were missing from Helmholtz's sequence of ratios. When these harmonics are included we get the sharps and flats that are found on a piano keyboard. The dotted line shows the equitempered scale. The points are the location of the notes when we round to the nearest harmonic. Note the familiar pattern of tones and semitones on this scale.
To the right of the plot are the frequencies, f, that Helmholtz gives for the notes and their equitempered equivalents f'. Below f' are the corresponding harmonic values h'. Rounding these to the nearest integer we get the set of harmonic numbers, h''', that we found previously. One sees that Helmholtz's scale is a fairly good approximation to the equitempered scale and it may be the best that one can do with a set of simple ratios. Note that the missing harmonics would also give good values for the frequencies of the flats and sharps on this scale.
The good fit indicates that the sequence of factors involving the steps a, b and c in Helmholtz's sequence of ratios is probably better than that of Ptolemy's.