Comparison of Ptolemy and Helmholtz Ratios

For a better comparison of the steps in frequency with each other one can plot the log of the relative frequency, logf, versus the number of a note. The formula for logf used to compare the steps in relative frequency is,

logf = 12·log₂(f/f₀)




Ptolemy's ratios seem to alternate above and below the equitempered line a bit better and may be better balanced but the ear is the ultimate judge. One also needs to take into consideration the sound of the chords.

Supplemental (Jan 5): Note that the intervals of the scales are determined by the lower C of the scale and only A is affected by switching from Ptolemy's ratios to Helmholtz's. If A is fixed at A = 440 Hz for both scales then only A would be unchanged by switching scales and all the other notes would end up being out of tune. One can see how the standard for A causes problems for the older scales and music based on them. It becomes a nuisance. I'd rather not get into an involved discussion of world standards and regimentation at this time but I do think that musicians, and others in addition, should be granted more latitude. Unnecessary constraints run contrary to liberty and tend to be repressive. We should value those who reduce the level of chaos over those who would add to it or redistribute it.