Background to the musical scale, continued . . .

Nature does not always oblige in having phenomena behave with what we consider to be arithmetical or geometrical perfection. The Earth is not a perfect sphere; the planets do not orbit in perfect circles. Nor do perfect integer ratios necessarily always produce perfectly consonant musical intervals. The tones of perfectly, acoustically consonant intervals do not necessarily exhibit perfect whole number ratios in their relationships. Nevertheless, there is a very strong relationship between whole number ratios and musical intervals in strings and pipes, that cannot be dismissed. 

This relationship is found in two other important 'musical' areas, which are related. The first is in the nature of musical tone, and the second is in the nature of musical consonance. To see how the 'Divine ratios' appear in tone and consonance, we need to consider The Chord of Nature