is generated by multiplying the vibration number of the root regarded as unity by 2, 3, and 5. In this way the number 7 cannot be produced, but 6 3/4 is the cube of 3, namely 27, divided by 2 twice to show it two octaves lower.1 In a similar way the number 11, supposed to be F#, is too low for a proper F#; if supposed to be F, is to high. The number here should have been 11 1/4, the proper number for F# in the Key of G; 11 1/4 is derived from the square of 3 multiplied by 5, namely 45, and twice divided by 3 to show it two octaves lower.1 The other two numbers 13 and 14, supposed to be A and B?, are both too low for these notes, and should be replaced by the number 13 1/2, which is the cube of 3, namely 27, divided by 2 to show it an octave lower; this is correctly A for the scale of G. In short, the musical ratios and the numbers 7, 11, 13, and 14 are like the iron and the clay of Nebuchadnezzar's great image, they do not mingle to adhere.
Another source of error was in considering some of the intervals "perfect" and others "imperfect." Some fifths and thirds are less than other fifths and thirds; "Jove's satellites are less than Jove;" but they are not on that account imperfect. The authors of this erroneous nomenclature did not see the use of a fifth, which is a comma less than 2:3; nor a minor third a comma less than the ratio of 5:6.2 Had the whole system been seen at one view, it would have been seen that the fifth from D to A, and the third from D to F, must necessarily be a comma less than the ratio of 2:3 and the ratio of 5:6; and that to make them what would be considered perfect as intervals would really be to make the system imperfect. It would have been seen that each interval fitted exactly its own place; that the one developed the other; and that they were all equally perfect.
1 The division by 2 is simply to bring the notes down by octaves from where they are generated, and has no other effect on the numbers; it has nothing to do with the generation of the notes. -Editor.
2 In the third edition of the Encyclopedia Britannica, the writer of the article on music sees no reason why the minor third from D to F should be a comma less than the other minor thirds. He is puzzled, but he slips out of the difficulty by saying that "music must always be a kind of metaphysico-mathematical science." The effects of music may be metaphysical, but the causes are as physical as the law of gravitation can make them.