built the one upon the other in order to form the first scale, and as the notes are built up to form the chords.
In related scales there is no case in which a note serves for more than four major and three minor, or four minor and three major keys;1 but Mr. Taylor makes B with the number 495 serve for six major scales. Forming the scales as he did from the notes of C, instead of the proper progression of scales, makes all the notes of the scales of A, E, and B major a comma too low for the scales of F, C, G, and D; and likewise, all the notes of the scales of D, G, C, and F minor a comma to high for the scales of A, E, and B. All the minor sevenths are a comma too low for their tonic notes. But as D, G, C, and F are a comma to high, if the tonic notes were right these four sevenths would be right with their present numbers. The other three scales, A, E, and B, which begin on their proper numbers, have their sevenths a comma too low.
Had the scales been properly derived from each other, the tonic chord D would have been the subdominant of A, and the tonic chord of A would have been the dominant of D, and so on; which is not the case in Mr. Taylor's major and minor scales.
Again, had the scales been properly derived from each other, the sharp of the one would always have been higher than the flat of the other;2 and instead of the flat being two commas, less the apotome minor, higher than the sharp, the sharp would have been the apotome minor, the ratio of 8192:8201 1/4, higher than the flat; and there would never have been more than the apotome minor between the sharp of the one and the flat of the other, except where the scales were nine and the notes themselves were twelve fifths apart, and then the difference would have been a comma and the apotome minor.
Mr. Taylor's work has not been referred to because it is more erroneous than other works on the same subject, but because, being recently published, it might be supposed to contain the knowledge of music up to the time.3
1 See Plates VIII, and XXVII
2 Ibid.
3 Of course, when this work is said here to be recent, it must be remembered that this is in relation to the time when the author is writing. -Editor.