whether veering round, or advancing and retreating in musical clef. I next tried the major keys which develope flats, and I thought that G? would develope a perfect harmony, but found that it must be F#, and that in this one harmony E# must be used in place of F?; on reference, I found that thus the twelve keys developed correctly in succession, the thirteenth being the octave, or first of a higher series.
I had forgotten all the minor keys, except that A is the relative minor of C major; but although I had only faint hopes of success, I determined to try, and I gained the twelve keys correctly, with the thirteenth octave. I found also that E? was usually printed as a minor key-note, Nature's laws having shown that it must be D#.
In a few remarks on "Tones and Colours," inserted in the Athenæum of February 24, 1877, I alluded to the great loss I had sustained by the sudden death of Dr. Gauntlett. I often retrace with grateful remembrance the kind manner in which he examined this scheme when it was but crude and imperfect; with a very capacious intellect, he had a warm and generous heart, causing him to think over with candour any new ideas placed before him. He was of the greatest use to me, by corroborating the points which I had gained. I remarked to him one day, "I find that, of the double tones, F# is a key-note and G? a root." He replied, "You must have a right foundation to work upon, or you would never have ascertained the necessity of the two poles; you have gained the double tones correctly, and the development of harmonies without limit. On this point I have always felt the failure of the laws followed by the musician."
I add quotations from the first letter I received from him. "I have read the MS., and there are some very curious coincidences—exceedingly so—here and there. Whether it will clear out into a demonstrable system, I cannot say at present. If we can get our harmonical start, I think all will come out plainly, for there is so much that is consistent in sequence. There has been nothing at all like it at present, and some of it squares singularly with the old Greek notions." "I am more than half a disciple of your theory of the six tones, and am inclined to imagine that it would do away with much complication, and keep the mind bent on a smaller circle. We can only see things in patches, and hear in trinities, and every single sound is a trinity."
There is amazing grandeur, united with simplicity, in the working of Nature's laws in the development of harmonies of sound, so that the smallest conceivable point has its complementary and corresponding gradation, which renders it capable of development into its peculiar harmony, causing the "multequivalency of harmonies" in endless variety, whether veering round, to and fro, ascending or descending, or advancing and retiring in musical clef.