the artificial system must not be mixed up. The wonders of Nature's laws in the developments of harmonies, consist in the beautiful adaption of keyed and all other musical instruments to a range commensurate with human powers. The chromatic scale of twelve notes (the thirteenth being the octave) is not the scale of Nature. To construct a musical instrument upon real divisions of musical tones, each of them being in correct ratio with the others, it would be necessary to have a larger number of tones to the octave. In the development of harmonies on the natural system, we trace the perfect adaptation of means to ends, meeting the intricacies of every musical instrument, including that most perfect of all— the human voice.
If the laws which I shall endeavour to explain develope the twelve major harmonies, with each note in succession expanding its six tones from within itself; and if each of these is found to be a lower development, which leads the ear to a corresponding higher expansion of the twelve major key-notes, and the six tones of each ascending and descending in an unbroken sequence from any twelve consecutively, the thirteenth being the octave of the first, which commences a higher or a lower series; and if the twelve minor harmonies are also gained by the same laws from their twelve relative key-notes (the thirteenth again being octave): if, again, all other notes are shown to be but higher or lower repetitions of these twenty-four harmonies—may we not consider the problem as in some measure solved? especially as the harmonies proceed in geometric as well as harmonical ratio, and an accurate parallel can be traced between the development of notes and colours, which latter correspond with all the intricacies of harmonic sounds.
In the diagrams the circles are not drawn as interlacing into each other, from the difficulty of representing them accurately as rising spirally in geometric progression. If we endeavour to realise the development of harmonies, both in geometric order, and at the same time advancing and retiring, as in musical clef, we must imagine a musician having the physical power of striking all the notes on a circular keyed instrument of seven octaves, linked to a lower series of seven octaves, and a corresponding series of seven higher. But in fact the depth of the lower series, and the height of the higher, are alike unfathomable to our present powers. C, the first note of the seven octaves, sounds the four lowest tones, F, G, A, B of the lower series; and B, the last and highest note of the seven octaves, sounds in its harmony C♮ and D# of the higher series of sevens.