SPHERES OF SOUND
(Sequence of wave vibrations on the harmonic diagonal)

The tone C is the whole, a ratio of 1:1, therefore C encloses the total space. The tone G*, a ratio of 2:3 or 12 in inverse, falls next in the sequence, so becomes 2 of C. Subsequent tones of F, 3:4; A, 3:8; E, 5:6; B, 9:16; D, 8:9, all become 2 of the previous tone and form comma-like wave motions when drawn, so that an extended line does not twist when it meets the diagonal. It is thus easy to see an expansion or contraction. An expansion goes away from the diagonal, while a contraction moves toward a diagonal. By reducing the scale , the whole takes on the familiar form of the PROGRESSION BY SQUARES, and is likened to a falling body.
* Note: For the drawings the ratios are in wavelengths. Some of the wavelengths are in inverse relationship to the frequencies of the notes. Therefore 2:3 in frequencies in music is an F, while 2:3 is a G in wavelength.


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