“The principle of the discontinuity of vibration means the definite and necessary characteristic of all vibrations in nature, whether ascending or descending, to develop not uniformly but with periodical accelerations and RETARDATIONs. This principle can be formulated still more precisely if we say that the force of the original impulse in vibrations does not act uniformly but, as it were, becomes alternately stronger and weaker. The force of the impulse acts without changing its nature and vibrations develop in a regular way only for a certain time which is determined by the nature of the impulse, the medium, the conditions, and so forth. But at a certain moment a kind of change takes place in it and the vibrations, so to speak, cease to obey it and for a short time they slow down and to a certain extent change their nature or direction; for example, ascending vibrations at a certain moment begin to ascend more slowly, and descending vibrations begin to descend more slowly. After this temporary RETARDATION, both in ascending and descending, the vibrations again enter the former channel and for a certain time ascend or descend uniformly up to a certain moment when a check in their development again takes place. In this connection it is significant that the periods of uniform action of the momentum are not equal and that the moments of RETARDATION of the vibrations are not symmetrical. One period is shorter, the other is longer. Fragments: Seven

“In order to determine these moments of RETARDATION, or rather, the checks in the ascent and descent of vibrations, the lines of development of vibrations are divided into periods corresponding to the doubling or the halving of the number of vibrations in a given space of time. Fragments: Seven

“It has been found and established that in this interval of vibrations, between the given number of vibrations and a number twice as large, there are two places where a RETARDATION in the increase of vibrations takes place. One is near the beginning but not at the beginning itself. The other occurs almost at the end. Fragments: Seven

“The laws which govern the RETARDATION or the deflection of vibrations from their primary direction were known to ancient science. These laws were duly incorporated into a particular formula or diagram which has been preserved up to our times. In this formula the period in which vibrations are doubled was divided into eight unequal steps corresponding to the rate of increase in the vibrations. The eighth step repeats the first step with double the number of vibrations. This period of the doubling of the vibrations, or the line of the development of vibrations, between a given number of vibrations and double that number, is called an octave, that is to say, composed of eight. Fragments: Seven

“The principle of dividing into eight unequal parts the period, in which the vibrations are doubled, is based upon the observation of the non-uniform increase of vibrations in the entire octave, and separate ‘steps’ of the octave show acceleration and RETARDATION at different moments of its development. Fragments: Seven

“The differences in the notes or the differences in the pitch of the notes are called intervals. We see that there are three kinds of intervals in the octave: 9/8, 10/9, and 16/15, which in whole numbers correspond to 405, 400, and 384. The smallest interval 16/15 occurs between mi and fa and between si and do. These are precisely the places of RETARDATION in the octave. Fragments: Seven

“What precisely does happen at the moment of the RETARDATION of vibrations? A deviation from the original direction takes place. The octave begins in the direction shown by the arrow: “But a deviation takes place between mi and fa; the line begun at do changes its direction and through fa, sol, la, and si it descends at an angle to its original direction, shown by the first three notes. Between si and do the second ‘interval’ occurs — a fresh deviation, a further change of direction. Fragments: Seven