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68 VIBRATION-FRA CTIONS. [ 11. § 35.
of an assigned sound [§ 28], we may express our three results as follows:—
When two sounds form with each other the intervals of an Octave, a Fifth or a Fourth, their vibration-numbers are to each other, in the first case as 2 to 1, in the second as 3 to 2, in the third as 4 to 3.
A ratio is most easily expressed by a fraction. Thus we may regard the fraction f as denoting the interval of a Fifth. It may be taken as an abbreviated statement of the fact that, when two sounds form a Fifth with each other, the more acute makes 3 vibrations while the graver makes 2.
35. By suitable experiments, similar numerical relations to those already established may be obtained for all the intervals already considered. A fraction can thus be determined for each interval, in the manner exemplified in the case of the Fifth. We will call this fraction the vibration-fraction of the interval in question. The accompanying table gives, in the second column, the vibration-fractions corresponding to the intervals named in the first; and in the third, states the consonant or dissonant character of each interval.
It is noticeable that the dissonant intervals involve higher numbers in their vibration-fractions than the consonant intervals do; the latter, with the