MELODIC RELATIONS OF SOUNDS OF THE SCALE. 45 

Returning to the Syren: if the 8 and 12 "hole circles" be opened together, we hear two sounds at an interval of a Fifth, and as in the case of the octave, this is the fact, whatever the velocity of rotation. The same result is obtained on opening the 10 and 15, or the 12 and 18 circles. When, therefore, two sounds are at an interval of a Fifth, for every 8 vibrations of the lower sound, there are 12 of the upper, or for every 10 of the lower there are 15 of the upper, or for every 12 of the lower there are 18 of the upper. But
Therefore two sounds are at the interval of a Fifth when their vibration numbers are as 2 to 3; that is when 2 vibrations of the one are performed in exactly the same time as 3 vibrations of the other. This may be conveniently expressed by saying that the vibration ratio or vibration fraction of a Fifth is 3 : 2 or Similarly the vibration ratio of an Octave is 2 : 1 or
Again, on opening the circles of 8 and 10 holes, two sounds are heard at the interval of a Major Third. The same interval is obtained with the 12 and 15 circles. Now 8 : 10 :: 4 : 5 and 12 : 15 :: 4 : 5. Therefore two sounds are at the interval of a Major Third, when their vibration numbers are as 4 : 5; or more concisely, the vibration ratio of a Major Ihird is

