§ 9 Definition of an Octave
Opening the innermost and outermost series of the orifices of our siren, and sounding both of them, either together or in succession, the musical ears present at once detect the relationship of the two sounds. They notice immediately that the sound which issues from the circle of sixteen orifices is the octave of that which issues from the circle of eight. But for every wave sent forth by the latter, two waves are sent forth by the former. In this way we prove that the physical meaning of the term “octave” is, that it is a note produced by double the number of vibrations of its fundamental. By multiplying the vibrations of the octave by two, we obtain its octave, and by a continued multiplication of this kind we obtain a series of numbers answering to a series of octaves. Starting, for example, from a fundamental note of 100 vibrations, we should find, by this continual multiplication, that a note five octaves above it would be produced by 3,200 vibrations. Thus:
| 100 | Fundamental note. |
| 2 | |
| —— | |
| 200 | 1st octave. |
| 2 | |
| —— | |
| 400 | 2d octave. |
| 2 | |
| —— | |
| 800 | 3d octave. |
| 2 | |
| —— | |
| 1600 | 4th octave. |
| 2 | |
| —— | |
| 3200 | 5th octave. |
This result is more readily obtained by multiplying the vibrations of the fundamental note by the fifth power of two. In a subsequent chapter we shall return to this question of musical intervals. For our present purpose it is only necessary to define an octave.