A Dictionary of Music and Musicians/Octave
OCTAVE. An octave is the interval of eight notes, which is the most perfect consonance in music. The ratio of its sounds is 1 : 2; that is, every note has twice the number of vibrations of its corresponding note an octave lower. The sense of identity which appears to us between notes of the same name which are an octave or more apart, arises chiefly from the upper octaves and their harmonics corresponding with the most prominent harmonics of the lower note. Thus Helmholtz says, 'when a higher voice executes the same melody an octave higher, we hear again a part of what we heard before, namely the even-partial tones of the former compound tones, and at the same time we hear nothing that we had not previously heard. Hence the repetition of a melody in the higher octave is a real repetition of what has been previously heard, not of all of it, but of a part. If we allow a low voice to be accompanied by a higher in the octave above it, the only part-music which the Greeks employed, we add nothing new, we merely reinforce the even-partials. In this sense, then, the compound tones of an octave above are really repetitions of the tones of the lower octaves, or at least of part of their constituents.'
Irregularly consecutive octaves are forbidden in music in which the part-writing is clearly defined. The prohibition is commonly explained on the ground that the effect of number in the parts variously moving is pointlessly and inartistically reduced; at the same time that an equally pointless stress is laid upon the progression of the parts which are thus temporarily united either in octaves or unison. Where however there is an appreciable object to be gained by uniting the parts, for this very purpose of throwing a melodic phrase or figure into prominence, such octaves are not forbidden, and small groups or whole masses of voices, or strings, or wind instruments, are commonly so united with admirable effect.
The interval of an augmented octave, exceeding the octave by a semitone, is occasionally met with; as in the following example from the first subject ot the Overture to Don Giovanni:—
