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Inasmuch as these Fundamentals are Simple Tones, all that has been said above about the latter apply to the former; moreover, it will be noted, that in the case just taken, the number of beats due to the 2nd partial, viz. 2, is the same as that due to the Combination Tone of the 1st order (see page 174), and it is evident that this must always be the case.
An Octave between Compound Tones, therefore is defined,
1st, by the coincidence of the Differential Tone, generated between their two Fundamentals, with the lower of the Fundamentals; and
2nd, by the coincidence of one of the Fundamental Tones with the 2nd partial of the other.
These coincidences it is plain can only occur when the vibration numbers of the Fundamentals are in the exact ratio of 2 : 1. Consequently, this explains why this exact ratio is necessary to the perfection of this interval.
To tune the Octave is thus a very easy matter: the mere mechanical process, of altering the pitch of one tone, till all beats vanish. As this interval is so well defined, great accuracy in its tuning is necessary, the slightest error becoming evident to the ear in the form of beats.
Fifth. Let the vibration numbers of two Compound Tones at this interval be Sn and 2n respectively. Then
3rd partial.... 6ra-----------6«.... 2nd partial
Fundamental.... 2w
the 2nd partial of the former will exactly coincide in pitch with the 3rd of the latter ; or musically