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176                    HAND-BOOK OF ACOUSTICS.
flattening the middle note, till these beats vanish, we may obtain a perfect fifth.
Similarly, to rectify a mistuned Fourth, 301 and 400, for example, we may take a tone, 200, an octave below the higher one, and proceed as in the above case.
Again, in the case of a false Major Third, say 400 and 501, tune a Simple Tone 600 a perfect Fifth from the lower tone. Then 501 — 400 = 101 .. Differential of 1st order, 600 — 501 = 99 ..                     1st
101 — 99 = 2 beats being heard between Differentials of 1st order. Tune as before till the beats disappear.
Similarly in the case of the mistuned Minor Third, 600 and 501, take a third tone 400, a true Fifth below the higher tone, and proceed as above.
We come now to the definition of Intervals between Compound Tones, and in the first place we shall assume the Compound Tones in question to be such as are produced by the Human Voice, Harmonium, Piano, and stringed instruments in general; that is to say, we shall suppose them to consist of, at least, the first six partials.
Octave. Let the vibration numbers of the fundamentals of two Compound Tones, at the interval of an octave, be n and In respectively. Then the 2nd partial of the former will be 2n which will thus coincide with the other fundamental; or, in musical language
2nd partial.. d'------------d1 • • Fundamental.
Fundamental.. d
H the ratio of the vibration numbers be not exactly as 2 : 1, beats will be heard between the 2nd partial of one tone and the funda­mental of the other. Suppose, for example, that the vibration numbers are 200 and 99. Then
2nd partial. .198------------200. .Fundamental
Fundamental.. 99 and thus 200 — 198 = 2 beats per second will be heard.