HANDBOOK OF ACOUSTICS - online book

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220                    HAND-BOOK OF ACOUSTICS.
Thus, as far as the partials go, the balance is largely in favour of No 1. The differentials tell the same tale; for four of those in No. 1 are identical with partial tones, and the other rather improves the chord than otherwise. On the other hand the vta.> in No 2 is decidedly detrimental, for not only is it foreign to the chord, but, being at such a low pitch, it will produce audible beats with d|.
In order to obtain some general results concerning these four-part chords, it will be better to apply the results that were arrived at in the case of the Triads.
Taking the Major Tetrads first, the following may be selected as a typical chord.
The rule in the case of the Major Triads was, to avoid Thirteenths and Minor Tenths. In transposing the tones of the above chord, therefore, and keeping within the compass of two Octaves, the "g" must not be more than a Minor Third above the " n," otherwise a Minor Tenth will be formed; nor must the "g" be transposed more than a Sixth below the " n," or a Major Thirteenth will occur; lastly the " d " must not be more than a Minor Sixth above " n," if we wish to avoid making a Minor Thirteenth, but it may be placed as far below it as we please. These rules may be briefly enunciated in the words of Helmholtz :—" Those Major chords are most harmonious, in which the Root or the Fifth does not lie more than a Sixth above the Third, or the Fifth does not He more than a Sixth below it."
It follows from this, that the "n" and "g" must not be duplicated by the Double Octave. There is another reason for this rule, however, namely that the differentials thus generated, will interfere with the other tones of the chord; thus—
In the first case the "t" will beat with the " d1" and the