212 HAND-BOOK OF ACOUSTICS.
We see from the above tables that the Differentials of the Major Triads are not only harmless to their respective chords, but actually improve them, supplying as it were a natural and true bass. On the other hand, in the Minor Triads, we find Differential Tones (f and s above) that are entirely foreign to the chords. They are not indeed close enough to beat, nor are they sufficiently distinct to destroy the harmony, but " they are enough to give a mysterious, obscure effect to the musical character and meaning of these chords, an effect for which the hearer is unable to account, because the weak differential tones on which it depends are concealed by other louder tones, and are audible only to a practised ear. Hence minor chords are especially adapted to express mysterious obscurity or harshness."
It must be remembered, that on tempered instruments, the differentials will not be exactly of the pitch given above, consequently those of the Major Triads will not fit in so well to the chords. Hence the superiority of Major to Minor chords, though still perceptible on tempered instruments, is not so marked as when the intervals are justly intoned.
It was shown in the last Chapter, that either tone of a Consonant Interval may be raised or lowered by an Octave, not indeed without somewhat altering the degree of its harmoniousness, but without losing its consonant character. By thus raising or lowering one or more of their tones, the Consonant Triads may be obtained in a great variety of distributions. We shall proceed to ascertain theoretically, the more harmonious of these distributions, in which the extreme tones of the Triad are within the compass of two Octaves.
In order to ascertain the more harmonious of these distributions of the six fundamental Triads, we shall in the first place have to bear in mind the rules, concerning the enlargement of an Interval by an Octave, which we obtained in the last Chapter (see page 203); and in the second place, we shall have to note the effect of Differential Tones. These two considerations will be sufficient to guide us in this enquiry.
With regard to the first, it will be convenient to briefly recapitulate the essential part of the results on page 203, viz.— Minor Tenths are inferior to Minor Thirds. Elevenths ,, ,, Fourths.
Thirteenths ,, ,, Sixths.
but the Fifth and Major Third are improved by being enlarged by an Octave.