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which we see from the table on page 159 is on the verge of the beating distance. G- = 384 may be taken therefore, as the limit above which a Major Sixth is a perfectly smooth interval, and below which it gradually deteriorates.
On the other hand the Minor Sixth at this pitch has still an element of roughness, for
which forms a harsh dissonance in this region.
On comparing the relative smoothness of the Fourth, Major Sixth, and Major Third, no valid reason appears for the precedence, which is usually granted to the first-named interval over the other two; and in fact no such precedence can be assigned to the Fourth, if the three intervals be judged by the ear alone, under similar circumstances. As Helmholtz remarks "the precedence given to the Fourth over the Major Sixth and Third, is due rather to its being the inversion of the Fifth, than to its own inherent harmoniousness."
In the Diminished Fifth and Augmented Fourth or Tritone the 2nd, 3rd, and 4th partials of the upper tone dissonate at the interval of a semitone with the 3rd, ith, and 6th partials of the lower tone ; after mentioning which elements of roughness, it is scarcely worth while to point out the tone dissonance between the 4th and 5th partials.
Although the above results apply fairly well to all instruments, the tones of which consist of the 1st six partials, to such instruments for example, as the pianoforte, harmonium, open pipes, of organs, and the human voice ; yet it will not do to apply them, in a hard and fast manner, to any instrument whatever. The low tones of the harmonium, for example, especially if the instrument be loudly played, contain more than the 1st six partials, while those of open organ pipes, gently blown, often consist of fewer. Again, the tones of the human voice vary wonderfully in their constitution, not only in different voices, but also, and chiefly, according to the particular vowel sound produced. The influence which the vowel sounds have in modifying the roughness or smoothness of an interval, can best be realized by making a few experiments with men's voices. Let such intervals as the Major and Minor Thirds bc«