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HARMONIOUSNESS OF CONSONANT INTERVALS. 201
dissonances are at the same interval in each; in the latter, it is the dissonance of a 3rd partial against a 2nd, while in the former it is the dissonance of a 3rd partial against a Fundamental. Similarly the Major and Minor Thirteenths are inferior to the Major and Minor Sixths.
The Major Tenth however is greatly superior to the Major Third, the 3rd and 4th partial dissonance of the latter, being absent in the former. "With regard to the Minor Tenth and Minor Third, although in the former, the 3rd and 4th partial dissonance
of the latter has disappeared, yet the semitone dissonance being now between 5th and 2nd partials, will be much more prominent in the former than in the latter, where it only occurs between 5th and 4th partials. Under these circumstances it is difficult to say which is the better interval of the two. Helmholtz holds that the Minor Third is the superior, and so obtains the following symmetrical rule to meet all cases :
" Those intervals, in which the smaller of the two numbers expressing the ratios of the vibration numbers is odd, are made worse by having the upper tone raised an Octave; " while
"Those intervals, in which the smaller of the two numbers expressing the ratios of the vibration numbers is even, are improved by having the upper tone raised an Octave."
We conclude the present Chapter, by giving in fig. 84, Helm-holtz's, graphic representation of the relative harmoniousness of musical intervals.
Fig. 84.
In this figure, the intervals are represented by the horizontal distances CEt?, CE, &c, measured from the point 0; while the