to have recourse to the second order. Thus, proceeding as in the case of the octave, we have.

The Fifth is, thus, a fairly well-defined consonance, though decidedly less sharply bounded than the octave, owing to the feebleness of the C. T. of 2nd order. For the Fourth we have

The 3rd-order tone being excessively weak, the interval of a Fourth can scarcely be said to be defined at all. Still less can the remaining consonant intervals of the scale, by the evanescent beats of still higher orders of combination-tones.

92.

With simple-tones, then, the case stands thus. The intervals of a Second and a Major-Seventh are palpably dissonant, owing to the beats of the primaries, in the former, and of a first-order combination-tone with a primary, in the latter. There is a certain amount of dissonance in intervals slightly

narrower or slightly wider than a Fifth, but of a feebler kind than in the case of the octave, inasmuch as it is due to only a second-order combination-tone. Whatever dissonance may exist near the Fourth is practically imperceptible. All other intervals are free from dissonance. Accordingly all intervals from the Minor Third nearly up to the Fifth, and from a little above the Fifth up to the Major Seventh, ought to sound equally smooth. This conclusion is probably very inconsistent with the views of musical theorists, who regard concord and discord as entirely independent of quality, but it is strictly borne out by experiment. With the tones of tuning-forks the intervals lying between the Minor and Major Thirds, and between the Minor and Major Sixths, though sounding somewhat strange, are entirely free from roughness, and, therefore, cannot be described as dissonant. As a further verification of this fact, Helmholtz advises such of his readers as have access to an organ to try the effect of playing alternately the smoothest concords and the most extreme discords which the musical scale contains, on stops yielding only simple-tones, such, e. g., as the flute, or stopped diapason. The vivid contrasts which such a proceeding calls out on instruments of bright timbre, like the pianoforte and harmonium, or the more brilliant stops of the organ, such as principal, hautbois,

trumpet, &c., are here blurred and effaced, and every* thing sounds dull and inanimate, in consequence. Nothing can show more decisively than such an experiment that the presence of over-tones confers on music its most characteristic charms.

Thus the remark put into the mouth of a supposed objector in § 89 turns out to be no objection whatever to Helmholtz's theory of consonance and dissonance, but, so far as it represents actual facts, to be valid against the prevalent views of musical theorists.

93. It may we well to advert briefly, in this place, to a point connected with combination-tones which may otherwise occur as a difficulty to the reader's mind. When two clangs coexist, combination-tones are produced between every pair which can be formed of a tone from one clang with a tone from the other. These intrusive tones will usually be very numerous, and, for aught that appears, may interfere with those originally present, to such an extent as to render useless a theory based on the presence of partial-tones only. Helmholtz has removed any such apprehension, by showing that, in general, dissonance due to combination-tones produced between overtones, never exists except where it is already present by virtue of direct action among the overtones themselves. Thus the only effect attri

butable to this source is a somewhat increased roughness in all intervals except absolutely perfect concords. No modifications, therefore, have to be introduced, on this score, in the conclusions of SS 81-86.

CHAPTER IX.

ON CONSONANT TRIADS.

94. In the ensuing portion of this enquiry we shall have to make more frequent use than hitherto of vibration-fractions. It may, therefore, be well to explain the rules for their employment, in order that the student may acquire some facility in handling them. The vibration-fraction of an assigned interval expresses the ratio of the numbers of vibrations performed in the same time by the two notes which form the interval. The particular length of time chosen is a matter of absolute indifference. The upper note of an octave, for instance, vibrates twice as often as the lower does in any time we choose to select, be it an hour, a minute, a second, or a part of a second. It is often convenient to determine the vibration-fraction of an interval from the vibration-numbers of its constituent notes: in such a case we choose one second as our time of comparison, and in this way vibration-fractions were