perfectly pure. The roughness must increase both when the interval widens and when it contracts, so that the Octave, in simple tones, is a well-defined concord bounded on either side by decided discords. This result may be easily verified experimentally by taking two tuning-forks forming an Octave with each other, and throwing the interval slightly out of tune by causing a pellet of wax to adhere to a prong of one of them. On vigorously exciting the forks the beats will be distinctly heard'.

The Octave is the only interval which is defined by the beats of a combination-tone of the first order with one of the primary tones. For the next smoothest concord, that of the Fifth, we are obliged to have recourse to the second order. Thus, proceeding by successive subtraction, we have :—

1 As, however, this mode of treatment produces very perceptible overtones [§ 48, note], the experimenter must be on his guard against attributing to combination-tone beats an effect which is really due to beats of overtones. There is special risk of being thus led astray in the case of the Octave, where the first-order combination-tone coincides in pitch with the lower of the two primaries. The result of my own observation is that with mounted forks excited in the gentlest possible way there is but slight dissonance even in an impure Octave.

The Fifth is, thus, a fairly well-defined concord, though decidedly less sharply bounded than the Octave, owing to the feebleness of the C. T. of the

second order. For the Fourth we have :–

The third-order tones 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 be regarded as defined by beats due to combination-tones of yet higher orders.

92. With two moderately loud simple tones, then, the case stands thus. The interval of a Second is rendered palpably dissonant by direct beats of the primaries; that of a Major Seventh slightly so by beats of a first-order combinationtone with one of the primaries. 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 one of the two Combination-Tones producing it is of the second order. 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, however inconsistent with the views of Musical theorists, who are apt to regard concord and discord as entirely independent of quality, is strictly borne out by experiment. 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.

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 approximately simple tones, such e.g. as the flute or stopped diapason. The vivid contrasts which such a proceeding calls out on instruments of bright quality, 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 everything sounds dull and inanimate in consequence. Nothing can show more decisively than such an experiment that the presence of overtones 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 a view commonly acted on by Musical theorists.

93. A point connected with combination-tones, which might otherwise occur as a difficulty to the reader's mind, shall here be briefly noticed. When two clangs coexist, combination-tones are produced between every pair which can be formed of a partialtone from one clang with a partial-tone from the other. These intrusive sounds will usually be very numerous, and, for aught that appears, might be thought likely to 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 generally that 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 attributable 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 into the conclusions of §§ 81-86.

CHAPTER IX.

ON CONSONANT TRIADS.

94. In the ensuing portion of this inquiry we shall have to make more frequent use than hitherto of vibration-fractions. It may, therefore, be well to explain at this point the rules for their employment, in order that the student may acquire the requisite 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. In like manner the vibrationfraction indicates that while the lower of two notes 岳 forming a Major Third makes four vibrations the

higher of them makes five. Therefore while the lower makes one vibration the higher makes ths of a vibration. The same reasoning being equally