Combination-tones of the second order may be treated as if they were first-order tones produced between one or other of the primaries and the combination-tone of the first order, Similarly we may regard each combination-tone of the third order as due to a second-order tone, paired either with one of the primaries, with the first-order tone or with its own. fellow of the second order. Successive subtraction will therefore enable us to determine the vibrationnumber of a combination-tone of any order from the vibration numbers of the two primaries.

Combination-tones grow rapidly feebler as their order becomes higher. Those of the first order are usually distinct enough, and those of the second to be heard with a little trouble. The third order is only recognizable when entire stillness is secured, and the greatest attention paid. It is a moot point whether the fourth-order tones can be heard at all.

91, We can now show that the existence of combination-tones prevents intervals formed by two simple tones from altogether lacking the characteristic differences of consonance and dissonance, though those differences are far less marked than in the case of composite sounds. To begin with the octave, Let us suppose that we have two simple tones forming nearly this interval, but that the higher of them

is a little sharp, so that the octave is not strictly in tune, is in fact slightly impure. Let the lower tone make 100, the higher 201, vibrations per second. They will give rise to a combination-tone making 101 vibrations per second (p. 177), and this with the lower primary will produce one beat per second. If the higher primary had been flat, instead of sharp, making, say, 199 vibrations per second, we should have had 99, as combination-tone, giving rise, with 100, to beats of the same rapidity as before. These beats cannot be got rid of except by making the vibration-ratio exactly 1:2, i. e. the octave 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 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, pro

ceeding 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