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 SS 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 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 applicable to all other cases, it follows that the vibration-fraction of any interval denotes the number of vibrations and parts of a vibration made by the higher of the two notes which form that interval while the lower of them is making a single vibration. We will next investigate rules for determining the vibration-fractions of the sum and difference of any two intervals whose vibration-fractions are known. 95. Suppose that, starting from a given note, we sound a second note, a Fifth above it, and then a third note, a Major Third above the second. What will be the vibration-fraction of the interval formed by the first and third notes, i.e. of the sum of a Fifth and a Major Third? We will, for shortness, call the three notes (1), (2), (3), in order of ascending pitch. The vibration-fractions being, for (1)-(2), , and, for (2)-(3), 4, we proceed thus: While (2) makes 4 vibrations, (3) makes 5 vibrations. Therefore, while (2) makes 1 vibration, (3) makes vibrations. Therefore, while (2) makes 3 vibrations, (3) makes 3 × vibrations. But while (2) makes 3 vibrations, (1) makes 2 vibrations. Therefore, while (1) makes 2 vibrations, (3) makes 3 x vibrations. Therefore, while (1) makes 1 vibration, (3) makes × vibrations. Our result, then, is the two vibration-numbers multiplied together. The reasoning is perfectly general, and gives us the following rule. To find the vibration-fraction of the sum of two intervals, multiply their separate vibration-fractions together. 96. Next, take the opposite case. Let (2) be a Major Third above (1), and (3) a Fifth above (1), and let the vibration-fraction for the interval (2)—( be required. While (1) makes 4 vibrations, (2) makes 5 vibrations. Therefore, while (1) makes 1 vibration, (2) makes vibrations. But, while (1) makes 2 vibrations, (3) makes 3 vibrations. Therefore, while (1) makes 1 vibration, (3) makes Hence, while (2) makes § vibrations, (3) makes Therefore, while (2) makes of a vibration, (3) makes Therefore, while (2) makes 1 vibration, (3) makes × vibrations. × of a vibration. vibrations. The result here is the quotient resulting from the division of the larger vibration-fraction by the smaller hence we have this general rule :— To find the vibration-fraction of the difference of two intervals, divide the vibration-fraction of the wider by that of the narrower interval. Thus multiplication and division of vibrationfractions correspond to addition and subtraction of intervals1. 1 By simply reducing the numerical results, obtained in $$ 95, 96, the student will establish the following propositions : 'A Major Third added to a Fifth gives a Major Seventh.' 'A Major Third subtracted from a Fifth leaves a Minor Third.' 97. One of the simplest cases of our second rule occurs when an interval has to be 'inverted.' By the 'inversion' of any assigned interval narrower than an Octave is meant the difference between it and an Octave, i.e. the interval which remains after it has been subtracted from an Octave. Thus to find the vibration-fraction for the inversion of the Minor Third we merely have to divide 2 by g, or in other words invert the vibration-fraction of the interval and multiply by 2. This applies to all cases. In the particular example selected the result is ; the inversion of the Minor Third is therefore the Major Sixth. The relation between an interval and its inversion is obviously mutual, so that each may be described as the inversion of the other. Accordingly the inversion of the Major Sixth is the Minor Third. The following table' shows the three pairs of consonant intervals narrower than an Octave which stand to each other in the mutual relation of inversions. Minor Third ()—Major Sixth (§) 98. A combination of musical sounds of different The last two results of this table will be easily verified by the student. |