HISTOEY.] MUSIC 89 his treatise Harmonic Universelle (1636) first enunciated the fact that a string yields other notes than that to which its entire length is tuned. The discovery was extended by William Noble and Thomas Pigot, respectively of Merton and Wadham Colleges, Oxford, to the perception of the mode in which a string vibrates in sections, each section sounding a different note. The ancient musicians tested by calculation the few phenomena of sound then discovered rather than by observation of the principles these exemplify. The measurement of major and minor tones was, after the distinction of perfect intervals, the subject dearest to their consideration, and it seems the furthest limit to which their knowledge attained. All the laws for melody, all the rules for counterpoint, were founded on this mathematical method. The step or the leap of stated intervals was prescribed ; combinations of sounds Avere reckoned by intervals from a named note, as 5th, or 6th, or 3d, not as constituting com plete chords traceable to a common source, and intervals which are discordant were permissible only if softened in effect by the previous sounding of their discordant note ; the canons for the progression of a single part and for the union of several parts were arbitrarily devised, peremptorily fixed, and rigidly enforced. Mouton and Monteverde found the good effect of musical combinations for which there was no account in the theory of their time, and employed them in their works ; the innovation was stigmatized by musical grammarians, but it gave delight to the public and was adopted by subsequent composers. No explanation was, however, given of the natural source of fundamental har monies, as chords of this class are now defined, and their employment was still exceptional, still an act of daring. In 1673 the two Oxonians above named, simultaneously, but independently, noticed the beautiful fact that a stretched string yields a different sound at every one of its nodal divisions, and the same is true of a column of air passing through a tube. The sounds so generated received from Sauveur 1 the name of harmonics, by which they were known for nearly two centuries, but they have of late been renamed partial tones or over-tones. 2 Here is a table of seventeen of the series : 14 15 16 17 The figures under the notes show the number of each harmonic, counting from the generator or prime as the 1st. The notes marked * differ in intonation from the corresponding notes in our tempered scale, the 7th and 14th, and also the 13th, and likewise the 17th being slightly flatter, and the llth being slightly sharper than our con ventional notes ; but the matter of temperament must rest for later consideration. The 8th above any note is double the number of that note ; thus every higher C is double the number of the C below it, namely, 1, 2, 4, 8, 16 ; and so with every higher G, namely, 3, 6, 12; again with the higher E, namely, 5, 10 ; and with the higher bB, namely, 7, 14. The number of each harmonic is the same as that of its relative number of vibrations in any given time as compared with those of the variously- numbered harmonics, namely, the 8th above has two vibrations to each one of the note from which the interval is reckoned, the 5th has three vibrations to two, and so forth throughout the series. From bB to E, the 7th and 10th, is the interval of the augmented 4th, which was shunned in classic times, ignored by the Chinese, the Mexicans, and the Scots, ruled against by contrapuntists, and avoided in melody and harmony until employed by 1 See Poggendorff, Geschichte d. Physik, p. 808. 2 See Helmholtz, Die Lehre von den Tonempfindungen. the Fleming and the Italian with such good effect that the world accepted it under the conditions of accompani ment with which those men employed it, and felt that a new element of beauty had been incorporated in the re sources of the artist. The occurrence, in the harmonic series, of the two notes that are separated by this interval accounts for the discord they produce when sounded together, not needing the artifice of preparation which is required to mitigate the harshness of other discords ; they are brought into being when the generator is sounded, and their assign ment to voices or instruments in performance is but to make more articulate, or, so to speak, to confirm what nature prepares in fact, what is induced by the generator. As light comprises all the colours and every gradation between each colour and the next, but yet seems spotless, so every musical sound comprises all other sounds, but yet seems to be one single note ; the blue, or the red, or the yellow, or any other ray is separated from its prismatic brotherhood and seems then a complete and independent object to the vision, and so any sound is separated from the harmonic column and then seems all in all to the sense of hearing. Let the reader observe in the musical example that the intervals become closer and closer as they rise, and that when the 8th or double of a note occurs, if there be any break in the numerical succession between such 8th and the note that would, by example of the lower octave, stand next below it, then some new harmonic appears whose number adjusts the broken order ; between the lowest C and the next is no break ; between this C and the one above it, 2 and 4, what would else be a blank is filled by G, the third harmonic; between 4 (C) and 6 (G) what would be a blank is filled by E, the fifth harmonic, and so on throughout the series. No division of an interval is ever equal, the lower portion being always the larger ; the interval between 2 and 4 is divided into a 5th and a 4th, that between 4 and 6 is divided into a major 3d and a minor, that between 6 and 8 by an interval less than a minor 3d and a 2d, and that between 8 and 10 by a major tone and a minor tone. It may be well to pause at this point, as it is the natural justification of what Ptolemy calculated, but Pythagoras failed to perceive. Thus much having been noticed, readers may be left to trace the same principle of larger and smaller division throughout the series. Beyond the 17th harmonic (the note known as the minor 9th when forming part of a chord) the series continues on the same principle of ever lessening distance, ever finer gradation, until the intervals become so small as to be almost impossible of articulation and of perception. What has here been adduced of the natural preparation of the discord of the harmonic 7th applies as truly to the discords of the major 9th, the llth, the major 13th, the minor 9th, and the minor 13th, which last is too high in the harmonic series for convenient exemplification by gradual ascent in this place, and these notes are now all used in combination by composers. Scientific discovery has seldom been made singly. When Art the time has been ripe for the revelation of a phenomenon, P recur - several observers have coincidently witnessed its existence, g[ e ce and simultaneously or nearly so displayed if not explained it to the world. In the instance under consideration, art foreran science, and its votaries continued the employment of harmonies which as yet could alone be justified by their beautiful effect, and even musical theorists did not for ages to come perceive the important, the all-powerful bearing of the principle of harmonics upon the subject they treated. What Mouton first ventured to write must be styled the starting-point of the modern in music, and one cannot too much marvel at the strong insight into the beautiful which those after -minds possessed, that, with no theory to guide, without star or compass, they XVII. 12
