Page:The American Cyclopædia (1879) Volume XII.djvu/92

 MUSIC same notes in the octave below. Thus we have a rational explanation of the fact that each succeeding octave repeats the impression made by the one which preceded it. The fun- damental tone of the twelfth is really the third harmonic of the tonic, and its second and third harmonics coincide with the sixth and ninth harmonics of the tonic; but the affinity be- tween the tonic and its twelfth is evidently far less than that existing between the tonic and its octave. In diminishing degrees of affinity follow the fifth, fourth, major third, and minor third. The nearest affinities dominated in the earlier periods of music. Thus, in the poly- phonic chanting of the middle ages the fifths were most in vogue, while the thirds and sixths are typical of modern music, and are charac- teristic of the early developments of harmony. According to Helmholtz, there is an affinity of the first degree between two sounds when they have at least one harmonic in common; an affinity of the second degree when the^two sounds have a harmonic in common with a third sound. From these premises he deduces the construction of the diatonic scale with notes which have for the tonic affinities of the first and second degrees. The immediate affin- ities of the tonic are composed of the notes O a, G, F, A, E, and E|>, if we confine ourselves to the first six harmonics, the others being too feeble to determine an affinity. We thus have the gamuts :Q__E_F_G_A__C a ; or better, _ _ E a |> _ F _ G _ A _ _ 2, for we cannot place in the same gamut notes so near to each other as E and Eu. In this series there are two intervals which are too large, and in order to divide them we must recur to the affinities of G, which are 0, D, Ej,, B, C 2. The D and the B are thus found to be related to by an affinity of the second degree ; on interpolating them in the above gamuts, we obtain the diatonic gamut 0, D, E, F, G, A, B, Ca ; which becomes the minor ascending gamut if we place E[, in the place of E. The D which we find in the affinity of F differs by a comma from D as determined by G. These examples will serve to show the method followed by Helmholtz. " In studying the rules of harmony we finally perceive that the accords, considered as complex sounds, contain the same relations of affinity as the notes of the gamut, by reason of the coincidence of some of their notes. The important function of the tonic in modern music, or what M. Fetis calls the principle of tonality, is also explained by the properties of the harmonics of the tonic. These principles, so clear and so simple, have afforded Helmholtz the means of deducing from considerations in some respects mathematical the fundamental rules of musical composition. Nevertheless, we cannot but be of the opinion that the last word on the theory of music has not been said, for all of the deductions of Helmholtz are noi beyond criticism. Thus, Arthur von Oettingen has criticised with much reason the explanation which Helmholtz gives of the difference be- tween the major and minor modes, for the jhenomenon of the harmonics is sometimes i>arely perceptible. Yon Oettingen finds that difference in the reciprocal principles of tonicity and of phonicity. The tonicity of an interval or of an accord consists in the possibility of con- sidering it as a group of harmonics of the same 'undamental sound. It is thus that the major accord is formed by the fourth, fifth, and sixth Harmonics of the tonic or fundamental. 1. Phonicity is the inverse property of having a larmonic in common ; the minor accord ^, , A has the sound 1 as common harmonic or phonic. The major accord has the phonic 60 ; the minor accord has for tonic -^5-. These re- lations can be expressed as follows : 4-5- 60 Tonic. Accord Phonic, (minor). F A-C-E E Tonic. Accord Phonic, (major). C C-E-G B Musicians call C the tonic and G the domi- nant of the gamut of C major, which can be written thus : OD 1 EFGA I I I I B Yon Oettingen calls E the phonic and A the dominant of A minor, and writes the above gamut as follows: E F G A ! f B C f I D E 4 1 By the development of this dualism he obtains the parallel construction of the major and mi- nor modes." (Radau, Acoustique.} Whenever music is written for parts, the laws of harmony necessarily come into play, and the skill of the composer is required, not only to have the harmonics correct, but that the parts shall be distinct and clear. This polyphonic style re- quires very intricate laws, and hence persons capable of creating lovely melodies, and wri- ting them in combination with other themes, are as rare as great poets. In harmonious treatment of music, the following are a few of the radical laws. In the regular progression of harmonics the fundamental bass note falls a fifth to whatever note, or rises a fourth to the octave above it ; but this law has many exceptions. If in the treble or soprano part the procession of notes is upward, say C D E G, the bass cannot give the same notes, but must use others, such iterations being intoler- able to the musical ear. Accordingly, it is a rule in harmony or part writing that contrary motion is best between the extreme parts ; or that when one goes upward the others go downward, and the reverse. The parallel mo- tion, as it is called, is in use between extreme parts, but then the notes must be different. Thirds or sixths move harmoniously together. When the parts are in octaves, the law against identical notes moving up or down together