In just such a manner, then, tone is based upon a manifold, however much we hear and perceive it as something entirely ultimate; a varied nature, which, for the reason that musical tone comes into being by means of the vibration of a body, and thereby together with its vibrations is subject to temporal condition, is deducible from the numerical relation of this oscillation in time, in other words from the determinate number of vibrations in a given period. I propose to draw attention merely to the following points in respect to this deduction.
Tones that accord in the fullest sense, and on hearing which a distinction is not perceptible as opposition, are those in the case of which the numerical relation of their vibrations is of the simplest character; those on the contrary which are not so out and out accordant possess proportionate numbers more complex. As an example of the first kind we have octaves. In other words, if we tune a string, where we shall have the keynote given us by a definite number of vibrations, and then halve the same; in that case this second half will give us in the same time precisely the same number of vibrations as the previous entire string[447]. Similarly in the case of fifths we have three vibrations to two of the keynote; in the case of thirds we have five to four of the keynote. In the case of seconds and sevenths we have a different kind of proportion; here to eight vibrations of the keynote we have in the former case nine and in the latter fifteen.
(ββ) Inasmuch then—we have already referred to this—as these relations cannot be posited as we like, but disclose an ideal necessity for their particular aspects[448], no less than the totality they together constitute, for the like reason the particular intervals, which are fixed by such numerical relations, do not persist in their relation of indifference to each other, but are inevitably comprised together in and as a whole. The first form of this totality of notes thus created is, however, as yet no concrete concord of different notes, but an entirely abstract series of a system, a series of notes related under the most elementary mode to each other, and their position within the totality thus comprised. This is no other than the simple series of notes known as scales. The fundamental basis of this is the tonic, which repeats itself in its octave, and is extended through the remaining six notes placed between these limits, which by virtue of the fact that the keynote directly falls into unison with its octave makes a return upon itself. The remaining notes of the scale either harmonize completely[449] with the keynote, as is the case with the fifth and the third, or possess a more fundamental distinction of sound in conflict with it, as is the case with seconds and sevenths, and take their place consequently in a definite series, which, however, I do not now propose to discuss or explain further.
(γγ) Thirdly, in these scales we find the source of different keys. In other words, every note of the scale can, in its turn, be posited as the keynote of a fresh series of notes, which is co-ordinated precisely as the first is. With the development of the scale through an increase of notes the number of keys has correspondingly increased. Modern music avails itself of a larger variety of keys than that of the ancients. Further, inasmuch as generally the different notes of the scales, as already observed, are related to one another in unobstructed harmony, or a relation that deviates from such immediacy in a more fundamental way, it follows that the different series which arise from these notes, taken severally as keynotes, either display a closer relation of affinity, and consequently permit of a passage readily from one to another, or, on account of their alien character, do not so admit of this. Add to this that the keys are divided from each other by the distinction of hardness and softness, that is, as major or minor tonality; in conclusion they possess, in virtue of their key-note, from which they are generated, a definite character, which of itself responds to a particular kind of emotion, such as lamentation, joy, mourning, and so forth. In this particular even writers in ancient times have anticipated much on the subject of distinction between the keys, and applied their theory in many ways to actual composition.
(γ) The third important matter, with the discussion of which we may conclude our brief remarks upon the theory of harmony, is concerned with the simultaneous concord of the notes themselves, in other words, the system of chords.
(αα) We have no doubt already seen that the intervals constitute a whole; this totality, however, is in the first instance comprised in the scales and the keys merely in the form of an associated series, in the succession whereof each note is asserted separately in isolation. In consequence the tonal sound remained abstract, because we find here that it is only one particular and determinate tone that is asserted. In so far, however, as the notes in fact are what they are[450] merely in virtue of their relation to one another, it follows necessarily that their tonal modality should attain also an existence as this concrete body of tone itself, in other words different notes will have to coalesce in one and the same body of tone. In this conjoint fusion, in the composition of which, however, the mere number of notes capable of such coalescence is not the essential point, for we may have a unity of this kind with merely two[451], we possess our definition of chords. For inasmuch as the different notes are not definable for what they are as a result of caprice or chance, but are necessarily regulated by virtue of an ideal principle and co-ordinated in their actual succession, it follows that a regularity of similar character will have to declare itself in the chords, in order that we may determine what kind of associations will be adapted to musical composition, and what on the contrary must be excluded. It is these rules which first give us the theory of harmony in the full sense; and it is according to this we find again that the chords are embraced in an essentially regulated system.
(ββ) In this system chords are particularized and distinguished in their passage from one to another, inasmuch as it is clearly defined notes which thus sound together. We have consequently to consider as an immediate fact a totality of separately distinguishable chords. In attempting the most general classification of these we shall find that the original distinctions we cursorily alluded to in our discussion of intervals, scales, and keys will once again serve us.
In other words the first kind of chords are those in which notes come together, which are completely consonant. In the musical effect of these consequently there is no opposition, no contradiction perceptible; the consonance remains completely undisturbed. Such is the case in the so-called consonant chords, the foundation of which is supplied by the triad. This confessedly is generated from the key-note, the third, the mediant[452] and the fifth or dominant. In these we find the notion of harmony expressed in its simplest form, or rather the intrinsic idea of harmony generally. For we have a totality of distinct notes under consideration, which assert this distinction while they also declare an undisturbed unity.
We have here an immediate identity, which moreover is not without the element of separation and mediation, albeit this mediation is not at the same time limited by the self-subsistency of different tones[453], satisfied with the mere transitional passage from one note to another in the relation of a series, but the unity is here an actual one and a return in immediacy upon itself.