major third. 4 major sixth.
fourth. 2 octave or eighth.
We have not yet, however, got a sufficiently agreeable scale, and the reasons why the ear will not be contented with the preceding most simple concords, must be derived from observation, frown which it appears 1. That a frequent repetition of sounds very near to one another in not pleasing to the uncultivated ear. Now the interval from the minor to the major third is as follows : the first makes 1 of a vibration while the second makes 1, or the first makes 1 vibration while the second makes 4 x a, or 11. This is much too near to a unison for con tinual repetition.
2. That a frequent repetition of sounds too fax from each other is not pleasing to the ear, after a little cultivation. If we look at the intervals from the fourth to the fifth, and from the fifth to the sixth, we find I and V for their representatives, while from the fundamental note to the minor third, and also from the sixth to the octave, the interval is 2, much larger than the preceding intervals.
Both these defects, as must easily be seen arithmetically, and as the ear finds out for itself, may be remedied by inserting a note between C and E in place of E b, which shall make a better division of the interval C E, and by placing an additional note between A and C'. But how are we to choose these additional notes If we cannot have any more very simple consonances with the fundamental note, we must take those tones which make the simplest consonances with other notes, and the more they make the better. We have already a repetition of some consonances ; for instance, Interval F Is 2 = 4 = 4, or a fifth. Interval G Cl is 2 ÷ 4 = 4, or a fourth.
Interval F A Is = 4, or a major third.
Now since Ix 4=1, we see that a note 4, or one which makes 9 vibrations while the fundamental note C makes 8, will be a fourth below G, and 1 divides C and E well, the three notes 1,1, giving the intervals 1, 1?, already found in another part of the scale. This _note is D. Again, observe the interval from E to F, or +2, and take a fifth above E, or x 4 or y : this fraction falls between n and 2, and looking at the intervals of and 2, we find 4 and +2, both of them intervals already found. This note or which makes 15 vibrations while the fundamental note makes 8, is B, and the usual scale of civilised nations, called the diatonic scale, is now complete in the following 1 4 4 4 4 2This diatonic scale seems then to be the scale of the simplest con cords of the fundamental note, with one alteration on account of the too great proximity of two concordant notes, and one interpolation on account of the too great distance of two others. If we examine all
its intervals, we shall find both repetition and variety as follows (C D standing for the interval from C to D, &e.), some new appel lations being added : • We observe hero the consonances mentioned before, two inhar monious intervals, a new species of consonance (the flat seventh) standing as it were between the more perfect consonances and the others, and new varieties of a tone, of a minor third, and of a fifth, from those already described, and flatter by the interval This interval Pi called a comma, and though the ear can distinguish a difference between the tones of two strings, one of which vibrates 81 times while the other vibrates 80. yet the difference is so slight as to produce no prejudicial effect. With regard to the comparatively harmonious character of the flat seventh, observe that is very nearly equal to I, differing only by the interval 31.
We have also the diatonic semitone, /4, which is incorrectly named, since, if beginning with 1, we repeat the interval of a semitone twice, we have 42x or which is very near to 4, sharper (that is, higher, na flatter means lower) than a major tone by the interval 4 and than a minor tone by it very nearly.
We shall presently resume the diatonic scale, but we now proceed to mention two varieties of it. It seems to have been offensive to the ears of rude nations to hear any semitones at all. If we deprive the diatonic scale of F and B, the notes which make semitones with their nearest neighbours, we have C, D, E, (1, A, C, for all the Bonn& which remain in the octave. This unfinished scale, as we should call it, is the original scale of the Chinese, Avans, Hindus, and Eastern Islands, the northern nations of Europe, &c. It is the well-known scale of the old Scotch and Irish mush: ; it is said to have been found in Wales and Cornwall, in various parts of Africa, and even in old Italian music. The Chinese, who never change, have preserved it in absolute perfec• tion, though the modern form of most ancient airs iu other countries has been relaxed. We copy the notes of a Chinese air given by Laborde :—