Common-chords of more than three constituent sounds can only be formed by adding to the consonant triads notes which are exact octaves above or below those of the triads.

104. The marked distinction existing, for every musical ear, between the bright open character of major, and the gloomy veiled effect of minor chords, is attributed by Helmholtz to the different way in which combination-tones enter in the two cases. The positions of the first-order combination-tones, for each of the six consonant triads, are shown in crotchets in the appended stave, the primaries being indicated by minims. Each interval gives rise to its own combination-tone, but, in the cases of the fundamental position and second inversion of the C-Major triad, two combination-tones happen to coincide. The reader will at once notice that in the major group no note extraneous to the harmony is brought in by the combination-tones. In the minor group this is no longer the case. The fundamental position, and the first inversion, of the triad, both bring in an Ab, which is foreign to the harmony, and the second inversion involves an additional extraneous note, Bb. The position of these adventitious sounds is not such as to produce dissonance, for which they are too far apart from each other and from the notes of the triad; but they cloud the transparency of the har

mony, and so give rise to the effects characteristic of the minor mode.

The unsatisfying character of Minor, compared with Major, triads, comes out with peculiar distinctness on the harmonium; as indeed, from the powerful combination-tones of that instrument, we should naturally have anticipated.

CHAPTER X.

ON PURE INTONATION AND TEMPERAMENT.

105. The vibration-fractions of the intervals formed by the successive notes of the Major scale with the tonic, are, including the octave of the tonic, these:

§, 4, 1, 2, %, §, V,
15 2.

The intervals between successive notes of the scale are determined by dividing each of these fractions by that which precedes it. Thus the consecutive intervals of the Major scale come out as follows:

C D E F G A B

C"

Only three different intervals are obtained. g is slightly wider than ; 1 decidedly narrower than the other two. and are called whole tones, 1a half-tone or semi-tone, though, strictly speaking, two intervals of this width added together somewhat exceed the greater of the two whole tones; since 8×18 or 58 is to g in the ratio of 2048 to 2025.

Suppose we had a keyed instrument containing a number of octaves, each divided into seven notes, forming the ordinary scale as above any music could be played on it which did not introduce notes foreign to the key of C Major. But now, suppose we wanted to be able to play in another Major key as well as in that of C, for instance G. It would be necessary for this purpose to introduce two new notes in every octave of the key-board. If G is the new tonic, A will not serve as the second of its scale, because the step between tonic and second is, not, but . Hence we must have a fresh note lying between A and B. Further, F will not do for the seventh of the scale of G, as it is separated from G by g, instead of 1g. This necessitates a second additional note lying between F and G. If we take, as our original octave, that from middle-C upwards, we have the following vibration-numbers:

C D E F G A B C" 264 297 330 352 396 440

495

528

The new notes, being respectively above, and 18 below G, have for their vibration-numbers & × 396 and 15 × 396, i.e. 445 and 3714. The other notes of the scale of G Major can be supplied from that of C Major. Hence these two scales are closely connected with each other. Another key nearly related

to the key of C is that of F. Its Fourth is x 352, or 4693, which falls between A and B. Its Major Sixth is × 352, or 586, which is clearly not an exact octave of any note between C and C. The corresponding note in our octave, found by. division by 2, is 2933, which comes between C and D. Thus, two more new notes in the octave must be introduced, to make the key of F major attainable.

seven

106. In order that the reader may see, at a glance, the variety of sounds which are requisite to supply complete Major scales for the keys of C, D, E, F, G, A and B, the vibrationnumbers for all the notes of these scales are calculated out and exhibited in the following table.

Reducing those notes which lie beyond the