They may also be defined in the language of Thorough Bass, which refers every chord to its lowest note in accordance with the mode adopted in (a), (b), (c); (a), (B), (y). Thus the triads (a'), (b′), (c') would be indicated by the figures ğ, §, & respectively, and so would the triads (a') (B′) (y); the differences between Minor and Major Thirds and Sixths being left to be indicated by the key-signature.

5 6 6

The positions (a') and (a') are regarded as the fundamental ones of each group; (b′), (c′) and (B′), (y') being treated as derived from them by successive inversion.

103. The fundamental triads bear the name of their lowest notes, thus (a') and (a') are called respectively the Major and Minor common chords of C.

The remaining members of each group are not named after their own lowest note, but after that which was lowest in the fundamental position of these chords; thus (b'), (c') and (B'), (y) are respectively Major and Minor common chords of C in their first and second inversions.

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 connected by Helmholtz with the different ways 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, B. 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 harmony, and so give rise to effects characteristic of the Minor mode.

The imperfect nature of Minor compared with Major triads comes out with peculiar distinctness on the harmonium; as indeed, from the powerful combination-tones of that instrument (§ 90) was to be anticipated.

CHAPTER X.

105.

ON PURE INTONATION AND TEMPERAMENT.

The vibration-fractions of the intervals formed by the notes of the Major scale with the tonic are, including the Octave of the tonic, these :8, 4, 3, 2, 5, 5, 2.

The intervals between the successive notes of the scale are determined by dividing each of these fractions by that which precedes it (§ 96). 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 10; 1 decidedly narrower than the other two. and 10 are called whole Tones, 1 a 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 1 × 18 or 258 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, so that any Music could be played on it not involving 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 in 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 10 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 instead of 18. This necessitates a second additional note lying between F and G. If we take as our original Octave that from middle Cupwards, we have the following vibration-numbers : C D E F G A B C'

264 297 330 352 396

440

The new notes, being respectively

495 528

above and 18 below G, have for their vibration-numbers & × 396 and 1 × 396, i.e. 445 and 3711. 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 × 352, or 4693, which falls between A and B. Its Major