MUSIC 81 a b c a |3 V Minor third. Fourth. Major third. Major third. Fourth. Minor third. Hence the two groups may be written as below : ,,. j Minor third.
- ' ] Major third.
,,( Major third. 1 '1 Minor third. 03') Fourth. Minor third. Fourth. Major third. Major third. Fourth. Minor third. Fourth. It can now be shown that the triads of each group are closely connected. Take (a), and form from it another triad, by causing its bot- tom note to ascend one octave, the other two remaining where they were. The middle will then become the bottom note, the top the mid- dle note, and the octave of the former note the top note. Hence the lower interval of the new triad will be the upper interval of the old triad, i. e., a major third. The upper interval of the new triad will necessarily be the inver- sion of the interval which separated the ex- treme notes of the old triad. This interval is a fifth (see (a) ), and its inversion by the table already given is a fourth. Hence the new triad is JMnor h third [ which is identical with (b'). If we modify (b^) in the same way, the new interval is the inversion of the minor sixth, i. e., the major third, and the resulting triad, viz., j FoTJ;h tllird [' is identical with (c'). This triad, when similarly treated, brings us back to (a'), and the cycle of changes is complete. By an extension of the word " in- version," it is usual to call the triads (b') and (c') the first and second inversions of the tri- ad (a'). Exactly similar relations hold be- tween the members of the second group of triads ; (/?') and (y') are accordingly called the first and second inversions of the triad (a). The proof is exactly like that just given, and will be easily supplied by the reader. If we choose as the bottom note of (a') and (a'), the major and minor groups will be expressed in musical notation by 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), (/3), (y). Thus the triads (a'), (b'), (c') would be indicated by the figures %, f, % respectively, and so would the triads (a'), (p'), and (y'); the differences be- tween minor and major thirds and sixths be- ing left to be indicated by the key signature. The positions (a 7 ) and (a') are regarded as the fundamental ones of each group, (b'), (c ), and (/?'), (y') being treated as derived from them respectively by inversion. The fundamental triads bear the name of their lowest notes; thus (a') and (a r ) are called respectively the major and minor common chords of C. The remaining members of each group are not named after their lowest note, but after that of their fundamental inversion ; thus (b'), (c ; ), and (p'), (y 7 ) are respectively the major and minor common chords of G in their first and second inversions. The reason of this, as far as the major group is concerned, follows di- rectly from Helmholtz's theory of consonance and dissonance. The notes of the triads (a'), v^O) ( c/ ) are all coincident with individual har- monics of a composite sound whose funda- mental tone is the low for (a') and (b'), and the octave above that note for (c') ; hence they may be regarded as form- ing a part of the composite vibration of a C sound, and therefore each triad may be ap- propriately called by its name. With the mi- nor triads this is not so completely true, be- cause the E|, in (a ; ), (/?'), and (y') is not coinci- dent with an overtone of C. The other two notes, however, are in each case leading har- monics of C, and" therefore these triads belong at any rate more to than to any other note, Common chords of more than three constitu- ent sounds can only be formed by adding to the consonant triads notes which are exact oc- taves above or below those of the triads. The bright open character of the major and the gloomy veiled effects of minor chords are at- tributed by Helmholtz to the different way in which combination tones enter in the two cases. The positions of the first order of com- bination tones, for each of the six consonant triads, are shown in crotchets in the appended stave, the primaries being indicated by minims : v & IT /L 22 22 II g -41^ ! t-^:- 4-.. 4 2 ^ ^ t- ~H~ i m
