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.

ON PURE INTONATION AND TEMPERAMENT.

105. 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:

કું, '

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:

Only three different intervals are obtained. is slightly wider than ; 1 decidedly narrower than the other two. and 10 are called whole Tones, 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 18 or 8 is to in the ratio of 8 × 2048 to 2025.

225

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 g. 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 264 297 330 352 396 440

495

C'

528

above and

The new notes, being respectively 18 below G, have for their vibration-numbers & × 396 and 1 × 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 the
exact Octave of any note between C and C'. The
corresponding note in that Octave, found by division
by 2, is 293, 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.

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 seven keys
of C, D, E, F, G, A and B, the vibration-numbers
for all the notes of these scales are calculated out
and exhibited in the following table.

Reducing within the compass of an Octave
those notes which lie beyond it, by dividing their
vibration-numbers by 2, and arranging the results

in order of magnitude, we have eleven notes foreign to the scale of C Major, the positions of which, with reference to the notes of that scale, are as follows:

C, 275, 278, 2931, D, 3093, E, 3341, F, 366, 3714, G, 412, A, 445, 464, 469, B.

107. If it is desired to be able to play in the several Minor modes of each of the seven keys, as well as in the Major, additional notes will be called for. Each scale must contain four notes making with the Tonic the intervals Minor Second, Minor Third, Minor Sixth, and Minor Seventh respectively. The following subsidiary table exhibits the vibrationnumbers of the sounds forming these intervals with the successive key-notes.