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232                    HAND-BOOK OF ACOUSTICS.
key will be of different pitches to those in the original key C. We have supposed here, that the music passes gradually through the keys of G, D and A; of course, if the change be a sudden one from C to E, the case would be somewhat different; The E of the central column would become the d of key E, the A would become its f, the B its s; only five new tones being therefore required for this key. It may be also noticed, that the E and B of the centre
column are not of exactly the same pitch as theof last column
on the right, which are derived by transition through the inter­mediate keys ; the latter being one comma higher than the former. Similarly each transition to the left of the central key requires
Thus to perform music, which modulates through the major keys of fig. 89, in the major mode only, requires a very large number of tones to the Octave. If to this, the minor mode be also added, a still larger number is necessary. Moreover, there are many more keys than those of fig. 89 used in modern music, so that the student will readily perceive that the number of tones to the Octave, thus required in modern music, is very large indeed.
All this presents no difficulty in the case of the voice, which is capable of producing tones of every possible gradation of pitch within its compass, and which, governed by the ear, readily forms the tones necessary to perfect harmony. Nor does it present any real difficulty in the violin class of instruments, which also may be made to emit tones of every gradation of pitch within their compass. The real difficulty is met with in such instruments as the Organ, Harmonium, Piano, &c, which have fixed tones, and consequently only possess a certain limited number of notes. On these instru­ments, which have but few notes to the Octave (generally only twelve), it is obviously impossible to execute music written in various keys and modes, in true intonation. The only thing that can be done is so to tune the fixed notes of the instrument, that the imperfections shall be as small as possible. The problem therefore is:—how so to tune an instrument, with but twelve tones to the Octave, as to be able to play in various keys and modes, with the smallest amount of imperfection. Any system of tuning by which this is brought about, is called a Temperament, and the false intervals thus obtained are termed tempered intervals.