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X.§§108,109.] REQUISITES FOR PURE INTONATION. 203
Reducing these to one Octave, as before, we find seven notes not included in the previous list, occupying the following positions :—
108. Hence, to play perfectly in tune in both Major and Minor modes of the seven keys C, D, E, F, G, A, B, it is necessary to have a key-board with twenty-Jive notes in every Octave. This number, large as it is, by no means includes all necessary notes. Modern music is written in sharp and fiat keys, i.e. in such whose tonics are not coincident with any one of the notes CDE...B. Moreover, the sharp and flat key-notes are different from each
sharper note. As the seven keys which have been already examined require 25 notes in the Octave, we may anticipate that the ten additional sharp and flat keys will bring in a still larger number.
109. It is, however, needless to institute a detailed inquiry into these scales, as the facts already established amply suffice to show how serious are the imperfections of tune which inevitably beset instruments of fixed sounds, such as the pianoforte, harmonium and organ, containing only twelve notes in