The formation of keys by the unrestricted use of hemitone displacements of the notes of the ancient diatonic order A to G resulted, therefore, in a system of fourteen. To these the ancient scale itself became added as a fifteenth key, illustrating the limiting case in which the displacements were zero in number.
The nomenclature of this system of modulation is in the first place individual. We speak of the keys of one flat or five sharps, implying that the other notes of the order are neither flatted nor sharped, and thus asserting a relation of hemitone
the same reason as before. A fourth, fifth, sixth, and seventh derivatives result from the displacement respectively of D, A, E, and B, the displacement of each being impossible without that of the foregoing note of the series. All the notes of the scale have now been displaced. In like manner, the displacement downward of one and only one note (B) of the diatonic order results in the same sequence of tones and hemitones. The B of this first derivative is the E of the original order, the displacement of which gives a second derivative; as before, the displacement downward of E is impossible without that of B. The third to the seventh derivatives are formed by the additional displacements downward of A, D, G, C, F, respectively, that of each demanding that of the previous note of the series. The formation of these fourteen derivative orders is given in the following scheme:
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No other sets of displacements than the fourteen here specified will reproduce the diatonic order. For consider any such set consisting of n displacements: either it contains the nth letter from the right in one or other of these two series; in which case it will not form a diatonic order unless the other n-1 letters are those to the right of this in the series in question; or it contains one of the letters to the left of this nth letter, and, lacking this, does not therefore form a diatonic order.
The determination of the fundamental scale as Pythagorean, Harmonic, or Tempered, gives three varieties of this system of keys into whose discussion we need not here enter.
