these two signs, b and B, which had already come to be written b and #.
The key system of modern European music rose out of this conception of the displacement of notes of the diatonic order. Since each one of the new notes introduced into the previous scale A to G was looked upon and named as a derivative from one of its notes, each of the new embodiments of the diatonic order which it became possible to form with their aid carried with it a reference to this fundamental series, and became therefore a key.
It proves that in the series of alternate groups of two and three tones separated by hemitones, which constitutes the diatonic scale, there is a certain note (viz. that below the group of three tones, or F), and only one, whose hemitone displacement upward will leave the order of the diatonic form; and one, and only one (viz. that above the group of three, or B), whose displacement downward will have the same result; further, that the diatonic character of the order will be left unchanged when a certain pair, a certain group of three, etc., up to a certain group of six, are all displaced upward, and when a certain other pair, group of three, etc., up to a group of six are all displaced downward; and that no other hemitone displacements will result in the diatonic order except that of all seven notes up and that of all seven notes down. The number of possible different embodiments of the diatonic scale which can be obtained from a given one by such hemitonic changes in its notes is therefore fourteen: those in which a certain one, two, etc., up to all seven of its notes, are moved upward a hemitone, or, as we say, sharped, and those in which a certain one, two, etc., up to all seven of its notes, are moved downward a hemitone, or, as we say, flatted.[1]
- ↑ The only hemitone displacement upward of a single step in the interval order
THTTHTT
ABCDEFGAwhich will result in the same sequence of tones and hemitones is that of F. The F of the new order is the C of the original one, and its displacement creates a second derivative formed from the original by two displacements upward, F and C. The displacement upward of C is impossible without that of F if the derivative order is to remain diatonic, for it results in the separation of two hemitones by a single tone only. The F of the second derivative is the G of the original, and its displacement gives a third derivative produced from the first by three displacements, F, C, G. The displacement of G without that of C is impossible for
