Text: The division of the octave. The problem is, to divide the octave into a number of intervals such, that the notes which separate them shall be suitable in number and arrangement for the purposes of practical harmony. The simplest form of temperament is that in common use, which divides the octave into twelve equal semitones. It is most convenient to express all intervals in terms ofthese semitones. The perfect fifth contains 7.019550008654 semitones; the perfect third 3.863137138649 semitones. Five places are enough for all practical purposes. In all harmonious music the fifth to any given note may re required at any time. Hence all systems provide series of fifths of a more or less complete character. If a series of perfect fifths be constructed starting from c, octaves being disregarded, it will not return exactly to c again. Taking the fifth to be 7.01955 semitones, each note in the series lies futher than the last from the equal temperament note of the same name; and this departure increases by .01955 for eachstep in the series, the equal temperament fifth being seven semitones. Thus teh c to which we return after twelve fifths is higher than that from which we started by 12 x .01955 = .23460 of a semitone. This interval is called the comma of Pythagoras. Regular systems consist of notes which form a continuous series of fifths, e.g., the system of perfect fifths. Regular cyclical systems consist of notes which form a continuous series of fifths, and divide the octave into a certain number of equal intervals. Primary regular cyclical sustems are those in which the departure of twelve of the approximate fifths of the system from the starting point is equal to one unit ofthe system. Secondary systems are those in which the departure of twleve fifths from the starting point is two units, and so on. Positive systems have fifths sharper than equal temperament fifths; negative systems have fifths flatter than equal temperament fifths. The division of the octave into 53 equal intervals furnishes an important primary positive system; into 118, a secondary positive system; into 31, a primary negative system; into 50, a secondary negative system.

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