combinations of dissonance, rests, sounding neither scale nor chords. Dissonance does not express opposition or separation, for there is no principle in musical tones which is productive of contraries; the dissonances follow the attraction of the tonic, or key-note, and the neutralization of the musical disturbance is implied in the disagreement in their motion with the repose of the unit, or key-note. So far is this from producing separation, that the apparent discord is simply a preparation for growth, the life of harmony causing an inherent tendency towards closer union.

We here trace the twelve harmonies developing in succession. Notice how exactly they all agree in their mode of development; also the use of the chasms between E and F, B and C. Remark also the beautiful results from the working of the double tones, especially C#-D?, and E#-F?, causing the seven tones of each harmony, when ascending, to rise one tone, and, descending, to reverse this movement. F#-G? is the only double tone which acts as F# when a key-tone, and G? when the root of D?. The root of each harmony is the sixth and highest tone in each succeeding harmony, rising one octave; when it is a double tone, it sounds according to the necessity of the harmony. The intermediate tones are here coloured, showing gradual modulation. The isolated fourths (sounding sevenths) were the previously developed key-tones; these also alter when they are double tones, according to the necessity of the harmony. Beginning with B, the isolated fourth in the harmony of C, the tones sound the twelve notes of a keyed instrument, E# being F?, and the double tones, some flats, some sharps.

Examine C# in musical clef as an example of double tones only developing each one major harmony. C# sounds neither B nor E, but C and C#, F and F#.

The only exception is the double tone F#-G?, which is a curious study. F# as a harmony takes the double tones as sharps, and F? is E#. G? is also a harmony sounding the same tones, by taking the double tones as flats, and B? as C?. F# therefore takes the imperfect tone of E#, and G? the imperfect tone of C?. (See here the harmony of G? in musical clef.)

In the same way are written the two last primaries of a series of twelve, which began with C. A higher series of twelve follows, and the first two primaries of a still higher series of twelve. The secondaries are written below the primaries.

We find that on a keyed instrument each primary sounds the same tones as the secondaries of each third harmony below, but in a different order, and the double tones are altered sharp or flat as the harmony requires. For example, the secondaries of B are sharps; when primaries of D?, they are flats. In order to trace this quickly, the sharps and flats are written to each note.

In any series of twelve, the primaries of the two first key-notes repeat the secondaries of the two last of a lower series of twelve; and the two last secondaries of the twelve in development are sounded as the two first primaries of a higher series of twelve. The three series are thus linked into each other.