Taking the fifth as the basis, it will be evident that if any finger be placed horizontally across two strings, in any part of the instrument, a perfect fifth will be produced. To accomplish this the student will have to depart from the rule respecting the stopping of notes with the tips of the fingers, as it is impossible to produce a fifth except with "flat" fingering, or of course the open strings.

From this it will be an easy matter to settle the fingering of a major sixth; in the first position, a major sixth may be produced by using a lower open string, and the first finger on a higher string, thus in any position the fingering of major sixths is obtained on the same principle, viz., the higher note is fingered a whole tone in advance of the lower.

The same method may be taken to obtain the fingering of a perfect fourth. In the first position, using the open string, a fourth in double-stops is produced with the open string for the higher note, and the first finger on the next lower string for the lower note; here the lower note is fingered as far as position is concerned a whole tone in advance of the higher note. Major thirds in the neck positions, have to be played with the "set" fingering of fourth and first; using the fourth finger for the lower note and the first on the next higher string for the higher note.

A little time expended in the study of Ex. 39 will clearly illustrate the preceding remarks, and at the same time will prove conclusively, that if double-stops are only treated according to their harmonic worth, that is as perfect fifths, fourths, major and minor sixths, thirds, etc., the difficulties of extreme keys, and positions, will in a great measure be cleared away.

Thus in any position the relation of two notes (providing the harmonic value is the same) remains undisturbed, the only difference in the fingering being in the lower positions where the open strings are utilized, and the only modification being in the very high positions, where the gradually decreasing distances of the intervals render the use of the fourth finger unnecessary.

Before attempting anything great in the matter of double-stopping, the student should practice various scales in sixths, thirds, etc. (both notes being sounded together) and also any possible arpeggi in double-stopping, after the manner of Ex. 40.

The student will not find any published arrangement of arpeggi in this form, but he may easily construct them for himself, using only the tonic, mediant, and dominant of the key. It will be found that a little practice in this direction will amply repay, as to accomplish the playing of arpeggi in double-stopping without scrambling and with the intervals correctly in tune, means a command of the positions truly remarkable.

In reading works composed chiefly of double-stops, it is at first rather difficult to decide in which position any passage should be played. The progression of both "melodies" has to be watched, as well as the intervals which separate any given two part chord, the fingering is also affected by the preceding and the following chord, as well as by the introduction of passing notes, or such ornamentation as trills, turns, etc., in either of the parts.