The results presented in the preceding section form the statement of but one half the evidence of higher rhythmical synthesis afforded by the material of the present investigation.  We turn now to the second set of results.  It deals, in general, with the quantitative relations of rhythmic forms which find expression through finger reactions.  Portions of this evidence have already been presented, through motives of economy, in connection with the discussion of the phases of differentiation in intensity and duration which such beaten rhythms manifest.  The burden of it, however, is contained in the results of an analysis, form by form, of the proportional mean variations which characterize these types of rhythmic expression.  This method has been applied to a study (a) of the characters of the constituent intervals of the unit, in their relation to accentuation and position; (b) of the simple group which these elements compose; and (c) of the forms of higher synthesis manifested by the variations in successive groups.  The first of these relations concerns, indeed, only the internal organization of the simple group, and has no direct bearing on the combination of such groups in higher syntheses; but, again for the sake of economy, the items are included with the rest of the material.

The application of such a method, as in all treatment of material by mean variations, involves much labor,[12] and on that account alone the lack of its employment to any considerable extent in previous investigations may be excused; but to this method, as it seems to me, must the final appeal be made, as an indisputable means by which all questions concerning the refined features of rhythmical organization, the definition of units and the determination of the forms in which they enter into larger rhythmic quantities, are to be settled.

[12] In connection with this work some 48,000 individual measurements were made (for the transcription of which I am indebted to the patient assistance of my wife).  Half of these were measurements of the intensity of the successive reactions; the other half, of the intervals which separated them.  The former series has been employed in obtaining the averages which appear in the section on the distribution of intensities; the latter in that on the distribution of durations.  The determination of mean variations was made in connection with the second series only (24,000).  These quantities were combined in series of single groups, and in series of two, four, eight and ten groups, and for each of these groupings severally the mean variation of the series was computed.

Of all the possible forms of rhythmic apprehension or expression, the material for such a statistical inquiry is most readily obtainable in the form of a series of finger reactions, and to such material the application of the method in the present investigation has been restricted.