| Ratio of Beat-interval | Value in Seconds of Average Interval, | |||
|---|---|---|---|---|
| to Group-interval. | 5/12 | 3/12 | 2/12 | |
| 1.000: 1.400 | 3.5 | 5.3 | 9.0 | |
| 1.000: 1.000 | 4.0 | 5.4 | 9.6 | |
| 1.000: 0.714 | 5.2 | 8.4 | 10.8 | |
The quantitative relations presented by these figures are consistent throughout. For every rate of speed the average rhythmic group is smallest when the interval separating the successive groups is at its maximum; it is largest when this interval is at its minimum; while in each case a median value is presented by the relation of uniformity among the intervals. In the second as well as the first of the ratios included in the foregoing table the interval which separates adjacent groups is felt to be distinctly longer than that internal to the group; in the third the relative durations of the two intervals are those which support psychological uniformity. In the latter case, in consequence of the freer passage from group to group, the continuity of the rhythmical series is more perfectly preserved than in the former, and the integration of its elements into higher syntheses more extended.
The extension of the numerical limits of the rhythm group in subjective rhythm which appear in consequence of progressive acceleration in the rate of succession is given for a series of six different values of the separating intervals in the following table, the figures of which represent the average for six observers:
TABLE V.
| HIGHEST UNITS WHICH APPEAR. | ||||||
|---|---|---|---|---|---|---|
| Value of interval in secs.: | 12/12 | 7/12 | 5/12 | 3/12 | 2/12 | 1/12 |
| No. of el's in rhythm group: | 2.5 | 3.0 | 4.0 | 7.0 | 9.0 | 11.0 |
| Average duration of group: | 2.500 | 1.750 | 1.666 | 1.750 | 1.500 | 0.917 |
| SIMPLE UNITS. | ||||||
| No. of els. in simplest group: | 2.5 | 2.3 | 2.9 | 3.7 | 4.7 | 5.0 |
| Duration of simplest group: | 2.50 | 1.34 | 1.21 | 0.92 | 0.78 | 0.41 |
The rate of increase here presented in the number of elements is not sufficiently rapid to counterbalance the acceleration of speed and maintain a constancy in the duration of the group. The greatest value of this period is coördinated with the slowest rate of succession, the lowest with the most rapid. As the speed increases, the duration of the rhythmic unit is shortened. Its average duration for all rates here included is 1.680 sec., or, without the first of the series (one-second intervals, at which only two of the observers received the impression of rhythm), 1.516 sec. These values are not for the simplest combinations, but for the highest synthetical unit which was immediately apprehended in the series of stimulations. This compounding becomes more pronounced as the rate of succession is accelerated, but even at intervals of 5/12 and 7/12 sec. it is the characteristic mode of apprehension.
The number of elements in the simple groups of which these higher units are composed, and their average duration, are also given in the table. These likewise show a progressive increase in number, but of a much slower rate than that manifested by the total synthesis of elements. That is to say, in subjective rhythm as well as in objectively figured series, subordinate rhythmical differences in the material sink out of consciousness less rapidly than the inclusion of fresh elements takes place; in other words, the organic complexity of the rhythmic unit increases with every acceleration in the rate of succession. The duration of these simple structural groups, as may be inferred, decreases with such acceleration, but at a much more rapid rate than is the case with the total reach of rhythmical apprehension, the value of that unit which appears in connection with the highest speed here included being less than half a second. The 'liveliness' of such rapid measures is thus a resultant of several factors. It is not a consequence solely of the more rapid rate at which the individual stimuli succeed one another, but depends also on the shortening of the periods of both these rhythmical units and on the progressive divergence of the simple from the complex group.
The influence of the rate of succession on the rhythmical unit is not confined to its segregation from adjacent groups, but affects the internal configuration of the measure as well. With every acceleration in rate the relative preponderance of the interval following the accented element (in rhythms having initial stress) increases; as the rate is retarded, smaller and smaller degrees of difference in the values of accented and unaccented intervals are discriminated. In this regard the influence of reduction in the absolute value of the separating intervals is analogous to that of increased accentuation within the group. In fast tempos and with high degrees of emphasis the interval following the initial accent is relatively longer, that following the unaccented relatively shorter, than at slow tempos and with weak emphasis. This is but another way of expressing the fact that as the elements of the auditory series succeed one another more and more slowly the impression of rhythm fades out and that as their succession increases in rapidity the impression becomes more and more pronounced. The following table presents these relations in a quantitative form for trochaic rhythm. The figures represent the number of times the second, or group interval, was judged to be greater than, equal to, or less than the first or internal interval of the group. Three rates were compared together, having average intervals of 5/12, 3/12 and 2/12 sec. Six observers took part, but only a small number of judgments was made by each, to which fact is probably to be attributed the irregularities of form which appear in the various curves:
TABLE VI.
| Ratio of 1st to 2d Interval. | 5/12 | 3/12 | 2/12 | ||||||
|---|---|---|---|---|---|---|---|---|---|
| + | = | - | + | = | - | + | = | - | |
| 1.000: 1.057 | 95.0 | 0.0 | 5.0 | 100.0 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 |
| 1.000: 1.000 | 94.7 | 5.3 | 0.0 | 86.0 | 10.5 | 3.5 | 87.5 | 12.5 | 0.0 |
| 1.000: 0.895 | 40.0 | 60.0 | 0.0 | 46.2 | 49.6 | 3.3 | 74.1 | 18.5 | 7.4 |
| 1.000: 0.846 | 41.0 | 50.0 | 9.0 | 39.4 | 54.6 | 6.0 | 40.0 | 52.0 | 8.0 |
| 1.000: 0.800 | 20.0 | 60.0 | 20.0 | 13.0 | 70.0 | 17.0 | 53.8 | 46.2 | 0.0 |
| 1.000: 0.756 | 29.4 | 23.5 | 47.1 | 21.8 | 43.4 | 34.8 | 28.0 | 72.0 | 0.0 |
| Av. for all ratios, | 53.3 | 33.1 | 13.5 | 51.1 | 38.0 | 10.8 | 63.9 | 33.5 | 2.6 |