Indian Antiquary Vol 41

Author(s): Richard Carnac Temple, Devadatta Ramkrishna Bhandarkar
Publisher: Swati Publications

Page 263

NOVEMBER, 1912.]
ANCIENT HINDU MUSIC
259
and a seritone respec-ively, which is only an approximation and not accurate, for, the exact ratios are 1.8214 ...: 1.625 : 1, and not 2:2: 1. The approximation may be justified thus : 1.8214 ... is nearly 2; and 1.8214: 1.625=1.1223: 1, i. e., 1:] nearly. But there is another way also of looking at these ratios : 1.8214: 1= 2:1 approximately, as before ; but 1.625 : 1=1.5: 1, more nearly than 2:1; in other words, the three intervals are in the ratio 4: 3: 2 approximately. It is this approximation which has been used by Sanskrit writers. It will be seen that the two approximations agree as regards the ratio between a major tone and 4 semitone; and if the Enropean approximation is more accurate as regards the ratio of a major to a minor tone, the Hindu approximation has the advantage of greater accuracy in the ratio of a minor tone to a semitone. The latter possesses the further advantage of indiciting that there are three kinds of intervals, whereas the former reduces these to only two. It was probably owing to this European custom of calling the inajor tone, the minor tone, and the diatonic semitone by the terms a tone, a tone, and a semitone that Mr. Hipkins overlooked the possibility of the Hindu' approximation being quite as good, if not better, for the purpose of expressing the actual ratios, and was led to misinterpret the intervals of the Hinda scale.
Having thus determined the valves of the intervals in the Hindu scale, it will be interesting to consider now the converse problem of what cycles can possibly be employed to express the same. The conditions of the problem evidently are:
(1) There must be three kinds of intervals.
(2) The octave to consist of three intervals of the greatest magnitude and two of each of the others.
(3) Intogers only to be used in expressing the intervals.
It is easily seen that no cycle of less than 15 degrees can satisfy all these conditions. The cycle of 53 with the three intervals in the ratio of 9: 8: 5 can express the scale with remarkable accuracy. If we now examine all possible cycles cousisting of 15 to 53 degrees, which satisfy these conditions, only the following ones make an approach to the scale for which we wish to find an expression :
Table C. Degrees in
Latios of the thro Degrees in the Cents in the Degrees in Conts in the oyale. intervale.
Major Third. the Fifth, the Fifth 22 4: 3:2
382
13 709 5:4:8
872
704 32 6:4:3
875
712.5 6:5: 8
20 706 7: 6:4
880-1 24. 702 8: 7: 4
391
704 9:8:5
385
702 Scale under consideration 1.8214 . : 1.625:1 ...
386
702 Thus the cyclo of 22 is the smallest that can be used for expressing the given scale; that of 29 gives the fifth more accurately, but the third is much worse; that of 32 is decidedly worse; the rest are all better, that of 53 being the best. We thus seo that assuming the value of the scale, which we have found from other consideration, to be correct, it could not have been better expressed than by means of a cycle of 22, unless the ancient Hindu writers had resorted to 34 degreesor more. This consideration, therefore, gives farther indirect support to the value we have assigned to the scale. Why cycles of 81 degrees or more were not used so as to secure a greater accuracy will be discussed presently; but we must first consider an apparently formidable objection. In the section " On the svaras and árutis " it has been mentioned that, according to Bharata, in order to convert the shadjagrama into the madhyamagrama, tbe pañchama must be lowered.
29
388