428 REPORT OF NATIONAL MUSEUM, 1900. 



an embouchere at the middle and holes S3'^mmetiically placed on each 

 side dividing the whole length into thirds, quarters, and sixths; so, if 

 the whole length is called 12, the month hole is at 6 and the linger 

 holes at 2, 3, -1, 8, 9, and 10. Mahillon copied th(* instrmnent, but did 

 not close the ends, and reports the scale as a chromatic one from E to A #. 

 Most of the old P^uropean wood wind instruments figured by Prjeto- 

 rius* (1618) arc conspicuously of this type, as the appended Plate 5 

 shows without necessity of description, and various similar instru- 

 ments of the Museum collections are figured in Plate 6. 



IV. INSTRUMENTS OF THE RESONATOR TYPE. 



1. The next group includes a variety of instruments of the resonator 

 type, a type that is widely distributed and conforms to a law hitherto 

 unrecognized as capable of furnishing a scale; though Sondhaus in 

 1850 stated the law and tried a few rough experiments. The mathema- 

 ticians" have proved that a mass of air in a confined space with a very 

 small nearly circular opening, as a short-necked bottle or a whistle, 

 has a frequency of vibration proportional to the square root of the 

 fraction which expresses the diameter of the hole divided by the volume 

 of the cavity; and if there are two such openings so placed that the 

 flow of air through one does not interfere with that through the other, 

 the numerator of the fraction will be the sum of the two diameters. Now 

 extend the same principle, and one may have a series of sounds rising 

 in pitch as one after another of several holes in the wall is opened; 

 and provided the character of the vibration is not essentially changed, 

 the frequencj^ of vibration of these notes will increase as the square 

 root of the sum of the diameters of the holes opened. Suppose, for 

 example, that a vessel has one mouth-hole of diameter 2 and several 

 properly placed finger-holes of diameter 1; then on successively open- 

 ing these a scale may be produced having vibration frequencies in the 

 ratio of the square roots of 2, 3, -1, 5, etc. A moment's consideration will 

 show that in such a scale the intervals between successive sounds 

 become less and less as the pitch rises, instead of becoming greater as 

 is the case with strings or flutes where the spacing of frets or holes is 

 uniform. 



The most ehiborate and beautiful illustrations M instruments of this 

 t3'pe are from graves in Central and South America. (See Plate 7.) 

 The United States National Museum has many whistles from Chiriqui 

 in Colombia, most of them giving but a single high note; these difl'er 

 substantially, it will be noticed, from stopped organ pipes, since in the 

 latter the mouth extends the full width of the tube. Whistles with 

 one or two finger-holes have come from Mexico and San Salvador, but 

 the most complete and perfect are from Costa Rica. Of these the one 



'Syntagma Musicum, pis. ix and x. 



*Rayleigh, Theory of Sound, II, 1878, Chap. xvi. 



