HISTORY OF MUSICAL SCALES. 437 



theoretical principle of scale determination is certainly and con.scioush'^ 

 embodied in any instrument anywhere; but some instruments in the 

 National Museum and some drawings in books make the assumption 

 seem plausible, that the primitive type of this instrument is a series of 

 bars, supported at points about one-tifth or one-fourth of their lenoth 

 from their ends, and decreasing- in lenoth by equal linear amounts. 



It is evident that these composite instruments are of minor impor- 

 tance in this study; but in the light of the theoretical laws here 

 suggested perhaps travelers mav learn something of the intention of 

 a savage who cuts his Pandean pipes or bars to form a musical 

 instrument. 



VII. CONCLUSIONS. 



There have now been considered all the types of instruments in 

 which several notes of different pitch are produced from the same 

 vibrating bod}" — whether string, column of air, or mass of air. 



(1^/.) There have been found examples from various parts of the world 

 of the intentional location of the stopping points of a vibrating string at 

 equal linear distances; since with all stringed instruments the finger- 

 ing- will cause a slight increase of tension, the ec|uivalent length of the 

 string is less than the actual length. 



(Ih.) There have been found numerous examples of wind instru- 

 ments pierced with holes in one or two groups spaced at equal linear 

 distances; since these holes are never sufficienth' large to allow the air 

 to flow through them with perfect freedom (unless in some Chinese 

 flutes) the equivalent length of the vibrating- column of air is greater 

 than the actual distance from the mouthpiece to the hole. 



(2.) There have been found instruments of the Marimba type with 

 bars of regularly decreasing lengths. 



(3.) There have been found many forms of instrmnents of the resona- 

 tor type embodj'ing a. series of equal and similarly-located holes; in 

 these, thickening the wall is equivalent acoustically to making the holes 

 smaller; while locating the hole nearer the point where the vibrating 

 air has its maximum change of density is equivalent to enlarging the 

 hole. 



Three simple laws give to the first approximation the scales of these 

 several instruments, namely: 



(1) The law of inverse lengths. 

 {'2) The law of inverse squares of lengths. 



(3) The law of the square roots of a series of numbers propor- 

 tional to .sums of diameters. 



The first and sec-ond laws give scales whose intervals increase as 

 the pitch rises; scales based on the third law have decreasing inter- 

 vals. Some results are shown in a table in the ai)pendix and graphi- 

 callv in Plate 10. 



