440 REPOBT OF NATIONAL MUSEUM, 1900. 



theoretical scales, diatonic and nondiatonic, are doubtless its direct 

 descendants, though at present it is not known what the influence was 

 that so transformed them and made them depend on ratios, not on dif- 

 ference of lengths. Possibly the theory of numbers bewitched musi- 

 cians thois as it has sometimes since, though the converse speculation 

 is a plausible one — that the recognized musical ratios gave a mj^stical 

 meaning to numbers. It is curious to note that Aristoxenus had some- 

 how got far enough to compUin that flutes distort most of the inter- 

 vals (p. 42, Mb.), and if his lost treatise on boring flutes should be 

 found it might throw light on this history. The Arab ''stepbj^step" 

 method is apparently a late descendant of the equal linear divisions, 

 appearing after men had learned to recognize the equalitj' of intervals 

 as well as of spaces. But the Chinese cjxle of fifths must be explained 

 and determined on entirely different physical principles, and the vari- 

 ous European scales as defined bv theorists or rendered In' the best 

 violinists or fixed by good tuners, when properly examined, reveal 

 elements as diverse as the elements of our language or our population. 

 The principle in question is therefore presented only as the simplest, 

 earliest, and most primitive principle of scale-building. 



