106 PHYSICAL SCIENCES IN ANCIENT GREECE. 



other things being equal, not as the weight, but as the square root of 

 the weight. But he is right as to the essential point, that those ratios 

 of 2 to 1, 3 to 2, and 4 to 3, are the characteristic ratios of the Oc- 

 tave, Fifth, and Fourth. In order to produce these intervals, the 

 appended weights must be, not as 12, 9, 8, and 6, but as 12 6f , 5 J, 

 and 3. 



The numerical relations of the other intervals of the musical scale, 

 as well as of the Octave, Fifth, and Fourth, were discovered by the 

 Greeks. Thus they found that the proportion in a Major Third was 5 

 to 4 ; in a Minor Third, 6 to 5 ; in a Major Tone, 9 to 8 ; in a Semi* 

 tone or Diesis, 16 to 15. They even went so far as to determine the 

 Comma, in which the interval of two notes is so small that they are in 

 the proportion of 81 to 80. This is the interval between two notes, 

 each of which may be called the Seventeenth above the key-note ; the 

 one note being obtained by ascending a Fifth four times over ; the other 

 being obtained by ascending through two Octaves and a Major Third. 

 The want of exact coincidence between these two notes is an inherent 

 arithmetical imperfection in the musical scale, of which the conse- 

 quences are very extensive. 



The numerical properties of the musical scale were worked out to a 

 very great extent by the Greeks, and many of their Treatises on this 

 subject remain to us. The principal ones are the seven authors pub- 

 lished by Meibomius. 1 These arithmetical elements of Music are to the 

 present day important and fundamental portions of the Science of 

 Harmonics. 



It may at first appear that the truth, or even the possibility of this 

 history, by referring the discovery to accident, disproves our doctrine, 

 that this, like all other fundamental discoveries, required a distinct and 

 well-pondered Idea as its condition. In this, however, as in all cases 

 of supposed accidental discoveries in science, it will be found, that it 

 was exactly the possession of such an Idea which made the accident 

 possible. 



Pythagoras, assuming the truth of the tradition, must have had an 

 exact and ready apprehension of those relations of musical sounds, 

 which are called respectively an Octave, i Fifth, and a Fourth. If he 

 had not been able to conceive distinctly this relation, and to apprehend 

 it when heard, the sounds of the anvil would have struck his ears to 

 no more purpose than they did those of the smiths themselves. He 



Antiques, dfusicce Scriptores septem, 1652. 



