118 NOTES TO BOOK II. 



the appended weights must be, not as 12, 9, 8, and 6, 

 but as 12, 6|, 51, 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 Semitone or 

 Diesis 16 to 15. They even went so far as to deter- 

 mine 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 coincidence between 

 these two notes is an inherent arithmetical imperfection 

 in the musical scale, of which the consequences 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 published by 

 Meibomius*. These arithmetical elements of Music are 

 to the present day important and fundamental portions 

 of the Science of Harmonics. 



* Antiques Musicce Scriptores septem, 1652. 



