or the Division of the Octave. 



509 



Name of interval in 

 general case. 



2-fifths tone 



5-fifths semitone. . . . 

 7-fifths semitone. . . . 

 Third by 4 fifths up. 



Third by 8 fifths 

 down. 



Name when fifths are 

 perfect. 



Major tone 



Pythygorean semitone. 



Apotome Pythagorica. . 



Dissonant, or Pythago- 

 rean third 



(Approximately perfect 

 third.) 



Series of intervals. 



c-d-e- . . . 

 /c-/c#-d-eb-\e. . . 

 \c-c#-d-/et?. . . 



c-e 



c 



/cfr-/ c . . . 

 \e-\ob-\\c... 



The last example shows us that three perfect thirds fall short 

 of the octave by two commas approximately. (The departure 

 of 12 perfect fifths is a Pythagorean comma = "23460 of anE.T. 

 semitone. The real comma is '21506. These are regarded as 

 approximately equal for present purposes.) 



Regular Systems. 



The importance of regular systems arises from the symmetry 

 of the scales which they form. 



Theorem 1. In any regular system five 7-fifths semitones + 

 seven 5-fifths semitones make an exact octave ; for the depar- 

 tures (from E.T.) of the first set are due to 35 fifths up, and 

 those of the second set to 35 fifths down, leaving twelve E. T. 

 semitones, which form an exact octave. 



Theorem 2. In any regular system the difference between the 

 7-fifths semitone and the 5-fifths semitone = departure of 1 2 

 fifths, having regard to sign; or 



7-fifths semitone — 5-fifths semitone = departure of 12 fifths; 

 for the 7-fifths semitone up from c is got by the fifths, 



c-g-d-a-b-/f$-/c$ ; 

 and the 5-fifths semitone down from /c% is got by the fifths, 



/-c%-/g%-/&%-/a%-/f-yc ; 

 whence the difference is the interval c—/c ; and this is the de- 

 parture of 12 fifths by definition. 



N.B. Here w T e extend our notation to fifths of any kind, nega- 

 tive as well as positive. 



Regular Cyclical Systems. 



The importance of regular cyclical systems arises from the in- 

 finite freedom of modulation in every direction, which is possible 

 in such systems when properly arranged ; whereas in non- 

 cyclical systems required modulations are liable to be impos- 

 sible, owing to the demand arising for notes outside the mate- 

 rial provided. 



Theorem 3. In regular cyclical systems of the rth order the 

 difference between the 7-fifths semitone and the 5-fifths semi- 

 tone is r units of the system, having regard to sign. 



