4 Phillips, Hmmonic Tetrachords of Claudius Ptolemy. 



canon, the ratio will be sufficiently exact. Let this inter- 

 cept be subdivided into six equal parts ; multiplying by 

 6 to reduce it to sub-units we have : — 



438, 439, 440, 441, 442, 443, 444. 



These being equal units of length, the six intervals 

 decrease slightly as the string is lengthened ; the first 

 being larger than the mean, the last smaller ; on the 

 whole the step 440, 441 will be extremely near to the 

 one-sixth of the Pythagorean comma. 



Now as 440 is a multiple of 8, we can denote the 

 ratio 8:9 as 440:495; and, cutting off 440:441, the 

 remainder, 441 :49s, is the equal tempered ratio required. 

 It reduces to 49 : 55 on division by 9. I shall show that 

 the other of Claudius's harmonic tetrachords enables us to 

 arrive at these very numbers in another way. 



The Harmonic Tetrachords of Claudius Ptolemy. 



In his first book, Claudius mentions and comments 

 on several systems of older writers, in the diatonic, 

 chromatic, and enharmonic (otherwise harmonic) genera ; 

 the smallest subdivision of any being the ratio 40 : 39, 

 which there is good reason to believe is somewhat too 

 small for a melodic step. The smallest intervals so used 

 were probably about a quarter of a tone, say 36 : 35 ; 

 anything smaller being only added on to or subtracted 

 from other intervals, not taken alone. Thus were formed 

 the " three quarter " and *' five quarter " tones mentioned 

 by Aristides Quintilianus ; which still exist in Arabic 

 music. Hence such a ratio as 46 : 45 cannot represent a 

 melodic interval, but must be a something to augment 

 or diminish some melodic step of considerably greater 

 magnitude. Much more must this be the ease with the 

 ratio 56 : 55. 



