6 Phillips, Harmonic Tetrachords of Claudius Ptolemy. 



T X represents 46 145. Then M T will be a mean tone. 

 The point T can be determined geometrically by inter- 

 polating two harmonic means between OX and 0N\ 

 arithmetically by dividing NX into 23 equal parts, and 

 taking 8 from the end. For the numbers 15, 16, thus 

 treated, give 345, 368 ; and taking 8 from 368 we have 

 345, 360, 368, which are in the required ratios : 23 : 24 

 and 45 : 46. 



In the second Book, without a word of explanation, 

 this harmonic tetrachord is replaced by another form : — 

 56 22 5 4 



— X — X - = — 



55 21 4 3 

 to the still greater mystification of Dr. Wallis. Treating 

 the interval 8 : 7 as before, but with this new factor, we 

 have 



7 56 49 

 the numbers already obtained for the equal tempered tone. 

 The construction turns out to be simpler than before ; 

 keeping to the last figure it is only necessary to divide 

 NX into 7 equal parts, and to take two from the end. 

 For, multiplying 15, 16 by 7, we have 105, 112; and 

 deducting 2 there result 105, no, 112, which are in the 

 ratios 21 : 22, 55 : 56. 



This last section provides the means for dividing an 

 octave into six equal tempered tones. 



M N TX 



Let the letters of this figure have the same significa- 

 tion as before, M T being an equal tempered tone. Draw 

 P 7^ equal to M T and at right angles thereto. Join P O. 



