2 Phillips, Harmonic Tetrachords of Claudius Ptolemy. 



of the problem by ear. This is an error which is incon- 

 sistent with the words of Aristoxenus ; and which an 

 acquaintance with Claudius Ptolemy's writings would 

 render impossible. 



All the existing evidence tends to show that the 

 enharmonic of Aristoxenus is identical with that of 

 Eratosthenes, of the so-called pseudo-Euclid, and of 

 Aristides himself It thus appears that the Pythagoreans, 

 the Alexandrians, and later mathematicians have 

 attempted the problem how best to divide a canon, 

 lute, or guitar fretboard into twenty-four practically equal 

 quarter-tones, with varying degrees of success. Among 

 all known writers, whether ancient or modern, who has 

 solved the question with the nearest approach to mathe- 

 matical accuracy ? Beyond comparison, Claudius Ptolemy! 

 But he has not revealed his intentions, and so far as is 

 known none have guessed the meaning of his harmonic 

 tetrachords, which have come down through seventeen 

 centuries as an insoluble enigma. 



The assertion that mean and equal tempered tones 

 can be marked off on a canon or monochord by means of 

 the harmonic tetrachords of Claudius Ptolemy, with a 

 degree of accuracy never attained by other means, is 

 capable of very easy proof This in itself would be a 

 strong argument that the tetrachords were designed for 

 the purpose in question ; but many would be inclined to 

 doubt whether Claudius, in the absence of modern mathe- 

 matical tables and methods, could be acquainted with the 

 ratios of the intervals with the accuracy requisite to 

 found on them the appropriate geometrical constructions. 



Instead of adopting the short, modern methods, it is 

 therefore preferable to make use of a longer, but more 

 elementary, more easily intelligible demonstration, founded 

 on the reasoning of the Pythagoreans. This deals with the 



