* VOL. XLIV.] PHILOSOPHICAL TRANSACTIONS. ' 2/1 



intervals are, a semitone major, succeeded by another semitone major ; and lastly 

 the complement of these 1 to the 4th, commonly called a superfluous tone. The 

 4th species is the chromaticum sesquialterum, which is constituted by the pro- 

 gression of a semitone major, a semitone minor, and a third minor. This is 

 mentioned by Ptolemy, as the chromatic of Didymus. Examples among the 

 modems are frequent. The 5th species is the chromaticum molle. Its intei*vals 

 are two subsequent semitones minor, and the complements of these 1 to the 4th; 

 that is, an interval compounded of a 3d minor, and an enharmonic diesis. This 

 species is never met with among the moderns. The 6th and last species is the 

 enharmonic. Salinas and others have determined this accurately. Its intervals 

 are, the semitone minor, the enharmonica, diesis and the third major. 



Examples of 4 of these species may be found in modem practice. But he 

 does not know of any theorist who ever yet determined what the chromaticum 

 toniaeum of the ancients was : nor have any of them perceived the analogy be- 

 tween the chromaticum sesquialterum and our modem chromatic. The enhar- 

 monic, so much admired by the ancients, has been little in use among our 

 musicians as yet. As to the diatonicum intensurn, it is too obvious to be mis- 

 taken. 



Aristoxenus and others often mention the tone is divided into 4 parts, and the 

 semitone into 1; thus making 10 divisions or dieses in the 4th. And this is 

 tme, if we consider these sounds in one tension ; that is, either ascending or 

 descending; but, accurately speaking, when we consider all the dieses or divi- 

 sions of the 4th, both ascending and descending, we shall find 13; 5 to each 

 tone, and 3 to the semitone major. But then it is to be observed, that some of 

 these divisions will be less than the enharmonic diesis; for if we divide the semi- 

 tone major into the semitone minor, and enharmonic diesis, ascending, for in- 

 stance, E, «E, F, and then divide in like manner descending, f, ''f, e, we shall 

 have the semitone major divided into 3 parts thus, e, "^f, «:e, f; where the in- 

 terval between ''f and «e is less than the enharmonic diesis between e and ''f, 

 and between «e and f, as is easily proved. 



Now, if we suppose these small intervals equal, by increasing the least divi- 

 sion, and diminishing the true enharmonic diesis, we shall then have a 4th divided 

 into 13 equal parts; and consequently the octave divided into 3 such equal 

 parts; which gives us the celebrated temperature of Huygens, the most perfect 

 of all. 



From this it appears, that the division of the octave into 31 parts, was neces- 

 sarily implied in the doctrine of the ancients. The first of the modems who 

 mentioned such a division was Don Vincentino, in his book L'Antica Musica 

 ridotta alia modema Prattica, printed at Rome, 1555, folio. An instrument had 

 been made according to his notion ; which was condemned by Zarlino and Salinas^ 



