ON TEMPERAMENT. 159 



itself or one by the other any number of times, to produce 

 the same result as by multiplying any other of those numbers 

 by itself any number of times." The ratios, you recollect, 

 of the different notes of the octave to one another were 

 briefly mentioned yesterday, and I have left the diagram up 

 there ; of course those ratios can be put in the form of frac- 

 tions by putting the antecedent as numerator, and the conse- 

 quent as denominator, so you easily form -|, J, &c., as stated in 

 Mr. Ellis's book. It is perhaps the simplest way of making you 

 understand this incommensurability, to take a case. If we 

 divide the octave into 12 equal semitones, of course the 5th 

 ought to be seven of these ; but it was found out very early 

 in the history of music that the 5th is a little more than 

 seven of these. As a matter of fact a 5th is 7 '01 955, and 

 consequently, taking 12 of these 5ths, they give rather 

 more than 7 octaves. They do not come back again to the 

 corresponding octave of the note from which you started. 

 This difference, or departure as it is termed, is the former 

 figure multiplied by twelve. I will give you the multiplica- 

 tion by twelve for simplicity's sake. We have "23460 of a 

 semitone as the excess. This is an old discovery generally 

 attributed to Pythagoras, and the figure is commonly called 

 " the comma of Pythagoras." What a comma is, I shall 

 presently show you. But it is questionable, as I mentioned 

 before, whether Pythagoras deserves entirely the credit of 

 this discovery, or whether he merely imported it from Egypt 

 or Babylon. At any rate the Greeks knew, as I told you 

 yesterday, of the monochord, the ratios to be derived from it, 

 and of the divisions of the scale. Euclid wrote a work 

 called the Sectio Canonis, or the division of the string, which 

 contains all these given in very full detail. The 3rd of their 

 scale was made in a similar way by four 5ths taken upwards, 

 and that is still called a Pythagorean 3rd. There is then an 

 incommensurability between the octave and the 5th which is 

 in nature, and this incommensurability when multiplied gives 

 the interval we term the comma. The error of the Pytha- 

 gorean 5th has been said, and is still said by some persons, to 

 be too trifling to be noticeable ; that human ears are unable 

 to appreciate it, and that you can overlook it. I believe I 

 can show you distinctly the opposite in two ways. I can 

 show it on this harmonium where I can play two notes differ- 

 ing by a comma, or by a smaller interval which I shall have 



