ON TEMPERAMENT. 163 



different ways of indicating the equal semitone system. We 

 may take decimals, but this involves long and unwieldy 

 figures. Of course there is no reason why long and unwieldy 

 figures should be unmanageable, but some people are afraid of 

 them. I will write out in full that Pythagorean comma of 

 which I gave you before the first few figures. It is 

 7-019550008654. Taking the third founded on that, it is 

 3-863137138649. That would be rather difficult to recoUect, 

 though we have worse numbers than this to deal with ; for 

 instance, the value of TT which is not only larger, but goes 

 on to all eternity and never stops. No doubt the first five 

 figures of decimals would be sufficient for many purposes, 

 but that is only an approximation. The question arises 

 wh3ther we cannot use smaller divisions than 12. Several 

 methods have been adopted which are very convenient. 24 

 has been used, 31 is good, and also 50; 53 is remarkably 

 good, and so is 118. I mean that instead of dividing the 

 octave into twelve divisions, we may divide it into a larger 

 number, and these are the several denominators or consequents 

 of ratios which produce the best results. 53 is so good that 

 I thought it worth while to make a diagram of it, and here 

 is a scale of 53 divisions to the octave from C to c : 



5 c octave 53 



9 b , seventh 48 



8 a sixth 39 



9 g fifth 31 



5 / fourth 22 



8 e third 17 



9 d second 9 



c 



53 



By adding these together you get the other intervals. For 

 instance, if you take 9 and 8, and 5 and 9 together, they 

 make 31, that is the 5th ; or take 9 and 8 = 17, and that is 

 the 3rd, the whole octave being divided into 53. This has 

 another advantage, that you can show by it, in a very simple 

 way, the comma ; you will find that the comma comes out to 

 be just one of these divisions, -^-rd of the octave. If you 

 want to go to greater nicety, you must take a larger number 

 of divisions, and of that I have also given some illustrations. 



M2 



