164 LECTURES TO SCIENCE TEACHERS. 



One of the best numbers to choose is 30103, which, in the 

 decimal form, is really the logarithm of the number 2. We 

 need not consider it as a logarithm, but simply treat it as a 

 common number. If you separate the octave into this large 

 number of very small divisions you can show all intervals 

 with great accuracy, and even get the interval of a 

 schisma. Taking the octave as 30 103, the fifth will be 17609 ; 

 the comma 539, and the schisma will be 49. Some put it at 

 48, but 49 has this advantage, that it is just y T th of the 

 comma. The table above is framed on the 30103 system. Now 

 I am in a position to show you the comparison between the 

 new and the old temperaments a little more closely. Here 

 are three columns, one the just intonation, the second the old 

 organ tuning, and the other the modern. 1 The old organ 

 tuning had one advantage, that it keeps the 3rd perfectly 

 correct, 9691, the same as in just intonation, but you see how 

 terribly that is thrown out by the equal tuning being raised 

 to 10034. On the other hand, the fifth is a little wrong; it 

 is more out than in the equal temperament. The mode of 

 working this out deserves a little consideration. You divide 

 ttie octave into any number of intervals, of aliquot parts, 

 which we may call m ; v of these make a fifth, t represent the 

 major 3rd, and q represent the comma ; taking the logarithm 

 of 2 which I have given in the form of a whole number, 

 and dividing it by these, we get the closest possible cyclic 

 approximation to just intonation. The cycle of 53, which 

 has the advantage of simplicity, was first proposed by 

 Mercator, who is known as the inventor of a plan for charts. 

 This system was employed by Perronet Thompson, and has 

 been fully carried out in that beautiful harmonium of 

 Mr. Bosanquet's, which many of you may have seen in the 

 galleries of the Exhibition on the other side. I did not 

 think it desirable, as it is a bulky, heavy, and delicate 

 instrument, to bring it over here, but you will probably have 

 heard it played and explained by Mr. Bosanquet himself. 

 For practical use there is no doubt that this 53 scale is the 

 most perfect we can get without running on to a most 

 impracticable number of divisions in the octave. Whether 

 it is the best for performance upon a playing instrument is 

 an entirely ulterior consideration of which I shall speak 

 presently. This 53 scale gives you an opportunity of seeing 

 1 See table on page 161. 



