162 LECTURES TO SCIENCE TEACHERS. 



They were the first to find tempered instruments, which 

 established the possibility of going round from key to key, 

 wandering in beautiful modulations wherever they wished. 

 This perhaps led them into the practice, and the desire for 

 it became so strong that they sacrificed a certain amount of 

 accuracy to obtain freedom of motion. 



Before, however, we are in a position to compare the 

 different systems of temperament, we ought to fix on a 

 standard of comparison. You should understand that we 

 are not dealing with ratios in speaking of this standard of 

 comparison, but with absolute numbers ; I gave you the 

 ratios yesterday. A ratio may remain large, fixed, and 

 simple, whilst the component numbers upon which that ratio 

 is founded dwindle down by degrees to infinitesimal 

 smallness and fractional complexity, or they may rise to 

 equally large values at the other extreme. The ratio and 

 the number are different things altogether. It is somewhat 

 singular that in these days the mistake of confusing these 

 two should be made. But it has been made, and is continu- 

 ally being made, therefore I feel bound to give you a 

 warning against it. In dealing with absolute numbers we 

 may employ two principal methods of estimating them. We 

 may use the geometrical method and compare them as 

 magnitudes, or the numerical, and compare them as numbers. 

 The geometrical method can be shown very well in a large 

 diagram kindly lent me by Mr. Ellis. This long column 

 contains the four forms of temperament marked at the head of 

 each column by their first letters. Here is the hemi-tonic, or 

 equal semitone system; the just, the meso-tonic, or old organ 

 tuning, and the Pythagorean systems follow. You will see that 

 the length of the black, blue, red, and yellow, is made to 

 designate each note, but as you go up and down the scale those 

 lengths do not at all agree in the same place ; one overtops, 

 another, and in another place falls short, thus exactly measur- 

 ing the inherent error of the scale which we have somehow or 

 other to get rid of. If they were all accurate to one another, 

 we should not have that locking in of one with the other. 

 That is the geometrical method of showing differences of 

 temperament, but we may do it by means of numbers. That 

 table admits of translation into numbers. I have one 

 which I shall be happy to lend to any one who is interested 

 .to copy. By the numerical method there are a good many 



