158 LECTURES TO SCIENCE TEACHERS. 



treating the subject, and since my original remarks at the 

 conference I have had a little conversation with Dr. Stainer. 

 Dr. Stainer, who, I need not say, is not only a most able, but 

 a most unbiassed judge in the matter, quite admits that he 

 does not hold those views so strongly as he did ; that he is 

 beginning to think there is a possibility of just intonation, 

 although he sees very clearly the mechanical difficulties which 

 we all admit, and which stand in the way of its production. 



Going into detail as to what temperament is, we may define 

 the object of it as being the division of the octave into a 

 number of intervals, such that the notes which separate them 

 shall be suitable in number and arrangement for the purposes 

 of practical harmony. This will be probably hew to many 

 persons. The old form of harmonium, piano, and every keyed 

 instrument, is so engraven on our minds from use, that most 

 persons are quite unaware that there is any other possible ar- 

 rangement. They may perhaps in a museum have occasionally 

 seen a strange-looking instrument, stranger even than the one 

 I have here, but they have passed it by, under the impression 

 that it was incomprehensible or worse. Now the usual instru- 

 ment which we are accustomed to, has of course its own 

 system of temperament, and that temperament although not 

 the oldest is certainly the simplest, and is generally called the 

 equal temperament. It divides the octave, as you see in the 

 harmonium, into 12 equal parts, or semitones. If it so hap- 

 pened that the octave could be divided into 1 2 equal semitones, 

 such that the other divisions, the 5th and the 3rd, should be in 

 tune, it would be a very great boon, but unfortunately nature 

 has not so ordained it, and the first point which I wish to 

 insist upon is what probably has not been conceived by every- 

 body, that the discrepancy, the difficulties, the errors which 

 we have to get over, lie, not in our system of music, but in 

 nature itself. Just as the diameter and the circumference of 

 a circle are not commensurable to one another ; so the 5th, the 

 3rd, and the octave are not commensurable. They do not 

 come to actual agreement in an arithmetical way, and this is so 

 very well given in Mr. Ellis's translation of Helmholtz's great 

 work, that I will ask your leave to quote a few words. When 

 speaking of temperament he says, " It is impossible to form 

 octaves by just 5ths or just 3rds or of both combined, or to 

 form just 3rds by just 5ths, because it is impossible by mul- 

 tiplying any one of the numbers f , or J- by two, or either by 



