60 . Scientific Proceedings, Royal Dublin Society. 



deyiation from nature, and is to be appreciated as artistic excel- 

 lence : if it transcend those limits it is perceived to be exaggeration, 

 and offends. 



The principles which underlie these remarks apply with singular 

 force to music. Judging from the literature on the subject, it may- 

 be said that people of an arithmetical turn of mind are sometimes 

 apt to forget that artistic correctness has been fully attained when 

 the outstanding errors have been reduced below what the cultivated 

 ear can perceive, and both the arithmeticians and those who are 

 devoted to the simpler forms of melody and harmony are apt to 

 overlook what one would think is very obvious — that some sacrifice 

 of one artistic effect may be welcomed when it procures for us new 

 artistic effects of sufficient value, and especially where the new field 

 of effects is of such vast range as it is in instrumental music, and 

 the effects themselves of a kind to be keenly relished by those who 

 have the good fortune to be gifted with a comprehensive apprecia- 

 .tion of harmony. 



Of this kind are the advantages acquired by tuning pianos on 

 the system of equal temperament. In this method of tuning, each 

 octave is divided into twelve equal semitones. This method of 

 tempering plainly puts it within the power of the composer to in- 

 dulge in absolutely unlimited transition from key to key, for. it 

 provides the same intervals in all the keys. 



It was for a piano tuned in this way that Sebastian Bach com- 

 posed the unrivalled preludes and fugues of his " Wohlte'mperirte 

 Clavier," and many of these, as well as a great deal of the best 

 modern music, would be impossible without the unrestricted power 

 of modulation which is thus provided. This immense advantage 

 ought to be candidly admitted by those who recommend other 

 methods of tempering. 



On the other hand, there are effects and exquisite effects, which 

 cannot be attained with instruments tuned in this way, which wiU 

 only come forth when suitable music is executed in the natural 

 scale, i. e. the scale indicated by the mathematical theory, or in a 

 scale very closely approximating to it. This natural scale does not 

 in its fifths differ sensibly from the scale of equal temperament, , 

 but its thirds and sixths are perceptibly smoother ; and it has two 

 supertonics to be used respectively according as this note is asso- 

 ciated with dominant or subdominant harmony. I would wish 



