1864.] 



of Instruments with Fixed Tones. 



419 



the Hemitonic system is universally adopted in intention. It is, however, 

 so difficult to realize by the ordinary methods of tuning, that equal tem- 

 perament," as the Hemitonic system is usually called, has probably never 

 been attained in this country, with any approach to mathematical precision. 



In Table XIV. I have given a detailed comparison of the Mesotonic and 

 Hemitonic temperaments with each other and with just intonation, for the 

 system of C (Proceedings, vol. xiii. p. 98), from which the great superiority 

 of the Mesotonic over the Hemitonic both in melody and harmony becomes 

 apparent. But this comparison rests upon the preceding calculations, which 

 were founded upon the beats that arise from rendering the conjunct har- 

 monics pulsative. It was therefore assumed that the qualities of tone em- 

 ployed were such as to develope these beats. The result will consequently 

 be materially modified when the requisite harmonics either do not exist or 

 are very faint. Now 



for the Vth the conjunct harmonics are 2 and 3, 

 „ 4th „ „ 3 and 4, 



„ Vlth „ 3 and 5, 



„ Ilird „ „ 4 and 5, 



,, 3rd ,, „ 5 and 6, 



„ 6th „ ,, 5 and 8. 



If then only simple tones are used, as in the wide covered pipes of organs, 

 or such qualities as develope the second harmonic only, such as tuning-forks, 

 to which we may add flutes, which have almost simple tones, no beats will 

 be heard, and any system of temperament may be used in which the ear 

 can tolerate the interval errors. Now Delezenne's experiments show {loc. 

 cit.) that a good ear distinguishes 



in the unison an interval error of 0*2807^;, 

 „ Vlllve „ „ 0-3U, 

 „ Vth „ „ 0-146U, 



„ Ilird „ „ 0-284^, 



„ Vlth „ „ 0-299^, 



and an indifferent ear perceives an error of 0*56 1^ in the Vlllve, and 

 0-292^ in the Vth. We may say, therefore, generally that the ear just 

 perceives an interval error of in the Vth, and in the other intervals. 

 Now in the Mesotonic system the interval error of the Vth is —\k, and 

 therefore just perceptible, but in scarcely any other interval does it exceed 

 \k. Thus it is —\k in the Vllth, in the Ilird, and +\k in the Vlth, 

 and it is therefore in those intervals imperceptible. In the Hemitonic 

 system the error of the Vth is —^k, and hence quite imperceptible, but 

 the errors of the Vllth, Ilird, and Vlth are respectively f^k, -Jjk, and 

 -f^k, and therefore perfectly appreciable. It is only in the Vllth that this 

 error is at all agreeable. The sharpness of the liird and Vlth is univer- 

 sally dishked. Hence in those qualities of tone which are most favourable to 

 the Hemitonic system, it is much inferior to the Mesotonic. In Table XV. 



VOL. XIII. 2 H 



