1884.] some existing Non-harmonic Musical Scales. 



309 



tempered Semitones, or parts of such Semitones, are contained in the 

 interval, so that he can realise it somewhat, as compared with the 

 notes of a modern piano, which are intended to be tuned in equal 

 temperament.* This transformation is easily effected by the follow- 

 ing brief table, premising that for brevity I use cent for the hundredth 

 part of an equally tempered Semitone, of which there are twelve to 

 the Octave. 



To convert tabular logarithms into cents, and conversely 



take the logarithm of the interval ratio and seek the next least in 

 the first column of the table ; then the next least to the difference, and 

 so on, taking the cents opposite. Generally it suffices to take to the 

 nearest cent, as that expresses an insensible interval. Thus, if the 

 numbers of vibrations are 440 and 528, the difference of their logs, 

 is '07918 ; the next least in the first column, '07520, gives 300 cts., 

 with remainder, '00392 ; the next least to which in the second column, 

 '00251, gives 10 cts., and remainder, '00141 ; the next least to which 

 in the third column, '00125, gives 5 cts., and remainder, '00016, which 

 in the fourth column gives '6 ct. Hence the interval is 315'6 cts., 

 for which usually 316 cts. is sufficient to write. Now this shows that 

 the interval contains 3 equal Semitones, and 16 hundredths of a Serni- 



* The first person to propose the measuring of musical intervals by equal Semi- 

 tones was, I believe, de Prony, but I have not been able to see his pamphlefe ; the 

 next was the late Professor de Morgan (" Cam. Phil. Trans.," x, 129), from whom I 

 learned it, and I employed it in the Appendix of my translation of Helmholtz, by 

 the advice of Mr. Bosanquet. Having found that two places of decimals sufficed for 

 most purposes, I was led to take the second place, or hundredth of an equal Semi- 

 tone as the unit, and [ have extensively employed this practice, here for the first 

 time published, with the greatest advantage. In fact, I do not know how I could 

 have expressed the results of the present investigation in any other biief and precise, 

 and at the same time suggestive, method. 



