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XVIII.— ON SOME CUEIOUS NUMEEICAL RELATIONS OP 

 THE MUSICAL SCALE. By JOCELYN H. THOMSON, 



Lieutenant, Eoyal Artillery. 



[Read, June 18, 1883.] 



The accompanying Table represents an octave divided into forty- 

 seven parts, in sucli a manner as to give twenty-four complete 

 major keys with perfect intervals, namely, those whose tonics are 

 marked with an asterisk. All these keys, except those of Abb2, 

 C bi, and E b \)i, have their minor scales also complete. 



Colmnn I. of the Table gives the names of the notes, the figure 

 below each serving to distinguish it when there is more than one 

 of the same name. Column II. gives the different keys to which 

 each note belongs. Column III. gives the fractions representing 

 the ratio of the number of vibrations of the note to that of Ci. 

 Column ly. gives the same ratio expressed as a logarithm. From 

 the principle of logarithms it is evident that this column gives the 

 numerical value of the interval from Ci. Column YI. gives the 

 logarithms, and, consequently, the numerical values of the inter- 

 vals in a scale of equal temperament ; i. e. a scale in which the 

 octave is divided into twelve equal parts, or semitones. Most 

 modern keyed instruments are tuned to equal temperament, which 

 system offers so many advantages, as to counterbalance the disad- 

 vantage that no interval is absolutely perfect. 



Since Column VI. gives the numerical value of the interval in 

 a scale of equal temperament, and Column IV. that of the true 

 interval, the difference between these two will give a measure of 

 the discord between the notes on the two scales. This difference 

 is given in Column VII. Now, on an inspection of this column, 

 the following points will be noticed : — First, that the numbers on 

 either side of the centre of the scale are equal, but of opposite sign. 

 This is merely due to the fact that in both scales each interval from 

 the lower C has a corresponding interval from the upper ; or, in 

 other words, both scales are symmetrical. Next, it will be noticed 



