€hi Musical Temperament, 33 



zinth regard to their successive tomes, as these have with regard 

 to C. Whenever an interval occurs, aflGected with a new flat 

 or sharp, it is to be considered as the commencement of a new 

 succession of products. The Illd C«E*, for example, does 

 not occur at all till we come to the key of two sharps, and even 

 then only in occasional modulations, corresponding to the 

 Hid on B in the natural key, whose multipUer is 10. In the 

 key of 3 sharps it becomes another accidental chord, answer- 

 ing to the Hid on E in the key of C, and consequently has 40 

 for its multiplier. It is «nly in the key of 6 sharps, that it be- 

 comes a constituent chord of the key ; when if that key were 

 ever used, it would correspond to the Hid GB on the domi- 

 nant of the natural key. 



After all the products have been taken and reduced to their 

 proper places, in the manner exemplified above, a similar 

 operation must be repeated with the numbers in the second 

 column of Table HI. and those in the second columns in the 

 three first divisions of Table H. 



The necessity of keeping the major, and its relative minor 

 key, distinct, will be evident, when we consider that the 

 several keys in the minor mode do not follow the same law of 

 frequency as in the major ; as is manifest from the observa- 

 tions in Schol. Prop. HI. and as clearly appears from an inspec- 

 tion of Table HI. 



Butin order to discover the relative frequency of the differ- 

 ent chords on every account, the results of the two forego- 

 ing operations must be united. Now, as the numbers in the 

 two columns of Table II. at a medium, are as 3 : 1. and those 

 in Table III. are in the same ratio, although the factors are to 

 each other in only the simple ratio of the relative frequency 

 of the two modes, yet their products will, at a medium, be in 

 the duplicate ratio of that frequency. Hence, to render the 

 two sets of results homologous, so that those which correspond 

 to the same interval may be properly added, to express the 

 general chance of occurrence for that interval in all the major 

 and minor keys in which it is found, this duplicate ratio must 

 be reduced to a simple one, either by dividing the first, or by 

 multiplying the last stories of results, by *^. We will do the 



Vol. I. ...No. 1. 3 " 



