On Musical Temperament. 31 



Besides the following chromatic intervals : 



C 8 extreme sharp 5ths on C 



Major mode. < 1 D 



( 1 extreme flat 7th G* 



I 4 extreme sharp 6ths on F 



Minor mode. < 4 extreme flat 7ths on C« 

 1 3 G* 



It was thought best to exhibit a complete table of all the 

 consonances which occurred in the 200 scores examined ; 

 although (Prop. II.) only the concords in the upper half of the 

 table can be regarded in forming a system of temperament. 

 For the more frequent consonances, this table may be regarded 

 as founded on a sufficiently extensive induction to be tolerably 

 accurate. For the more unfrequent chords, and especially for 

 those which arise from unusual modulations, it expresses the 

 chance of occurrence with very little accuracy ; and it is 

 doubtless the fact that a more extensive investigation would 

 include some chords not found at all in this list. But it must 

 be recollected, on the other hand, that the influence of these 

 unusual chords on the resulting system of temperament would 

 be insensible, could their chance of occurrence be determined 

 with the greatest accuracy. 



But none of the numbers in the foregoing table by any means 

 expresses the chance that a given interval will occur, consi- 

 dering all the keys in which it is found. For example, the 

 Vth CG on the tonic of the natural key, in music written on 

 this key, is the one of most frequent occurrence, its chance 

 being expressed by 1807 ; but in the key of two flats, it be- 

 comes the Vth on the supertonic, and its chance of occurrence 

 is only as 197. Hence the problem can be completed only by 

 finding a set of numbers which shall express, with some degree 

 of accuracy, the relative frequency of different signatures. 



An examination of 1600 scores, comprising four entire col- 

 lections of music for the organ and voice, by the best Euro- 

 pean composers, besides many miscellaneous pieces, afforded 

 the results in the following table : 



