Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 143 



Q is a system in which the change of temperament of the fifth, in passing from a key to 

 that of its dominant, is always of the same amount, one way or the other. That is, the tem- 

 peraments of the fifths in the keys of C, G, D, A, E, B, F|^, are m, 2m, 3m, 4-m,, 5m, 6m, Im ; 

 while those in the keys of C^, G||, D|^, A|f, F are 6m, 5m, 4m, Sm, 2m. Here 4m must be 

 the temperament of the fifth in the equal system. I have described this system in the article 

 Tuning in the Penny Cyclopcedia. 



R is a system in which all the major thirds are equally tempered : and the variety of the 

 fifths in passing from key to key is made as great as, consistently with this condition, it can be. 



5* is a system in which all the minor thirds are equally tempered, the varieties of the fifths 

 being made as great as they can then be. 



In the article cited above, I have exhibited all the relations of the temperaments in tlie 

 form of three theorems, including 25 equations, as follows. The temperament of fifths and 

 minor thirds is considered positive when they are tuned flat : that of major thirds is positive 

 when they are tuned sharp. 



1. The sum of the temperaments of the fifths in all the 12 keys must be •2346 of a mean 

 semitone. 



2. The keys being arranged dominantly, that is, in the order C, G, D, A, E, B, F|f, Cijif, 

 ^% 1^1^' -^tt' 1^' C' ^' I>,...the temperament of the major third in any key together with the 

 temperament of the fifth in that key and the three succeeding keys will always amount to 

 a comma, or "SlSl of a mean semitone. 



3. The temperament of the minor third in any key, together with the temperaments of 

 the fifths in the three preceding keys, will always amount to a comma. 



Thus in all systems, the temperament of AC^, together with those of AE, EB, BFj:f, 

 F^C|f, will make a comma. And the temperaments of AC, together with those of CG, GD, 

 DA, will make a comma. 



If then the temperaments of the fifths go in cycles of four, that is, if the twelve keys, 

 dominantly arranged, have the temperaments p, q, r, s, p, q, r, s, p, q, r, s, in their fifths, the 

 temperament of every major third will be p+q+r+s less than a comma, or -0782 of a mean 

 semitone less than a comma. In the system R, I have taken p=0, q=-0391, r=0, s=-039i : 

 that is, the dominantly consecutive fifths are alternately perfect and tempered as much again 

 as in equal temperament. This is the way of satisfying the condition 3 (p+q+r+s) = -2346, 

 which gives most variety of key. The temperaments of the minor thirds in dominantly con- 

 secutive keys are alternately '1369 and •136.9 + -0391, equal temperament giving -1564 to all. 



If the temperaments of the fifths run in cycles of three, as in p, q, r, p, q, r, p, q, r, p, q, r, 

 it follows that the temperament of every minor third is p + q+r less than a comma. And 

 p+q+r must be '0587. In system S I have made p=0, 5'=-01955 as in equal temperament, 

 r=2q; which satisfies 4(p+g' + r) = '2346. The temperaments of the major thirds in dominantly 

 successive keys are -1564, •1564-5', •1564-2g': that is, the major third is never more tempered 

 than the minor third in equal temperament. 



