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Mr. A. J. Ellis on Musical Chords. 



[June 16, 



In addition to the chords already noticed, we have now the twenty-seven 

 tetrad, 1, 3, 5, 27, or F CAB, and the twenty-seven triad, 1, 5, 27, or 

 F AD, and all the discords derived from 1, 3, 5, 9, 15, 25, 27, 45. But 

 for those derived from 7 and 17 substitutes must be employed. These are 

 obtained as follows. The chord 9, 27, 45, 1 . 64 is 9 times 1, 3, 5, 7^, so 

 that GDBF approximates to 1, 3, 5, 7 in a manner already tested. 

 Again, 1 . 32, 3 . 32, 5 . 32, 225 is 32 times 1, 3, 5, 73V, whence FCAXB'^ 

 gives the second and closer approximation to 1, 3, 5, 7 already considered. 

 When 7^ is used for 7 it will be better to use 1 , 3, 5, 7^, 8f , or one-ninth 

 of 9, 27, 45, 1 . 64, 5 . 16, that is G B D FA, in place of 1, 3, 5, 7^, 9 or 

 one-ninth of 9, 27, 44, 1 . 64, 81, that is G B DFfA, to avoid the disso- 

 nance 5 . 7^, 4 . 9, or 35|^, 36. This will therefore replace the seven-nine 

 pentad 1, 3, 5, 7, 9. 



The chord 45, 135, 225, 5 . 64^, 3 . 256 is 45 times 1, 3, 5, 7-^, 17-^, or 

 B XD^ A C, and it forms an excellent substitute for the seven-seventeen 

 pentad 1, 3, 5, 7, 17. Again, the chord 3 . 16, 5 . 16, 15 . 16, 135, or 16 

 times 3, 5, 15, 1 6-J, that is C AE F% is a sufficiently close approximation 

 to the rough discord 3, 5, 15, 17. 



It has already been shown that the alterations in the discords thus pro- 

 duced will be slight, and, under certain circumstances, improvements. The 

 omission of 7, 17 in the base 1,3, 5 is therefore justified. Their insertion 

 would embarrass the performer and composer by an immense variety of 

 tones very slightly differing from each other, as 64, 63 ; 135, 136 ; 255, 256. 

 As it is, the distinction between 81, 80 is the source of much difficulty, 

 and separates chords such as 81, 243, 405, and 5 . 16, 15 . 16, 25 . 16, or 

 80, 240, 400, that is, fA fE C^ and AEXC% which composers desire to 

 consider as identical. It was shown in my former paper (Proceedings, 

 vol. xiii. p. 98) that the use of 1, 3, 5 as a basis requires 72 different tones, 

 exclusive of octaves. The introduction of 7 in the base would increase this 

 number by 45, and the introduction of 17 by 30, while the mental effect 

 produced would be very slightly different. On the other hand, if instead 

 of 1, 3, 5 as a base, we took 1, 2v, AT, where v, T are ratios differing 

 slightly from 3 : 2 and 5 : 4, we might avoid the ratio 81 : 80, reduce the 

 number of tones to 27, and greatly increase the relations of chords. How 

 to effect this important result with the least dissonant effect will be con- 

 sidered in the following paper on Temperament. 



The three major triads 1, 3, 5 ; 3, 9, 15 ; 9, 27, 45 are so related as to 

 form two major pentads, 1, 3, 5, 9, 15 and 3.1,3.3, 3.5, 3.9, 3.15. 

 Hence the middle triad forming part of both pentads connects the three 

 triads into a whole, closely related to the middle triad, and therefore to its 

 root. These are called the tonic chord and tonic tone, and the connexion 

 itself is termed tonality. If octaves of these tones be taken, thus, 

 1 . 32, 3.8, 5 . 8 or C ^, 

 3.8, 9.4, 15.2 CGE, 

 9.4, 27, 45 GBB, 



