1864.] 



Mr. A. J. Ellis 07i Musical Chords. 



401 



an octave of the root of the minor triad, so that e, ff, b or \0, \2, 15 is con- 

 sidered by them as derived from IJ^ instead of C^. 



Chords will evidently be related to each other when one or more of their 

 constituents are identical, and natural qualities of tone will be related which 

 have one or more identical harmonics, or which form parts of related chords. 

 Transitions between related chords and compound tones will be easy and 

 pleasing. Hence, in forming a collection of compound tones for use either 

 as natural qualities of tone (in melody) or as constituents of artificial 

 qualities of tone, that is, chords (in harmony), it is important to select such 

 as will have numerous relations, and will produce the concordant dyads 

 and triads, and the least dissonant discord, the minor triad. Hence, taking 

 the concordant major triad 1, 3, 5 as a basis, we must possess its products 

 by 2, 3, and 5. There must be abundant multiples by 2 in order to take 

 the several forms of the triad and to introduce the duplications. The 

 products by 3. and 5 give 3, 9, 15 and 5, 15, 25. We have then the 

 tones 1, 3, 5, 9, 15, 25, and their octaves. These give three concordant 

 major triads, 1, 3, 5 ; 3, 9, 15 ; and 5, 15, 25, each of which has one con- 

 stituent in common with each of the others. We have also the major 

 pentad 1, 3, 5, 9, 15, the nine-tetrad 1, 3, 5, 9, the major tetrad I, 3, 5, 

 15, and the minor tetrad 3, 5, 9, 15, whence, by omissions, result the nine- 

 triad 1, 3, 9 and minor triad 3, 5, 15. Each of these triads is related to 

 two of the three major triads. The minor triad is intimately related to all 

 three major triads by having two constituents in common with each of 

 them. The discords involving 7 and 17 would evidently require 1, 3, 5, 

 7, 17 to be taken as a basis. Neglecting these discords for the present, 

 the above results show that we should obtain a useful series of tones by 

 multiplying 1 , 3, 5 successively by 3, and each product by 5, taking octaves 

 above and below all the tones thus introduced. We thus find 



IVIAJOE. 



Minor. 



Majob. 



MA JOE. 



Minor. 



Major. 



1, 3, 5 



3, 9, 15 

 9, 27, 45 

 27, 81, 135 

 81, 243, 405 



3, 5, 15 

 9, 15, 45 

 27, 45, 135 



5, 15, 25 

 15, 45, 75 

 45, 135, 225 



1, 3, 5 



3, 5, 15 



5, 15, 25 



EGA 

 C G E 

 G D B 



DfA n 



tA tEC5 



c a e 

 g e h 



AE 



EB JGS 

 BFjfJDj 



Any of the smaller numbers may be multiplied by 2, 4, 8, 1 6, 32, 64, 

 128, 256, in order to compare them with the larger numbers. Such mul- 

 tiplications are presumed to have been made in the columns of notes. 

 Hence t^:^=81 : 5.16, or t = 81 : 80, 



F^:F=\3b'. 1.128, orlf=135 : 128, 

 JCS: C=25 : 3.8, or J±t=25 : 24, whence ^ = 80 : 81. 

 And in the same way the other ratios in ' Proceedings,' vol. xiii. p. 95, are 

 reproduced. 



