1864.] 



Mr. A. J. Ellis on Musical Chords, 



403 



and the results be taken in order of pitch, we find, on supplying the second 

 octave 3.16, 



24, 27, 30, 32, 36, 40, 45, 48 

 C, G, A, B, c. 



In this series any two consecutive tones, except 40, 45 or Ay By belong to 

 the same major pentad, and these are therefore eminently adapted for suc- 

 cessions of chords. Even 40, 45, or 5 . 8, 45, belong to two related discords ; 

 for 1, 3, 5, 9, or Fy (7, A, G, and 1, 3, 5, 27, or FCADy have each two 

 constituents in common with 9, 27, 45, 1 . 64, or G DB F. The discord 

 3, 5, 15, 45, or C AE B, contains both the tones in question. These con- 

 siderations justify the major diatonic scale. 



The last discord contains a minor triad, 3, 5, 15. These minor triads, 

 from their relations to three major triads, are evidently peculiarly adapted 

 to introduce successions of harmonies. Taking then the three minor triads 

 and forming their octaves, thus 



3.64, 5.32, 15.8 orc2«^, 

 9 . 16, 15 . 8, 45.4 g e by 

 27.8, 45 .2, 135 dnf% 



we may extend them into a scale, 



120, 135, 144, 160, 180, 192, 216, 240 

 e, fi, 9> a, by c/s^ e-, 



where the chordal relations are even more intimate than before, and by 

 means of the chord 45, 135, 22.5, 5 . 64, 3 . 256, or B F^ JD|: a c, already 

 noticed, the major triad, 45, 135, 225, or B F^XD^yis brought into close 

 connexion with the minor triad, 3, 5, 15, ov c a e. Practically the use of 

 the minor scale consists of a union of four major triads, 1, 3, 5 ; 3, 9, 15 ; 

 9, 27, 45 ; 27, 81, 135, forming two major scales, with three other major 

 triads, 5, 15, 25 ; 15, 45, 75 ; 45, 135, 225, forming a third major scale, 

 by means of three minor triads, 3, 5, 15 ; 9, 15, 45 ; 27, 45, 135, the roots 

 of which, 1, 3, 9, are the same as the roots of the first three major triads. 

 There are therefore seven roots to all the chords introduced, namely 1, 3, 

 9, 27, and 5, 15, 45, or Fy C, G, D and A, Ey By and these seven roots 

 form a major diatonic scale. From these relations spring all the others 

 which distinguish the minor scale together with all its various forms and 

 its uncertain tonality, which is generally assumed to be the relation of the 

 chords to 15 or E, the tonic of the last three major triads, but which evi- 

 dently wavers between this and 3, 9 or C, G, the tonics of the first four 

 major triads, and these three tonics, 3, 9, 15, or CGEy form a major triad. 



By extending this system of chords up and down, right and left, we 

 arrive at the perfect musical scale in Table V. (Proceedings, vol. xiii» p. 1 08), 

 which is therefore entirely justified on physical and physiological grounds, 

 without any of those metaphysical assumptions or mystical attributes of 

 numbers which have hitherto disfigured musical science. In that Table the 



VOL. XIII, 2 G 



