398 



Mr. A. J. Ellis on Musical Chords. 



[June 16, 



complete or imperfect, and of strong discords, and a smaller letter gives the 

 root of weak discords, a number pointing out the family. In the minor 

 triad the characteristic number is omitted; thus c is written for 15c, 

 meaning the minor triad g eh, which is the major tetrad 15 0, or G G E B, 

 with its root C omitted, and is usually called "the minor chord of e," a 

 nomenclature which conceals its derivation. 



Although chords of the same type have the same general character, this 

 is so much modified by the particular forms which they can assume, that 

 it is necessary to examine these forms in detail. They may be distinguished 

 as simple and duplicated. In the former the number of constituents is the 

 same as in the type ; thus 4, 5, 6 ; 2, 3, 5 are simple forms of the type 

 1, 3, 5. In the latter, the number of constituents is increased by the 

 higher octaves of some or all of them ; thus 1, 2, 3, 5 ; 2, 4, 5, 6 are dupli- 

 cated forms of 1, 3, 5 and 2, 5, 6, as they contain the octaves 1, 2 and 2, 4. 



The mode in which the effect of any or all of these combinations may be 

 calculated is shown in Table VII., which consists of two corresponding 

 parts, each commencing with a column containing the " No. of J. H.," or of 

 the joint harmonics resulting from the combination of the harmonics of the 

 constituent compound tones. The next columns are headed by the relative 

 pitch of the constituent tones, and contain their harmonics, never extending 

 beyond the 8th, arranged so that their pitch is opposite to the corresponding 

 number of the joint harmonic. It is thus seen at a glance which harmonics 

 of the constituents are conjunct or tend to reinforce each other, and produce 

 a louder joint harmonic, and also which are disjunct and pulsative. In the 

 second part of the Table the extent of each harmonic of each constituent is 

 given on the assumptions already explained. To find the extent of the 

 joint harmonic, we add the extents of the generating conjunct harmonics, 

 and thence find the intensity by squaring and dividing by 100. The dif- 

 ferential tones must then be found by subtracting the pitches of the pri- 

 maries (or in exceptional cases of higher and louder harmonics). The in- 

 tensity of these differential tones may be called 1 for a single tone, and 4 

 for two concurrent tones, and this number may be subscribed to the inten- 

 sity of the corresponding joint harmonic, as 0,, 25^. 



The beat intervals have next to be noted, and the beat factors, which are 

 usually the reciprocal of the relative pitch of the lower pulsative harmonic. 

 Thus for the dyad 3, 4 the beat interval is f , and the beat factor From 

 this factor, or 1 : /, we calculate the range P :p=70f: 10/=210 : 30 in 

 the present case. This must not be reduced, as' it shows that the interval 

 is dissonant when the pitch of the lower tone is between 30 and 210. To 

 find the intensity, we add and subtract the extents of the pulsative joint 

 harmonics ; in this case 50 and 33 are the extents of the 8th and 9th 

 joint harmonics, and their sum and difference are 83 and 17. Then we 

 take the ratio of their squares, each divided by 100, which gives 69 : 3. 

 This result must not be reduced, as it gives not only the relative loudness 

 of the swell and fall, but also the loudness of these in relation to the other 



