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Mr. A. J. Ellis on Musical Chords. [June 16, 



1, 3, 5 may be converted into 1, 2, 3, 5, and by thus strengthening the 2, 4, 

 and 8 joint harmonics the finest form of concord is produced. In this 

 way the series of duplications in Table X. was produced. In this Table an 

 example has been added to each form to facilitate trial ; but the great im- 

 perfection of the major third in the ordinary system of tuning pianos and 

 harmoniums materially deteriorates the effect of the chords, which ought 

 to be taken on some justly tuned instrument. 



The discords may be deduced from Tables VII. and VIII., when pro- 

 perly extended, by supposing 7, 9, 15, 17, 25, 27, 45 to be used in the first, 

 and their effect allowed for in the second. The additional discordant effect 

 of 7 will be necessarily least felt where 7 occurs as a differential tone, but 

 these are not the best forms of either triad or tetrad. In the better forms 

 the dissonances 6, 7 and 7, 8 will always be well developed, and as the 

 latter is sharper, the omission of 8, at least as a constituent tone, is 

 suggested. If 7^ is used instead of 7, the omission of 8 becomes more 

 urgent, while 6, 7^ will beat less sharply than 6, 7, and therefore almost 

 inaudibly. The real beats of the constituents 6, 7^ arise from the har- 

 monics Q . b . or 36, 35f , which are, however, not so much felt as 

 those of 6 . 6, 5 . 7 arising from 6, 7, because 36 : 35f= 1*0125 is further 

 from 16: 15 = 1-0667 than is 36 : 35= 1-0286. Hence, when 8 is omitted, 

 the dissonance arising from 7^ is less than that arising from 7 itself. 

 When 8 is present, 7 or is superior to 7^. The use of 1 7-^ 1 ^ 

 would hardly create any perceptible alteration of roughness when 18 is 

 absent, and when 18 is present 18 : \7Y^=l'0i>AS is further from 16 : 15 

 than is 18 : 17=1 "0588, and therefore the roughness is not quite so great. 



Of all discords the least dissonant is the minor triad 3, 5, 15, which is 

 formed from the tetrad 1, 3, 5, 15 by omitting the root 1, to avoid the dis- 

 sonance 15, 16. When the differential tones derived from the primaries 

 of the constituents are deeper than the primaries, and therefore merely 

 indicate the presence of a pulsative tone, which is only faintly realized by 

 the differential tones derived from the upper harmonics of the primaries, 

 and when the dissonant intervals of the minor tenth and major thirteenth 

 5, 12 and 3, 10 are not present in the constituent tones, this chord may 

 be treated as a concord. But in most positions the minor triad is sensibly 

 dissonant, as shown in Table XI., where an attempt has been made to 

 arrange its 20 forms in order of sonorousness. The pitches of the differ- 

 ential tones are added, and examples subjoined. The effect of the minor 

 chord is very much injured by the usual tuning of harmoniums, &c. A 

 peculiar character of these and other discords, when the pulsative con- 

 stituent is not the highest, consists in the quality of tone being due to very 

 high joint harmonics, except such as are due to differential tones. The 

 root will be consequently extremely deep when the constituent tones are 

 taken at a moderate absolute pitch. This great depth renders its recogni- 

 tion by the ear difficult. Hence probably the disputes of musicians con- 

 cerning the roots of certain discords, and their error in considering 5 to be 



