396 



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



these numbers multiplied by various powers of 2, that is, as 1, 3, 5, or their 

 octaves. 



If any of the constituents is pulsative the chord will generally have 

 beats, but may be exceptionally without beats. Such chords are termed 

 discords. Their character consists in having two or more of the pitches of 

 their constituents as 1, 3, 5, or their octaves, and at least one of them as 

 7, 9, or some other pulsative tones, or their octaves. What pulsative tones 

 should be selected depends on the sharpness of the dissonance which it is 

 intended to produce, and therefore on the interval of the beat which is 

 created. Thus, since 7 : 6= 1- 16667 and 8 : 7=1*14286 are both near the 

 limit 6 : 5=1*2, the discord arising from 7 would be slight. Some writers 

 have even considered the chord 1, 3, 5, 7 to be concordant. Again, 

 9:8=1*125 is rather rough, but 10:9=1*11111 is much rougher. Hence, 

 if 9 is introduced, 1 should be avoided, that is, the octave of 5 should be 

 omitted, which generally necessitates the omission of 5 itself, as in the 

 chord 1,3,9. But 11 : 10=1*1 and 12: 11 = 1-09091 are both so sharply 

 dissonant, that if 11 is used neither 10 nor 12 should be employed. Now 

 10 is the octave of 5, and 12 is both the 3rd harmonic of 4 and the 4th 

 harmonic of 3, and would therefore be produced from 3 and 4. Hence 

 the use of 1 1 would forbid the use of 3, 4, and 5, that is, of the best disjunct 

 tones. Hence 11 cannot be employed at all. Similarly, 13 : 12=1*08333 

 and 14 : 13=1*07692 are both extremely harsh. The latter is of no con- 

 sequence, because 7 can be easily omitted. But even 15 : 13=1*15384 is 

 more dissonant than 7:6. Hence 13 would also beat with the harmonics 

 of 3, 4, and 5. Consequently 13 must be also excluded. All combina- 

 tions in which the differential tones 11 and 13 are developed will also be 

 extremely harsh. As we therefore suppose that 14 : 13=1*07692 never 

 occurs, and as 14 : 12=7 : 6, the mildest of the dissonances, 14 may be 

 used if 15 is absent, and thus 15 : 14= 1*07143 avoided. When 14 and 15 

 are developed as harmonics of 7 and 5, and not as the primaries of con- 

 stituent tones, their intensity will be so much diminished that the discord 

 will not generally be too harsh. When 15 is used as a constituent, 14 

 and 16 should be avoided ; that is, 7, and 1, 2 and 4, of which 14 and 16 

 are upper harmonics, should be omitted to avoid 15: 14=1*07143 and 

 16 : 15 = 1*06667, which may be esteemed the maximum dissonance. By 

 omitting 16 and 18, and thus avoiding 17:16=1*0625 and 18:17= 

 1 '05882 (that is, by. not using 4, 8, or 9 as constituent tones), 17 becomes 

 useful; for 17 : 15=1*13333 is milder than 9 : 8 = 1*125, which is by no 

 means too rough for occasional use. The other pulsative harmonics, which 

 are represented by prime numbers, are not sufficiently harmonious for use ; 

 but those produced from 2, 3, 5 (such as 25, 27, 45) may be sometimes 

 useful, provided that the tones with which they form sharp dissonances are 

 omitted. 



The result of the above investigation is that the only pulsative tones 

 suitable for constituents are 7, 9, 15, 17, 25, 27, 45, and their octaves. 



