"STY - S zing variety of other intervals (except those near to any of these concords; as above mentioned) within these se ven octaves, are discords.
Wherein it is observable, in comparing the terms of the ratios of all these concords in the lowest line, ing to the intervals major and minor, as expressed above them in the first line, that the numerator, or least term of the ratio, never exceeds 5 ; and that in the third and all succeeding octaves in ascending, the numerators are I, 5, 1 3, I, 5 3 and 1. That 4 never appears as a numerator but in the first octave, and 2 only in the two first octaves. That the denominators, or largest term of the ratios of the concords in the above seven taves, constitute the following scrics, when arranged, 1, 2, viz. I 1 1 , 1 2 2 2 4 4 II 1 160, 192, 8 8 8 16 16 26 32 32 32 1 320, 384, and 512. .
64 64 64 '28 The small fi re gures p fixed, denoting the number of times that these occur as denominators, in these seven octaves. All the numbers in the above series will be found included in one of the following three forma, viz. 2x, 2xX3, or 2xx5 ; where 2'=1, 2' =2, &c. or the powers of 2 arc indefinite, while only the first power of 3 or of 5, enter into any of the largest terms of the ratios of concords. If we examine the differences in the above series of numbers, it will he observed, that they arc powers of 2, viz. 2° (or 1), 2', 2', and and that after the number 3, or third term of the scrics, they proceed by three of each of these, in succession ; the consequence of three dif ferent forms being combined in this one scrics, as above.
In the middle line of the first octave, the intervals of the original concords therein, (as such are called,) are set down, viz. 2 S-f- s, d,s,s+s, S 0' and 2 Si-S, in the CHROMATIC Elements (sec that article); by which it appears, that the octave is similarly divided by the original concords, into two similar parts, but reversed; or, the progression is the very same in proceeding from both its extremities towards the middle of the first or original octave, and so of all the superior ones. It thus also appears, that where the numeral designation of the concords differ two, as between I and 3, and VI and VIII, the difference is S-{-S, or the 3rd; where the same differ one, as between III and 4th, and V and 6th, the difference is S; and where the numerals arc the same. only major and minor, as between 3rd and III, and 6 and VI, the difference is j: these last have, by I)r Callcott. Dr Busby, and many other writers, most
improperly and unnecessarily been called imperfect con cords, merely because they arc sometimes major and sometimes minor, and the VIII, V, and 4th perfect, be cause each of them have hut one numeral designation . whereas imperfect concords should always mean tempered or altered concords, as above mentioned.
These several concords are not equally harmonious, satisfactory, or pleasing to the ear, either considered or compared altogether, or in groups, within each succes sive octave, respectively ; but it seems agreed by Dr Robert Smith, I)r Robison, and others of the best mo dern writers on the subject, that their order of simpli city, or smoothness of effect on the car, in the Ist oc tave, is 1, VIII, V, 4th, VI, III, 3rd, and 6th ; or 4, !, 2 3 3 3 and • which ratios form series, increasing with the degree of comparative roughness or want of pleasing effect in the concord, as above, whether we con template the numerators, the denominators, or the sum of these, viz. 2, 3, 5, 7, 8, 9, II, and 13. If we ar range all these several concords in seven octaves, ac cording to the sum of the terms of their respective ra tios, they will stand as follows, viz.
sums of the terms in the concords expressed above, and the lower lint the original concords, of which many of the same are compounded, by the addition of octaves. In comparing the first 16 terms of this complete series of concords, in the order of the sums or their terms, with the similar series above, when only the eight ori ginal concords in the first octave are considered ; it will appear, that the XII or octave of the VIII is here inter posed between the VIII and V ; and the XV and XVII, or 2 VIII and VIII+III, are interposed between the V and 4th as they stood in the original concords, which superior simplicity of the doubled concords XII and XVII, to their respective originals, seems to point out one reason of these and the ch,ublc octave being the only Tartinian sounds that are heard to accompany a note ; since no other double concords but XII and XVII, have a less sum to the terms of their ratio, than their originals have. It is further observable, that X and XIX or VII I+ III and 2 VIII+ V, are interposed here, be tween the 4th and VI of the original series ; that XXII or 3 VIll is interposed between 111 and 3d ; and 1 1 th and XXIV or VIII+4th and 3 VIII+111 between 3d and 6th.