103 



CONCORD. 



Concord, and discover beforehand whether they would prove eon- 



V *^"Y"™"'' cords or discords. 



The concurrent experience of all musicians, and of 

 others who have experimented on musical sounds and 

 consonances, is, that the following intervals are con- 

 cords, viz. 



2S + S 



3rd 



4th 



s+sls 



T 



¥ r 



10th x 11th xn 13th xin XV 17th xvn 18th 



S z 3 i. _5 3 1_ 5 i 3 



ix S 8 3 l~S I 0~ 4 "*T 7" Tff 



fVIII 



I I 



fxix 20th xx XXII 24th xxiv 25th xxvi 27th xxvn 



"1 I S 3 i S i 5 



l 'S' T» *o" 3 T3" To T" 



("XXIX 31st xxxi 32d xxxm 



1 « f » 3 ' 12 



I "TBT 97 "o 'ST i? 



>4th 



tt ^j- 4-3- 



xxxiv XXXVI 



i i 



To T^ 



J 38th xxxviii 39th xl 41st xn XLIII 45th xlv 46th 



? 



3T T5T ¥o 



and that all the ama- 



f xlvii 48th XLViii L, &c. 

 1 I s 3 1 &« 



zihg 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, be- 

 longing 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 1, 5, 1 3, 1, 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 

 octaves, constitute the following series, when arranged, 



viz. 



fl, 2, *% Hi "5, 2 6, 38, 

 ■j 1 1 1 1 1 2 



10, 2 12, 316, 2 20, 2 24, 



i 4 4 4 



' 3 32, 2 40, 2 48, 



'64, 



i(5 



2 S0, 2 96, 



H28, 



3s- 



160, 



3J 



192, 



3* 



* 



2 256, 320, 384, and 512.7 r™ ,, -. 



, 64 64 64 »8 } • TllS SmaU fi g UreS P re - 



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 forms, viz. 2 X , 2 X X 3, or 2 X X 5; where 

 2°=:1, 2' =2, 2 2 =4, &c. or the powers of 2 are 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 be 

 observed, that they are powers of 2, viz. 2° (or 1), 2 1 , 

 2 l , 2 3 , 2 4 , 2 5 , and 2 6 ; and that after the number 3, or 

 third term of the series, they proceed by three of each 

 of these, in succession ; the consequence of three dif- 

 ferent forms being combined in this one series, 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 + 8, cf , S, S + §>, S d and 2 S -f- 8, 

 in the Chromatic Elements (see 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 T and S, and VI and Concord. 

 VIII, the difference is S-fS, or the 3rd; where the "* "Y"""' 

 same differ one, as between III and 4th, and V and 6th, 

 the difference is S; and where the numerals are the 

 same, only major and minor, as between 3d and III, 

 and 6 and VI, the difference is e? : these last have, by 

 Dr Callcott, Dr Busby, and many other writers, most 

 improperly and unnecessarily been called imperfect con- 

 cords, merely because they are sometimes major and 

 sometimes minor, and the VIII, V, and 4th perfect, be- 

 cause each of them have but one numeral designation ; 

 whereas imnerfect concord* 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, Dr Robison, and others of the best mo- 

 dern writers on the subject, that their order of simpli- 

 city, or smoothness of effect on the ear, in the 1st oc- 

 tave, is I, VIII, V, 4th, VI, III, 3rd, and 6th ; or i, §, 

 t> f ' f > r> ?' an d f '> 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, 11, 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. 



I VIII XII V and XV XVII 4th, X and XIX VI 

 2, 3 , 4 , 5 , 6 , 7 ,3 



V VIII III HI V 



IIIandXXH3d,llth,andXXIV6th,XIIIandXXVI 

 9 11 13 



VIII 4th III VI V 



10th and XXIX 18th 13th and XXXI XX XXXIII 

 17 , 19 , 21 , 23, 25 , 



3d VIII 4th 6th III VI V 



17th XXXVI 25th 20th XXXVIII XXVII XL 



29 , 33 , 35 , 37 , 41 , 43 , 49, 

 3d VIII 4th 6th III VI V 



•24th XLIII 32d 27th XLV XXXIV XLVII 31st 

 53 , 65 , 67, 69, 81 , 83 , 97 , 101, 

 3d VIII 4th 6th III VI V 3d 



L 39th 34th XLI 38th 46th 41st XLVIIf 

 129, 131, 133, 163, 197, 259, 261, 323 , 

 VIII 4th 6th VI 3d 4th 6th VI 



!45th and 48th 1 

 389 , 517 J- . Where the middle line shews the 



3d 6th 3 



sums of the terms in the concords expressed above, and 

 the lower line 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 of 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 Vth is here 

 interposed 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 



