€ O M 



ComraeUna, ticuiar, as to the invention and use of these apparently 

 Commeu- Common measures of musical ratios ; for which, see that 

 surable &tt [ c \ e ( f ) 

 nI^T^ COMMELINA, a genus of plants of the class Tri- 

 andria, and order Monogynia. See Botany, p. 96. 



COMMENSURABLE Intervals, in Music, are such 

 only, -whose ratios in numbers, have both their terms, the 

 game power of some numbers respectively ; as for instance, 

 .both the numerator and denominator of the fraction, or 

 terms of the ratio, squares, both cubes, both biqua- 

 drates, numbers, &c. : in which case, the interval is an 

 exact multiple of some other smaller interval : thus, in 

 the Table in Plate XXX. Vol. II., the ratios answering 

 to 4 £ and to 3 /are the only ones, whose numerator 

 and denominator in col. 3. are both biquadrates, cubes, 

 squares, or any other power, of any whole number 

 whatever. In like manner, the indices of the primes 2, 

 3, and 5, in col. 4, and also the numbers of Z,f, and m, 

 in col. G. respectively, are prime to each other, or have 

 no common measure but unity, in any of the lines of 

 the Table, except those of 4 £ and 3 f, above mention- 

 ed. And the same thing will hold, in the terms of any 

 notation by 3 intervals, in which these or other inter- 

 vals may be expressed. So that except the few inter- 

 vals that are thus obviously multiples of some others, 

 -and ought not to have separate names, but be called 

 double, triple, quadruple, &c. of their component inter- 

 vals, as No. 22. in our Table of Commas (see that arti- 

 cle) is the double minor comma, No. 24 is the double 

 major comma, &c. 



All useful or practicable musical intervals, are prime, 

 or the terms of their ratios are irrational, or surd, to- 

 wards each other ; and strictly speaking, such a thing 

 as a Common Measure (see that article) among musi- 

 cal intervals, is impossible. This general incommensu- 

 rability of musical intervals arises from the very nature of 

 the prime numbers of which their ratios are composed; 

 of which primes it is demonstrable, that no finite power 

 of one prime number can exactly equal any finite power 

 whatever of any other prime number, nor can the multi- 

 ples of any two produce a third prime number. And so, 

 no number of any interval, can exactly equal any number 

 of another prime or incommensurable interval, as most of 

 those in practice are : for instance, no number of 3ds, 

 Illds, 4ths, Vths, 6ths, Vlths, &c. can exactly make up 

 an octave,or any number of such; no number of 3ds can 

 make any number of Vths, nor can any number of 1 1 Ids 

 do the same thing, &c. When it is said, that two intervals, 

 or their ratios, are incommensurable, or prime to each 

 other, it is by no means meant that all the numbers 

 composing such ratios we prime numbers; indeed, that 

 is seldom the case. Thus, if |- and 44 are compared, all 

 the numbers concerned are either squares or cubes, and 

 yet these ratios are absolutely prime to each other, (g) 

 COMMENSURABLE Systems of musical Intervals; 

 are such tempered systems of 7 notes or septaves, whose 

 intervals being T, T, L, T, T, T, and L, (where 5 T 

 + 2L = VIII), T has a finite or commensurable ra- 

 tio to L ; as suppose, T : L : : 8 : 5, then such an oc- 

 tave consists of 5x8 + 2x5 = 50 equal parts, 

 whereof the tone = 8, and the limma = 5 parts. 



So in a system of 12 notes or douzeave, /, L, L, I, L, 

 I, L, /, L, L, I, and L, {where T — L = I, and 7 L + 

 5 I — VIII), if L is to I in any finite ratio, this also is 

 a commensurable system; if for example, L : / : : 5 : 3, 

 then 7x5 + 5x3 = 50 parts in the octave, as 

 above. 



. The 12 intervals above, being considered as between 

 the notes C, C*, D, Eh, E, F, F* G.G| A, Bb B 



21 COM 



and c, constitute a regular douzeave ; to which the notes 

 of the octave next above, c*, d, e\), e,f,f%, g, g * a, 

 &c. being joined, (each an exact octave above those in 

 the given octave), then the consonances that can be 

 taken true, or according to the intended system of tem- 

 perament, are, as in the following Table, in part ex- 

 tracted from the Philosophical Magazine, vol. xxxix. p. 

 414, viz. 



Coaimen' 



surabie 

 Systems. 



Conson- 

 ances. 



II 



3 

 III 



4 



IV 

 V 



6 



VI 



7 



VII 



Bass and Treble Notes. 



Kegnlar 

 Tempera- 

 ments. 



C*D,DEb, EF,F*G,G*A,ABb 

 and Be 



CD, DE, EbF, EF*, EG, F* G*, 

 GA, AB, Bbc, and Be* 



CEb, C*E, DF, EG, F*A, GBh, 

 G*B, Ac, and Brf 



CE, DF*, EpG, EG*, FA, GB, 

 Ac*, and B^ 



CF, C* F*, DG, EA, FBb, F*B, 

 Gc, G*c*, Ad, Bb ch, and Be 



CF*, DG*, Eh A, FB,Gc*,andB,be 



CG, C*G*, DA, EbBt). EB, Fe 

 F* c*, Grf, Ae, B\)f, and B/* . 



C*A, DBb, Ec, F^d, Gcb, G*e, 



A/, find Bg 

 CA, DB, Ebc, Ec*, Fd, Ge, Aft, 



Bbg, andBg* 



CBb, C*B, Dc, Ed, Feb, F*e, Gf 



G*/*, Ag, andBa 



CB, Dc*, EK, Fe, G/*, Ag*, and 



Bb« 



L+Z 



2L+/ 



L + 2 f 



3L+ 21 

 3L+3/ 



4L + 31 



5L+ 31 



5L+ 4Z 



6L+4Z 



6L+51 



And the consonances that will result, or form wolves, 

 on a douzeave instrument, for want of more than 12 

 strings or pipes, are as follows, viz. 



Conson 

 ances. 



2 



II 



3 

 III 



4 

 JV 



V 



6 

 VI 



7 

 VII 



Bass and Treble Notes. 



CC*, Eb E, FF*, GG*, and Bb B 



C* Eb, andG* Bb 



Eh F*, FG*, and Bbc* 



C* F, F* Bb, G*e, and B eh . . 



EhG* 



C*G,EBh,F*c,G*rf,Aeh,andB/ 



G*eh 



CG*,EhB, Fc*, andBb/*. . 

 C*Bb, F*eb, andG*/ . • . . 



Ebc*, andBbg* ■• • ■ 



C*c, Eeb, F*/;G*g, and Bib 



Wolves. 



/ 

 2L 



L + 2Z 

 3L -j- / 

 2L -j- 31 

 4L + 2/ 

 5L + 21 

 4 L -j- 4 1 

 6' L -j- 3 1 

 5L + 51 

 7L + 4Z 



The above Tables, to which we shall frequently have 

 occasion to refer, when treating on Tempered Systems, 

 exhibits all the relations of the notes in a regularly tem- 

 pered system, that is, wherein all the fifths, in modula- 

 ting, either by flats or by sharps, from C (to G* eb, 

 when there are but 12 notes,) are alike tempered, as far, 

 at 1 least, as the numbers of sounds in the octave will ad- 

 mit ; and shews, that when L and / have a finite or 

 commensurable ratio, all the intervals that can arise in 

 the scale, being expressed in those terms, as in the last 

 columns, are also finitely or commensurately related, 

 in all commensurable systems. In the example of M. 

 Henfling's system of 50 equal parts in the octave re- 

 ferred to above, since L = 5 and I = 3, the regularly 

 tempered consonances 2, II, 3, III, &c. will be 5, 8, 13, 

 16, &c. of such equal parts; and the wolves of this 

 system, on defective instruments, 2, II, 3, III, &c. will 



