VOL. LXXVIII.] PHILOSOPHICAL TKANSACTIONS. 447 



different sounds ; which shows why an octave comprehends neither more nor less 

 than 13 notes. Without dwelling any longer on the names or number of those 

 notes, Mr. C. proceeds to find out the temperament. 



It appears by the above divisions, that the length of the string for the last 5th 

 is shorter than the length of the last octave, and also that one of them must 

 necessarily be taken for both purposes; but here we must consult nature, exa- 

 mining by the ear which of the 2 is least disagreeable. This however is soon de- 

 cided; for imperfect octaves are quite insufferable, whereas a certain degree of 

 imperfection in the 5ths is tolerable; therefore we are necessitated to leave the 

 octaves perfect, and to let the 7th octave serve for the 5th of f. In this case it 

 is evident that each of the notes in the succession of 5ths is a perfect 5th to its 

 preceding note, excepting the last, which would be by much too flat, and there- 

 fore it is necessary to divide the imperfection equally among them all. For this 

 purpose it must be considered, that as the 12 successive 5ths, together with the 

 whole string or first note, are each -f- of its preceding note ; they form a geome- 

 trical series, the ratio of which is -f, its extremes are 13286025, the first length, 

 and 102400, the 12th fifth, and the number of terms is 13. But because in- 

 stead of 102400, which is the last 5th, we must take the number 103797, viz. 

 the length nearly of the 7th octave, for the last term of the series; therefore the 

 problem is reduced to the finding out of 1 1 mean proportionals between the two 

 numbers 13286025 and 103797- Now by the nature of a geometrical pro- 



13286025 



gression, -j^j^ = ^^^ = 128, consequently r = 'J/128 = 1.4983 the ratio of 

 the series. 



The ratio having been ascertained, the succession of tempered 5ths is thus 

 easily determined; viz, divide the length of the whole string by this ratio, and 

 the quotient gives the 1st tempered 5th; divide this 5th by the same ratio, and 

 the quotient gives the 2d tempered 5th; divide this 2d 5th by the same ratio, 

 and so on till the last 5th, which comes out equal to 103797-8Vj which is so nearly 

 equal to the length of the 7th octave, that the difference is truly insignificant. 

 The divisions, thus ascertained, form a series of notes, in which the octaves only 

 are perfect; but all the 5ths, all the 3ds, and in short all the chords of the same 

 denomination, are equally tempered throughout; so that whichever of them is 

 taken for the key-note, its 5th, 6th, &c. will have always the same proportion to 

 it, and consequently will always produce the same harmony when sounded with it. 

 It is evident that, besides this, there can be no other temperament capable of 

 producing equal harmony ; for when the extremes of a geometrical series and 

 number of mean proportionals are given, there can be only one set of those 

 means. If, on the other hand, we endeavour to find a better temperament by 

 introducing more than 1 3 notes within the limits of an octave, we shall find it 



