446 '. PHILOSOPHICAL TRANSACTIONS. [aNNO 1788. 



octaves by taking its double, or its half, or the double of the double, &c.: we 

 may find its octave below by taking twice 90, viz. ISO, or the octave of this 

 octave, which is 36o, viz. equal to twice 180, or to four times 9O; and, on 

 the other side, we may find the octave above of the given note by taking its half, 

 which is 45, &c. 



Mr. C. now shows why within the octave there are admitted only 13 different 

 notes, viz. 8 principal ones, and 5 others, called sharps and flats. He assumes 

 a line to represent a musical string, the length of which is supposed to be di- 

 vided into a certain number of equal parts, suppose 13286025. On one side of 

 this line are set the divisions of 7 successive octaves, viz. the half of it, a 

 quarter of it, &c.; and on the other side are the divisions of a series of 5ths, 

 viz. the 5th of the whole string, the 5th of this 5th and so on, which are 

 found by taking ^ of the whole string, then -|- of those -f^, and so on. Here 

 notice is taken only of the octaves and 5ths, because they are the principal and 

 the best concords; so that a temperament being required, it is' necessary first to 

 take care that these concords be not rendered insufferable to the ear, the rest 

 admitting of a greater latitude in the temperament or deviation from the perfect 

 state. Besides, all the other notes are derived from the series of successive 5ths. 

 In whatever key a piece of music is performed, its 5th is the most predominant 

 of its concords ; and as the notes of music must be so ordered as that, for the 

 sake of modulation, any note may be considered as the key-note; therefore 

 having found the 5 th of the whole string by taking f of its length, which gives 

 a note called g, we must suppose, that this g may be considered as the key-note, 

 consequently must find its 5th, which gives d, and so on, until we find one of 

 those successive 5ths, which coincides with one of the successive octaves; for 

 after that, to find more successive 5ths would be only repeating the same thing 

 over again. 



Indeed, if we carry the succession of octaves and of 5ths indefinitely far, we 

 shall find, that no one of the 5ths ever coincides perfectly with one of the 

 octavevS, and therefore the division would have no end. However, as the length 

 of the 7 th octave comes so very near to the 12th fifth, we must be contented 

 with taking this 7 th octave for the 5th of f, the difference between them being 

 about the 100th part of its length; whereas, if we carry on the succession of 5ths 

 and of octaves, we shall find, that among 30 and more 5ths none comes nearer 

 to one of the octaves than the above-mentioned one. Hence the number of 

 5ths in this series is 12; and as, when the division expressing a certain note has 

 been assigned in any part of a string, we may easily find all its octaves above and 

 below, it follows, that by finding all the octaves of those 12 divisions, we shall 

 have 12 distinct notes within half the string, viz. within the first octave of the 

 whole string; to which, if the sound of the whole string be added, we have 13 



