50 Temperaments of different musical Systems. 



t 5 55 5 



or 5?z = 23/, «» (l 7,-=-^; a » d T£" s + "£J m or 



— c is the flat temperament of the fifth. In theorem 



3 17 



3, we get 18 -,-S + IfirW tne sharp temperament of 



12 22 



the major third; and in theorem 6, 20 ---- -f 1 — m 



the sharp temperament of the major sixth. 



This is the syftem for defective scales which Dr. Smith 

 describes, and recommends, p.icres 219, 215, 1S9> 211, 

 and 212: and of which Mr. Atwood has (but without 

 acknowledgement) given the lengths of strings in his 

 " Rectilinear Motion, &c." 



A system, wherein the octave is divided into 74 

 equal parts, to be found in M. Sauveur's table, and 

 where the temperament is V — 2*38382, differs but 

 very little from V — 2-39301 52 m this system. 



Scholium 10. If a douzeave be required, wherein the fifth 

 (}) and the major sixth (f ) shall heat equally quick, 

 the former flat and the latter sharp : we have from 

 , ' r lis— 3r 



temperament of the fifth: which in theorem 3, gives 



v _ — m or — c f or t_ ne fl a t temperament of 



9 9 9/ [ 



the major third: and in theorem 6, gives -^-2 -f- 



. m or -~ re, the sharp temperament of the sixth. 



This is the famous System of Equal Harmony in 

 3 octaves, invented by Dr. Robert Smith. See his 

 Harmonics, pages 216, 191, 206, 212, 214, &c. 

 And differs but little from M. Ilenfling's system, 

 (Mem.de l'Acad. 1711, 16mo, p. 40S), wherein the 

 octave is divided into 5o equal parts, as Dr. Smith 

 shows in his Harmonic?, p, 157, and states its fifth to 



be 



