92 Colours and their Relations. [January, 



vibrations referred to those of do as unity, and with the 

 logarithms of these numbers, the common ratio of the pro- 

 gression being the twelfth root of 2. 





Ideal Scale. 



" 



Names of 

 Notes. 



Ratios of 

 Vibrations. 



Logarithms. 



Com. Dif. 



Do . . 



. I 



O'OOOOOOO 



0*025085 



Do# re!? 



. 1-059463 



0*0250858 





Re . . 



. 1*122462 



0*0501716 





Re# mib 



. 1*189207 



0-07525:4 





Mi . . 



. I*25992I 



0*1003432 





Fa . . 



• 1*334839 



0*1254290 





Fa# solb 



• I'4i42i3 



0*1505148 





Sol . . 



. 1*498306 



0*1756006 





Sol# lab 



. 1*587400 



0*2006864 





La . . 



. 1*681792 



0*2257722 





Laif sib 



. 1*781796 



0*2508580 





Si . . 



• 1-887747 



0*2759438 





Do 2 . . 



. 2 



0*3010300 





Thus constituting a regular geometrical progression. 



Had the earliest musicians been also mathematicians it 

 is not improbable that this is the scale the3 T would have 

 adopted ; while so great are the powers of habit and inhe- 

 ritance on man's mind and organisation that it would, in the 

 course of time, have come to be regarded as the true scale, 

 the succession of its notes as perfect melody, their combi- 

 nations as perfect harmony. The state of the fact, how- 

 ever, is quite otherwise. Melody and harmony have become, 

 to a certain extent, dissociated, and the scale which is re- 

 garded as yielding the most perfect melody differs from that 

 which is regarded as yielding the most perfect harmony. — 

 neither of them, however, forming regular geometrical pro- 

 gressions, consequently both differing considerably from the 

 ideal scale. 



In the Pythagorean scale, which yields the most perfect 

 melody, the sol is regarded as occupying the exact middle 

 point between do and its octave do 2 ; consequently the ratio 

 of its vibrations referred to do as unity is 1*5. From these 

 three, do, sol, do 2 , all the other members of the scale are 

 derived by multiplication or division. The principal notes 

 are found thus : — Sol 2 -*- do 2 = re, re 2 = mi, sol -*- re=fa, 

 mi xfa = la, mi xsol = si. The sharps thus : — Fa-^mi = do$, 

 sol -=- mi = reft, reft 1 =fa$> re% xfa = sol%, far = la$. The flats 



