1873-1 



Colours and their Relations. 



93 



thus : — Mi -*- re$ = reb , la -f- jfajj: = mb , re 3 = sob , re* = lab , 

 re 5 =sib . 



This scale, with its logarithms and their differences, 

 stands as follows : — 



Pythagorean Scale. 



Names of 

 Notes. 



Ratios of 

 Vibrations. 



Logarithms. 



Differences. 



Differences. 



Do . 



. I 







0*0226335 





Do#. 



• 1*0535 



0*0226335 





0*0058856 



Reb . 



1*067872 



0*0285191 



0*0226335 





Re . 



1*125 



0*05II526 



0-0226335 





Re#. 



. 1-185185 



0*0737861 





0-0058856 



Mib . 



. 1*201356 



0*0796717 



0*0226335 





Mi . . 



1*265635 



0*I023052 



0*0226335 





Fa . . 



*'33' 



0*1249387 



0*0226335 





Fa*. 



1-404663 



0*I475722 





0*0058856 



Solb . 



. 1*423829 



0*1534578 



0*0226335 





Sol . 



. i*5 



0*1760913 



0*0226335 





Sol#. 



. 1*580207 



0*1987248 





0*0058856 



Lab . 



. 1-601808 



0*2046104 



0*0226335 





La . 



. 1-6875 



0*2272439 



0*0226335 





La#. 



. 1-77' 



0*2498774 





0*0058856 



Sib . 



. 1*802034 



O-255763O 



0*0226335 





Si . 



. 1*898438 



0*2783965 



0*0226335 





Do 2 . 



. 2 



0*30I0300 







This is the scale according to which the violin is tuned and 

 played, except when it accompanies a keyed instrument. It 

 will be observed that the sharps and flats are here separated, 

 and this distinction is recognised by all good violinists. 

 The geometrical progression in this scale is far from perfect, 

 but the irregularities are recurrent and nearly symmetrical. 

 It has been shown, moreover, by MM. Cornu and Mercadier, 

 that this scale agrees very closely with observation, when a 

 violin is made to register automatically the vibrations of its 

 strings. (See *' Nature,'* vol. iii., 75). 



It is found however, in practice, that the mi of this scale, 

 when struck along with the do, does not produce perfect 

 harmony, and that, to obtain a harmony free from beats, the 

 mi must be lowered by a small interval called a comma, its 

 value being 81-^-80. This alteration in the value of mi 

 involves an alteration in several other notes of the scale, in 

 order to obtain good harmony ; while it is also found most 

 convenient to throw the adjacent sharps and flats together 



