94 Colours and their Relations. [January, 



into one note, as the}' exist in the ideal scale, for better 

 adaptation to keyed instruments. 



In the construction of this harmonic scale the- same three 

 notes, do, sol, do 2 , are, as in the former case, assumed as a 

 basis, re and fa being derived from them in the same manner 

 as before. But the other notes are deduced from these on a 

 different principle from that which is followed in the pre- 

 ceding case. There are formed three arithmetical progres- 

 sions — do mi sol, do fa la, re sol si — the first having a common 

 difference of i-4th, the second of i-3rd, and the third of 3-8ths, 

 while from these mi, la, and si are respectively derived 

 thus: — 



Mi = do + , la = do +J- , si = 2 sol — re. 



2 2 



The chromatic members of the scale are found thus : — 

 Fa -s- mi = dojjf. or reb , sol -=- mi = re$ or mfo , re x mi = fajjf. 

 or solb , fa x mh = soljfe or lab , fa 2 = lajfc or sib • The fol- 

 lowing is the scale thus constructed, with its logarithms and 

 their differences : — 



Harmonic Scale, . 



Names of Ratios of Logarit hms. Differences. Differences. Differences. 

 Notes. Vibrations. to 



Do . . 1 o* 0*0280287 



Do$ reb ro66' 0*0280287 0*0231238 



Re . . 1*125. °" 5 I][ 525 ©'0280287 



Re# mib 1*2 0*0791812 0*0177288 



Mi . • 1*25 0*0969100 0*0280287 



Fa . . 1*33' 0*1249387 0*0231238 



Faftsoll? 1*40625 0*1480626 0*0280287 



Sol . . 1*5 0*1760913 0*0280287 



Sol*(j: lab i*6 0*2041200 0*0177287 



La . . i*66' 0*2218487 0*0280287 



La# sib 1*77' 0*2498775 0*0231238 



Si . . 1*875 0*2730013 0*0280287 



Do 2 . . 2 0*3010300 



While the departure from a geometrical progression is in 

 this case somewhat greater than in the Pythagorean scale, 

 there is here more simplicity in the relation which the 

 vibrations of each note bear to those of the tonic, whence 

 probably its greater harmonic power. There is another 

 result following from the departure from a regular geome- 

 trical progression, both in the case of the Pythagorean and 

 the harmonic scale. According to the ideal scale, in which 



