to what Extent, and how most readily attainable P" 43 



the conjoint operation of tlie otiier three : but this being unat- 

 tainable from any duple series commencing with 7miiy, we must 

 necessarily resort to the comparatively imperfect series of .3, 6, 

 and 12, whose geometrical proportions, if the unit were prefixed, 

 would be totally destroyed, the numbers 1. .3. 6. and 12 being 

 in no acknowledged proportion at all. 



We must submit then to this comparative imperfection ; and 

 changing the numbers 4, 3 and 2 into their representatives 12, 

 9, and 6, let us multiply the extremes together, and divide the 

 product by the mean ; as thus. 

 Base = 12 

 X by Octave = 6 

 ' -=- bv Fourth =9) 72 (8 : which 8 is the fourth num- 



ber required ;'and being equal to 2-3ds of the whole string 12, 

 is equivalent to that interval in music called the Jifth. The four 

 numbers therefore obtained bv these various procedures, are 

 12 . 9 '. S .6 

 In music = Base 4th, 5th. bth. 

 Were it still further intended to create the number next in 

 value to the 5th from our materials 12. 9. 8. 6(12 and 6 for the 

 harmonical creation of numbers being equivalent), we should, in 

 my opinion, generate neither from 12 and 9, nor from 8 and 6*, 

 whose proportions are as 4 to 3 ; while the simpler, and conse- 

 quently more perfect generative proportions of 3 to 2 are to be 

 found ; viz. in 12 to 8, or in 9 to 6. Now, 12 and 6 being in 

 this instance equal, our choice must rest between the numbers 

 9 and 8 ; to the former of which, viz. 9, as the better concord, 

 we must give precedence; — and therefore the minor 6th which 

 is equal to 74- f (the mean between 9 and 6) must be held, in 

 harmonical value, as the successor of the fftk. 



To retrace our subject : — The octave springs of necessity fr(5m 

 unity, called in music the fundamental; from the fundamental 

 and its octave in conjunction proceeds the fourth ; from the con- 

 joint operation of the fundamental, its octave and its fourth, pro- 

 ceeds the fifth; and from the octave and the fourth alone is ge- 

 nerated the minor 6th. 



These few and very simple operations are, in my mind, a suf- 

 ficient guidance for the analysis of our musical numbers ; — and 

 hence therefore 1 shah venture, in my next letter, upon the for- 

 mation ofu table which shall comprehend every individual inter- 

 val within our octave. 



[To be continued.] 



» Intervals approximating in value to our thirds and sixths may be ge- 

 nerated between these, — but are not in modern use. 



•7 1 5 



t That is -^ equal to — of the whole string 12. 



IX. On 



