I8S On Musical Temperament. 



the same theorem be applied as before, till no value of x can 

 be obtained, and the temperaments for that scale will be the 

 best adjusted possible. 



But as the scale which contains but 13 degrees, or 12 inter, 

 vals, to the octave,' is in much more general use than everj' 

 other, we shall content ourselves with stating how the problem 

 may be solved for scales contaimng any intei-mediate number 

 of degrees, and proceed directly to the consideration of that 

 which is so much the most practically important. 



Lemma. 



No arrangement of the intervals in the common scale of 12 

 degrees, which renders none of the Vths or 3ds sharp, and 

 none of the Illds flat, can make any change in the aggre- 

 gate temperaments of all the concords of the same name. 



We will conceive the 12 Vths of the Douzeave scale to be 

 arranged in succession, as CG, GD, DA, &.c. embracing 7 

 octaves. Let them at first be all equal : they wiJi each be 

 flattened 49. I say that no change in these Vths which pre- 

 serves the two extreme octaves perfect, and renders none of 

 them sharp, can alter the sum of their temperaments. Let 

 a, b, c, &c. he any quantities, positive or negative, by which 

 the points C, G, D, &c. may be conceived to be raised above 

 the corresponding points, lielonging to the scheme of equal 

 Vths. Then as the mean temperament Vth==V — 49, the first 

 Vth in the supposed arrangement will be V — 49 -fa. The dis- 

 tance from C to D will be, in Kke manner, 2. V — 49+6 ; and 

 consequently the Vth GD will be V — 494-6 — a. In the 

 same manner the third Vth DE will be V — 49-f-c~6, &c. 

 Hence the temperament of CG=: — 49-[-a, of GD= — 49+6 

 — a, of DA=— 49+c — 6, &c. Adding the 12 temperaments 

 together, we find their sum= — 12X49+a+6+&c. — a — 6 — 

 &c. in which all the terms except the first destroy each 

 other, and leave their sum =— 1^ X49 which is the aggregate 

 temperament of the twelve equal Vths in the scheme of equal 

 semitones. 



The same reasoning holds good if we bring these Vths 

 within the compass of an octave : .-ince, if the octave be kepi 



