408 



Mr. A. J. Ellis on the Temperament [June 16, 



of ^ and 7c. If for Jc we put its value •0053950, these last expressions 

 become 



2e'=-0009314 - 1-I43 7400a7+ 420^^ 

 S/3'=-00043659-5-8158100^+1998^^. 



Hence Table XII. suffices to give complete information respecting the 

 effect of any system of temperament when x is known. The following are 

 some of the principal conditions on which it has been proposed to found 

 a system of temperament. I shall first determine the value of x and log v 

 on these conditions, and then compare the results. 



A. Harmonic Systems of EauAL Temperament. 



I. Systems with two concords perfect. 



No. 1 (45) *. System of perfect 4ths and Vths. 

 Here c^=0, log ^=-1760913. 



This is the old Greek or Pythagorean system of musical tones, more 

 developed in the modern Arabic scale of 17 tones. No nation using it has 

 shown any appreciation of harmony. 



No. 2 (2). System of perfect Illrds and 6ths. 



Here e for III, or ^-4^=0, ^=i^= -00134875, log v=- 17474255. 

 Hence log m=log '048455 1=|- log f =A (log f +1 V), so that 

 the tempered mean tone is an exact mean between the just major and 

 minor tones. Hence this is known as the System of Mean Tones, or the 

 Mesotonic System, as it will be here termed. It was the earliest system 

 of temperament, and is claimed by Zarlino and Salinas. See also Nos. 

 13 and 19. 



No. 3 (23). System of perfect 3rds and Vlths. 



Here e for 3, or — ^ + 3a?=0, (c=^ ^='0017983, log y=-1742930. 



II. Systems in which the harmony of two concords is equal. 



No. 4 (20). The Ilird and Vth to the same bass; beat equally and in 

 opposite directions f. 

 Here ^ for III + /5 for V= 0, or {hh - 20 ^) - 3^= 0, x^^^ >5; = -001 1 725, 

 log «;= 1749188. 



No. 5 (15). The 6th and Ythbeat equally, and in the same direction J. 



Here /3 for 6=^ for Y, or -8^+32^=— 3^, ^=^>^=-0012331, 

 log 1748582. 



* The number preceded by No. points out the order of the system in the 

 present classification. The number in a parenthesis shows the position of the 

 system in the comparative Table XV., which is explained hereafter (p. 418). 



t That iS; one interval is too great, or beats sharp," and the other too 

 small, or " beats flat." 



\ That is, both beat sharp " or both " beat flat." 



