348 On the SlaJihope and other Tetnperanieiits. 



In column 4 of the preceding table, the 2nd ( = 51 i: -{- 

 f -f 4^\ m) will be found to be the difference or interval 

 between each adjoining note, and consequently every 3rd, 

 IFI, 4lh, V, 6th or VI, as well as VIII in this system are 

 equal, in whatever pail of the scale the same are taken j 

 and if VIII (012 S + 12 f+ 53 m) be added to the 2nd, II, 

 3rd, III, &c., in order to obtain a second octave above the 

 one in the table, this will at once appear, by subtracting 

 the value of any note from the value of another, situate the 

 proper number of half-notes above it. 



A superior octave being in like manner supplied to each 

 of the other syslcnis, in columns 5 and 6, and column 6, 

 vol. xxviii. p. 141, if the 12 several minor thirds (Srds) in 

 each system be taken out, and if the difference between each 

 ot these and a true or perfect 3rd (16 1 2 + 3 f -[- 14 m) be 

 found and placed in order in a table, with the sign -f to 

 every sharp temperament of interval larger than a 3rd, and 

 the sign — to every flat temperament or interval smaller 

 than a true 3rd, then these several temperaments will be 

 found very unequal among themselves, except in the Isoto- 

 nic (where each is 8 2 + |- m) ; but on adding up the whole 

 of them, paying proper attention to the signs + and — , it 

 ^vill be found, that the sum of the temperamctits of all the 

 Srds, in each system, is 96 S +9 m, or 3 Semitones mini- 

 mum, as mentioned in my engraved table, vol. xxviii. In 

 like manner, if all the major thirds (III) in each of (he four 

 systems be taken out, and their temperaments or imperfec- 

 tions (or difteience from 197 S + 4 f + 17 m) arc found, 

 the sum of all these will in each system be found equal 

 to 84 - + 8 ni, or 4 Enharmonic Diesis ; also if all the tem- 

 peraments of the 4ihs, (or differences from 2 54 S + 5 f 

 -f 22 m) be summed up, they will in each system amount 

 to 12 I^ + m, or the Diaschisma : the sum of the tempera- 

 ments of the Vths (or difference? from 358 S + 7 f -|- 31 m) 

 will be found 12^ + m ; of the 6ths (or diflerenccs from 

 4!52 + 8f + 36 m) 84 IS + 8 m ; and of the Vlihs (or 

 differences from 451 S -f- 9 f + 39 m) 96+9 m, as 

 nientioncd in the engraved table above referred to. 

 With such tablfs as these before us, calculated for each and 



every 



