180 



On Musical Temperament. 



The sign plus denotes that the degree to which it belongs is 

 to be raised, and minus, that it is to be depressed. The cor- 

 rections in each succeeding operation are to be added to those 

 in the preceding. The errors, in the 3d approximation, are 

 so trifling, that a 4th would be wholly useless. 



Note. The foregoing calculations will be rendered much 

 more expeditious and sure, by reducing the theorem, in some 

 sense, to a diagram, as in the first of the following figures ; 

 and by applying the successive corrections to the circum- 

 ference of a circle divided into parts proportioned to the 

 intervals of the enharmonic scale, as in the second. 



•V«i 



-a 



I 

 P 



±c 



+a 



V Sec- 



Hr" 



j». 



-6 



+5 





J-v 



tk 



Proposition VII. 



To determine the temperaments and beats of all the con- 

 cords, together with the values of the diatonic and chro- 

 matic intervals, and the lengths and vibrations per second 

 of a string producing all the sounds, of the system result- 

 ing from the last proposition. 



The temperaments of all the concords are easily deduced, 

 from Table V. The Vth CG, for example, has its lower 

 extremity lowered 12, and its upper extremity 14. Hence 

 it is flatter by 2 than at first, and consequently its tempera- 

 ment's 166.. The temperaments of all the concords, thus 



