510 Musical Intonation and Temperament. 



The reasons for the precise intervals of the Diatonic scale can be 

 best understood by examining the effect of hearing two or more 

 pitches at the same time. These, when agreeable to each other, 

 form Harmon*. When Do, Mi and Sol, for instance, are heard at 

 once, their vibrations being as 4, 5, and C, must often coincide thus : — 



Sol 



Mi 



Do " . 



Here every other vibration of Do corresponds with one of Sol, and 

 every fourth pulsation of Do corresponds with one both of Sol and of 

 Mi. Do is found to harmonize with Mi 4:5, with Fa 3 : 4, with 

 Sol 2 : 3, and with La 3 : 5. Re harmonizes with Fa 35$ : 42$ 

 :: 320 : 384 :: 5 : 6, with Fan 4 : 5, with La 86$ : 53$ :: 2 : 3, and 

 with Sir 5:8. Re harmonizes with Sol 3 : 4, and with Si 3 : 5. Mi 

 harmonizes with Sol 5:6, with La 3:4, with Si 2:3, with Son 



4 : 5, and with Do 5:8. Fa harmonizes with La 4:5, with Sir 

 3:4, with Do 2: 3, and with Re 3 : 5. Fan harmonizes with La 



5 : 6, and with Re 5 : 8. Sol harmonizes with Si 4 : 5, with Do 3 : 4, 

 with Re 2 : 3, and with Mi 3 : 5. Sdn harmonizes with Si 5 :6, and 

 with Mi 5:8. La harmonizes with Do 5 : 0, with Re 3:4, with Mi 

 2:3, with Fan 3:5, and with Fa 5:8. Sir harmonizes with Fa 

 2 : 3, and with Re 4:5. Si harmonizes with Re 5 : 6, with Mi 3 : 4, 

 with Sol 5:8, and with Son 3:5. 



It is obvious that the corresponding degrees of every other key 

 harmonize with each other in the same way as those of Do. From 

 this we see that the precise intervals of the Diatonic scale are not ar- 

 bitrarily but exactly such as will secure the greatest amount of harmony. 



We see here only six ratios of vibration which produce harmony ; 

 2 : 3 called a Fifth (Vth) ; 3 : 4 a Fourth (IVth) ; 3 : 5 a Major 

 Sixth (Vlth); 5:8 a Minor Sixth (6th); 4:5 a Major Third 

 (IJIrd); and 5:6 a Minor Third (3d). We need go no farther 

 into the subject of harmony — it constitutes a department of science 

 comparatively well cultivated. 



In a composition one single train of sounds must be such as to be 

 of itself agreeable. This train is called tho Air, and its agreeable 

 quality is Melody. The melody of pieces of a certain cast often 

 requires the Sharp 4th and 5th. The harmony of these same pieces 

 often contains minor thirds and sixths, where others would have major 

 thirds and sixths. Hence they are said to be in the Minor Mode, 

 and other pieces are said to be in the Major Mode. The Diatonic 



