Musical Intonation and Temperament 507 



If Sol be the first note of a key, the second must be La. Now, 

 the interval between the first and second of the scale must be a Tone. 

 That between Sol and La is only a Tone. If Sol, 24, La must be, 

 not 26§, but 27. Now, 26| : 27 :: 80 : 91. The interval of 80 : 81 

 is called Comma. From La to Si is a Tone, as it should be be- 

 tween the second and third ; from Si to Do, a Hemitone ; Do to Re, 

 a Tone ; Re to Mi a Tone ; but from Mi to Fa only a Hemitone. 

 This will not do — we must have a Tone there. Mi a has 40 vibra- 

 tions, 8 : 9:: 40 : 4"). The new sound is to Sol 3 as 45 : 43 :: 15 : 16 ; 

 so this interval is, us it should be, a Hemitone. The intervals now, 

 from Sol' to Sol 3 , are Tone, Tone, Hemitone, Tone, Tone, Tone, 

 Hemitone ; consequently it is a true Diatonic scale — the scale of the 

 Key of Sol. This key has no Fa in it ; its place is therefore left 

 vacant for the new pitch. Occupying the same place, we will give 

 it a name by adding an n to Fa and call it Fan. Fa is to Fan as 

 42= : 15 :: 128:135. The interval is less than a Hemitone ; it is call- 

 ed Major Limma. This rendering a pitch limma higher is called 

 Sharping it. 



The same transposition which we have performed on the key of 

 Do, we may repeat on that of Sol. Taking its 5th (Re) for tonic, 

 raising its 2nd (Mi) comma, and sharping its 4th (Do) into Don, we 

 have the Key of Re. This may be repeated again and again. It is 

 obvious this process maybe reversed. The Key of Re may be trans- 

 posed to that of Sol, by making its 7th (Don) limma lower, (which is 

 called Flatting it,) and making its 2nd comma lower. By the 

 same process the key of Sol can be transposed to that of Do, and that 

 again to the key«of Fa ; in each case reducing the 2nd comma, and 

 the 7th to its proper place, and the 4th becomes the Tonic. Begin- 

 ning with Fa, the first two intervals are Tone and Tone, but instead 

 of Si we use a pitch which is to La as 15 : 10. This is written in the 

 same place with Si. We will call it Sir. Now, La: Sir :: 15: 16 

 :: 26| : 28$, and Sir : Do :: 9 : 10 :: 284 : 32. 



This process has no assignable limits. A few keys only are 

 needed to write pieces in, but music often passes from one key to an- 

 other, in the midst of a strain, for the effect of the change. This 

 change is called Modulation. 



In the table at the close, will be found all the pitches of 18 keys, 

 arranged in their order, with the number of their pulsations in 

 the lowest octave. The acute and grave accents imply sounds 



