Musical Intonation and Temperament. 505 



sound more acute than this is utterly inaudible to human ears. Each 

 of these numbers is just double the preceding. 



Let us go backwards a moment, halving the numbers. I said 

 12S pulsations a second make Do 4 ; 64, Do 3 ; 32, Do" ; and 16, Do 1 . 

 If 8 vibrations per second make a sound, it must be named Do ; 4, 

 Do- 1 ; 2, Do- 2 and 1 vibration a second, Do- 3 . From this imaginary 

 point let us ascend again. 1 vibration should produce Do- 3 ; 2, Do- 2 ; 

 and 4, Do- 1 . Now, "> vibrations should produce a sound between Do- 4 

 and Do-' ; let its name be Sol- 2 . Double this, and 6 pulsations should 

 make Sol- 1 ; and 12, Sol- ; 2 4 make Sol 1 ; 48, Sol' J ; 90, Sol'; 

 192, Sol% &c. Thus between each Do and the one above there is a 

 Sol. The interval between Sol and the Do below is 2 : 3 ; that be- 

 tween Sol and Do above is 3 : 4. Thus Do<, is 123 ; Sol 4 is 192. 

 Now, 12S : 192 : : 2 : 3. And 192 : 250 (Middle Do,) : : 3 : 4. 



Again, 5 pulsations a second should form a theoretical sound be- 

 tween Do- 1 and Sol- 1 . The name is Mi- 1 . Doubling this, we have 

 Mi', 10 ; Mi 1 , 20 ; Mi 2 , 40 ; Mi 3 , SO ; Mi*, 100 ; Mi 5 , 320 ; &c. 

 So between each Do and the Sol above is a Mi, which is to the Do 

 a- 5 : 4, and to the Sol as 5 : 0. Thus Do 4 is to Mi 1 as 128 : 160 :: 4 : 5, 

 and Mi 4 to Sol 4 as 160 : 192 : : 5 : 6. Observe, 

 now, from these data how to calculate the in- 4:5 



terval from Do to Sol. It cannnot be done by 5:6 



addition. We must compound the ratios 4:5 10 J 20 : 30 

 and 5:6; multiplying, we have 20 : 30, and 2 : 3 



dividing by 10, 2 : 3, as in the margin. 



Thus far we see 1 pulsation per second should yield Do- 3 ; 2, Do- 2 ; 

 3, Sol- 2 ; 4, Do- 1 ; 5, Mi- 1 ; 0, Sol- 1 ; 8, Do- . A new pitch, named Re , 

 should arise from 9 vibrations ; Re', 18 ; Re 2 , 30 ; &c. 15 vibrations 

 would be Si : 30, Si 1 : &c. No other even vibrations per second 

 yield a pitch to which we need now give a name. Between these 



. -s there are two large intervals. One is between Mi and Sol, 

 of 5 : 0. We must have an intermediate pitch, 1 will call its name 

 e will take it as much above Do 1 as Do* is above Sol 1 , that is 

 3 : 1. Now 3:4:: I i : Ji£, which is our Fa 1 . Here is a disagree- 

 able fraction which follows us through every Fa, as Fa*, 12 j-j ; Fa 3 , 

 852 ; F»\ 17 0§; &*.' We find the interval from Mi 1 to Fa"' to be 

 20 : 21£, which is *. ; : 64, which i ; IS : 16, the same as from Si to 

 Do. .'... I the interval from Fa 1 to So; 1 is 21j : 21 :: 64 : 72 :. 8 : 9, 



