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«; 64, Do 3 ; 32, Do 1 ; 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-'; 

 and 4, Do-'. Now, 8 vibrations should produce a sound between Do-' 

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

 make Sol-'; and 12, Sol- ; 24 make Sol'; 48, Sol' ; 96, Sol'; 

 192, Sol 4 , &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 128 ; Sol 4 is 192. 

 Now, 128 : 192 : : 2 : 3. And 192 : 256 (Middle Do,) : : 3 : 4. 



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

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

 Mi , 10 ; Mi', 20 ; Mi', 40 ; Mi 3 , 80 ; Mi*, 160 ; Mi s , 320 ; &c. 

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

 as 5 : 4, and to the Sol as 5 : 6. Thus Do 1 is to Mi* as 128 : 160 :: 4 : 5, 

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

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



tervaJ 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-' ; 

 3, Sol-' ; 4, Do- 1 ; 5, Mi-' ; 6, Sol-' ; 8, Do-°. A new pitch, named Re", 

 should arise from 9 vibrations ; Re 1 , 18 ; Re', 36 ; &c. 15 vibrations 

 would be Si° : 30, Si' ; &c. No other even vibrations per second 

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

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

 ,of 5 : 6. We must have an intermediate pitch, and will call its name 

 Fa. Wo will take it as much above Do' as Do' is above Sol 1 , that is 

 3 : 4. Now 3 : 4 :: 16 : 21&, which is our Fa 1 . Here is a disagree- 

 able fraction which follows us through every Fa, as Fa', 42 § ; Fa», 

 85*5 ; Fa', 170§ ; &c. We find the interval from Mi 1 to Fa 1 to be 

 20 : 21i, which is 60 : 64, which is 15 : 10, the same as from Si to 

 Do. And the interval from Fa 1 to So! 1 is 2l£ : 24 :: 64 : 72 :: 8 : 9, 



