516 Musical Intonation and Temperament. 



tliat it is more difficult than the old way. Experience alone can an- 

 swer this objection, and the results will a little surprise those who, 

 knowing it to bo better, expect to find it also slower than the old modes. 



But some of our best teachers are firmly of the opinion that any 

 system of names used in singing will prove a serious impediment to 

 vocalization, and compel the singer who has once used it to apply 

 names in every difficult place, before he can apply the words. This 

 is a necessary consequence of names transposed with the change of 

 key, and it is barely possible that the inconvenience might result 

 from a ri^'id perseverance in the use of fixed names long after the oc- 

 casion for them had passed, but their moderate use by beginners, 

 like spelling words to learn to pronounce them, or beating in order 

 to keeping time, will prove a great aid at first, and, if duly discon- 

 tinued, of not the least inconvenience afterwards. 



The other perfect instruments need no further notice. AVc pass 

 to imperfect instruments, and first to Keyed Instruments, as the Organ 

 and Piano Forte. These instruments almost universally have 12 

 fixed sounds in each octave. These sounds supply imperfectly the 

 various pitches of all the scales in which we play. The difference 

 between the true pitch and that used for it is Temperament. To 

 examine this subject, we will suppose the 12 intervals to bo exactly 

 equal. This is called EauM, Temperament. To divide the oc- 

 tave into 12 equal intervals, wo must find a ratio which multiplied 12 

 times into itself, will produce the ratio of 1:2. This ratio is 

 "</l: 'V2, or 1 : a </2. To extract the 12th root of 2, we begin 

 by extracting its square root. This, we know, cannot be expressed 

 in figures, and of the 12th root is equally incommensurable. If two 

 strings, T ' ? of an octave apart, vibrated once together at the Creation, 

 their vibrations would not again coincide till the Resurrection Morn ! 

 The problem, however, like squaring the circle, can be solved near 

 enough for all practical purposes, and tho vibrations will be 

 1:1.059463. To see how these intervals will fit our purpose, we 

 will call the lowest of 12 pitches Do, tho second Don and Rer, the 

 third Re, &c. When we first look at tho 02 sounds in the 18 scales 

 we have given, we are ready to despair ofany accommodation of them 

 to 12 fixed pitches, but we know, as the intervals of every scale are 

 exactly similar, that an instrument of equal temperament will fit one 

 key as well as another, since it matters not with which of tho twelve 

 sounds you begin. By comparing the scale of the key of Do with the 



