814 Transactions of the American Institute. 



the second interval (a minor tone), 40 ; and as the measure of the 

 third interval (a so-called semitone), 24. Substituting these values 

 in the remainder of the series, we have the septave intervals mea- 

 sured by the following whole numbers, 45, 40, 24, 45, 40, 45, 24 ; 

 and the sum of these numbers, 263, measures the interval from 

 the tonic to its octave. The interval of the major third is 

 measured by 85 ; the fourth by 109 ; the fifth by 154, and 

 the sixth by 194. Each of these sounds harmonizes with 

 the tonic. If C is taken as the tonic, the intervals made 

 by D, E, F, G-, A, B, are measured by 45, 40, 24, 45, 40, 45, 24 ; 

 but these sounds of the natural key cannot all be used when the 

 position of the tonic is changed. In the progress of modulation, 

 the major and minor intervals are subdivided by flats and sharps, and 

 finally other sounds are introduced in place of those indicated by 

 letters. In order to obviate this difficulty, and to use the smallest 

 number of sounds possible in modulations, the true notes are tem- 

 pered so as to permit the same sound to be used in several keys. By 

 this method, the twelve so-called semitone intervals, made on keyed 

 instruments, fulfill the required conditions. When the temperament 

 is isotonic, these intervals are of the same length. The amount of 

 temperament thus required is shown by the following comparison of 

 the several intervals : True scale 45, 40, 24, 45, 40, 45, 24 = 263. 

 Tempered scale 44, 44, 22, 44, 44, 44, 22=264. The fourths and fifths 

 are seen to be nearly correct in every key, while the thirds are very 

 discordant. The temperament of the fifth is one-twelfth of a comma 

 where fifty-three such commas measure the septave, and twelve-fifths 

 measure the same distance as seven octaves, while in the true system, 

 the sum of the intervals being 263, seven octaves are measured by 

 1841, and twelve fifths by 1848. The same point may be illustrated 

 by the use of the mono-chord. Let a homogeneous musical string 

 be of such diameter and length as to make the sound C designated by 

 the note on the second leger line below the bass staff, two-thirds of 

 its length will give the fifth above, which may be indicated by f, and 

 one-third an octave above. A series of upward and downward 

 measurements may be made, the sum of which is expressed by f|4, 

 and a B q is thus obtained which is a little higher than the original 

 sound C. If the string were divided to 531, 441 parts, the sound 

 W would be made by 524, 283 of those parts ; its acuteness as 

 compared with C would be measured by 7,153 parts. It is evident 

 that the B^ thus generated is not the B3 used in the key of C^ 



