MUSIC 



fifth is the smoothest, and after it the fourth, then 

 the major third and major sixth. On the border- 

 land between consonance and dissonance lie the 

 minor third and minor sixth. The minor seventh is 

 nearly as smooth as the minor sixth. The tone, 

 the major seventh, and the semitone, are intervals 

 which must be pronounced to be decided discords. 

 This statement as to the comparative smoothness 

 of the different intervals is based upon an exam- 

 ination of the relative position of the partial tones 

 in clangs which each contain the first five over- 

 tones. Between clangs differently constituted, the 

 character of the interval varies very considerably. 

 Differentials. It is at once apparent, from the 

 definition of dissonance, that there is no direct 

 discordance in the interval even of a major 

 seventh, if the v notes composing it are simple 

 tones ; and a question naturally arises as to 

 what is the cause of the dissonance that does 

 undoubtedly accompany this interval, even when 

 its constituent sounds are as nearly simple as any 

 rounds can be the sounds of a pair of tuning- 

 forks, for instance. The cause of this lies in the 

 fact, discovered more than a century ago, but 

 forgotten until Helmholtz revivified it, that when 

 any two notes, simple or compound, are sounded 

 together, a third sound is formed, the vibration- 

 number of which is the difference between the 

 vibration-numbers of the two original sounds. 

 This third sound, which is called a difference- 

 tone or differential, is, in the cases just mentioned, 

 within beating distance of the lower tone, and the 

 dissonance which is heard is the result of the 

 beats between it and that lower tone. Thus, 

 suppose the lower simple tone to have 256 for its 

 vibration-number that is, to make 256 vibrations 

 per second and the upper one to be its octave ; 

 the vibration-number of the latter will then be 

 256 X 2 = 512, and the differential will vibrate 

 512 256= 256 times per second. It will thus 

 merely strengthen the lower primary tone. But 

 if the upper tone be a major seventh above the 

 lower one, instead of an octave, its vibration- 

 number will be 256 X V = 480 ; and the vibra- 

 tion-number of the corresponding differential will 

 be 480 256 = 224, the tone corresponding to 

 which is scarcely more than two semitones below 

 the first primary, and therefore will beat disagree- 

 ably with it. Other differentials are produced 

 between this first differential and the primary 

 tones, but th^y are less audible and of less 

 importance. The differential of a minor seventh 

 is a little more than a minor third below the 

 lower primary, and therefore beyond beating 

 distance at least in all but the lower octaves ; 

 and we find, correspondingly, that when this 



interval occurs between tones that are as nearly 

 simple as we caft make them, it has lost nearly 

 the whole of its roughness, and sounds almost as 

 agreeable as a major sixth. We may reasonably 

 inter that, if the sounds were absolutely free from 

 over-tones, this interval would no longer have 

 even the slightest suspicion of dissonance! 



MUSICAL NOTATION. 



In order to be complete, a musical notation 

 must so define the music written in it that it can 

 be sung in only one way. For this purpose, it is 

 necessary that the following points should be 

 determined namely, the absolute pitch of the 

 notes, their absolute and relative duration, their 

 rhythmic accent, and the degree of force with 

 which they are to be sung. We shall take these 

 points in the order in which we have mentioned 

 them. 



i. Absolute Pitch. The absolute pitch of a 

 musical note is determined by its position upon 

 five parallel lines, called the staff, in conjunction 

 with a sign called a clef 

 (Fr., from Lat clavis, a 

 key), placed upon them. 

 The two commonest forms 

 of clef are the G and F 

 clefs (fig. 3), so called be- 

 cause the lines upon which 

 the clef signs are made 

 stand for G and F respect- 

 ively. The G clef is put 

 upon the second line, count- 

 ing upwards, in the upper 



C G 



Fig- 3- 



staff, and the F clef upon the fourth line in 

 the lower one ; and when two staves having 

 these clef-marks are bracketed together, it is 

 necessary to assume that there is one note 

 between a note placed over the upper line of the 

 lower (or bass) staff and one placed under the 

 lower line of the upper (or treble) staff. This 

 note is the middle C of the pianoforte (the highest 

 C in a man's voice, and the lowest in a woman's), 

 and is placed upon a line between the treble and 

 bass staves, as in fig. 3. It will be seen from this 

 that if the two staves of five lines each had 

 another line placed between them, they might be 

 combined into one 'great staff' of eleven lines, 

 the notes still retaining the same significance as 

 at present This would, however, be practically 

 inconvenient, for obvious reasons. There is a 



third clef-sign O which always denotes that the 

 line upon which it is placed is middle C It is 



EfE 



Violin or Treble stave. 



Soprano stave. 



Alto stave. 

 Fig. 4. 



Tenor stave. 



Bau ture. 



used frequently in orchestral music, and often in 

 the middle parts of choral music ; and, according 

 to its position, the staff on which it is gets the 

 name of soprano, alto, or tenor staff. The ordi- 

 nary treble and bass staves may be considered to 

 be simply the upper and lower sections of the 



' great staff' already mentioned, and these other 

 staves to be slices cut out of the middle of it. 

 Their relative position will be better understood 

 from fig. 4, in which the eleven lines are dotted 

 throughout, and the different sections shewn in 

 full, with a clef-mark on each. 



