v THE SENSE OF HEAEING 247 



can be distinguished by the ear. The total number of single 

 notes that can be perceived by the normal ear is thus very con- 

 siderable. 



But we have already seen that in music the extreme perceptible 

 tones, whether the highest or the lowest, are not utilised, so that 

 the musical tone-range does not usually exceed seven octaves, 

 commonly known as the contra- octave, great octave, small octave, 

 once-, twice-, thrice-, and four-times accented octaves. More- 

 over, the very small intervals that can be distinguished by a good 

 ear between two adjacent notes are not counted as true musical 

 tones, these being confined to intervals of which the vibrations 

 stand to each other in a given ratio. Thus, the once accented 

 octave, which lies in the centre of the tone-range, is divided, not 

 into 1200 tones, but only into 12, the least interval recognised 

 between them being a semitone. The same applies to the octaves 

 above and below. Intervals of less than a semitone are used in 

 oriental vocal music, but in our system of music, when we do not 

 employ instruments with the tempered scale (infra), we only 

 make use of the so-called enharmonic comma, for which, however, 

 there is no specific notation. 



The most important musical intervals are the octave, in which 

 the ratio of the vibrations of the two notes is as 1 : 2 ; the fifth, 

 2:3; the fourth, 3:4; the major third, 4:5; the minor third, 

 5:6; the major sixth, 3:5; the minor sixth, 5:8; the second, 

 8:9; the seventh, 8 : 15. To a musical ear these can always be 

 distinguished from other intervals, whatever their position on the 

 scale, so long as they are not at the extreme upper or lower end 

 of the appreciable tones : their value, however, is not absolute, 

 but relative. 



From these intervals is derived the natural musical scale : 



24 : 27 : 30 : 32 : 36 : 40 : 45 : 48 

 c : d : e : f : g : a : b : c 1 



9_ 5_ 4 3^ 5_ 15 

 : 8 : 4 : 3~ : 2 : 3 : 8' 



The scale is continued above and below in the other octaves, 

 the same intervals being repeated (Fig. 101). Between each two 

 notes of this scale (0 major), however, there is a semitone only 

 between e and /, and between & and c. Semitones are obtained 

 by adding to the notes the chromatic intervals of sharps and 

 flats, i.e. notes higher or lower than those named in the ratio of 

 about 25 : 24. C sharp has f of the vibrations of c ; d flat f f of d. 



Clearly the single intervals between one note and the next 

 are not absolutely identical; for instance, the interval between 

 c and d in the scale of C major is not perfectly identical with 

 that between d and e, although both are intervals of a whole tone. 

 For while d represents f of the vibrations of c, e does not exactly 



