144 Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 



The tables here seen are described in the following paragraphs : — 



The vibrations are calculated from the formula 



log itf =log iV+-J^ log 2 X iP, 



where M and N are the vibrations in the higher and lower notes, and x the number of mean 

 semitones in the interval. The beats are calculated from the formula mN—nM', for the fifths 

 SN—ZM. The beats are those of each note with the fifth above it : thus A^ F' (the octave 

 above F) l)eats 86-8 times in a minute in equal temperament (P). 



The vibrations are taken as in the pitch frequently used for organs, when not wanted to 

 combine with the orchestra, that is, a diatonic semitone (15 : 16) below the ordinary concert- 

 pitch of our day, in which C (on the first line below the treble) gives 256 double vibrations 

 per second. In tuning to the concert-pitch, each number in the lower table, be it of vibra- 

 tions or of beats, must be increased by its 15th part. For an octave above, the number of 

 beats must be doubled : for an octave below, it must be halved. Thus, CG beating 48*8 

 times in a minute, C,Gi beats 24-4 times, and C'G' beats 97-6 times in a minute. 



