Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 141 



sharp; and the same of the rest. Let g be the greatest common measure of ml and mn. 

 Then 



m + w/3' = gB. 



From this we may obtain such theorems as the following. The beats of a minor third 



exceed those of the following major third by twice the beats of the whole fifth which they 



make up. Twice the beats of a minor third exceed three times the beats of the major third 



which it follows by five times the beats of the fifth they make up. 



a 

 Smith's beats themselves have a long inequality whenever - is not an integer; of which I 



u 



suppose (though I am by no means sure) the ear could hardly be made sensible. The theory 



of the beats of a consonance of more than two notes would offer no difficulty, if there be any 



thing presented to the ear which it would be of any interest to explain. 



A. DE MORGAN. 



Univbrsitt College, London, 

 August n, 1857. 



POSTSCRIPT. 



A FEW observations on tuning and on temperament will not be out of place. The method 

 of tuning employed in this country at present is simply adjustive. In equal temperament, 

 for example, the tuner gets one octave into tune, with its adjacent parts so far as successions 

 of fifths up and octaves down require him to go out of it; and the notes thus tuned are 

 called the bearings: all the rest is then tuned by octaves from the bearings. The method 

 of tuning the bearings, after taking a standard note from the tuning-fork, consists merely 

 in tuning the successive fifths a little flat, by the estimation of the ear, making corrections 

 from time to time, as complete chords come into the part which is supposed to be in tune, by 

 the judgment of the ear upon those chords. Proceeding thus, if the twelfth fifth appear to 

 the ear about as flat as the rest, the bearings are finished : if not, the tuner must try back. 

 The system generally used is the equal temperament: when any other is adopted, beats are 

 sometimes, but not always, employed, that is, counting the beats. For the ordinary tuner, 

 even in equal temperament, learns to help himself by a perception of the rapidity of the 

 beating : but without numerical trial. 



Now it appears to me that there is in this a loss of time and a loss of accuracy. Difl'erent 

 tuners, however excellent their ears, do not agree in their results. Two men, tuning different 



