NATURE 



{May lo, 1872 



ON THE MEASUREMENT OE MUSICAL 

 INTERVALS* 



IN a series of communications to the Acadi'mie dcs 

 Sciences (February 8 and 22, 1S69, July 17, 1S71, and 

 January 29, 1872), M. Cornu and I have shown that musi- 

 cal impressions arc based upon several systems of musical 

 intervals. We were also able to announce, as a pre- 

 liminary result of experiments not yet completed, the 

 following propositions, which, while they show clearly the 

 origin of discussions that have gone on for more than two 

 thousand years, appear to be capable of putting an end 

 to these discussions, by reconciling the two contrary 

 opinions which have always been entertained upon this 

 subject. 



I. The musical intervals formed by the successive 

 sounds of a melody without modulation, belong to the 

 Pythagorean scale, the degrees of which are represented 

 by the following ratios, containing only the factors 2 

 and %\ : — 



2. The intervals formed by the simultaneous sounds of 

 the concords, wliich are the bxsis of harmony, belong to 

 very different systems, dcjending upon the complexity of 

 th; cords. Those v/hich form part of the simpler cor.- 



cords of two or three sounds, thirds, sixths, perfect con- 

 cords, &c., may be included in the scale given in all 

 treatises on physics, the degrees of which are represented 

 by the following ratios formed by the factors 2. 3, and 5 : 



fa 



I 9 5 i 3 1 15 ,. 



84323 s "■ 



To demonstrate these propositions several conditions 

 require to be fulfilled. 



In the first place, in the two scales above s^iven, the 

 three different intervals, viz., the major third (/l' — ;«/, the 

 sixth do — /ii, and the seventh do — si, differ from one 

 another b/ the inter\ral ciUed a"commV'the value of 



8[ 

 which is IT- , as will be found on dividing oneb / the other 

 bo' 



the fractions which represent these intervals on the two 

 scales. Now this value of the comma is very small, 

 though very perceptible to the car ; to demonstrate it we 

 must, therefore, seek the assistance of skilled m.usicians, 

 and employ apparatus of considerable delicacy. 



Secondly, to measure the intervals formed by successive 

 sounds it is best to study these intervals not separately, 

 but as they occur in the actual course of a melody. Con- 

 sequently, if we employ as our means of measurement 

 the process wiiich consists in causing the sounding body 

 to trace out its own vibrations (and in the present state 



of science no better method can be adopted), we must 

 have the means of inscribing continuously the sounds 

 which constitute fragments of melodies as they are exe- 

 cuted upon an instrument. 



Lastly, it is clearly necessary that the registration of 

 the vibrations shall be automatic, and independent of the 

 volition of the observers. The player must have nothing 

 to do with it ; he must not even see it going on, so that 

 his attention may be entirely devoted to the music which 

 he is playing. 



After many trials we have succeeded in fulfilling these 

 conditions. The apparatus which we use is very simple, 

 the elements of it being found in every physical cabinet. 

 It will, therefore, be useful to describe it. 



Experiment shows that a metallic wire of steel, copper, 

 brass, &c., without tension, and merely supported in such 

 a manner that its vibrations may be executed freely, trans- 

 mits to one of its extremities, by transverse vibrations, 

 the sounds emitted by a sonorous body fixed to the other 

 extremity. To show this it is sufficient to take two 

 tuning-forks having mirrors attached to them, and tuned 

 exactly in unison. Fix the end of a wire to one of them, 

 and attach to the other end a feather carrying a shining 

 point placed in front of the mirror of the second tuning- 

 fork. On setting one of the forks in vibration, and pro- 



n/ (A' Physyjue. 



:), Stir I'Histoire de VAcou^tiqiic Musicalc. 



perly adjusting the feather, the shining point is seen to 

 describe an ellipse characteristic of the unison, and vary- 

 ing in form when a weight however small (a little piece of 

 wax for example) is attached to the tuning-fork fixed to 

 the wire. 



A wire five, six, eight, ten, &c., metres long, suspended 

 by narrow strips of caoutchouc, is soldered at one end to 

 a small plate of brass, L, placed between the sounding- 

 board of a stringed instrument and the foot of the bridge, 

 the other end being slightly clasped to a heavy stand S. 

 Near the fixed point a small piece of tinsel [c; is soldered 

 on, and to this is attached a feather {b), by means of 

 a little soft wax (by this arrangement a greater amplitude 

 of vibration is attained than if the feather w^-re directly 

 attached to the wire). The musician standi in such a 

 position that the wire may not impede the nvn-ements of 

 his bow, and plays fragments of simple meloiies in slow 

 time (each note lasting at least a second). TIil- vibrations 

 of the strings are transmitted to the bridg ■. the metal 

 plate, the wire, and, lastly, to the feather, which vibrates 

 synchronously. It only remains to trace these vibrations. 



The registering instrument is composed of a metal cylin- 

 der, M, the axis of which is furnished with a s/rew moving 

 a double nut, firmly fixed to a table or to the wall. This 

 cylinder is covered with a sheet of paper, which is 

 blackened by making it revolve over the smoky flame of 

 an oil-lamp. A tuning-fork, D, making from 300 to 500 



