HARMONY 469 The dissonance of this interval is greater than in the case of the fifth, because the harmonics 3-2 are both vibrations of intensity, and there- fore give louder beats than the pairs 3-^ and 3-5 of the fifth. In the fourth we have also the additional beats of pairs 6-4 and 6-5. THE MAJOE THIED AND THE MAJOE SIXTH. ie major third and the major sixth are writ- ,en together as they are about equally conso- aant, for the dissonance caused by the beats of pair 3-2, separated by a tone, in the sixth, about equals that of the weaker beating pair 4-3, -eparated by a semitone, in the major third. THE MINOE SEVENTH. ie minor seventh is the smoothest of that class of chords sometimes denominated dis- cords, and is less dissonant than the minor sixth. Besides the beats of the harmonics ex- isting as described in the above intervals, we have also the influence of the beats of the re- sultant tones, which are the products of the combined vibrations of the fundamental notes and of their harmonics. These resultant tones can produce beats either with harmonics or with other resultant tones. These resultant tones are of two kinds, viz. : difference tones and summation tones. Difference tones were discovered by Sorge in 1740, and their pitch is equal to the difference of the two vibrations of the sounds producing them. Summation tones were discovered by Helmholtz, and their pitch is equal to the sum of the vibrations of the two sounds producing them. It will be ob- served that Helmholtz's work is to a great ex- tent merely qualitative ; and although he indi- cates the existence of beats as the cause of dis- cord, yet he does not give laws capable of quan- titative expression, by which to determine be- forehand the degree of consonance or dissonance existing in any given chord. The recent re- search of Prof. Mayer of Hoboken, N. J., " On the Experimental Determination of the Law connecting the Pitch of a Note with the Dura- tion of the Residual Sensation it produces in the Ear" (American Journal of Science, 1874), first gave the duration in absolute time of the sensation of sounds after the exciting vibrations had ceased to exist outside the ear, and thus afforded the means of determining with quan- titative exactness the smallest number of beats that two sounds must produce in order that they form a consonant interval. This latter condition will of course be fulfilled when the beats become just rapid enough in their succes- sion to produce a continuous sensation in the ear. The following is the important law dis- covered by Prof. Mayer : If N equal the num- ber of vibrations producing any given note, and D equal, in the fraction of a second, the dura- tion of the residual sensation (that is, the time during which the sensation remains after the vibrations outside the ear have ceased), then D ss-(!jj+tf}-0qpl. The denominator of the (vulgar) fraction thus determined will be the smallest number of beats per second which one simple sound must make with another in order that harshness or dissonance shall entirely dis- appear from the interval. Thus the simple note giving the middle C of the piano makes 264 vibrations per second, and the residual sen- sation of its sound remains on the ear -fa of a second ; therefore the note which will make 48 beats per second with this C will form an in- terval free from all harshness. The number of vibrations of this note will be 264 + 48, or 312, which is D, and forms with C the interval of the minor third. Hence the nearest note to this C which will form with it a harmonious combination is its minor third. If we in like manner calculate the nearest interval to form .08 > .01 "' IZS Zl* 3*4 532 640 768 Curve showing the Relation of Pitch and Duration. a consonance with the C below the middle C, we shall find it to be the major third. This nearest consonant interval contracts as the pitch ascends, so that for the C of the fifth oc- tave above the middle C (the highest octave used in music) the interval has contracted to
