522 Prof. A. M. Mayer's Researches in Acoustics. 



6. The Curve of a Musical Note formed by combining the sinus- 

 oids of its first six harmonics, and the curves formed by com- 

 hining the curves of musical notes corresponding to various con- 

 sonant intervals. 



We have already seen that any composite vibration which 

 produces in us the sensation of a musical note, can always be 

 reproduced by the simultaneous production of a certain number 

 of the simple sounds of an harmonic series, provided these simple 

 sounds have the proper relative intensities. Therefore, to ob- 

 tain the resultant curve corresponding to a musical note, we draw 

 on one axis its harmonic components with their proper wave- 

 lengths and amplitudes ; and the algebraic sums of their corre- 

 sponding ordinates are the ordinates of the required resultant 

 curve. 



Fig. 3 is the curve of a musical note, being the resultant of 

 the simple vibrations of its first six harmonics. The first six 

 harmonics having been drawn on a common axis, I erected five 

 hundred equidistant ordinates, and extended these ordinates some 

 distance below the axis on which I desired to construct the re- 

 sultant. The algebraic sum of the ordinates, passing through 

 the harmonic curves, were transferred to the corresponding ordi- 

 nates of the lower axis ; and by drawing a continuous line through 

 these points, I formed the resultant curve. The first six har- 

 monics are alone used in the combinations which I have given, 

 because the seventh, ninth, eleventh harmonics, and the major 

 number of those above the twelfth, form dissonant combinations 

 with the lower and more powerful harmonics. Indeed the harmo- 

 nics above the sixth are purposely eliminated from the notes of the 

 piano by striking the string in the neighbourhood of its seventh 

 nodal point. The amplitudes of the harmonics of fig. 3 are 

 made to vary as the wave-length — not that this variation repre- 

 sents the general relative intensities in such a composite sound, 

 but they were so made to bring out strongly the characteristic 

 flexures of the resultant. To simplify the consideration of the 

 curves, they are all represented with the same phase of initial 

 vibration. Of course the resultants have an infinite variety of 

 form, depending on the difference in the initial phase and on the 

 amplitudes of the harmonic elements. 



In figs. 4, 5, and 6 I have drawn the resultant curves formed 

 by combining the curves of musical notes corresponding to the 

 various consonant intervals indicated below the curves. As 

 these curves are the resultants formed by the combination of the 

 composite vibrations of musical notes, it follows that the com- 

 ponents of these curves are not simple harmonics, as in the case 

 of fig. 3, but are derived from the resultant of fig. 3 by reducing 



