554 EXPERIMENTS RESPECTING SOUND AND LIGHT. 



Plate 5. Fig. 44. and 45. Two different positions of a major third, composed of similar 

 vibrations, as represented by figures of sines- 



Fig. 46. A contracted representation of a series of vibrations. A, a simple uniform sound. 

 B, the beating of two equal sounds nearly in unison, as derived from rectilinear figures. C, 

 the beats of two equal sounds, derived from figures of sines. D, a musical consonance, 

 making by its frequent beats a fundamental harmonic. E, the imperfect beats of two unequal 

 sounds. F, the beats of two equal sounds, supposing the strength of the sound to be as the 

 square of the velocity : this figure agrees much better with the audible effect of a beat than 

 the former. 



Fig. 47. Various forms of the orbit of a musical chord, when inflected, and when struck. 



Fig. 48. Forms of the orbit, when the sound is produced by means of a bow. 



Fig. 49. Epitrochoidal curves, formed by combining a simple rotation or vibration with 

 other subordinate rotations or vibrations. 



Fig. 50 and 51. The successive forms of a tended chord, when inflected and let go, ac- 

 cording to the construction of De la Grange and Euler. 



Fig. 52. The appearance of a vibrating chord which had been inflected in the middle, the 

 strongest lines representing the most luminous paits. 



Fig. 53. The appearance of a vibrating chord, when inflected at any other point than the 

 middle. 



Fig. 54. Tlie appearance of a chord, when put in motion by a bow applied nearly at one 

 third of the length from its end. 



Fig. 55. The method of tuning recommended for common use. 



Plate 6. Fig. 56. A comparative view of different s^'stems of temperament. The whole cir- 

 cumference represents an octave. The inner circle L isdividedinto 30103 parts,corresponding 

 with the logarithmical parts of an octave. The next circle R shows tlie magnitude of the 

 simplest musical and other ratios. Q is divided into twelve equal parts, representing the semi- 

 tones of the equal temperament described by Zarlino, differing but little from the system of 

 Aristoxenus, and warmly recommended by Marpurg and other late writers. Y exhibits the 

 system proposed in this paper as the most desirable ; and P the practical method nearly ap- 

 proaching to it, which corresponds with the eleventh method in Marpiwg's enumeration, 

 except that, by beginning with C instead of IS, the practical effect of tlie temperament is 

 precisely inverted. This system differs little from that which was formerly proposed by 

 Romieu. K is the system of Kirnberger and Sulzer; which is derived from one perfect 

 third, ten perfect, and two equally imperfect fifths. M is the system of mean tones, the 

 sistema participato of the old Itahan writers, still frequently used in tuning organs, approved 

 also by Dr. Smith for common use. S shows the result of all the calculations in Dr. Smith's 

 harmonics, the system proposed for his changeable harpsichord, but neither in that nor 

 any other form capable of practical application. 



