Polytechnic Association Proceedings. 771 



1. If the ends that are played are at the angle, the pipes rein- 

 force each other. 



2. If the other ends are at the angle, they tend to silence each 

 other. 



3. If dissimilar ends are at the angle, they reinforce each other. 

 In all these experiments the pipes employed were open at both 



ends. 



Now that science is in possession of this delicate optical method, 

 which requires for its success no nice musical ear, other problems, 

 heretofore settled by assumption, may be brought within the range 

 of demonstration. 



n. ox A METHOD OF MEASURING MUSICAL INTEKVALS UPON A SPIRAL 



PROJECTION. 

 By Samuel D. Tillman, of New York City. 



Vibrations, producing the different sounds of the diatonic scale, 

 have fixed numerical relations. Starting from the first or lowest 

 sound, and proceeding upward, the eighth sound is found to har- 

 monize most perfectly with the first; proceeding still upwai'd, the 

 fifteenth is found to harmonize just as perfectly with the eighth, 

 and the twenty-first with the fifteenth. Each of these four sounds 

 are, therefore, regarded as the first in a distinct, but similar, series, 

 embracing seven sounds. The vibrations producing each sound of 

 the first series are, to those causing a similar sound in the second, 

 as 1 to 2. This natural division of musical sounds into septaves is 

 not clearly shown when the characters or signs representing these 

 sounds are arranged in one right line, as found in all popular 

 systems of instruction; moreover, the ear, in demanding the addi- 

 tion of the octave, as the terminal note of the diatonic scale, helps 

 to mislead. 



To obviate this difficulty, I represent the progress of pitch, from 

 the lowest to the highest musical sound, perceptible to the human 

 ear, by a line passing spirally around a cone, as seen in Fig. 1. 



Each ring measures a septave. A right liue drawn from the 

 apex to the base of the cone cuts the spiral at each point where 

 there is an apparent return to the tonic by a repetition of the octave. 

 All the sounds known as C are shown. On the right is the number 

 of vibrations producing each, assuming that the middle C has 512, 

 or is the ninth octave of an imaginary sound, having one vibration 

 per second. The French standard pitch requires for this sound 522 

 vibrations per second. On the left are the positions of all the notes 

 C on the staff, as fixed by the base and treble clefs. 



