Polytechnic Association. 813 



Mr. Danks — The cylinder is about six feet in diameter on the out- 

 side and four and a half feet long. The whole weighs about five tons. 

 "We have a pair of engines for each furnace. Several velocities are 

 necessary, and also a reverse. The axis rests upon four rollers, so 

 that the friction is not very great. "When we are refining, a slow 

 motion is necessary. 



The President — Have you built any in England ? 



Mr. Danks — The iron and steel manufacturers sent a commission, 

 who brought about sixty tons of material, and have tested the process 

 and made a favorable report. 



The President read the following paper 



On Tkue Musical Intonation. 

 By Samuel D. Tillman, LL.D. 



The pleasing effect of harmony arises from simultaneous impres- 

 sions made upon the organs of hearing. Air-waves producing such 

 impressions have a definite relation to each other, expressed by ratios 

 which contain no higher prime number than seven. Taking any 

 musical sound as the tonic, we find it is produced by a fixed number 

 of air-waves within a given time, and that the octave or eighth 

 sound above in the diatonic scale is produced by just double the 

 number of waves or pulsations in the same time. These two series 

 of waves whose relation is expressed by 1:2, produce sounds so 

 nearly resembling each other that they form the limits of the diatonic 

 scale. The tonic is the first of a series of seven sounds, and the 

 octave is the first of similar series of seven sounds, each of which is 

 an octave above the corresponding sound in the first series. On com- 

 paring six sounds of the septave with the first or fundamental sound, 

 we obtain the following ratios: tonic and dominant or the fifth, 

 2:3; tonic and subdominant or the fourth, 3:4; tonic and mediant 

 or the third, 4:5; tonic and submediant or the sixth, 3:5; tonic 

 and supertonic or the second, 8:9; tonic and subtonic or the 

 seventh, 8 : 15. 



When heard in their proper order each sound has a relation to the 

 next, which enables us to indicate the seven intervals by the follow- 

 ing series : 



.2. 10 16 9 10 9 16 

 8 T TT T V ¥ XT' 



Taking the first three, we find their differences indicated by ^-, % 

 and ^ ; these fractions reduced to a common denominator give, 

 tWo"> tVVu-j Trfir '•> dividing the numerators by 3 we obtain as the ■ 

 measure of the first interval (a major-tone), 45 ; as the measure of 



