328 
IOWA ACADEMY OP SCIENCE 
These considerations coupled with the fundamental law of 
string vibrations, to the effect that, for a string of given weight 
and tension, the frequency of a vibrating segment is inversely 
proportional to its length, suggest a simple method of finding 
those string lengths which will give the successive tones of the 
tempered scale. 
Draw two intersecting straight lines including any conven- 
ient angle (see accompanying diagram, figure 26) . From the point 
of intersection lay off on one line any convenient length Oc=L, 
on the other a length OC=L-f-1.05946. Join the points Cc by 
a straight line. Locate the corresponding points B and and 
join by a dotted straight line. Now draw the series Cd, dD, De, 
etc., and the dotted series, parallel to Bc;^ and cC. By this means 
the points c^, d, d^, e, etc., are detemined at which a string of 
length L (=:Oc) must be stopped to give the successive tones of 
the tempered (chromatic) scale. This will be evident from the 
construction of the figure in which Oc/OC=OC/Od=Od/OD= 
etc., the value of this ratio being 1.05946 by construction.* 
If this diagram is drawn on the top of a sonometer, or a table- 
top across which a string is stretched, and bridges are placed 
under the string opposite 0 and c, it forms a complete finger 
board for running the major, minor and chromatic scales. 
The device lends itself to the demonstration of the following 
relations : 
(1) Comparison of the major and minor scales. (2) Com- 
parison of the major and minor chords. (3) To show that on 
the tempered scale any note may be taken as key note, and all 
scales are equally good. For this purpose choose any point as 
starting point, calling it point 1. Number the points from point 
1 upward. Sound in succession the tones given by the string 
when stopped at points 1, 3, 5, 6, 8, 10, 12 and 13. (4) Com- 
parison of just and tempered scales. Lay off from O on Oe 
lengths equal to 8/9, 4/5, 3/4, 2/3, 3/5 and 8/15 of L. The points 
so determined are those at which the string should be stopped 
to give the tones of the just scale. A glance at the board will 
now show to what extent each interval of the tempered scale is 
falsified. 
Physics Laboratory, 
Iowa State College. ■ , 
